solutions to end of chapter problems
TRANSCRIPT
Chapter 2 2.1
Nominal rate(%)(NR) 5 10 20 60Inflation rate(%) ( IR) 2 4 10 40Real rate by the rule of thumb(%)= NR - IR
3 6 10 20
Correct real rate (%)=(1+NR)/(1+IR) -1
2.94 5.77 9.09 14.29
Error from using the rule of thumb(%) 0.06 0.23 0.91 5.71
Chapter 3FINANCIAL STATEMENTS, TAXES AND CASH FLOW
3.1.
(a)Classified cash flow statement
For the Period 01.04.20 X 0 to 31.03.20 X 1 (Rs. in million)------------------------------------------------------------------------------------------------------------
A. Cash flow from operating activities- Net profit before tax and extraordinary items 150- Adjustments for
Interest paid 30Depreciation 30
- Operating profit before working capital changes 210- Adjustments for
Inventories (20)Debtors (20)Trade creditors 20
- Cash generated from operations 190Income tax paid (30)
- Cash flow before extraordinary items 160Extraordinary item (60)
- Net cash flow from operating activities 100B. Cash flow from investing activities
- Purchase of fixed assets (50)- Net cash flow from investing activities (50)
C. Cash flow from financing activities\- Additional share capital 20- Proceeds loans 10- Interest paid (30)- Dividends paid (40)Net cash flow from financing activities (40)
D. Net increase in cash and cash equivalents 10- Cash and cash equivalents as on 31.03.2000 20- Cash and cash equivalents as on 31.03.2001 30
(b)
A. Cash flow from assets- Operating cash flow 120- Net capital spending (50)- Change in net working capital (30)- Cash flow from assets 40
B. Cash flow to creditors- Interest paid 30- Net new borrowing (10)- Cash flow to creditors 20
C. Cash flow to shareholders- Dividends paid 40- Net new equity raised (20)- Cash flow to shareholders 20
We find that (A) = (B) + ( C)
i.e., Cash flow from assets = Cash flow to creditors + Cash flow to shareholders
3.2. (a)
Classified cash flow statementFor the Period 01.04.20 X 0 to 31.03.20 X 1 (Rs. in million)
------------------------------------------------------------------------------------------------------------ A. Cash flow from operating activities
- Net profit before tax and extraordinary items 100
- Adjustments forInterest paid 30Depreciation 20
- Operating profit before working capital changes 150- Adjustments for
Inventories 10Debtors (10)Trade creditors 10Provisions (5)Increase in other assets (5)
- Cash generated from operations 160- Income tax paid (20)- Cash flow before extraordinary items 140
Extraordinary item (50)- Net cash flow from operating activities 90
B. Cash flow from investing activities- Purchase of fixed assets (30)- Net cash flow from investing activities (30)
C. Cash flow from financing activities- Repayment of term loans (15)- Interest paid (30)- Dividends paid (20)Net cash flow from financing activities (65)
D. Net increase in cash and cash equivalents (5)- Cash and cash equivalents as on 31.03.2000 20- Cash and cash equivalents as on 31.03.2001 15
Note It has been assumed that “other assets” represent “other current assets”.
(b)
B. Cash flow from assets- Operating cash flow 80- Net capital spending (30)- Change in net working capital (5)- Cash flow from assets 45
B. Cash flow to creditors- Interest paid 30- Repayment of debt (5)- Cash flow to creditors 25
C. Cash flow to shareholders- Dividends paid 20- Net new equity raised 0- Cash flow to shareholders 20
We find that (A) = (B) + ( C)
i.e., Cash flow from assets = Cash flow to creditors + Cash flow to shareholders
Chapter 4ANALYSING FINANCIAL PERFORMANCE
Net profit1. Return on equity =
Equity
= Net profit Net sales Total assets x x
Net sales Total assets Equity
1 = 0.05 x 1.5 x = 0.25 or 25 per cent
0.3
Debt EquityNote : = 0.7 So = 1-0.7 = 0.3
Total assets Total assets
Hence Total assets/Equity = 1/0.3
2. PBT = Rs.40 million PBIT
Times interest covered = = 6 Interest
So PBIT = 6 x InterestPBIT – Interest = PBT = Rs.40 million 6 x Interest = Rs.40 millionHence Interest = Rs.8 million
3. Sales = Rs.7,000,000Net profit margin = 6 per cent
Net profit = Rs.7000000 x 0.06 = 420,000Tax rate = 60 per cent
420,000 So, Profit before tax = = Rs.1,050,000
(1-.6)Interest charge = Rs.150,000
So Profit before interest and taxes = Rs.1,200,000 Hence
1,200,000 Times interest covered ratio = = 8
150,000
4. CA = 1500 CL = 600 Let BB stand for bank borrowingCA+BB
= 1.5CL+BB
1500+BB = 1.5
600+BB
BB = 120
1,000,0005. Average daily credit sales = = 2740
365160000
ACP = = 58.4 2740
If the accounts receivable has to be reduced to 120,000 the ACP must be:120,000
x 58.4 = 43.8days160,000
Current assets6. Current ratio = = 1.5
Current liabilities
Current assets - InventoriesAcid-test ratio = = 1.2
Current liabilities
Current liabilities = 800,000 Sales
Inventory turnover ratio = = 5 InventoriesCurrent assets - Inventories
Acid-test ratio = = 1.2 Current liabilities
Current assets InventoriesThis means - = 1.2
Current liabilities Current liabilities
Inventories1.5 - = 1.2
800,000
Inventories = 0.3
800,000
Inventories = 240,000
Sales = 5 So Sales = 1,200,000
2,40,000
7. Debt/equity = 0.60Equity = 50,000 + 60,000 = 110,000So Debt = Short term bank borrowing = 0.6 x 110,000 = 66,000Hence Total assets = 110,000+66,000 = 176,000Total assets turnover ratio = 1.5So Sales = 1.5 x 176,000 = 264,000Gross profit margin = 20 per centSo Cost of goods sold = 0.8 x 264,000 = 211,200Day’s sales outstanding in accounts receivable = 40 days
SalesSo Accounts receivable = x 40
360
264,000 = x 40 = 29,333
360
Cost of goods sold 211,200Inventory turnover ratio = = = 5
Inventory Inventory
So Inventory = 42,240
As short-term bank borrowing is a current liability as well,
Cash + Accounts receivableAcid-test ratio =
Current liabilities
Cash + 29,333 = = 1.2
66,000So Cash = 49867
Plant and equipment = Total assets - Inventories – Accounts receivable – Cash = 176,000 - 42240 - 29333 – 49867 = 54560
Pricing together everything we get
Balance SheetEquity capital 50,000 Plant & equipment 54,560Retained earnings 60,000 Inventories 42,240Short-term bank borrowing 66,000 Accounts receivable 29,333
Cash 49,867
176,000 176,000
Sales 264,000Cost of goods sold 211,200
8. For purposes of ratio analysis, we may recast the balance sheet into report form as under. Let assume that ‘Others’ in the balance sheet represents other current assets.
Liabilities and EquityEquity capital 10,000,000Reserves and surplus 22,500,000Long-term debt 12,500,000
Short-term bank borrowing 15,000,000Total 60,000,000
AssetsFixed assets (net) 30,000,000Current assets Cash and bank 5,000,000Receivables 15,000,000Inventories 20,000,000Pre-paid exp 2,500,000 Others 2,500,000 45,000,000 Less: Current liablilitiesTrade creditors 10,000,000Provisions 5,000,000 15,000,000Net current assets 30,000,000
Total 60,000,000
(i) Current ratio = Current assets/ Current liabilities
45,000,000 = = 1.5
30,000,000Note: Please note that for the purpose of calculation of current ratio and acid –test ratio, we have
to include short-term bank borrowings in current liabilities.
Current assets – Inventories 25,000,000(ii) Acid-test ratio = = = 0.8
Current liabilities 30,000,000
Long-term debt + Short-term bank borrowings (iii) Debt-equity ratio =
Equity capital + Reserves & surplus
12,500,000 + 15,000,000 = = 0.8 10,000,000 + 22,500,000
Profit before interest and tax (iv) Times interest coverage ratio =
Interest
15,100,000 = = 3.02
5,000,000
Cost of goods sold 72,000,000 (v) Inventory turnover period = = = 3.6
Inventory 20,000,000 365
(vi) Average collection period = Net sales/Accounts receivable
365 = = 57.6 days
95,000,000/15,000,000
(vii) Total assets =Equity + Total debt =( 10,000,000 + 22,500,000 ) +(12,500,000+15,000,000) = 60,000,000
Net sales 95,000,000Total assets turnover ratio = = = 1.6
Total assets 60 ,000,000
Profit after tax 5,100,000 (ix) Net profit margin = = = 5.4%
Net sales 95,000,000
PBIT 15,100,000 (x) Earning power = = = 25.2%
Total assets 60,000,000
Equity earning 5,100,000 (xi) Return on equity = = = 15.7%
Net worth 32,500,000
The comparison of the Omex’s ratios with the standard is given below
Omex StandardCurrent ratio 1.5 1.5Acid-test ratio 0.8 0.8Debt-equity ratio 0.8 1.5Times interest covered ratio 3.02 3.5Inventory turnover ratio 3.6 4.0Average collection period 57.6 days 60 daysTotal assets turnover ratio 1.6 1.0Net profit margin ratio 5.4% 6%Earning power 25.2% 18%Return on equity 15.7% 15%
9. We may rearrange the balance sheet figures in the report form as under, for purposes of ratio analysis. It is assumed that ‘Other assets’ are other current assets.
20X1 20X2 20X3 20X4 20X5Share capital 2.4 2.4 3 3 3.2Reserves and surplus 0.6 1 1.5 2 2.5Long-term debt 1.2 1.3 2 2.3 2.6Short-term bank borrowing 1.2 1.4 2.1 2.5 2.6
Total 5.4 6.1 8.6 9.8 10.9AssetsNet fixed assets 2.5 3.2 4.4 4.7 4.8Current assets Cash and bank 0.5 0.6 0.7 0.8 0.7 Receivables 1.5 1.6 2.3 2.6 3.2 Inventories 2 2.2 3 3.7 4.2Other assets 0.2 4.2 0.3 4.7 0.3 6.3 0.4 7.5 0.6 8.7Less: Current liabilities 1.3 1.3 1.8 1.8 2.1 2.1 2.4 2.4 2.6 2.6Net current assets 2.9 2.9 4.2 5.1 6.1
Total 5.4 6.1 8.6 9.8 10.9
20X1 20X2 20X3 20X4 20X5
Current ratio 1.68 1.47 1.50 1.53 1.67Debt-equity ratio 0.80 0.79 0.91 0.96 0.91
Total assets turnover ratio 0.74 1.00 0.91 0.93 1.03
Net profit margin(%) 5.00 6.56 3.85 5.49 6.25 Earning power (%) 9.26 18.03 11.63 13.27 18.35
Return on equity (%) 6.67 11.76 6.67 10.00 12.28
MINICASE
Solution:
(a) Key ratios for 20 X 5 12.4
Current ratio = ---------- = 0.67 6.7+11.7
3.8 + 11.7Debt-equity ratio = = 0.98
6.5 + 9.3
57.4Total assets turnover ratio = = 1.96
[(34 – 6.6) + (38 – 6.7)] / 2
3.0Net profit margin = = 5.2 percent 57.4
5 Earning power = = 17.0 percent
[(34 – 6.6) + (38 – 6.7)] / 2
3.0Return on equity = = 20.2 percent
(13.9 + 15.8) / 2
(b) Dupont Chart for 20 x 5
Net sales +/-Non-op. surplus
deficit 57.8
–
÷
÷
+
+
Return on total assets
10.2%
Net profitmargin5.2%
Total asset turnover
1.96
Net profit3.0
Net sales57.4
Net sales57.4
Average total assets29.35
Total costs54.8
Average fixed assets
21.4
Average net current assets 54.0
Average other assets
2.55
(c) Common size and common base financial statements
Common Size Financial Statements Profit and Loss Account
Regular (in million) Common Size (%)20 X 4 20 X 5 20 X 4 20 X 5
Net sales 39.0 57.4 100 100 Cost of goods sold 30.5 45.8 78 80 Gross profit 8.5 11.6 22 20 Operating expenses 4.9 7.0 13 12 Operating profit 3.6 4.6 9 8 Non-operating surplus / deficit
0.5 0.4 1 1
PBIT 4.1 5.0 11 9 Interest 1.5 2.0 4 3 PBT 2.6 3.0 7 5 Tax - - - - Profit after tax 2.6 3.0 7 5
Balance Sheet
Regular (in million) Common Size (%)20 X 4 20 X 5 20 X 4 20 X 5
Shareholders’ funds 13.9 15.8 51 50 Loan funds 13.5 15.5 49 50
Total 27.4 31.3 100 100 Net fixed assets 19.6 23.2 72 74 Net current assets 5.1 5.7 19 18 Other assets 2.7 2.4 10 8
Total 27.4 31.3 100 100
Common Base Year Financial Statements
Profit and Loss Account
Regular (in million) Common Base Year(%)20 X 4 20 X 5 20 X 4 20 X 5
Net sales 39.0 57.4 100 147 Cost of goods sold 30.5 45.8 100 150 Gross profit 8.5 11.6 100 136 Operating expenses 4.9 7.0 100 43 Operating profit 3.6 4.6 100 128 Non-operating surplus / deficit
0.5 0.4 100 80
PBIT 4.1 5.0 100 122 Interest 1.5 2.0 100 133 PBT 2.6 3.0 100 115 Tax - - 100 100 Profit after tax 2.6 3.0 100 115
Balance Sheet
Regular (in million) Common Base Year(%)20 X 4 20 X 5 20 X 4 20 X 5
Shareholders’ funds 13.9 15.8 100 114 Loan funds 13.5 15.5 100 115
Total 27.4 31.3 100 114 Net fixed assets 19.6 23.2 100 118 Net current assets 5.1 5.7 100 112 Other assets 2.7 2.4 100 89
Total 27.4 31.3 100 114
(d) The financial strengths of the company are:
Asset productivity appears to be good. Earning power and return on equity are quite satisfactory Revenues have grown impressively over 20 x 4 – 20 x 5
The financial weaknesses of the company are:
Current ratio is unusually low While revenues grew impressively, costs rose even faster: As a result profit margins
declined The company did not have any tax liability in the last two years. If the company has to
bear the burden of regular taxes, its return on equity will be adversely impacted
(e) The following are the problems in financial statement analysis
There is no underlying theory It is difficult to find suitable benchmarks for conglomerate firms Firms may resort to window dressing Financial statements do not reflect price level changes Diversity of accounting policies may vitiate financial statement analysis It is somewhat difficult to judge whether a certain ratio is ‘good’ or ‘bad’
(f) The qualitative factors relevant for evaluating the performance and prospects of a company are as follows:
Are the company’s revenues tied to one key customer? To what extent are the company’s revenues tied to one key product? To what extent does the company rely on a single supplier? What percentage of the company’s business is generated overseas? How will competition impact the company? What are the future prospects of the firm? What could be the effect of the changes in the legal and regulatory environment?
Chapter 5FINANCIAL PLANNING AND FORECASTING
1. The proforma income statement of Modern Electronics Ltd for year 3 based on the per cent of sales method is given below
Average per cent Proforma income statement of sales for year 3 assuming sales of
1020
Net sales 100.0 1020.0Cost of goods sold 76.33 778.57Gross profit 23.67 241.43Selling expenses 7.40 75.48General & administration expenses 6.63 67.63Depreciation 6.75 68.85Operating profit 2.90 29.58Non-operating surplus/deficit 1.07 10.91Earnings before interest and taxes 3.96 40.39Interest 1.24 12.65Earnings before tax 2.72 27.74Tax 1.00 10.20Earnings after tax 1.72 17.54Dividends (given) 8.00Retained earnings 9.54
2. The proforma income statement of Modern Electronics for year 3 using the combination method is given below:
Average per cent Proforma income statementof sales for year 3
Net sales 100.0 1020.0Cost of goods sold 76.33 778.57Gross profit 23.67 241.43Selling expenses 7.40 75.48General & administration expenses Budgeted 55.00Depreciation Budgeted 60.00Operating profit 50.95Non-operating surplus/deficit 1.07 10.91Earnings before interest and taxes 61.86 Interest Budgeted 12.0Earnings before tax 49.86 Tax 1.00 10.20Earnings after tax 39.66Dividends (given) Budgeted 8.00Retained earnings 31.66
3. The proforma balance sheet of Modern Electronics Ltd for year 3 is given below
Average of percent Projections for year 3 of sales or some based on a forecast other basis sales of 1400
Net sales 100.0 1020.0
ASSETSFixed assets (net) 40.23 410.35Investments No change 20.00
Current assets, loans & advances :Cash and bank 1.54 15.71Receivables 22.49 229.40Inventories 21.60 220.32
Prepaid expenses 5.09 51.92Miscellaneous expenditure & losses No change 14.00
961.70
LIABILITIES:
Share capital:Equity No change 150.00Reserves & surplus Proforma income 160.66
statement
Secured loans:Term loans No change 175.00Bank borrowings No change 199.00
Current liabilities:Trade creditors 17.33 176.77Provisions 5.03 51.31
External funds requirement Balancing figure 48.96
961.7
A L4. EFR = - S – m S1 (1-d)
S S
800 190 = - 300 – 0.06 x 1,300 (1-0.5)
1000 1000
= (0.61 x 300) – (0.06) x 1,300 x (0.5)
= 183 – 39 = Rs.144.
Projected Income Statement for Year Ending 31st December , 2001
Sales 1,300Profits before tax 195Taxes 117Profit after tax (6% on sales) 78Dividends 39Retained earnings 39
Projected Balance Sheet as at 31.12 2001
Liabilities Assets
Share capital 150 Fixed assets 520Retained earnings 219 Inventories 260Term loans (80+72) 152 Receivables 195Short-term bank borrowings 272 Cash 65(200 + 72)Accounts payable 182Provisions 65
1,040 1,040
A L5. (a) EFR = - S – m S1 (1 –d)
S S
150 30 = - x 80 – (0.0625) x 240 x (0.5)
160 160
= (60 – 7.5) = 52.5
(b) Projected Balance Sheet as on 31.12.20X1
Liabilities Assets
Share capital 56.25 Net fixed assets 90Retained earnings 47.50 Inventories 75(40 + 7.5)Term loans 46.25 Debtors 45Short-term bank 30.00 Cash 15borrowingsTrade creditors 37.50Provisions 7.50
225.00 225.00
(c) 20X0 20X1 i) Current ratio 1.50 1.80 ii) Debt to total assets ratio 0.53 0.54 iii) Return on equity 14.3% 14.5%
(d) A L
EFR 20X1= - S – mS1 (1 – d) S S
150 30 = - 20 – 0.0625 x 180 x 0.5
160 160
= 9.38
150 x (1.125) 30 x 1.125EFR 20X2 = - x 20 – 0.0625 x 200 x 0.5
180 180
168.75 33.75 = - x 20 –0.0625 x 220 x 0.5
180 180
= 8.75
168.75 x (1.11) 33.75 x (1.11)EFR 20X3 = - 20 – 0.0625 x 220 x 0.5
200 200
187.31 37.46= - x 20 – 6.88
200 200
= 8.11
187.31 x (1.1) 37.46 x (1.1) EFR 20X4 = - x 20 – 0.0625 x 240 x 0.5
220 220
= 7.49
Balance Sheet as on 31st December, 20X4
Liabilities Rs. Assets Rs.
Share capital 46.87 Net fixed assets 90.00(30+16.87) (60 x 240/160)Retained earnings Inventories(40.00+5.63+6.25+6.88+7.50) 66.26 (50x240/160) 75.00Term loans(20+16.87) 36.87 Debtors (30x240/160) 45.00Short-term bank borrowings 30.00 Cash (10x240/160) 15.00Trade creditors 37.50Provisions 7.50
225.00 225.00
6. EFR A L m (1+g) (1-d)= - -
S S S gGiven A/S= 0.8 , L/S= 0.5 , m= 0.05 , d= 0.6 and EFR = 0 we have,
(0.05)(1+g)(0.4)(0.8-0.5) - = 0
g (0.05)(1+g)(0.4)
i.e. 0.3 - = 0 g
Solving the above equation we get g = 7.14%
A L7. (a) EFR = - S – mS1 (1-d)
S S
320 70= - x 100 – (0.05) (500) (0.5)
400 400
= Rs.50
(b) Let CA = denote Current assetsCL = Current liabilities
SCL = Spontaneous current liabilities STL = Short-term bank borrowings FA = Fixed assets
and LTL = Long-term loans
i. Current ratio CA
i.e greater than or equal to 1.25 or CL
CA
STL +SCL
As at the end of 20X1, CA = 20x0 x 1.25 = 237.50SCL = 70 x 1.25 = 87.50Substituting these values, we get1.25 (STL + 87.5) 237.50or 1.25 STL x
or STL =
1.25i.e STL Rs.102.50
ii. Ratio of fixed assets to long term loans 1.25FA
LTL
At the end of 20X1 FA = 130 x 1.25 = 162.5 162.5
LTL or LTL = Rs.130 1.25
If STL and LTL denote the maximum increase in ST borrowings & LT borrowings , we have : STL = STL (20X1) – STL (20X0) = 102.50 – 60.00 = 42.50LTL = LTL (20X1)- LTL (20X0) = 130.00 – 80.00 = 50.00Hence, the suggested mix for raising external funds will be :
Short-term borrowings 42.50 Long-term loans 7.50 Additional equity issue --
50.00
A L8. EFR = - S – m S1 (1-d)
S S A S
Therefore, mS1(1-d) – - S represents surplus funds S S
Given m= 0.06, S1 =11,000, d= 0.6 , L= 3,000 S= 10,000 and surplus funds = 150 we have
A 3,000(0.06) 11,000 (1-0.6) - - 1,000 = 150
10,000 10,000
A – 3,000= (0.06) (0.4) (11,000) – 150 = 114
10
or A = (1,140 + 3,000) = 4,140
The total assets of Videosonics must be 4,140
9. m= .05 , d = 0.6 , A/E = 2.5 , A/S = 1.4
m (1-d)A/E .05 (1-0.6) 2.5
(a) g = = = 3.70 per cent A/S –m(1-d)A/E 1.4 -.05 (1-0.6) 2.5
.05 (1-0.6) x A/E(b) 0.5 = A/E = 3.33
2.4 - .05 (1-0.6) A/E
d = 0.466The dividend payout ratio must be reduced from 60 per cent to 46.6 per cent
.05 (1-0.6) x A/E(c) .05 = A/E = 3.33
1.4 -.05 (1-0.6) A/E
The A/E ratio must increase from 2.5 to 3.33
m (1-0.6) 2.5(d) .06 = m = 7.92 per cent
1.4 – m (1-0.6) x 2.5
The net profit margin must increase from 5 per cent to 7.92 per cent
.05 (1-0.6) 2.5(e) .06 = A/S = .883
A/S - .05 (1-0.6) 2.5
The asset to sales ratio must decrease from 1.4 to 0.883
10 m= .06 , b = 0.8 , S0/A= 1/0.9 =1.11
m ( S0 /A0 ) b 0.06 x 1.11 x 0.8g = ---------------------------- = ----- ------------------- = 0.0563 1 – m ( S0 / A0 ) b 1 – 0.06 x 1.11 x 0.8
or 5.63 percent
Chapter 6
TIME VALUE OF MONEY
1. Value five years hence of a deposit of Rs.1,000 at various interest rates is as follows:
r = 8% FV5 = 1000 x FVIF (8%, 5 years)= 1000 x 1.469 = Rs.1469
r = 10% FV5 = 1000 x FVIF (10%, 5 years)= 1000 x 1.611 = Rs.1611
r = 12% FV5 = 1000 x FVIF (12%, 5 years)= 1000 x 1.762 = Rs.1762
r = 15% FV5 = 1000 x FVIF (15%, 5 years)= 1000 x 2.011 = Rs.2011
2. Rs.160,000 / Rs. 5,000 = 32 = 25
According to the Rule of 72 at 12 percent interest rate doubling takes place dsxapproximately in 72 / 12 = 6 years
So Rs.5000 will grow to Rs.160,000 in approximately 5 x 6 years = 30 years
3. In 12 years Rs.1000 grows to Rs.8000 or 8 times. This is 23 times the initial deposit. Hence doubling takes place in 12 / 3 = 4 years.
According to the Rule of 69, the doubling period is:
0.35 + 69 / Interest rate
Equating this to 4 and solving for interest rate, we get
Interest rate = 18.9%.
4. Saving Rs.2000 a year for 5 years and Rs.3000 a year for 10 years thereafter is equivalent to saving Rs.2000 a year for 15 years and Rs.1000 a year for the years 6 through 15.Hence the savings will cumulate to:2000 x FVIFA (10%, 15 years) + 1000 x FVIFA (10%, 10 years)= 2000 x 31.772 + 1000 x 15.937 = Rs.79481.
5. Let A be the annual savings.
A x FVIFA (12%, 10 years) = 1,000,000A x 17.549 = 1,000,000
So, A = 1,000,000 / 17.549 = Rs.56,983.
6. 1,000 x FVIFA (r, 6 years) = 10,000
FVIFA (r, 6 years) = 10,000 / 1000 = 10
From the tables we find that
FVIFA (20%, 6 years) = 9.930FVIFA (24%, 6 years) = 10.980
Using linear interpolation in the interval, we get:
20% + (10.000 – 9.930) r = x 4% = 20.3% (10.980 – 9.930)
7. 1,000 x FVIF (r, 10 years) = 5,000FVIF (r,10 years) = 5,000 / 1000 = 5
From the tables we find that
FVIF (16%, 10 years) = 4.411FVIF (18%, 10 years) = 5.234
Using linear interpolation in the interval, we get:
(5.000 – 4.411) x 2% r = 16% + = 17.4%
(5.234 – 4.411)
8. The present value of Rs.10,000 receivable after 8 years for various discount rates (r ) are:r = 10% PV = 10,000 x PVIF(r = 10%, 8 years)
= 10,000 x 0.467 = Rs.4,670
r = 12% PV = 10,000 x PVIF (r = 12%, 8 years)= 10,000 x 0.404 = Rs.4,040
r = 15% PV = 10,000 x PVIF (r = 15%, 8 years)= 10,000 x 0.327 = Rs.3,270
9. Assuming that it is an ordinary annuity, the present value is:2,000 x PVIFA (10%, 5years)
= 2,000 x 3.791 = Rs.7,582
10. The present value of an annual pension of Rs.10,000 for 15 years when r = 15% is:10,000 x PVIFA (15%, 15 years)= 10,000 x 5.847 = Rs.58,470
The alternative is to receive a lumpsum of Rs.50,000.
Obviously, Mr. Jingo will be better off with the annual pension amount of Rs.10,000.
11. The amount that can be withdrawn annually is:100,000 100,000
A = ------------------ ------------ = ----------- = Rs.10,608 PVIFA (10%, 30 years) 9.427
12. The present value of the income stream is:1,000 x PVIF (12%, 1 year) + 2,500 x PVIF (12%, 2 years)+ 5,000 x PVIFA (12%, 8 years) x PVIF(12%, 2 years)
= 1,000 x 0.893 + 2,500 x 0.797 + 5,000 x 4.968 x 0.797 = Rs.22,683.
13. The present value of the income stream is:2,000 x PVIFA (10%, 5 years) + 3000/0.10 x PVIF (10%, 5 years)= 2,000 x 3.791 + 3000/0.10 x 0.621= Rs.26,212
14. To earn an annual income of Rs.5,000 beginning from the end of 15 years from now, if the deposit earns 10% per year a sum of
Rs.5,000 / 0.10 = Rs.50,000is required at the end of 14 years. The amount that must be deposited to get this sum is:
Rs.50,000 / FVIF (10%, 14 years) = Rs.50,000 / 3.797 = Rs.13,165
15. Rs.20,000 =- Rs.4,000 x PVIFA (r, 10 years)PVIFA (r,10 years) = Rs.20,000 / Rs.4,000 = 5.00
From the tables we find that:PVIFA (15%, 10 years) = 5.019PVIFA (18%, 10 years) = 4.494
Using linear interpolation we get:5.019 – 5.00
r = 15% + ---------------- x 3%5.019 – 4.494
= 15.1%
16. PV (Stream A) = Rs.100 x PVIF (12%, 1 year) + Rs.200 xPVIF (12%, 2 years) + Rs.300 x PVIF(12%, 3 years) + Rs.400 xPVIF (12%, 4 years) + Rs.500 x PVIF (12%, 5 years) +Rs.600 x PVIF (12%, 6 years) + Rs.700 x PVIF (12%, 7 years) +Rs.800 x PVIF (12%, 8 years) + Rs.900 x PVIF (12%, 9 years) +Rs.1,000 x PVIF (12%, 10 years)
= Rs.100 x 0.893 + Rs.200 x 0.797 + Rs.300 x 0.712 + Rs.400 x 0.636 + Rs.500 x 0.567 + Rs.600 x 0.507 + Rs.700 x 0.452 + Rs.800 x 0.404 + Rs.900 x 0.361 + Rs.1,000 x 0.322
= Rs.2590.9
Similarly,PV (Stream B) = Rs.3,625.2PV (Stream C) = Rs.2,851.1
17. FV5 = Rs.10,000 [1 + (0.16 / 4)]5x4
= Rs.10,000 (1.04)20
= Rs.10,000 x 2.191= Rs.21,910
18. FV5 = Rs.5,000 [1+( 0.12/4)] 5x4
= Rs.5,000 (1.03)20
= Rs.5,000 x 1.806= Rs.9,030
19 A B C
Stated rate (%) 12 24 24
Frequency of compounding 6 times 4 times 12 times
Effective rate (%) (1 + 0.12/6)6- 1 (1+0.24/4)4 –1 (1 + 0.24/12)12-1
= 12.6 = 26.2 = 26.8
Difference between theeffective rate and statedrate (%) 0.6 2.2 2.8
20. Investment required at the end of 8th year to yield an income of Rs.12,000 per year from the end of 9th year (beginning of 10th year) for ever:
Rs.12,000 x PVIFA(12%, ∞ )= Rs.12,000 / 0.12 = Rs.100,000
To have a sum of Rs.100,000 at the end of 8th year , the amount to be deposited now is: Rs.100,000 Rs.100,000
= = Rs.40,388PVIF(12%, 8 years) 2.476
21. The interest rate implicit in the offer of Rs.20,000 after 10 years in lieu of Rs.5,000 now is:Rs.5,000 x FVIF (r,10 years) = Rs.20,000
Rs.20,000 FVIF (r,10 years) = = 4.000
Rs.5,000
From the tables we find thatFVIF (15%, 10 years) = 4.046
This means that the implied interest rate is nearly 15%.I would choose Rs.20,000 after 10 years from now because I find a return of 15% quite acceptable.
22. FV10 = Rs.10,000 [1 + (0.10 / 2)]10x2
= Rs.10,000 (1.05)20
= Rs.10,000 x 2.653= Rs.26,530
If the inflation rate is 8% per year, the value of Rs.26,530 10 years from now, in terms of the current rupees is:
Rs.26,530 x PVIF (8%,10 years)= Rs.26,530 x 0.463 = Rs.12,283
23. A constant deposit at the beginning of each year represents an annuity due.PVIFA of an annuity due is equal to : PVIFA of an ordinary annuity x (1 + r)To provide a sum of Rs.50,000 at the end of 10 years the annual deposit should
be
Rs.50,000 A = FVIFA(12%, 10 years) x (1.12)
Rs.50,000 = = Rs.2544
17.549 x 1.12
24. The discounted value of Rs.20,000 receivable at the beginning of each year from 2005 to 2009, evaluated as at the beginning of 2004 (or end of 2003) is:
Rs.20,000 x PVIFA (12%, 5 years)= Rs.20,000 x 3.605 = Rs.72,100.
The discounted value of Rs.72,100 evaluated at the end of 2000 isRs.72,100 x PVIF (12%, 3 years)
= Rs.72,100 x 0.712 = Rs.51,335
If A is the amount deposited at the end of each year from 1995 to 2000 thenA x FVIFA (12%, 6 years) = Rs.51,335A x 8.115 = Rs.51,335A = Rs.51,335 / 8.115 = Rs.6326
25. The discounted value of the annuity of Rs.2000 receivable for 30 years, evaluated as at the end of 9th year is:
Rs.2,000 x PVIFA (10%, 30 years) = Rs.2,000 x 9.427 = Rs.18,854
The present value of Rs.18,854 is:Rs.18,854 x PVIF (10%, 9 years)
= Rs.18,854 x 0.424= Rs.7,994
26. 30 per cent of the pension amount is 0.30 x Rs.600 = Rs.180
Assuming that the monthly interest rate corresponding to an annual interest rate of 12% is 1%, the discounted value of an annuity of Rs.180 receivable at the end of each month for 180 months (15 years) is:
Rs.180 x PVIFA (1%, 180)
(1.01)180 - 1Rs.180 x ---------------- = Rs.14,998
.01 (1.01)180
If Mr. Ramesh borrows Rs.P today on which the monthly interest rate is 1%
P x (1.01)60 = Rs.14,998P x 1.817 = Rs.14,998
Rs.14,998P = ------------ = Rs.8254
1.817
27. Rs.300 x PVIFA(r, 24 months) = Rs.6,000PVIFA (r,24) = Rs.6000 / Rs.300 = 20
From the tables we find that:PVIFA(1%,24) = 21.244PVIFA (2%, 24) = 18.914
Using a linear interpolation21.244 – 20.000
r = 1% + ---------------------- x 1% 21.244 – 18,914
= 1.53%
Thus, the bank charges an interest rate of 1.53% per month.The corresponding effective rate of interest per annum is
[ (1.0153)12 – 1 ] x 100 = 20%
28. The discounted value of the debentures to be redeemed between 8 to 10 years evaluated at the end of the 5th year is:
Rs.10 million x PVIF (8%, 3 years) + Rs.10 million x PVIF (8%, 4 years) + Rs.10 million x PVIF (8%, 5 years)
= Rs.10 million (0.794 + 0.735 + 0.681) = Rs.2.21 million
If A is the annual deposit to be made in the sinking fund for the years 1 to 5, thenA x FVIFA (8%, 5 years) = Rs.2.21 millionA x 5.867 = Rs.2.21 millionA = 5.867 = Rs.2.21 millionA = Rs.2.21 million / 5.867 = Rs.0.377 million
29. Let `n’ be the number of years for which a sum of Rs.20,000 can be withdrawn annually.
Rs.20,000 x PVIFA (10%, n) = Rs.100,000PVIFA (10 %, n) = Rs.100,000 / Rs.20,000 = 5.000
From the tables we find thatPVIFA (10%, 7 years) = 4.868PVIFA (10%, 8 years) = 5.335
Thus n is between 7 and 8. Using a linear interpolation we get
5.000 – 4.868n = 7 + ----------------- x 1 = 7.3 years
5.335 – 4.868
30. Equated annual installment = 500000 / PVIFA(14%,4)= 500000 / 2.914= Rs.171,585
Loan Amortisation Schedule
Beginning Annual Principal RemainingYear amount installment Interest repaid balance------ ------------- --------------- ----------- ------------- ------------- 1 500000 171585 70000 101585 398415 2 398415 171585 55778 115807 282608 3 282608 171585 39565 132020 150588 4 150588 171585 21082 150503 85*
(*) rounding off error
31. Define n as the maturity period of the loan. The value of n can be obtained from the equation.
200,000 x PVIFA(13%, n) = 1,500,000PVIFA (13%, n) = 7.500
From the tables or otherwise it can be verified that PVIFA(13,30) = 7.500Hence the maturity period of the loan is 30 years.
32. Expected value of iron ore mined during year 1 = Rs.300 million
Expected present value of the iron ore that can be mined over the next 15 years assuming a price escalation of 6% per annum in the price per tonne of iron
1 – (1 + g)n / (1 + i)n
= Rs.300 million x ------------------------ i - g
= Rs.300 million x 1 – (1.06) 15 / (1.16) 15 0.16 – 0.06
= Rs.300 million x (0.74135 / 0.10)= Rs.2224 million
33 (a) PV = Rs.500,000 (b) PV = 1,000,000PVIF10%,6yrs = 1,000,000 x 0.564 = Rs.564,000 (c ) PV = 60,000/r = 60,000/0.10 = Rs.600,000 (d) PV = 100,000 PVIFA10%,10yrs = 100,000 x 6.145 = Rs.614,500 (e) PV = C/(r-g) = 35,000/(0.10-0.05) = Rs.700,000
Option e has the highest present value viz. Rs.700,000
34. (a) PV = c/(r – g) = 12/[0.12 – (-0.03)] = Rs.80 million
1+g n
1 - -------(b) 1+r
PV = A(1+g) ----------------- = 12 x 0.97 x 0.9725 / 0.15 = Rs.75.466 million r - g
35. It may be noted that if g1 is the growth rate in the no. of units and g2 the growth rate in price
per unit, then the growth rate of their product, g = (1+g1)(1+g2) - 1In this problem the growth rate in the value of oil produced, g = (1- 0.05)(1 +0.03) - 1 = - 0.0215
Present value of the well’s production =
1+g n
1 - ------- 1+r
PV = A(1+g) ----------------- r - g
= (50,000 x 50) x ( 1-0.0215)x 1 – (0.9785 / 1.10) 15 0.10 + 0.0215
= $ 16,654,63336.
The growth rate in the value of the oil production g = (1- 0.06)(1 +0.04) - 1 = - 0.0224
Present value of the well’s production =
1+g n
1 - ------- 1+r
PV = A(1+g) ----------------- r - g
= (80,000 x 60) x ( 1-0.0224)x 1 – (0.9776 / 1.12) 20 0.12 + 0.0224
= $ 30,781,328.93
37. Future Value Interest Factor for Growing Annuity,( 1+ i )n – ( 1 + g)n
FVIFGA = i - g
(1. 09)20 – ( 1.08)20
So the value of the savings at the end of 20 years = 100,000 x 0.09 – 0.08
= Rs. 9,434,536
38Assuming 52 weeks in an year, the effective interest rate is
0.08 52 1 + - 1 = 1.0832 - 1 = 8.32 percent
52
MINICASE
Solution:
1. How much money would Ramesh need 15 years from now?
500,000 x PVIFA (10%, 15years)+ 1,000,000 x PVIF (10%, 15years)= 500,000 x 7.606 + 1,000,000 x 0.239= 3,803,000 x 239,000 = Rs.4,042,000
2. How much money should Ramesh save each year for the next 15 years to be able to meet his investment objective?
Ramesh’s current capital of Rs.600,000 will grow to :
600,000 (1.10)15 = 600,000 x 4.177 = Rs 2,506,200
This means that his savings in the next 15 years must grow to :
4,042,000 – 2,506,200 = Rs 1,535,800
So, the annual savings must be : 1,535,800 1,535,800
= = Rs.48,338FVIFA (10%, 15 years) 31.772
3. How much money would Ramesh need when he reaches the age of 60 to meet his donation objective?
200,000 x PVIFA (10% , 3yrs) x PVIF (10%, 11yrs)
= 200,000 x 2.487 x 0.317 = 157,676
4. What is the present value of Ramesh’s life time earnings?
400,000 400,000(1.12) 400,000(1.12)14
46 1 2 15
1.12 15
1 – 1.08
= 400,000 0.08 – 0.12
= Rs.7,254,962
Chapter 7
VALUATION OF BONDS AND STOCKS
1. 5 11 100P = +
t=1 (1.15)t (1.15)5
= Rs.11 x PVIFA(15%, 5 years) + Rs.100 x PVIF (15%, 5 years)= Rs.11 x 3.352 + Rs.100 x 0.497= Rs.86.7
2.(i) When the discount rate is 14%7 12 100
P = +t=1 (1.14) t (1.14)7
= Rs.12 x PVIFA (14%, 7 years) + Rs.100 x PVIF (14%, 7 years)= Rs.12 x 4.288 + Rs.100 x 0.4= Rs.91.46
(ii) When the discount rate is 12%7 12 100
P = + = Rs.100t=1 (1.12) t (1.12)7
Note that when the discount rate and the coupon rate are the same the value is equal to par value.
3. The yield to maturity is the value of r that satisfies the following equality. 7 120 1,000
Rs.750 = + t=1 (1+r) t (1+r)7
Try r = 18%. The right hand side (RHS) of the above equation is:Rs.120 x PVIFA (18%, 7 years) + Rs.1,000 x PVIF (18%, 7 years)= Rs.120 x 3.812 + Rs.1,000 x 0.314= Rs.771.44
Try r = 20%. The right hand side (RHS) of the above equation is:Rs.120 x PVIFA (20%, 7 years) + Rs.1,000 x PVIF (20%, 7 years)= Rs.120 x 3.605 + Rs.1,000 x 0.279= Rs.711.60
Thus the value of r at which the RHS becomes equal to Rs.750 lies between 18% and 20%.
Using linear interpolation in this range, we get
771.44 – 750.00Yield to maturity = 18% + 771.44 – 711.60 x 2%
= 18.7%
4. 10 14 100
80 = + t=1 (1+r) t (1+r)10
Try r = 18%. The RHS of the above equation is
Rs.14 x PVIFA (18%, 10 years) + Rs.100 x PVIF (18%, 10 years)= Rs.14 x 4.494 + Rs.100 x 0.191 = Rs.82
Try r = 20%. The RHS of the above equation isRs.14 x PVIFA(20%, 10 years) + Rs.100 x PVIF (20%, 10 years)= Rs.14 x 4.193 + Rs.100 x 0.162= Rs.74.9
Using interpolation in the range 18% and 20% we get:
82 - 80Yield to maturity = 18% + ----------- x 2%
82 – 74.9
= 18.56%
5.12 6 100
P = +t=1 (1.08) t (1.08)12
= Rs.6 x PVIFA (8%, 12 years) + Rs.100 x PVIF (8%, 12 years)= Rs.6 x 7.536 + Rs.100 x 0.397= Rs.84.92
6. The post-tax interest and maturity value are calculated below:
Bond A Bond B
* Post-tax interest (C ) 12(1 – 0.3) 10 (1 – 0.3)=Rs.8.4 =Rs.7
* Post-tax maturity value (M) 100 - 100 -[ (100-70)x 0.1] [ (100 – 60)x 0.1]=Rs.97 =Rs.96
The post-tax YTM, using the approximate YTM formula is calculated below
8.4 + (97-70)/10Bond A : Post-tax YTM = --------------------
0.6 x 70 + 0.4 x 97
= 13.73%
7 + (96 – 60)/6Bond B : Post-tax YTM = ----------------------
0.6x 60 + 0.4 x 96
= 17. 47%
7.14 6 100
P = +t=1 (1.08) t (1.08)14
= Rs.6 x PVIFA(8%, 14) + Rs.100 x PVIF (8%, 14)= Rs.6 x 8.244 + Rs.100 x 0.341= Rs.83.56
8. Do = Rs.2.00, g = 0.06, r = 0.12
Po = D1 / (r – g) = Do (1 + g) / (r – g)
= Rs.2.00 (1.06) / (0.12 - 0.06)= Rs.35.33
Since the growth rate of 6% applies to dividends as well as market price, the market price at the end of the 2nd year will be:
P2 = Po x (1 + g)2 = Rs.35.33 (1.06)2
= Rs.39.70
9. Po = D1 / (r – g) = Do (1 + g) / (r – g)= Rs.12.00 (1.10) / (0.15 – 0.10) = Rs.264
10. Po = D1 / (r – g)
Rs.32 = Rs.2 / (0.12 – g)g = 0.0575 or 5.75%
11. Po = D1/ (r – g) = Do(1+g) / (r – g)Do = Rs.1.50, g = -0.04, Po = Rs.8So8 = 1.50 (1- .04) / (r-(-.04)) = 1.44 / (r + .04)
Hence r = 0.14 or 14 per cent
12. The market price per share of Commonwealth Corporation will be the sum of three components:
A: Present value of the dividend stream for the first 4 yearsB: Present value of the dividend stream for the next 4 yearsC: Present value of the market price expected at the end of 8 years.
A = 1.50 (1.12) / (1.14) + 1.50 (1.12)2 / (1.14)2 + 1.50(1.12)3 / (1.14)3 ++ 1.50 (1.12)4 / (1.14)4
= 1.68/(1.14) + 1.88 / (1.14)2 + 2.11 / (1.14)3 + 2.36 / (1.14)4
= Rs.5.74
B = 2.36(1.08) / (1.14)5 + 2.36 (1.08)2 / (1.14)6 + 2.36 (1.08)3 / (1.14)7 ++ 2.36 (1.08)4 / (1.14)8
= 2.55 / (1.14)5 + 2.75 / (1.14)6 + 2.97 / (1.14)7 + 3.21 / (1.14)8
= Rs.4.89
C = P8 / (1.14)8
P8 = D9 / (r – g) = 3.21 (1.05)/ (0.14 – 0.05) = Rs.37.45So
C = Rs.37.45 / (1.14)8 = Rs.13.14
Thus,Po = A + B + C = 5.74 + 4.89 + 13.14
= Rs.23.77
13. Let us assume a required rate of return of 12 percent. The intrinsic value of the equity share will be the sum of three components:
A: Present value of the dividend stream for the first 5 years when the growth rate expected is 15%.
B: Present value of the dividend stream for the next 5 years when the growth rate is expected to be 10%.
C: Present value of the market price expected at the end of 10 years.
2.00 (1.15) 2.00 (1.15)2 2.00 (1.15)3 2.00(1.15)4 2.00 (1.15)5
A = ------------- + ------------- +-------------- + ------------- + ------------- (1.12) (1.12)2 (1.1.2)3 (1.1.2)4 (1.12)5
= 2.30 / (1.12) + 2.65 / (1.12)2 + 3.04 / (1.12)3 + 3.50 / (1.12)4 + 4.02/(1.12)5
= Rs.10.84
4.02(1.10) 4.02 (1.10)2 4.02(1.10)3 4.02(1.10)4 4.02 (1.10)5
B = ------------ + ---------------- + ------------- + --------------- + --------------- (1.12)6 (1.12)7 (1.12)8 (1..12)9(1.12)10
4.42 4.86 5.35 5.89 6.48 = --------- + -------------- + --------------- + ------------- + -------------
(1.12)6 (1.12)7 (1.12)8 (1.1.2)9(1.12)10
= Rs.10.81
D11 1 6.48 (1.05)C = -------- x --------------- = ------------------- x 1/(1.12)10
r – g (1 +r)10 0.12 – 0.05
= Rs.97.20
The intrinsic value of the share = A + B + C= 10.84 + 10.81 + 97.20 = Rs.118.85
14. Terminal value of the interest proceeds= 140 x FVIFA (16%,4)= 140 x 5.066= 709.24
Redemption value = 1,000
Terminal value of the proceeds from the bond = 1709.24
Define r as the yield to maturity. The value of r can be obtained from the equation
900 (1 + r)4 = 1709.24r = 0.1739 or 17.39%
15. Intrinsic value of the equity share (using the 2-stage growth model)
(1.18)6
2.36 x 1 - ----------- 2.36 x (1.18)5 x (1.12) (1.16)6
= --------------------------------- + -----------------------------------0.16 – 0.18 (0.16 – 0.12) x (1.16)6
- 0.10801= 2.36 x ----------- + 62.05
- 0.02
= Rs.74.80
16. Intrinsic value of the equity share (using the H model)
4.00 (1.20) 4.00 x 4 x (0.10)= -------------- + ---------------------
0.18 – 0.10 0.18 – 0.10
= 60 + 20= Rs.80
17.
Po = D1
r – g
Po
=Rs. 8 = Rs. 266.7
0.15-0.12
Po = E1 + PVGOr
Po = Rs. 20 + PVGO0.15
Rs. 266.7 = Rs. 133.3 + PVGO
So, PVGO = Rs. 133.4
MINICASE
(a) The value of a bond is calculated using the formula n C M
P = + t=1 (1+r)t (1+r)n
where P is the value (in rupees), n is the number of years to maturity, C is the annual coupon payment (in rupees), r is the periodic required return, M is the maturity value, and t is the time when the payment is received
(b) Value of the bond = 100 PVIFA8% , 5years + 1000 PVIF8% , 5years
= 100 x 3.993 + 1000 x 0.681 = Rs.1080.30
100 + ( 1000 – 1060)/8 (c) Approximate YTM = = 8.93%
0.4 x 1000 + 0.6 x 1060
(d) 100 + ( 1050 – 1060)/2 Approximate YTC = = 9 %
0.4 x 1050 + 0.6 x 1060
(e) The general formula for valuing any stock is : Dt
P0 =
t=1 (1+r)t
(f) A constant growth stock is valued using the formula
D1
P0 = r - g
where D1 is the dividend expected a year hence, r is the required rate of return and g is the constant growth rate
(g)
(i) The expected value of the stock a year from nowD2 6 x (1+0.12)2
P1 = = = Rs.250.88 r - g 0.15 – 0.12
6 x 1.12(ii) Price of the stock at present, P0 = = Rs.224
0.15 – 0.12
Expected dividend in the first year = 6 x1.12 = Rs.6.72 6.72
Dividend yield = x 100 = 3 % 224
Expected capital gains in the first year = P1 –P0 = 250.88 – 224 = Rs.26.88 26.88
Capital gains yield = x 100 = 12 % 224
(h) Present value of the stock is :
1+g1 n
1 - 1+r D1 (1+g1)n-1 (1+g2) 1
P0 = D1 + r - g1 r - g2 (1+r)n
1.25 4
1 - 1.16 (10 x 1.25) x (1.25)3 x 1.10 1 = (10 x 1.25) + x
0.16 – 0.10 (1.16)4
0.16 – 0.25
= 48.38 + 447.59 /1.81 = Rs. 295.67
(i) Intrinsic value per share:
D0 [(1+gn) + H (ga-gn)]P0 =
r - gn
8[ 1.10 + 1.5 x (0.20 – 0.10] = = Rs. 250
0.14 – 0.10
-----------------------------------------------------------------------------------------------------------------------
Chapter 8RISK AND RETURN
1 (a) Expected price per share a year hence will be:
= 0.4 x Rs.10 + 0.4 x Rs.11 + 0.2 x Rs.12 = Rs.10.80
(b) Probability distribution of the rate of return is
Rate of return (Ri) 10% 20% 30%
Probability (pi) 0.4 0.4 0.2
Note that the rate of return is defined as:
Dividend + Terminal price-------------------------------- - 1
Initial price
2 (a) For Rs.1,000, 20 shares of Alpha’s stock can be acquired. The probability distribution of the return on 20 shares is
Economic Condition Return (Rs) ProbabilityHigh Growth 20 x 55 = 1,100 0.3Low Growth 20 x 50 = 1,000 0.3Stagnation 20 x 60 = 1,200 0.2Recession 20 x 70 = 1,400 0.2
Expected return = (1,100 x 0.3) + (1,000 x 0.3) + (1,200 x 0.2) + (1,400 x 0.2)
= 330 + 300 + 240 + 280= Rs.1,150
Standard deviation of the return = [(1,100 – 1,150)2 x 0.3 + (1,000 – 1,150)2 x
0.3 + (1,200 – 1,150)2 x 0.2 + (1,400 – 1,150)2 x 0.2]1/2
= Rs.143.18
(b) For Rs.1,000, 20 shares of Beta’s stock can be acquired. The probability distribution of the return on 20 shares is:
Economic condition Return (Rs) Probability
High growth 20 x 75 = 1,500 0.3Low growth 20 x 65 = 1,300 0.3Stagnation 20 x 50 = 1,000 0.2Recession 20 x 40 = 800 0.2
Expected return = (1,500 x 0.3) + (1,300 x 0.3) + (1,000 x 0.2) + (800 x 0.2) = Rs.1,200
Standard deviation of the return = [(1,500 – 1,200)2 x .3 + (1,300 – 1,200)2 x .3 + (1,000 – 1,200)2 x .2 + (800 – 1,200)2 x .2]1/2 = Rs.264.58
(c ) For Rs.500, 10 shares of Alpha’s stock can be acquired; likewise for Rs.500, 10 shares of Beta’s stock can be acquired. The probability distribution of this option is:
Return (Rs) Probability(10 x 55) + (10 x 75) = 1,300 0.3(10 x 50) + (10 x 65) = 1,150 0.3(10 x 60) + (10 x 50) = 1,100 0.2(10 x 70) + (10 x 40) = 1,100 0.2
Expected return = (1,300 x 0.3) + (1,150 x 0.3) + (1,100 x 0.2) + (1,100 x 0.2)
= Rs.1,175Standard deviation = [(1,300 –1,175)2 x 0.3 + (1,150 – 1,175)2 x 0.3 +
(1,100 – 1,175)2 x 0.2 + (1,100 – 1,175)2 x 0.2 ]1/2
= Rs.84.41d. For Rs.700, 14 shares of Alpha’s stock can be acquired; likewise for Rs.300, 6 shares of Beta’s stock can be acquired. The probability distribution of this
option is:
Return (Rs) Probability
(14 x 55) + (6 x 75) = 1,220 0.3(14 x 50) + (6 x 65) = 1,090 0.3(14 x 60) + (6 x 50) = 1,140 0.2(14 x 70) + (6 x 40) = 1,220 0.2
Expected return = (1,220 x 0.3) + (1,090 x 0.3) + (1,140 x 0.2) + (1,220 x 0.2) = Rs.1,165
Standard deviation = [(1,220 – 1,165)2 x 0.3 + (1,090 – 1,165)2 x 0.3 + (1,140 – 1,165)2 x 0.2 + (1,220 – 1,165)2 x 0.2]1/2
= Rs.57.66
The expected return to standard deviation of various options are as follows :
OptionExpected return
(Rs)Standard deviation
(Rs)Expected / Standard return deviation
a 1,150 143 8.04b 1,200 265 4.53c 1,175 84 13.99d 1,165 58 20.09
Option `d’ is the most preferred option because it has the highest return to risk ratio.
3.(a) Define RA and RM as the returns on the equity stock of Auto Electricals Limited a and Market portfolio respectively. The calculations relevant for calculating the beta of the stock are shown below:
Year RA RM RA-RA RM-RM (RA-RA) (RM-RM) RA-RA/RM-RM
1 15 12 -0.09 -3.18 0.01 10.11 0.292 -6 1 -21.09 -14.18 444.79 201.07 299.063 18 14 2.91 -1.18 8.47 1.39 -3.434 30 24 14.91 8.82 222.31 77.79 131.515 12 16 0-3.09 0.82 9.55 0.67 -2.536 25 30 9.91 14.82 98.21 219.63 146.877 2 -3 -13.09 -18.18 171.35 330.51 237.988 20 24 4.91 8.82 24.11 77.79 43.319 18 15 2.91 -0.18 8.47 0.03 -0.5210 24 22 8.91 6.82 79.39 46.51 60.7711 8. 12 -7.09 -3.18 50.27 10.11 22.55
RA = 15.09 RM = 15.18
(RA – RA)2 = 1116.93 (RM – RM) 2 = 975.61 (RA – RA) (RM – RM) = 935.86
Beta of the equity stock of Auto Electricals (RA – RA) (RM – RM)
(RM – RM) 2
= 935.86 = 0.96975.61
(b)Alpha = RA – βA RM
= 15.09 – (0.96 x 15.18)= 0.52
Equation of the characteristic line is
RA = 0.52 + 0.96 RM
4 The required rate of return on stock A is:
RA = RF + βA (RM – RF)= 0.10 + 1.5 (0.15 – 0.10)= 0.175
Intrinsic value of share = D1 / (r- g) = Do (1+g) / ( r – g)
Given Do = Rs.2.00, g = 0.08, r = 0.175 2.00 (1.08)
Intrinsic value per share of stock A = 0.175 – 0.08
= Rs.22.74
5. The SML equation is RA = RF + βA (RM – RF)
Given RA = 15%. RF = 8%, RM = 12%, we have
0.15 = .08 + βA (0.12 – 0.08)
0.07i.e.βA = = 1.75
0.04
Beta of stock A = 1.75
6. The SML equation is: RX = RF + βX (RM – RF)
We are given 0.15 = 0.09 + 1.5 (RM – 0.09) i.e., 1.5 RM = 0.195or RM = 0.13%
Therefore return on market portfolio = 13%
7. RM = 12% βX = 2.0 RX =18% g = 5% Po = Rs.30
Po = D1 / (r - g)
Rs.30 = D1 / (0.18 - .05)
So D1 = Rs.39 and Do = D1 / (1+g) = 3.9 /(1.05) = Rs.3.71
Rx = Rf + βx (RM – Rf)
0.18 = Rf + 2.0 (0.12 – Rf)
So Rf = 0.06 or 6%.
Original Revised
Rf 6% 8%RM – Rf 6% 4%g 5% 4%βx 2.0 1.8
Revised Rx = 8% + 1.8 (4%) = 15.2%
Price per share of stock X, given the above changes is
3.71 (1.04)= Rs.34.45
0.152 – 0.048
We know that:
Debt (1-tc) β equity = β assets 1 + Equity i.e
β equity 1.1 β assets = = = 0.71
Debt(1-tc) 4 1 + ----------- 1 + --- ( 1 – 0.30)
Equity 5
9
RB=138RB=13.8
RM=118RM= 11.8
(RB-RB)(RM-RM)=1223.6
(RM-RM)2=1191.6
RM-RM (RB-RB)(RM-RM)
= 132.4 CovB,M n-1 n-1
CovB,M 136Beta :
Alpha = B = RB - RM = 13.8 – 1.03 x 11.8 = 1.65% The characteristic line for stock B is : RB = 1.65 + 1.03 RM
Chapter 9RISK AND RETURN: PORTFOLIO THEORY AND ASSET PRICING MODELS
1. (a)E (R1) = 0.2(-5%) + 0.3(15%) + 0.4(18%) + .10(22%)
= 12%E (R2) = 0.2(10%) + 0.3(12%) + 0.4(14%) + .10(18%)
= 13%σ(R1) = [.2(-5 –12)2 + 0.3 (15 –12)2 + 0.4 (18 –12)2 + 0.1 (22 – 12)2]½
= [57.8 + 2.7 + 14.4 + 10]½ = 9.21%
σ(R2) = [.2(10 –13)2 + 0.3(12 – 13)2 + 0.4 (14 – 13)2 + 0.1 (18 – 13)2] ½
= [1.8 + 0.09 + 0.16 + 2.5] ½ = 2.13%
(b) The covariance between the returns on assets 1 and 2 is calculated below
State of nature
Probability Return on asset 1
Deviation of return on asset 1 from its mean
Return on asset 2
Deviation of the
return on asset 2 from its mean
Product of deviation
times probability
(1) (2) (3) (4) (5) (6) (2)x(4)x(6)1 0.2 -5% -17% 10% -3% 10.22 0.3 15% 3% 12% -1% -0.9%3 0.4 18% 6% 14% 1% 2.44 0.1 22% 10% 18% 5% 5
Sum = 16.7
Thus the covariance between the returns of the two assets is 16.7.
(c) The coefficient of correlation between the returns on assets 1 and 2 is:Covariance12 16.7
= = 0.85 σ1 x σ2 9.21 x 2.13
2. Expected rates of returns on equity stock A, B, C and D can be computed as follows:
A: 0.10 + 0.12 + (-0.08) + 0.15 + (-0.02) + 0.20 = 0.0783 = 7.83%6
B: 0.08 + 0.04 + 0.15 +.12 + 0.10 + 0.06 = 0.0917 = 9.17%6
C: 0.07 + 0.08 + 0.12 + 0.09 + 0.06 + 0.12 = 0.0900 = 9.00%
D: 0.09 + 0.09 + 0.11 + 0.04 + 0.08 + 0.16 = 0.095 = 9.50%
6
(a) Return on portfolio consisting of stock A = 7.83%
(b) Return on portfolio consisting of stock A and B in equalproportions = 0.5 (0.0783) + 0.5 (0.0917)
= 0.085 = 8.5%
(c ) Return on portfolio consisting of stocks A, B and C in equalproportions = 1/3(0.0783 ) + 1/3(0.0917) + 1/3 (0.090)
= 0.0867 = 8.67%
(d) Return on portfolio consisting of stocks A, B, C and D in equalproportions = 0.25(0.0783) + 0.25(0.0917) + 0.25(0.0900) +
0.25(0.095)= 0.08875 = 8.88%
3. The standard deviation of portfolio return is:
p= [w121
2 + w222
2 + w323
2 + 424
2 + 2 w1 w2 12 1 2 + 2 w1 w3 13 1 3 + 2 w1 w4
14 14 + 2 w2 w3 23 2 3 + 2 w2 w4 24 2 4 + 2 w3 w4 34 3 4 ]1/2
= [0.22 x 42 + 0.32 x 82 + 0.42 x 202 + 0.12 x 102 + 2 x 0.2 x 0.3 x 0.3 x 4 x 8 + 2 x 0.2 x 0.4 x 0.5 x 4 x 20 + 2 x 0.2 x 0.1 x 0.2 x 4 x 10 + 2 x 0.3 x 0.4 x 0.6 x 8 x 20 + 2 x 0.3 x 0.1 x 0.8 x 8 x 10 + 2 x 0.4 x 0.1 x 0.4 x 20 x 10]1/2
= 10.6%
MINICASEa. For stock A:
Expected return = (0.2 x -15) + (0.5 x 20) + (0.3 x 40) = 19
Standard deviation = [ 0.2 ( -15 -19)2 + 0.5 (20-19)2 + 0.3 (40 – 19)2 ] 1/2
= [231.2 + 0.5 + 132.3]1/2 = 19.08
For stock B:
Expected return = (0.2 x 30) + (0.5 x 5) + [ 0.3 x (-) 15] = 4Standard deviation = [0.2 ( 30 – 4)2 + 0.5 (5 -4)2 + 0.3 (-15–4)2]1/2
= (135.2 + 0.5 + 108.3) ½ = 15.62For stock C:
Expected return = [0.2 x (-5)] + (0.5 x 15) + (0.3 x 25)] = 14
Standard deviation = [0.2 (-5 – 14)2 + 0.5 (15 -14)2 + 0.3 (25-14)2] ½
= [72.2 + 0.5 + 36.3] ½ = 10.44
For market portfolio:
Expected return = [0.2 x (-)10] + (0.5 x 16) + (0.3 x 30) = -2 + 8 + 9 = 15
Standard deviation = [0.2 (-10-15)2 + 0.5(16-15)2 + 0.3 (30 – 15)2] ½
= ( 125 + 0.5 + 67.5 ) ½ = 13.89
b.
State of the Economy
Prob-ability (p)
Return on A (%) (RA)
Return B (%) (RB)
RA-E(RA) RB-E(RB) px [RA-E(RA)]x[RB-E(RB)]
Recession 0.2 -15 30 -34 26 -176.8Normal 0.5 20 5 1 1 0.5Boom 0.3 40 -15 21 -19 - 119.7
total = - 296.00
Covariance between the returns of A and B is (-) 296
State of the Economy
Prob-ability (p)
Return on A (%) (RA)
Return C (%) (RC)
RA-E(RA) RC-E(RC) px [RA-E(RA)]x[RC-E(RC)]
Recession 0.2 -15 - 5.0 -34 -19 129.2Normal 0.5 20 15.0 1. 1 0.5Boom 0.3 40 25.0 21 11 69.3
total = 199.0
Covariance between the returns of A and C is 199
(-) 296c. Coefficient of correlation between the returns of A and B = = (-) 1
19.08 x 15.62
199Coefficient of correlaton between the returns of A and C = = 1
19.08 x 10.44
d. Portfolio in which stocks A and B are equally weighted:
Economic condition Probability Overall expected returnRecession 0.2 0.5 x (-) 15 + 0.5 x 30 = 7.5Normal 0.5 0.5 x 20 + 0.5 x 5 = 12.5 Boom 0.3 0.5 x 40 + 0.5 x (-)15 = 12.5
Expected return of the portfolio = (0.2 x 7.5) + (0.5 x 12.5) + (0.3 x 12.5) = 0.7 + 6.25 + 4.5 = 11.5
Standard deviation of the portfolio = [ 0.2 (7.5 – 11.5)2 + 0.5 (12.5 – 11.5)2 + 0.3 (12.5 – 11.5)2]1/2
= [ 3.2 + 0.5 + 0.3] ½ = 2
Portfolio in which weights assigned to stocks A, B and C are 0.4, 0.4 and 0.2 respectively.
Expected return of the portfolio = (0.4 x 19.0) + (0.4 x 4) + (0.2 x 14)) = 12
For calculating the standard deviation of the portfolio we also need covariance between B and C, which is calculated as under:
State of the Economy
Prob-ability (p)
Return onB (%) (RB)
Return on C (%) (RC)
RB-E(RB) RC-E(RC) p x[RB-E(RB)]x[RC-E(RC)]
Recession 0.2 30 - 5.0 26 -19 (-) 98.8Normal 0.5 5 15.0 1 1 0.50Boom 0.3 (-)15 25.0 (-)19 11 (-) 62.7
total = (-)161
Covariance between the returns of B and C is (-)161
We have the following values:WA = 0.4 WB = 0.4 WC = 0.2 σA = 19.08 σB = 15.62 σC = 10.44 σAB = (-)296 σAC = 199 σBC = (-) 161
Standard deviation
= [ (0.4 x 19.08)2 + (0.4 x 15.62)2 + (0.2 x 10.44)2 + [ 2 x 0.4 x 0.4 x (-) 296 ] + + [2 x 0.4 x 0.2 x 199] + [2 x 0.4 x 0.2 x (-) 161]1/2
= (58.25 + 39.04 + 4.36– 94.72 + 31.84 – 25.76)1/2 = 3.61e.
(i) Risk-free rate is 6% and market risk premium is 15 – 6 = 9% The SML relationship is Required return = 6% + β x 9%
(ii) For stock A:Required return = 6 % + 1.2 x 9 % = 16.8 %; Expected return = 19 %Alpha = 19 – 16.8 = 2.2 %
For stock B:Required return = 6 % - 0.70 x 9 % = - 0.3 %; Expected return = 4 %Alpha = 4 + 0.3 = 4.3 %
For stock C:Required return = 6% + 0.9 x 9 % = 14.1 %; Expected return = 14%Alpha = 14 – 14.1 = (-) 0.1 %
f.
σ2
m = 218.8/4 = 54.7 Cov (D,M) = 335.6/4 = 83.9 ß = 83.9 / 54.7 = 1.53
Interpretation: The change in return of D is expected to be 1.53 times the expected change in return on the market portfolio.
h.
CAPM assumes that return on a stock/portfolio is solely influenced by the market factor whereas the APT assumes that the return is influenced by a set of factors called risk factors.
Chapter 10OPTIONS AND THEIR VALUATION
1. S = 100 u = 1.5 d = 0.8
E = 105 r = 0.12 R = 1.12
The values of ∆ (hedge ratio) and B (amount borrowed) can be obtained as follows:
Cu – Cd
∆ =(u – d) S
Cu = Max (150 – 105, 0) = 45
Cd = Max (80 – 105, 0) = 0
45 – 0 45 9∆ = = = = 0.6429
0.7 x 100 70 14
u.Cd – d.Cu
B =(u-d) R
(1.5 x 0) – (0.8 x 45)=
0.7 x 1.12
-36= = - 45.92
0.784
C = ∆ S + B= 0.6429 x 100 – 45.92= Rs.18.37
Value of the call option = Rs.18.37
2. S = 40 u = ? d = 0.8R = 1.10 E = 45 C = 8
We will assume that the current market price of the call is equal to the pair value of the call as per the Binomial model.
Given the above data
Cd = Max (32 – 45, 0) = 0
∆ Cu – Cd R= x
B u Cd – d Cu S
∆ Cu – 0 1.10= x
B -0.8Cu 40
= (-) 0.034375
∆ = - 0.34375 B (1)C = ∆ S + B8 = ∆ x 40 + B (2)
Substituting (1) in (2) we get
8 = (-0.034365 x 40) B + B8 = -0.375 Bor B = - 21.33
∆ = - 0.034375 (-21.33) = 0.7332
The portfolio consists of 0.7332 of a share plus a borrowing of Rs.21.33 (entailing a repayment of Rs.21.33 (1.10) = Rs.23.46 after one year). It follows that when u occurs either u x 40 x 0.7332 – 23.46 = u x 40 – 45
-10.672 u = -21.54 u = 2.02
or
u x 40 x 0.7332 – 23.46 = 0u = 0.8
Since u > d, it follows that u = 2.02.Put differently the stock price is expected to rise by 1.02 x 100 = 102%.
3. Using the standard notations of the Black-Scholes model we get the following results:ln (S/E) + rt + σ2 t/2
d1 = t
= ln (120 / 110) + 0.14 + 0.4 2 /2 0.4
= 0.08701 + 0.14 + 0.08 0.4
= 0.7675
d2 = d1 - t= 0.7675 – 0.4= 0.3675
N(d1) = N (0.7675) ~ N (0.77) = 0.80785N (d2) = N (0.3675) ~ N (0.37) = 0.64431
C = So N(d1) – E. e-rt. N(d2)= 120 x 0.80785 – 110 x e-0.14 x 0.64431= (120 x 0.80785) – (110 x 0.86936 x 0.64431)= 35.33
Value of the call as per the Black and Scholes model is Rs.35.33.4
l (S/E) + (r + σ2 /2) td1 =
t
= ln (80 / 82) + [0.1503 + (0.2) 2 /2] 0.2
= -0.0247 + 0.1703 0.2
= 0.7280
d2 = d1 - t= 0.7280 – 0.2= 0.5280
N(d1) = N (0.7280). From the tables we have N(0.70) = 1- 0.2420 = 0.7580
and N(0.75)= 1- 0.2264 = 0.7736By linear extrapolation, we get N(0.7280) = 0.7580 + (0.7280 – 0.7000)(0.7736-0.7580)/0.05
= 0.7580 + 0.008736 = 0.7667N(d2) = N(0.5280) From the tables we have N(0.50) = 1- 0.3085 = 0.6915
N(0.55) = 1- 0.2912 = 0.7088By linear extrapolation, we getN(0.5280) = 0.6915 + (0.5280 – 0.5000)(0.7088 – 0.6915)/0.05
= 0.6915 + 0.009688 = 0.7012E/ert = 82/1.1622 = 70.5558C = So N(d1) – E. e-rt. N(d2) = 80 x 0.7667 -70.5558 x 0.7012 = 11.86
5
l (S/E) + (r + σ2 /2) td1 =
t
= ln (80 / 85) + [0.1503 + (0.2) 2 /2] 0.2
= -0.060625 + 0.1703 0.2
= 0.5484
d2 = d1 - t= 0.5484 – 0.2= 0.3484
N(d1) = N (0.5484).
From the tables we have N(0.50) = 1- 0.3085 = 0.6915and N(0.55)= 1- 0.2912 = 0.7088
By linear extrapolation, we get N(0.5484) = 0.6915 + (0.5484 – 0.5000)(0.7088-0.6915)/0.05
= 0.6915 + 0.0167 = 0.7082N(d2) = N(0.3484) From the tables we have N(0.30) = 1- 0.3821 = 0.6179
N(0.35) = 1- 0.3632 = 0.6368By linear extrapolation, we getN(0.3484) = 0.6179 + (0.3484 – 0.3000)(0.6368 – 0.6179)/0.05
= 0.6179 + 0.0183= 0.6362E/ert = 85/1.1622 = 73.1372C = So N(d1) – E. e-rt. N(d2) = 80 x 0.7082 -73.1372 x 0.6362 = 10.13P = C –S + E/ert
= 10.13 – 80 + 73.1372 = 3.27
Value of the put option = Rs.3.27
6. So = Vo N(d1) – B1 e –rt N (d2)
= 6000 N (d1) – 5000 e – 0.1 N(d2)
ln (6000 / 5000) + (0.1 x 1) + (0.18/2)d1 = ----------------------------------------------
0.18 x 1
ln (1.2) + 0.19=
0.4243
= 0.8775 = 0.88
N(d1) = N (0.88) = 0.81057d2 = d1 - t
= 0.8775 - 0.18= 0.4532 = 0.45
N (d2) = N (0.45) = 0.67364So = 6000 x 0.81057 – (5000 x 0.9048 x 0.67364)
= 1816
B0 = V0 – S0
= 6000 – 1816
= 4184
MINICASE b)
Call options with strike prices 280, 300 and 320 and put options with strike prices 340and 360 are in - the – money. Call options with strike prices 340 and 360 and put options with strike prices 280, 300 and 320 are out of – the – money.
c) (i) If Pradeep Sharma sells Jan/340 call on 1000 shares, he will earn a call premium of Rs.5000 now. However, he will forfeit the gains that he would have enjoyed if the price of Newage Hospitals rises above Rs.340. (ii) If Pradeep Sharma sells Mar/300 call on 1000 shares, he will earn a call premium of Rs.41,000 now. However, he will forfeit the gains he would have enjoyed if the price of Newage Hospital remains above Rs.300.
d) Let s be the stock price, p1 and p2 the call premia for March/ 340 and March/ 360 calls respectively. When s is greater than 360, both the calls will be exercised and the profit will be { s-340-p1} – { s-360-p2 } =Rs. 11 The maximum loss will be the initial investment , i.e. p1-p2 =Rs. 9 The break even will occur when the gain on purchased call equals the
net premium paid i.e. s-340 = p1 – p2
=9 Therefore s= Rs. 349
e) If the stock price goes below Rs.300, Mr. Sharma can execute the put option and ensure that his portfolio value does not go below Rs. 300 per share. However , if stock price goes above Rs. 340, the call will be exercised and the stocks in the portfolio will have to be delivered/ sold to meet the obligation, thus limiting the upper value of the portfolio to Rs. 340 per share. So long as the share price hovers between R. 300 and Rs. 340, Mr. Sharma will be gainer by Rs. 8 ( net premium received).
f). Other things remaining constant, value of a call option - increases when the current price of the stock increases. - decreases when the exercise price increases. - increases when option term to maturity increases. - increases when the risk-free interest rate increases. - increases when the variability of the stock price increases.
g). The three equations are E
C0 = S0 N(d1) - ------ N (d2) ert
S0 σ2
ln ------ + r + ----- E 2
d1 = σ t
d2 = d1 - σ √ t
S0 = 325 E =320 t =0.25 r = 0.06 σ =0.30
325 (0.30)2
ln + 0.06 + x 0.25 320 2d1 =
0.30 x 0.25
0
Profit
Stock price
·305 375340
Pay off
= ( 0.0155 + 0.02625) / 0.15 = 0. 2783
d2 = 0.2783 -0.30 √0.25 = 0.2783 – 0.15 = 0.1283 Using normal distribution table
N (d1) = 1 – [ 0.3821 + ( 0.4013- 0. 3821) ( 0.30 – 0.2783 ) /( 0.30 – 0.25) ]=1- [ 0.3821 + 0. 0192 x 0.0217 / 0.05 ] = 0.6096
N ( d2 ) = 1- [ 0. 4404 + ( 0. 4602- 0.4404) ( 0. 15 – 0. 1283 /( 0. 15- 0.10 ) ] = 1- [ 0.4404 + 0.0198 x 0.0217 / 0.05 ] = 0. 5510
E / ert = 320 / e0.06 x 0. 25 = 320 / 1. 0151 = 315. 24
C0 = 325 x 0.6096 – 315.24 x 0. 5510 = 198.12 – 173. 70 = Rs. 24.42
Chapter 11TECHNIQUES OF CAPITAL BUDGETING
1.(a) NPV of the project at a discount rate of 14%.
= - 1,000,000 + 100,000 + 200,000---------- ------------ (1.14) (1.14)2
+ 300,000 + 600,000 + 300,000 ----------- ---------- ---------- (1.14)3 (1.14)4 (1.14)5
= - 44837
(b) NPV of the project at time varying discount rates
= - 1,000,000
+ 100,000
(1.12)
+ 200,000
(1.12) (1.13)
+ 300,000
(1.12) (1.13) (1.14)
+ 600,000
(1.12) (1.13) (1.14) (1.15)
+ 300,000 (1.12) (1.13) (1.14)(1.15)(1.16)
= - 1,000,000 + 89286 + 158028 + 207931 + 361620 + 155871= - 27264
2. IRR (r) can be calculated by solving the following equations for the value of r. 60000 x PVIFA (r,7) = 300,000
i.e., PVIFA (r,7) = 5.000
Through a process of trial and error it can be verified that r = 9.20% pa.
3. The IRR (r) for the given cashflow stream can be obtained by solving the following equation for the value of r.
-3000 + 9000 / (1+r) – 3000 / (1+r) = 0
Simplifying the above equation we get
r = 1.61, -0.61; (or) 161%, (-)61%
NOTE: Given two changes in the signs of cashflow, we get two values for the IRR of the cashflow stream. In such cases, the IRR rule breaks down.
4. Define NCF as the minimum constant annual net cashflow that justifies the purchase of the given equipment. The value of NCF can be obtained from the equation
NCF x PVIFA (10,8) = 500000NCF = 500000 / 5.335
= 93271
5. Define I as the initial investment that is justified in relation to a net annual cashinflow of 25000 for 10 years at a discount rate of 12% per annum. The value of I can be obtained from the following equation
25000 x PVIFA (12,10) = Ii.e., I = 141256
6. Let us assume a discount rate of 15 %.
PV of benefits (PVB) = 25000 x PVIF (15,1)+ 40000 x PVIF (15,2)+ 50000 x PVIF (15,3)+ 40000 x PVIF (15,4)+ 30000 x PVIF (15,5)= 122646 (A)
Investment = 100,000 (B)
Benefit cost ratio = 1.23 [= (A) / (B)]
7. The NPV’s of the three projects are as follows:
Project P Q R
Discount rate
0% 400 500 6005% 223 251 312
10% 69 40 7015% - 66 - 142 - 135
25% - 291 - 435 - 46130% - 386 - 555 - 591
8 NPV profiles for Projects P and Q for selected discount rates are as follows:(a)
ProjectP Q
Discount rate (%) 0 2950 500 5 1876 20810 1075 - 2815 471 - 22220 11 - 382
b) (i) The IRR (r ) of project P can be obtained by solving the following equation for `r’.
-1000 -1200 x PVIF (r,1) – 600 x PVIF (r,2) – 250 x PVIF (r,3)+ 2000 x PVIF (r,4) + 4000 x PVIF (r,5) = 0
Through a process of trial and error we find that r = 20.13%
(ii) The IRR (r') of project Q can be obtained by solving the following equation for r'
-1600 + 200 x PVIF (r',1) + 400 x PVIF (r',2) + 600 x PVIF (r',3)+ 800 x PVIF (r',4) + 100 x PVIF (r',5) = 0
Through a process of trial and error we find that r' = 9.34%.
c) From (a) we find that at a cost of capital of 10%
NPV (P) = 1075NPV (Q) = - 28
Given that NPV (P) . NPV (Q); and NPV (P) > 0, I would choose project P.
From (a) we find that at a cost of capital of 20%
NPV (P) = 11
NPV (Q) = - 382
Again NPV (P) > NPV (Q); and NPV (P) > 0. I would choose project P.
d) Project P
PV of investment-related costs
= 1000 x PVIF (12,0)+ 1200 x PVIF (12,1) + 600 x PVIF (12,2)+ 250 x PVIF (12,3)
= 2728
TV of cash inflows = 2000 x (1.12) + 4000 = 6240
The MIRR of the project P is given by the equation:
2728 = 6240 x PVIF (MIRR,5)
(1 + MIRR)5 = 2.2874
MIRR = 18%
Project Q
PV of investment-related costs = 1600
TV of cash inflows @ 15% p.a. = 2772
The MIRR of project Q is given by the equation:
16000 (1 + MIRR)5 = 2772
MIRR = 11.62%
9.(a) Project A
NPV at a cost of capital of 12%= - 100 + 25 x PVIFA (12,6)= Rs.2.79 million
IRR (r ) can be obtained by solving the following equation for r.25 x PVIFA (r,6) = 100i.e., r = 12,98%
Project B
NPV at a cost of capital of 12%= - 50 + 13 x PVIFA (12,6)= Rs.3.45 million
IRR (r') can be obtained by solving the equation13 x PVIFA (r',6) = 50i.e., r' = 14.40% [determined through a process of trial and error]
(b) Difference in capital outlays between projects A and B is Rs.50 millionDifference in net annual cash flow between projects A and B is Rs.12 million.NPV of the differential project at 12%
= -50 + 12 x PVIFA (12,6)= Rs.3.15 million
IRR (r'') of the differential project can be obtained from the equation12 x PVIFA (r'', 6) = 50i.e., r'' = 11.53%
10.(a) Project M
The pay back period of the project lies between 2 and 3 years. Interpolating in this range we get an approximate pay back period of 2.63 years/
Project NThe pay back period lies between 1 and 2 years. Interpolating in this range we get an approximate pay back period of 1.55 years.
(b) Project MCost of capital = 12% p.aPV of cash flows up to the end of year 1 = 9.82PV of cash flows up to the end of year 2 = 24.97PV of cash flows up to the end of year 3 = 47.75PV of cash flows up to the end of year 4 = 71.26
Discounted pay back period (DPB) lies between 3 and 4 years. Interpolating in this range we get an approximate DPB of 3.1 years.
Project NCost of capital = 12% per annumPV of cash flows up to the end of year 1 = 33.93PV of cash flows up to the end of year 2 = 51.47
DPB lies between 1 and 2 years. Interpolating in this range we get an approximate DPB of 1.92 years.
(c ) Project MCost of capital = 12% per annumNPV = - 50 + 11 x PVIF (12,1)
+ 19 x PVIF (12,2) + 32 x PVIF (12,3)+ 37 x PVIF (12,4)
= Rs.21.26 million
Project NCost of capital = 12% per annumNPV = Rs.20.63 million
Since the two projects are independent and the NPV of each project is (+) ve, both the projects can be accepted. This assumes that there is no capital constraint.
(d) Project MCost of capital = 10% per annumNPV = Rs.25.02 million
Project NCost of capital = 10% per annumNPV = Rs.23.08 million
Since the two projects are mutually exclusive, we need to choose the project with the higher NPV i.e., choose project M.
NOTE: The MIRR can also be used as a criterion of merit for choosing between the two projects because their initial outlays are equal.
(e) Project MCost of capital = 15% per annumNPV = 16.13 million
Project NCost of capital: 15% per annumNPV = Rs.17.23 million
Again the two projects are mutually exclusive. So we choose the project with the higher NPV, i.e., choose project N.
(f) Project M Terminal value of the cash inflows: 114.47MIRR of the project is given by the equation
50 (1 + MIRR)4 = 114.47i.e., MIRR = 23.01%
Project NTerminal value of the cash inflows: 115.41MIRR of the project is given by the equation
50 ( 1+ MIRR)4 = 115.41i.e., MIRR = 23.26%
MINICASE(a) Project A
Year Cash flow
Cumulative net cash inflow
Discounting factor @12%
Present value
Cumulative net cash flow after discounting
0 (15,000) (15,000) 1.000 (15,000) (15,000)1 11,000 (4,000) 0.893 9,823 (5,177)2 7,000 3,000 0.797 5,579 402 3 4,800 0.712 3,418
Payback period is between 1 and 2 years. By linear interpolation we get the payback period = 1 + 4,000 /(4,000 + 3,000) = 1.57 years. Discounted payback period = 1 + 5,177 / ( 5,177 + 402) = 1.93 yearsProject B
Year Cash flow
Cumulative net cash inflow
Discounting factor @12%
Present value
Cumulative net cash flow after discounting
0 (15,000) (15,000) 1.000 (15,000) (15,000)1 3,500 (11,500) 0.893 3,126 (11,875)2 8,000 (3,500) 0.797 6,376 (5,499)3 13,000 9,500 0.712 9,256 3,757
Payback period is between 2 and 3 years. By linear interpolation we get the payback period = 2 + 3,500 /(3,500 + 9,500) = 2.27 years. Discounted payback period = 2 + 5,499 / ( 5,499 + 3,757) = 2.59 years
(b)Project A
Year Cash flow
Discounting factor @12%
Present value
0 (15,000) 1.000 (15,000)1 11,000 0.893 9,823 2 7,000 0.797 5,579 3 4,800 0.712 3,418
Net present value= 3,820
Project B
Year Cash flow
Discounting factor @12%
Present value
0 (15,000) 1.000 (15,000)1 3,500 0.893 3,126 2 8,000 0.797 6,376 3 13,000 0.712 9,256
Net present value= 3,758 Project C
Year Cash flow
Discounting factor @12%
Present value
0 (15,000) 1.000 (15,000)1 42,000 0.893 37,506 2 (4,000) 0.797 (3,188)3
Net present value= 19,318
(c) Project A
IRR is the value of r in the following equation.11,000 / (1+r) + 7,000 / (1+r)2 + 4,800 / (1+r)3 = 15,000Trying r = 28 %, the LHS = 11,000 / (1.28) + 7,000 / (1.28)2 + 4,800 / (1.28)3
= 15,155As this value is slightly higher than 15,000, we try a higher discount rate of 29%
for r to get 11,000 / (1.29) + 7,000 / (1.29)2 + 4,800 / (1.29)3
= 14,970By linear interpolation we get r = 28 + (15,155 – 15,000) / (15,155 – 14,970) =
28.84 %
Project B
IRR is the value of r in the following equation.3,500 / (1+r) + 8,000 / (1+r)2 + 13,000 / (1+r)3 = 15,000Trying r = 23 %, the LHS = 3,500 / (1.23) + 8,000 / (1.23)2 + 13,000 / (1.23)3
= 15,119
As this value is slightly higher than 15,000, we try a higher discount rate of 24% for r to get 3,500 / (1.24) + 8,000 / (1.24)2 + 13,000 / (1.24)3
= 14,844
By linear interpolation we get r = 23 + (15,119 – 15,000) / (15,119 – 14,844) = 23. 43 %
Project C
IRR rule breaks down as the cash flows are non conventional.
(d) Calculation of MIRR for the three projects.Project A Terminal value of cash flows if reinvested at the cost of capital of 12% is= 11,000 x (1.12)2 + 7,000 x 1.12 + 4,800 = 26,438MIRR is the value of r in the equation: 26,438 / (1+r)3 =15,000r = (26,438 / 15,000)1/3 -1 = 20.8%Therefore MIRR = 20.8%
Project B Terminal value of cash flows if reinvested at the cost of capital of 12% is= 3,500 x (1.12)2 + 8,000 x 1.12 + 13,000 = 26,350MIRR is the value of r in the equation: 26,350 / (1+r)3 =15,000r = (26,350 / 15,000)1/3 -1 = 20.7 %Therefore MIRR = 20.7 %
Project C Terminal value of cash flow if reinvested at the cost of capital of 12% is= 42,000 x 1.12 = 47,040Present value of the costs = 15,000 + 4,000 / (1.12)2 = 18,189MIRR is the value of r in the equation: 47,040 / (1+r)2 =18,189r = (47,040 / 18,189)1/2 -1 = 60.8 %Therefore MIRR = 60.8 %
Chapter 12
ESTIMATION OF PROJECT CASH FLOWS1.(a) Project Cash Flows (Rs. in million)
Year 0 1 2 3 4 5 6 7
1. Plant & machinery (150)
2. Working capital (50)
3. Revenues 250 250 250 250 250 250 250
4. Costs (excluding de- preciation & interest) 100 100 100 100 100 100 100
5. Depreciation 37.5 28.13 21.09 15.82 11.87 8.90 6.67
6. Profit before tax 112.5 121.87 128.91 134.18 138.13 141.1143.33
7. Tax 33.75 36.56 38.67 40.25 41.44 42.33 43.0
8. Profit after tax 78.75 85.31 90.24 93.93 96.69 98.77100.33
9. Net salvage value of plant & machinery 48
10. Recovery of working 50 capital
11. Initial outlay (=1+2) (200)
12. Operating CF (= 8 + 5) 116.25 113.44 111.33 109.75 108.56 107.6 107.00
13. Terminal CF ( = 9 +10) 98
14. N C F (200) 116.25 113.44 111.33 109.75 108.56 107.67 205IRR (r) of the project can be obtained by solving the following equation for r
-200 + 116.25 x PVIF (r,1) + 113.44 x PVIF (r,2)
+ 111.33 x PVIF (r,3) + 109.75 x PVIF (r,4) + 108.56 x PVIF (r,5)+107.67 x PVIF (r,6) + 205 x PVIF (r,7) = 0
Through a process of trial and error, we get r = 55.17%. The IRR of the project is 55.17%.
2. Post-tax Incremental Cash Flows (Rs. in million)
Year 0 1 2 3 4 5 6 7
1. Capital equipment (120)2. Level of working capital 20 30 40 50 40 30 20 (ending)3. Revenues 80 120 160 200 160 120 804. Raw material cost 24 36 48 60 48 36 245. Variable mfg cost. 8 12 16 20 16 12 86. Fixed operating & maint. 10 10 10 10 10 10 10 cost7. Variable selling expenses 8 12 16 20 16 12 88. Incremental overheads 4 6 8 10 8 6 49. Loss of contribution 10 10 10 10 10 10 1010.Bad debt loss 411. Depreciation 30 22.5 16.88 12.66 9.49 7.12 5.3412. Profit before tax -14 11.5 35.12 57.34 42.51 26.88 6.6613. Tax -4.2 3.45 10.54 17.20 12.75 8.06 2.0014. Profit after tax -9.8 8.05 24.58 40.14 29.76 18.82 4.6615. Net salvage value of capital equipments 2516. Recovery of working 16 capital17. Initial investment (120)18. Operating cash flow 20.2 30.55 41.46 52.80 39.25 25.94 14.00
(14 + 10+ 11)19. Working capital 20 10 10 10 (10) (10) (10)20. Terminal cash flow 41
21. Net cash flow (140) 10.20 20.55 31.46 62.80 49.25 35.94 55.00 (17+18-19+20)
(b) NPV of the net cash flow stream @ 15% per discount rate
= -140 + 10.20 x PVIF(15,1) + 20.55 x PVIF (15,2)+ 31.46 x PVIF (15,3) + 62.80 x PVIF (15,4) + 49.25 x PVIF (15,5)+ 35.94 x PVIF (15,6) + 55 x PVIF (15,7)
= Rs.1.70 million
3.(a) A. Initial outlay (Time 0)
i. Cost of new machine Rs. 3,000,000ii. Salvage value of old machine 900,000iii Incremental working capital requirement 500,000iv. Total net investment (=i – ii + iii) 2,600,000
B. Operating cash flow (years 1 through 5)
Year 1 2 3 4 5
i. Post-tax savings in manufacturing costs 455,000 455,000 455,000 455,000 455,000
ii. Incremental depreciation 550,000 412,500 309,375 232,031 174,023
iii. Tax shield on incremental dep. 165,000 123,750 92,813 69,609 52,207
iv. Operating cash flow ( i + iii) 620,000 578,750 547,813 524,609 507,207
C. Terminal cash flow (year 5)
i. Salvage value of new machine Rs. 1,500,000ii. Salvage value of old machine 200,000iii. Recovery of incremental working capital 500,000iv. Terminal cash flow ( i – ii + iii) 1,800,000
D. Net cash flows associated with the replacement project (in Rs)
Year 0 1 2 3 4 5
NCF (2,600,000) 620000 578750 547813 524609 2307207
(b) NPV of the replacement project= - 2600000 + 620000 x PVIF (14,1)
+ 578750 x PVIF (14,2) + 547813 x PVIF (14,3) + 524609 x PVIF (14,4) + 2307207 x PVIF (14,5)
= Rs.2678494. Tax shield (savings) on depreciation (in Rs)
Depreciation Tax shield PV of tax shield
Year charge (DC) =0.4 x DC @ 15% p.a.
1 25000 10000 8696
2 18750 7500 5671
3 14063 5625 3699
4 10547 4219 2412
5 7910 3164 1573 ----------
22051 ----------
Present value of the tax savings on account of depreciation = Rs.22051
5. A. Initial outlay (at time 0)i. Cost of new machine Rs. 400,000ii. Salvage value of the old machine 90,000iii. Net investment 310,000
B. Operating cash flow (years 1 through 5)
Year 1 2 3 4 5i. Depreciation of old machine 18000 14400 11520 9216 7373
ii. Depreciation of new machine 100000 75000 56250 42188 31641
iii. Incremental depreciation ( ii – i) 82000 60600 44730 32972 24268
iv. Tax savings on incremental depreciation ( 0.35 x (iii)) 28700 21210 15656 11540 8494
v. Operating cash flow 28700 21210 15656 11540 8494
C. Terminal cash flow (year 5)
i. Salvage value of new machine Rs. 25000ii. Salvage value of old machine 10000iii. Incremental salvage value of new
machine = Terminal cash flow 15000
D. Net cash flows associated with the replacement proposal.
Year 0 1 2 3 4 5
NCF (310000) 28700 21210 15656 11540 23494
MINICASE Solution:
a. Cash flows from the point of all investors (which is also called the explicit cost funds point of view)
Rs.in million
Item 0 1 2 3 4 5
1. Fixed assets (15)2. Net working capital (8)3. Revenues 30 30 30 30 304. Costs (other than depreciation and interest) 20 20 20 20 205. Loss of rental 1 1 1 1 16. Depreciation 3.750 2.813 2.109 1.582 1.1877. Profit before tax 5.250 6.187 6.891 7.418 7.8138. Tax 1.575 1.856 2.067 2.225 2.3449. Profit after tax 3.675 4.331 4.824 5.193 5.46910. Salvage value of fixed assets 5.00011. Net recovery of working capital 8.000 12. Initial outlay (23)13. Operating cash inflow 7.425 7.144 6.933 6.775 6.65614. Terminal cash flow 13.000
15. Net cash flow (23) 7.425 7.144 6.933 6.775 19.656
b. Cash flows form the point of equity investors
Rs.in million
Item 0 1 2 3 4 5
1. Equity funds (10)2. Revenues 30 30 30 30 303. Costs (other than depreciation and interest) 20 20 20 20 204. Loss of rental 1 1 1 1 15. Depreciation 3.75 2.813 2.109 1.582 1.1876. Interest on working capital advance 0.70 0.70 0.70 0.70 0.707. Interest on term loans 1.20 1.125 0.825 0.525 0.2258. Profit before tax 3.35 4.362 5.366 6.193 6.8889. Tax 1.005 1.309 1.610 1.858 2.06610. Profit after tax 2.345 3.053 3.756 4.335 4.82211. Net salvage value of fixed assets 5.00012. Net salvage value of current assets 10.00013. Repayment of term term loans 2.000 2.000 2.000 2.000 14. Repayment of bank advance 5.00015. Retirement of trade creditors 2.00016. Initial investment (10) 17. Operating cash inflow 6.095 5.866 5.865 5.917 6.00918. Liquidation and retirement cash flows (2.0) (2.0) (2.0) 6.0019. Net cash flow (10) 6.095 3.866 3.865 3.917 12.009
Chapter 13RISK ANALYSIS IN CAPITAL BUDGETING
1.NPV of the project = -250 + 50 x PVIFA (13,10)
= Rs.21.31 million
(a) NPVs under alternative scenarios:(Rs. in million)
Pessimistic Expected Optimistic
Investment 300 250 200Sales 150 200 275Variable costs 97.5 120 154Fixed costs 30 20 15Depreciation 30 25 20Pretax profit - 7.5 35 86Tax @ 28.57% - 2.14 10 24.57Profit after tax - 5.36 25 61.43Net cash flow 24.64 50 81.43Cost of capital 14% 13% 12%
NPV - 171.47 21.31 260.10
Assumptions: (1) The useful life is assumed to be 10 years under all three scenarios. It is also assumed that the salvage value of the
investment after ten years is zero.
(2) The investment is assumed to be depreciated at 10% per annum; and it is also assumed that this method and rate of depreciation are acceptable to the IT (income tax) authorities.
(3) The tax rate has been calculated from the given table i.e. 10 / 35 x 100 = 28.57%.
(4) It is assumed that only loss on this project can be offset against the taxable profit on other projects of the company; and thus the company can claim a tax shield on the loss in the same year.
(b) Accounting break even point (under ‘expected’ scenario)Fixed costs + depreciation = Rs. 45 millionContribution margin ratio = 80 / 200 = 0.4Break even level of sales = 45 / 0.4 = Rs.112.5 million
Financial break even point (under ‘xpected’ scenario)
i. Annual net cash flow = 0.7143 [ 0.3 x sales – 45 ] + 25= 0.2143 sales – 7.14
ii. PV (net cash flows) = [0.2143 sales – 7.14 ] x PVIFA (13,10)= 1.1628 sales – 38.74
iii. Initial investment = 200
iv. Financial break even levelof sales = 238.74 / 1.1628 = Rs.205.31 million
2.(a) (i) Sensitivity of NPV with respect to quantity manufactured and sold:
(in Rs)Pessimistic Expected Optimistic
Initial investment 30000 30000 30000Sale revenue 24000 42000 54000Variable costs 16000 28000 36000Fixed costs 3000 3000 3000Depreciation 2000 2000 2000Profit before tax 3000 9000 13000Tax 1500 4500 6500Profit after tax 1500 4500 6500Net cash flow 3500 6500 8500NPV at a cost of capital of 10% p.aand useful life of 5 years -16732 - 5360 2222
(ii) Sensitivity of NPV with respect to variations in unit price.
Pessimistic Expected Optimistic
Initial investment 30000 30000 30000Sale revenue 28000 42000 70000Variable costs 28000 28000 28000Fixed costs 3000 3000 3000Depreciation 2000 2000 2000Profit before tax -5000 9000 37000Tax -2500 4500 18500Profit after tax -2500 4500 18500Net cash flow - 500 6500 20500
NPV - 31895 (-) 5360 47711
(iii) Sensitivity of NPV with respect to variations in unit variable cost.
Pessimistic Expected Optimistic
Initial investment 30000 30000 30000Sale revenue 42000 42000 42000Variable costs 56000 28000 21000Fixed costs 3000 3000 3000Depreciation 2000 2000 2000Profit before tax -11000 9000 16000Tax -5500 4500 8000Profit after tax -5500 4500 8000Net cash flow -3500 6500 10000NPV -43268 - 5360 7908
(b) Accounting break-even point
i. Fixed costs + depreciation = Rs.5000ii. Contribution margin ratio = 10 / 30 = 0.3333iii. Break-even level of sales = 5000 / 0.3333
= Rs.15000Financial break-even point
i. Annual cash flow = 0.5 x (0.3333 Sales – 5000) = 2000ii. PV of annual cash flow = (i) x PVIFA (10,5)
= 0.6318 sales – 1896iii. Initial investment = 30000iv. Break-even level of sales = 31896 / 0.6318 = Rs.50484
3. Define At as the random variable denoting net cash flow in year t.
A1 = 4 x 0.4 + 5 x 0.5 + 6 x 0.1= 4.7
A2 = 5 x 0.4 + 6 x 0.4 + 7 x 0.2= 5.8
A3 = 3 x 0.3 + 4 x 0.5 + 5 x 0.2= 3.9
NPV = 4.7 / 1.1 +5.8 / (1.1)2 + 3.9 / (1.1)3 – 10= Rs.2.00 million
12 = 0.41
22 = 0.56
32 = 0.49
12 2
2 32
2NPV = + + (1.1)2 (1.1)4 (1.1)6
= 1.00 (NPV) = Rs.1.00 million
4. Expected NPV 4 At
= - 25,000 t=1 (1.08)t
= 12,000/(1.08) + 10,000 / (1.08)2 + 9,000 / (1.08)3
+ 8,000 / (1.08)4 – 25,000
= [ 12,000 x .926 + 10,000 x .857 + 9,000 x .794 + 8,000 x .735] - 25,000
= 7,708
Standard deviation of NPV 4 t
t=1 (1.08)t
= 5,000 / (1.08) + 6,000 / (1.08)2 + 5,000 / (1,08)3 + 6,000 / (1.08)4
= 5,000 x .926 + 6,000 x .857 + 5000 x .794 + 6,000 x .735= 18,152
5. (a) Expected NPV 4 At
= - 10,000 …. (1) t=1 (1.06)t
A1 = 2,000 x 0.2 + 3,000 x 0.5 + 4,000 x 0.3= 3,100
A2 = 3,000 x 0.4 + 4,000 x 0.3 + 5,000 x 0.3= 3,900
A3 = 4,000 x 0.3 + 5,000 x 0.5 + 6,000 x 0.2= 4,900
A4 = 2,000 x 0.2 + 3,000 x 0.4 + 4,000 x 0.4= 3,200
Substituting these values in (1) we get
Expected NPV = NPV
= 3,100 / (1.06)+ 3,900 / 1.06)2 + 4,900 / (1.06)3 + 3,200 / (1,06)4
- 10,000 = Rs.3,044(b)
The variance of NPV is givenby the expression 4 2
t
2 (NPV) = …….. (2) t=1 (1.06)2t
12 = [(2,000 – 3,100)2 x 0.2 + (3,000 – 3,100)2 x 0.5
+ (4,000 – 3,100)2 x 0.3]= 490,000
22 = [(3,000 – 3,900)2 x 0.4 + (4,000 – 3,900)2 x 0.3
+ (5,000 – 3900)2 x 0.3]= 690,000
32 = [(4,000 – 4,900)2 x 0.3 + (5,000 – 4,900)2 x 0.5
+ (6,000 – 4,900)2 x 0.2]= 490,000
42 = [(2,000 – 3,200)2 x 0.2 + (3,000 – 3,200)2 x 0.4
+ (4,000 – 3200)2 x 0.4]= 560,000
Substituting these values in (2) we get490,000 / (1.06)2 + 690,000 / (1.06)4
+ 490,000 / (1.06)6 + 560,000 / (1.06)8
[ 490,000 x 0.890 + 690,000 x 0.792 + 490,000 x 0.705 + 560,000 x 0.627 ]
= 1,679,150NPV = 1,679,150 = Rs.1,296
NPV – NPV 0 - NPVProb (NPV < 0) = Prob. <
NPV NPV 0 – 3044
= Prob Z < 1296
= Prob (Z < -2.35)
The required probability is given by the shaded area in the following normal curve.
P (Z < - 2.35) = 0.5 – P (-2.35 < Z < 0)= 0.5 – P (0 < Z < 2.35)= 0.5 – 0.4906= 0.0094
(c)So the probability of NPV being negative is 0.0094
Prob (P1 > 1.2) Prob (PV / I > 1.2)Prob (NPV / I > 0.2)Prob. (NPV > 0.2 x 10,000)Prob (NPV > 2,000)
Prob (NPV >2,000)= Prob (Z > 2,000- 3,044 / 1,296)Prob (Z > - 0.81)
The required probability is given by the shaded area of the following normal curve:P(Z > - 0.81) = 0.5 + P(-0.81 < Z < 0)
= 0.5 + P(0 < Z < 0.81)= 0.5 + 0.2910= 0.7910
So the probability of P1 > 1.2 as 0.7910
6
YearCash Flow
Certainty Equivalent Factor: αt =1 - 0.06t
Certainty Equivalent value
Discount Factor at 8%
Present Value
0(30,000
) 1 (30,000) 1 (30,000.00)1 7,000 0.94 6,580 0.9259 6,092.59 2 8,000 0.88 7,040 0.8573 6,035.67 3 9,000 0.82 7,380 0.7938 5,858.48 4 10,000 0.76 7,600 0.7350 5,586.23 5 8,000 0.7 5,600 0.6806 3,811.27
NPV = (2,615.77)
MINICASE
Solution:1. The expected NPV of the turboprop aircraft
0.65 (5500) + 0.35 (500)NPV = - 11000 +
(1.12)
0.65 [0.8 (17500) + 0.2 (3000)] + 0.35 [0.4 (17500) + 0.6 (3000)] +
(1.12)2
= 2369
2. If Southern Airways buys the piston engine aircraft and the demand in year 1 turns out to be high, a further decision has to be made with respect to capacity expansion. To evaluate the piston engine aircraft, proceed as follows:
First, calculate the NPV of the two options viz., ‘expand’ and ‘do not expand’ at decision point D2:
0.8 (15000) + 0.2 (1600)Expand : NPV = - 4400 +
1.12
= 6600
0.8 (6500) + 0.2 (2400)Do not expand : NPV =
1.12= 5071
Second, truncate the ‘do not expand’ option as it is inferior to the ‘expand’ option. This means that the NPV at decision point D2 will be 6600
Third, calculate the NPV of the piston engine aircraft option.
0.65 (2500+6600) + 0.35 (800)NPV = – 5500 +
1.12
0.35 [0.2 (6500) + 0.8 (2400)] +
(1.12)2
= – 5500 + 5531 + 898 = 929
3. The value of the option to expand in the case of piston engine aircraftIf Southern Airways does not have the option of expanding capacity at the end of year 1, the NPV of the piston engine aircraft would be:
0.65 (2500) + 0.35 (800) NPV = – 5500 +
1.12
0.65 [0.8 (6500) + 0.2 (2400)] + 0.35 [0.2 (6500) + 0.8 (2400)]+
(1.12)2
= - 5500 + 1701 + 3842 = 43
Thus the option to expand has a value of 929 – 43 = 886
4. Value of the option to abandon if the turboprop aircraft can be sold for 8000 at the end of year 1
If the demand in year 1 turns out to be low, the payoffs for the ‘continuation’ and ‘abandonment’ options as of year 1 are as follows.
0.4 (17500) + 0.6 (3000)Continuation: = 7857
1.12
Abandonment : 8000
Thus it makes sense to sell off the aircraft after year 1, if the demand in year 1 turns out to be low.
The NPV of the turboprop aircraft with abandonment possibility is
0.65 [5500 +{0.8 (17500) + 0.2 (3000)}/ (1.12)] + 0.35 (500 +8000)NPV = - 11,000 +
(1.12)
12048 + 2975 = - 11,000 + = 2413
1.12
Since the turboprop aircraft without the abandonment option has a value of 2369, the value of the abandonment option is : 2413 – 2369 = 44
5 The value of the option to abandon if the piston engine aircraft can be sold for 4400 at the end of year 1:
If the demand in year 1 turns out to be low, the payoffs for the ‘continuation’ and ‘abandonment’ options as of year 1 are as follows:
0.2 (6500) + 0.8 (2400)Continuation : = 2875
1.12
Abandonment : 4400
Thus, it makes sense to sell off the aircraft after year 1, if the demand in year 1 turns out to be low.
The NPV of the piston engine aircraft with abandonment possibility is:
0.65 [2500 + 6600] + 0.35 [800 + 4400]NPV = - 5500 +
1.12
5915 + 1820 = - 5500 + = 1406
1.12
For the piston engine aircraft the possibility of abandonment increases the NPV from 929 to 1406. Hence the value of the abandonment option is 477.
Chapter 14THE COST OF CAPITAL
1(a) Define rD as the pre-tax cost of debt. Using the approximate yield formula, rD can be calculated as follows:
14 + (100 – 108)/10rD = ------------------------ x 100 = 12.60%
0.4 x 100 + 0.6x108
(b) After tax cost = 12.60 x (1 – 0.35) = 8.19%
2. Define rp as the cost of preference capital. Using the approximate yield formula rp can be calculated as follows:
9 + (100 – 92)/6rp = --------------------
0.4 x100 + 0.6x92
= 0.1085 (or) 10.85%
3. WACC = 0.4 x 13% x (1 – 0.35)+ 0.6 x 18%
= 14.18%
4. Cost of equity = 10% + 1.2 x 7% = 18.4%(using SML equation)
Pre-tax cost of debt = 14%
After-tax cost of debt = 14% x (1 – 0.35) = 9.1%
Debt equity ratio = 2 : 3
WACC = 2/5 x 9.1% + 3/5 x 18.4%
= 14.68%
5. Given0.5 x 14% x (1 – 0.35) + 0.5 x rE = 12%
where rE is the cost of equity capital.
Therefore rE – 14.9%
Using the SML equation we get
11% + 8% x β = 14.9%
where β denotes the beta of Azeez’s equity.
Solving this equation we get β = 0.4875.
6(a) The cost of debt of 12% represents the historical interest rate at the time the debt was originally issued. But we need to calculate the marginal cost of debt (cost of raising new debt); and for this purpose we need to calculate the yield to maturity of the debt as on the balance sheet date. The yield to maturity will not be equal to12% unless the book value of debt is equal to the market value of debt on the balance sheet date.
(b) The cost of equity has been taken as D1/P0 ( = 6/100) whereas the cost of equity is (D1/P0) + g where g represents the expected constant growth rate in dividend per share.
7. (a) The book value and market values of the different sources of finance are provided in the following table. The book value weights and the market value weights are provided within parenthesis in the table.
(Rs. in million)Source Book value Market valueEquity 800 (0.54) 2400 (0.78)Debentures – first series 300 (0.20) 270 (0.09)Debentures – second series 200 (0.13) 204 (0.06)Bank loan 200 (0.13) 200 (0.07)Total 1500 (1.00) 3074 (1.00)
(b) I would use weights based on the market value because to justify its valuation Samanta must earn competitive returns for investors on its market value
8.(a) Given
rD x (1 – 0.3) x 4/9 + 20% x 5/9 = 15%rD = 12.5%,where rD represents the pre-tax cost of debt.
(b) Given13% x (1 – 0.3) x 4/9 + rE x 5/9 = 15%rE = 19.72%, where rE represents the cost of equity.
9. Cost of equity = D1/P0 + g = 3.00 / 30.00 + 0.05 = 15%
(a) The first chunk of financing will comprise of Rs.5 million of retained earnings costing 15 percent and Rs.2.5 million of debt costing 14 (1-.3) = 9.8 per centThe second chunk of financing will comprise of Rs.5 million of additional equity costing 15 per cent and Rs.2.5 million of debt costing 15 (1-.3) = 10.5 per cent
(b) The marginal cost of capital in the first chunk will be :5/7.5 x 15% + 2.5/7.5 x 9.8% = 13.27%
The marginal cost of capital in the second chunk will be: 5/7.5 x 15% + 2.5/7.5 x 10.5% = 13.50%
Note : We have assumed that(i) The net realisation per share will be Rs.25, after floatation costs, and(ii) The planned investment of Rs.15 million is inclusive of floatation costs
9.1 (a) (i) The cost of equity and retained earningsrE = D1/PO + g
= 1.50 / 20.00 + 0.07 = 14.5%The cost of preference capital, using the approximate formula, is :
11 + (100-75)/10rP = = 15.9%
0.6 x 75 + 0.4 x 100The pre-tax cost of debentures, using the approximate formula, is :
13.5 + (100-80)/6rD = = 19.1%
0.6x80 + 0.4x100
The post-tax cost of debentures is 19.1 (1-tax rate) = 19.1 (1 – 0.5)
= 9.6%
The post-tax cost of term loans is 12 (1-tax rate) = 12 (1 – 0.5)
= 6.0%
The average cost of capital using book value proportions is calculated below:
Source of capital Component Book value Book value Product of Cost Rs. in million proportion (1) & (3) (1) (2) (3)
Equity capital 14.5% 100 0.28 4.06Preference capital 15.9% 10 0.03 0.48Retained earnings 14.5% 120 0.33 4.79Debentures 9.6% 50 0.14 1.34Term loans 6.0% 80 0.22 1.32
360 Average cost 11.99%capital
(ii) The average cost of capital using market value proportions is calculated below :
Source of capital Component Market value Market value Product of cost Rs. in million proportion (1) (2) (3) (1) & (3)
Equity capitaland retained earnings 14.5% 200 0.62 8.99Preference capital 15.9% 7.5 0.02 0.32Debentures 9.6% 40 0.12 1.15Term loans 6.0% 80 0.24 1.44
327.5 Average cost 11.90% capital
(b)The Rs.100 million to be raised will consist of the following:Retained earnings Rs.15 millionAdditional equity Rs. 35 millionDebt Rs. 50 millionThe first batch will consist of Rs. 15 million each of retained earningsand debt costing 14.5 percent and 14(1-0.5)= 7 percent respectively. Thesecond batch will consist of Rs. 10 million each of additional equity anddebt at 14.5 percent and 7percent respectively. The third chunk willconsist of Rs.25 million each of additional equity and debt costing 14.5percent and 15(1-0.5) = 7.5 percent respectively.The marginal cost of capital in the chunks will be as underFirst batch: (0.5x14.5 ) + (0.5 x 7) = 10.75 %Second batch: (0.5x14.5 ) + (0.5 x 7) = 10.75 %Third batch : (0.5x14.5 ) + (0.5 x 7.5) = 11 %
The marginal cost of capital schedule for the firm will be as under.Range of total financing Weighted marginal cost of( Rs. in million) capital ( %)0 - 50 10.7550-100 11.00Here it is assumed that the Rs.100 million to be raised is inclusive of floatation costs.
10(a) WACC = 1/3 x 13% x (1 – 0.3)
+ 2/3 x 20%= 16.37%
(b) Weighted average floatation cost= 1/3 x 3% + 2/3 x 12%= 9%
(c) NPV of the proposal after taking into account the floatation costs
= 130 x PVIFA (16.37, 8) – 500 / (1 - 0.09)= Rs.8.51 million
11. Required return based on SML Expected
Project Beta equation (%) return (%)
P 0.6 14.8 13Q 0.9 17.2 14R 1.5 22.0 16S 1.5 22.0 20
Given a hurdle rate of 18% (the firm’s cost of capital), projects P, Q and R would have been rejected because the expected returns on these projects are below 18%. Project S would be accepted because the expected return on this project exceeds 18%. An appropriate basis for accepting or rejecting the projects would be to compare the expected rate of return and the required rate of return for each project. Based on this comparison, we find that all the four projects need to be rejected.
MINICASE
Solution:
a. All sources other than non-interest bearing liabilities
b. Pre-tax cost of debt & post-tax cost of debt
10 + (100 – 112) / 8 8.5rd = = = 7.93
0.6 x 112 + 0.4 x 100 107.2
rd (1 – 0.3) = 5.55
c. Post-tax cost of preference
9 + (100 – 106) / 5 7.8 = = 7.53%
0.6 x 106 + 0.4 x 100 103.6
d. Cost of equity using the DDM
2.80 (1.10) + 0.10 = 0.385 + 0.10
80= 0.1385 = 13.85%
e. Cost of equity using the CAPM
7 + 1.1(7) = 14.70%
f. WACC0.50 x 14.70 + 0.10 x 7.53 + 0.40 x 5.55= 7.35 + 0.75 + 2.22 = 10.32%
g. Cost of capital for the new business
0.5 [7 + 1.5 (7)] + 0.5 [ 11 (1 – 0.3)]8.75 + 3.85 = 12.60%
Chapter 15CAPITAL BUDGETING: EXTENSIONS
1. Let us assume that the cost of capital is 12 percent.EAC(Plastic Emulsion) = 300000 / PVIFA (12,7)
= 300000 / 4.564= Rs.65732
EAC(Distemper Painting) = 180000 / PVIFA (12,3)
= 180000 / 2.402= Rs.74938
Since EAC of plastic emulsion is less than that of distemper painting, it is the preferred alternative.
2. PV of the net costs associated with the internal transportation system
= 1 500 000 + 300 000 x PVIF (13,1) + 360 000 x PVIF (13,2)+ 400 000 x PVIF (13,3) + 450 000 x PVIF (13,4)+ 500 000 x PVIF (13,5) - 300 000 x PVIF (13,5)
= 2709185
EAC of the internal transportation system
= 2709185 / PVIFA (13,5)= 2709185 / 3.517= Rs.770 311
3. EAC [ Standard overhaul]
= 500 000 / PVIFA (14,6)= 500 000 / 3.889= Rs.128568 ……… (A)
EAC [Less costly overhaul]
= 200 000 / PVIFA (14,2)= 200 000 / 1.647= Rs.121433 ……… (B)
Since (B) < (A), the less costly overhaul is preferred alternative.
4.(a) Base case NPV
= -12,000,000 + 3,000,000 x PVIFA (20,6)= -12,000,000 + 997,8000= (-) Rs.2,022,000
(b) Issue costs = 6,000,000 / 0.88 - 6,000,000
= Rs.818 182
Adjusted NPV after adjusting for issue costs
= - 2,022,000 – 818,182= - Rs.2,840,182
(c) The present value of interest tax shield is calculated below :
Year Debt outstanding at Interest Tax shield Present value of the beginning tax shield
1 6,000,000 1,080,000 324,000 274,590 2 6,000,000 1,080,000 324,000 232,697 3 5,250,000 945,000 283,000 172,538 4 4,500,000 810,000 243,000 125,339 5 3,750,000 675,000 202,000 88,513 6 3,000,000 540,000 162,000 60,005 7 2,225,000 400,500 120,000 37,715 8 1,500,000 270,000 81,000 21,546 9 750,000 135,000 40,500 9,133
Present value of tax shield = Rs.1,022,076
5.(a) Base case BPV
= - 8,000,000 + 2,000,000 x PVIFA (18,6)= - Rs.1,004,000
(b) Adjusted NPV after adjustment for issue cost of external equity
= Base case NPV – Issue cost= - 1,004,000 – [ 3,000,000 / 0.9 – 3,000,000]= - Rs.1,337,333
(c) The present value of interest tax shield is calculated below :
Year Debt outstanding at Interest Tax shield Present value of the beginning tax shield
1 5,000,000 750,000 300,000 260,880 2 5,000,000 750,000 300,000 226,830 3 4,000,000 600,000 240,000 157,800 4 3,000,000 450,000 180,000 102,924 5 2,000,000 300,000 120,000 59,664 6 1,000,000 150,000 60,000 25,938
Present value of tax shield = Rs.834,036
Chapter 18 RAISING LONG TERM FINANCE
1 Underwriting Shares Excess/ Credit Netcommitment procured shortfall shortfall
A 70,000 50,000 (20,000) 4919 (15081)
B 50,000 30,000 (20,000) 3514 (16486)
C 40,000 30,000 (10,000) 2811 (7189)
D 25,000 12,000 (13,000) 1757 (11243)
E 15,000 28,000 13,000
2. Po = Rs.180 N = 5a. The theoretical value of a right if the subscription price is Rs.150
Po – S 180 – 150 = = Rs.5
N+1 5+1
b. The ex-rights value per share if the subscription price is Rs.160 NPo + S 5 x 180 + 160
= = Rs.176.7 N+1 5+1
c. The theoretical value per share, ex-rights, if the subscription price isRs.180? 100?
5 x 180 + 180 = Rs.180
5+1
5 x 180 + 100 = Rs.166.7
5+1
Chapter 19CAPITAL STRUCTURE AND FIRM VALUE
1. Net operating income (O) : Rs.30 millionInterest on debt (I) : Rs.10 millionEquity earnings (P) : Rs.20 millionCost of equity (rE) : 15%
Cost of debt (rD) : 10%Market value of equity (E) : Rs.20 million/0.15 =Rs.13 million Market value of debt (D) : Rs.10 million/0.10 =Rs.100 millionMarket value of the firm (V) : Rs.233 million
2. Box Cox
Market value of equity 2,000,000/0.15 2,000,000/0.15 = Rs.13.33 million = Rs.13.33 million
Market value of debt 0 1,000,000/0.10=Rs.10 million
Market value of the firm Rs.13.33million =23.33 million
(a) Average cost of capital for Box Corporation13.33. 0
x 15% + x 10% = 15%13.33 13.33
Average cost of capital for Cox Corporation 13.33 10.00
x 15% + x 10% = 12.86%23.33 23.33
(b) If Box Corporation employs Rs.30 million of debt to finance a project that yields Rs.4 million net operating income, its financials will be as follows.
Net operating income Rs.6,000,000Interest on debt Rs.3,000,000Equity earnings Rs.3,000,000Cost of equity 15%Cost of debt 10%Market value of equity Rs.20 millionMarket value of debt Rs.30 millionMarket value of the firm Rs.50 million
Average cost of capital 20 30
15% x + 10% = 12% 50 50
(c) If Cox Corporation sells Rs.10 million of additional equity to retire Rs.10 million of debt , it will become an all-equity company. So its
average cost of capital will simply be equal to its cost of equity, which is 15%.
3. rE = rA + (rA-rD)D/E 20 = 12 + (12-8) D/E So D/E = 2
4. E D E D rE rD rA = rE + rD
D+E D+E (%) (%) D+E D+E
1.00 0.00 11.0 6.0 11.00 0.90 0.10 11.0 6.5 10.55 0.80 0.20 11.5 7.0 10.60 0.70 0.30 12.5 7.5 11.00 0.60 0.40 13.0 8.5 11.20 0.50 0.50 14.0 9.5 11.75 0.40 0.60 15.0 11.0 12.60 0.30 0.70 16.0 12.0 13.20 0.20 0.80 18.0 13.0 14.00 0.10 0.90 20.0 14.0 14.20
The optimal debt ratio is 0.10 as it minimises the weighted average cost of capital.
5. (a) If you own Rs.10,000 worth of Bharat Company, the levered company which is valued more, you would sell shares of Bharat Company, resort to personal leverage, and buy the shares of Charat Company.
(b) The arbitrage will cease when Charat Company and Bharat Company are valued alike
6. The value of Ashwini Limited according to Modigliani and Miller hypothesis is
Expected operating income 15 = = Rs.125 million Discount rate applicable to the 0.12 risk class to which Aswini belongs
7. The tax advantage of one rupee of debt is : 1-(1-tc) (1-tpe) (1-0.55) (1-0.05) = 1 - (1-tpd) (1-0.25)
= 0.43 rupee
Chapter 20CAPITAL STRUCTURE DECISION
1.(a) Currently No. of shares = 1,500,000 EBIT = Rs 7.2 million Interest = 0 Preference dividend = Rs.12 x 50,000 = Rs.0.6 million EPS = Rs.2
(EBIT – Interest) (1-t) – Preference dividend EPS = No. of shares
(7,200,000 – 0 ) (1-t) – 600,000 Rs.2 =
1,500,000
Hence t = 0.5 or 50 per cent
The EPS under the two financing plans is : Financing Plan A : Issue of 1,000,000 shares
(EBIT - 0 ) ( 1 – 0.5) - 600,000 EPSA = 2,500,000
Financing Plan B : Issue of Rs.10 million debentures carrying 15 per cent interest
(EBIT – 1,500,000) (1-0.5) – 600,000 EPSB =
1,500,000
The EPS – EBIT indifference point can be obtained by equating EPSA and EPSB
(EBIT – 0 ) (1 – 0.5) – 600,000 (EBIT – 1,500,000) (1 – 0.5) – 600,000 = 2,500,000 1,500,000
Solving the above we get EBIT = Rs.4,950,000 and at that EBIT, EPS is Rs.0.75 under both the plans
(b) As long as EBIT is less than Rs.4,950,000 equity financing maximixes EPS.When EBIT exceeds Rs.4,950,000 debt financing maximises EPS.
2. (a) EPS – EBIT equation for alternative A EBIT ( 1 – 0.5)
EPSA = 2,000,000
(b) EPS – EBIT equation for alternative B EBIT ( 1 – 0.5 ) – 440,000
EPSB = 1,600,000
(c) EPS – EBIT equation for alternative C (EBIT – 1,200,000) (1- 0.5)
EPSC = 1,200,000
(d) The three alternative plans of financing ranked in terms of EPS over varying Levels of EBIT are given the following table
Ranking of Alternatives
EBIT EPSA EPSB EPSC
(Rs.) (Rs.) (Rs.) (Rs.)
2,000,000 0.50(I) 0.35(II) 0.33(III) 2,160,000 0.54(I) 0.40(II) 0.40(II)
3,000,000 0.75(I) 0.66(II) 0.75(I) 4,000,000 1.00(II) 0.98(III) 1.17(I) 4,400,000 1.10(II) 1.10(II) 1.33(I)
More than 4,400,000 (III) (II) (I)
3. Plan A : Issue 0.8 million equity shares at Rs. 12.5 per share.Plan B : Issue Rs.10 million of debt carrying interest rate of 15 per cent.
(EBIT – 0 ) (1 – 0.6) EPSA =
1,800,000 (EBIT – 1,500,000) (1 – 0.6)
EPSB = 1,000,000
Equating EPSA and EPSB , we get (EBIT – 0 ) (1 – 0.6) (EBIT – 1,500,000) (1 – 0.6) = 1,800,000 1,000,000
Solving this we get EBIT = 3,375,000 or 3.375 million
Thus the debt alternative is better than the equity alternative when EBIT > 3.375 million
EBIT – EBIT 3.375 – 7.000 Prob(EBIT>3,375,000) = Prob >
EBIT 3.000
= Prob [z > - 1.21] = 0.8869
4. ROE = [ ROI + ( ROI – r ) D/E ] (1 – t )15 = [ 14 + ( 14 – 8 ) D/E ] ( 1- 0.5 )
D/E = 2.67
5. ROE = [12 + (12 – 9 ) 0.6 ] (1 – 0.6) = 5.52 per cent
6. 18 = [ ROI + ( ROI – 8 ) 0.7 ] ( 1 – 0.5) ROI = 24.47 per cent EBIT7. a. Interest coverage ratio =
Interest on debt
15 =
4 = 3.75
EBIT + Depreciation b. Cash flow coverage ratio =
Loan repayment instalment Int.on debt +
(1 – Tax rate) = 15 + 3
4 + 5
= 28. The debt service coverage ratio for Pioneer Automobiles Limited is given by : 5 PAT i + Depi + Inti) i=1 DSCR = 5
Inti + LRIi) i=1
= 133.00 + 49.14 +95.80
95.80 + 72.00
= 277.94 167.80
= 1.66
9. (a) If the entire outlay of Rs. 300 million is raised by way of debt carrying 15 per cent interest, the interest burden will be Rs. 45 million.
Considering the interest burden the net cash flows of the firm duringa recessionary year will have an expected value of Rs. 35 million (Rs.80 million - Rs. 45 million ) and a standard deviation of Rs. 40 million . Since the net cash flow (X) is distributed normally
X – 35
40 has a standard normal deviation Cash flow inadequacy means that X is less than 0. 0.35 Prob(X<0) = Prob (z< ) = Prob (z<- 0.875)
40 = 0.1909
(b) Since µ = Rs.80 million, = Rs.40 million , and the Z value corresponding to the risk tolerance limit of 5 per cent is – 1.645, the cash available from the operations to service the debt is equal to X which is defined as :
X – 80 = - 1.645
40 X = Rs.14.2 million
Given 15 per cent interest rate, the debt than be serviced is 14.2
= Rs. 94.67 million 0.15
MINICASE
(a) If the firm chooses the equity option, it will have to issue 2 crore shares and its interest burden will remain at the current level of Rs.20 crore. If the firm chooses the debt option, the interest burden will go upto Rs.36 crore, but the number of equity shares will remain unchanged at 14 crore. So, the EPS – PBIT indifference point is the value of PBIT in the following equation.
(PBIT -20)( 1- 0.3) (PBIT – 36)( 1-0.3) =
16 14
PBIT = Rs. 148 crore
(b) The projected EPS under the two financing options is given below
Projected Current Equity option Debt optionRevenues 800 1040 1040Variable costs 480 624 624Contribution margin 320 416 416Fixed operating costs 180 230 230PBIT 140 186 186Interest 20 20 36PBT 120 166 150Tax 36 55.33 50PAT 84 110.67 100No.of equity shares 14 16 14EPS 6 6.92 7.14
(c)
Contribution marginThe degree of total leverage (DTL) is defined as :
PBITSo, the DTL will be as follows:
DTL
Current = 320/140 = 2.67Equity option = 416/166 = 2.51Debt option = 416/150 = 2.77
Chapter 21DIVIDEND POLICY AND FIRM VALUE
1.(a) Payout ratio Price per share
3(0.5)+3(0.5) 0.15 0.5
0.12 = Rs. 28.13
0.12
3(0.7 5)+3(0.25) 0.15 0.12
0.75 = Rs. 26.56 0.12
3(1.00) 1.00 = Rs. 25.00 0.12
(b)
Dividend payout ratio
Price as per Gordon model P0 =E1(1-b)/(k-br)
25% = 3 x 0.75/(0.12 - 0.75x 0.15) =Rs. 300
50% = 3 x 0.50/(0.12 - 0.50x 0.15) =Rs. 33.33
75% = 3 x 0.25/(0.12 - 0.25x 0.15) =Rs. 9.09
2.P Q
Next year’s price 80 74 Dividend 0 6 Current price P Q Capital appreciation (80-P) (74-Q) Post-tax capital appreciation 0.9(80-P) 0.9 (74-Q) Post-tax dividend income 0 0.8 x 6 Total return 0.9 (80-P)
P= 14%
0.9 (74-Q) + 4.8Q
=14% Current price (obtained by solving the preceding equation)
P = Rs.69.23 Q = Rs.68.65
Chapter 22
DIVIDEND DECISION
1. a. Under a pure residual dividend policy, the dividend per share over the 4 year period will be as follows:
DPS Under Pure Residual Dividend Policy( in Rs.)
Year 1 2 3 4
Earnings 10,000 12,000 9,000 15,000Capital expenditure 8,000 7,000 10,000 8,000Equity investment 4,000 3,500 5,000 4,000Pure residualdividends 6,000 8,500 4,000 11,000Dividends per share 1.20 1.70 0.80 2.20
b. The external financing required over the 4 year period (under the assumption that the company plans to raise dividends by 10 percents every two years) is given below :
Required Level of External Financing (in Rs.)
Year 1 2 3 4
A . Net income 10,000 12,000 9,000 15,000
B . Targeted DPS 1.00 1.10 1.10 1.21
C . Total dividends 5,000 5,500 5,500 6,050
D . Retained earnings(A-C) 5,000 6,500 3,500 8,950
E . Capital expenditure 8,000 7,000 10,000 8,000
F . External financingrequirement 3,000 500 6,500 Nil(E-D)if E > D or 0 otherwise
c. Given that the company follows a constant 60 per cent payout ratio, the dividend per share and external financing requirement over the 4 year period are given below
Dividend Per Share and External Financing Requirement(in Rs.)
Year 1 2 3 4
A. Net income 10,000 12,000 9,000 15,00
B. Dividends 6,000 7,200 5,400 9,000
C. Retained earnings 4,000 4,800 3,600 6,000
D. Capital expenditure 8,000 7,000 10,000 8,000
E. External financing(D-C)if D>C, or 0 4,000 2,200 6,400 2,000otherwise
F. Dividends per share 1.20 1.44 1.08 1.80
2. Given the constraints imposed by the management, the dividend per share has tobe between Rs.1.00 (the dividend for the previous year) and Rs.1.60 (80 percent of earnings per share)
Since share holders have a preference for dividend, the dividend should be raised over the previous dividend of Rs.1.00 . However, the firm has substantial
investment requirements and it would be reluctant to issue additional equitybecause of high issue costs ( in the form of underpricing and floatation costs)
Considering the conflicting requirements, it seems to make sense to payRs.1.20 per share by way of dividend. Put differently the pay out ratio may beset at 60 per cent.
MINICASE
(a) Plausible Reasons for Paying Dividends
(i) Investor preference for dividends (ii) Information signaling(iii) Clientele effect (iv) Agency costs
Dubious Reasons for Paying Dividends
(i) Bird-in-hand fallacy (ii) Temporary excess cash
(b)(i) Funds requirement(ii) Liquidity(iii) Access to external sources of financing (iv) Shareholder preference (v) Difference in the cost of external equity and retained earnings(vi) Control(vi) Taxes(vii) Stability
(c)
Rs.in million 1 2 3 4 5 TotalEarnings 96 108 84 115 147 550Net investments 104 94 90 108 192 588Equity investment 69.33 62.67 60.00 72.00 128.00 392Pure residual dividends 26.67 45.33 24.00 43.00 19.00 158Dividends under fixed dividend payout ratio 28.8 32.4 25.2 34.5 44.1 165Dividends under smoothed residual dividend policy 30 30 30 34 34 158
(d)DPS for the current year : Dt = cr EPSt + (1-c) Dt-1
= 0.6 x 0.3 x 9 + (1-0.6) x 2 = Rs.2.42
(e)
Bonus Issue Stock Split The par value of the share is
unchanged The par value of the share is
reduced. A part of reserves is capitalised There is no capitalisation of
reserves The shareholders' proportional
ownership remains unchanged The shareholders' proportional
ownership remains unchanged The book value per share, the
earnings per share, and the market price per share decline
The book value per share, the earnings per share, and the market price per share decline
The market price per share is brought within a popular trading range.
The market price per share is brought within a more popular trading
range.
Chapter 23 Debt Analysis and Management
1. (i) Initial Outlay(a) Cost of calling the old bonds
Face value of the old bonds 250,000,000 Call premium 15,000,000 265,000,000
(b) Net proceeds of the new bondsGross proceeds 250,000,000 Issue costs 10,000,000
240,000,000(c) Tax savings on tax-deductible expenses
Tax rate[Call premium+Unamortised issue cost on the old bonds] 9,200,000 0.4 [ 15,000,000 + 8,000,000]Initial outlay i(a) – i(b) – i(c) 15,800,000
(ii) Annual Net Cash Savings(a) Annual net cash outflow on old bonds
Interest expense 42,500,000- Tax savings on interest expense and amortisation of issue expenses 17,400,0000.4 [42,500,000 + 8,000,000/10]
25,100,000(b) Annual net cash outflow on new bonds
Interest expense 37,500,000- Tax savings on interest expense and amortisation of issue cost 15,500,000
0.4 [ 37,500,000 – 10,000,000/8] 22,000,000
Annual net cash savings : ii(a) – ii(b) 3,100,000
(iii) Present Value of the Annual Cash Savings Present value of an 8-year annuity of 3,100,000 at a discount rate of 9 per cent which is the post –tax cost of new bonds 3,100,000 x 5.535 17,158,500
(iv) Net Present Value of Refunding the Bonds
(a) Present value of annual cash savings 17,158,500
(b) Net initial outlay 15,800,000(c) Net present value of refunding the bonds :
iv(a) – iv(b). 1,358,5002. (i) Initial Outlay
(a) Cost of calling the old bonds Face value of the old bonds 120,000,000 Call premium 4,800,000
124,800,000(b) Net proceeds of the new issue
Gross proceeds 120,000,000Issue costs 2,400,000
117,600,000 (c) Tax savings on tax-deductible expenses 3,120,000
Tax rate[Call premium+Unamortised issue costs onthe old bond issue] 0.4 [ 4,800,000 + 3,000,000]
Initial outlay i(a) – i(b) – i(c) 4,080,000
(ii) Annual Net Cash Savings(a) Annual net cash out flow on old bonds
Interest expense 19,200,000- Tax savings on interest expense and amortisation of issue costs 7,920,000 0.4[19,200,000 + 3,000,000/5]
11,280,000
(b) Annual net cash outflow on new bonds Interest expense 18,000,000- Tax savings on interest expense and amortistion of issue costs 7,392,000
0.4[18,000,000 + 2,400,000/5] 10,608,000
Annual net cash savings : ii(a) – ii(b) 672,000 (iii) Present Value of the Annual Net Cash Savings
Present value of a 5 year annuity of 672,000 at as discount rate of 9 per cent, which is the post-tax 2,614,080 cost of new bonds
(iv) Net Present Value of Refunding the Bonds (a) Present value of annual net cash savings 2,614,080
(b) Initial outlay 4,080,000
(c) Net present value of refunding the bonds : - 1,466,000iv(a) – iv(b)
3. Yield to maturity of bond P 8 160 1000
918.50 = + t=1 (1+r)t (1+r)8
r or yield to maturity is 18 percent
Yield to maturity of bond Q 5 120 1000
761 = + t=1 (1+r)t (1+r)5
r or yield to maturity is 20 per cent
Duration of bond P is calculated below
Year Cash flow Present Value Proportion of Proportion of bond’s at 18% bond’s value Value x Time
1 160 135.5 0.148 0.148 2 160 114.9 0.125 0.250 3 160 97.4 0.106 0.318 4 160 82.6 0.090 0.360 5 160 69.9 0.076 0.380 6 160 59.2 0.064 0.384 7 160 50.2 0.055 0.385 8 160 308.6 0.336 2.688
4.913
Duration of bond Q is calculated below
Year Cash flow Present Value Proportion of Proportion of bond’s at 20% bond’s value Value x Time
1 120 100.0 0.131 0.131 2 120 83.2 0.109 0.218 3 120 69.5 0.091 0.273 4 120 57.8 0.076 0.304 5 1120 450.2 0.592 2.960
3.886
Volatility of bond P Volatility of bond Q
4.913 3.886= 4.16 = 3.24
1.18 1.20
4. The YTM for bonds of various maturities is
Maturity YTM(%)
1 12.36 2 13.10
3 13.21
4 13.48
5 13.72
Graphing these YTMs against the maturities will give the yield curve
The one year treasury bill rate , r1, is
1,00,000 - 1 = 12.36 %
89,000
To get the forward rate for year 2, r2, the following equation may be set up :
12500 112500 99000 = +
(1.1236) (1.1236)(1+r2)
Solving this for r2 we get r2 = 13.94%
To get the forward rate for year 3, r3, the following equation may be set up :
13,000 13,000 113,000 99,500 = + +
(1.1236) (1.1236)(1.1394) (1.1236)(1.1394)(1+r3)
Solving this for r3 we get r3 = 13.49%
To get the forward rate for year 4, r4 , the following equation may be set up :
13,500 13,500 13,500100,050 = + +
(1.1236) (1.1236)(1.1394) (1.1236)(1.1394)(1.1349)
113,500 +
(1.1236)(1.1394)(1.1349)(1+r4)
Solving this for r4 we get r4 = 14.54%
To get the forward rate for year 5, r5 , the following equation may be set up :
13,750 13,750 13,750 100,100 = + +
(1.1236) (1.1236)(1.1394) (1.1236)(1.1394)(1.1349)
13,750 +
(1.1236)(1.1394)(1.1349)(1.1454)
113,750 +
(1.1236)(1.1394)(1.1349)(1.1454)(1+r5)
Solving this for r5 we get r5 = 15.08%
Chapter 24LEASING, HIRE PURCHASE, AND PROJECT FINANCE
1
Year 0 1 2 3 4 5
1Cost of the asset 1,500,000
2 Depreciation 499,950.00 333,316.67 222,222.22 148,155.55 98,775.31
3
Loss of depreciation tax shield
-166,633.34
-111,094.44 -74,066.67 -49,380.25 -32,921.81
4Lease payment -420,000 -420,000 -420,000 -420,000 -420,000
5
Tax shield on lease payment 139,986.00 139,986.00 139,986.00 139,986.00 139,986.00
6
Loss of salvage value
-300,000.00
7Cash flow of lease 1,080,000.
-446,647.34
-391,108.44
-354,080.67
-329,394.25
-192,935.81
NAL of lease 446,647.34 391,108.44 354,080.67 329,394.25 192,935.81
= 1,080,000 - - - - - 1.08 ( 1.08)2 ( 1.08)3 ( 1.08)4 ( 1.08)5
= 1,080,000 – 413,562.35 – 335,312.45 – 281,080.65 – 242,114.61 – 131,308.87= - 323,378.93
. 2. Under the hire purchase proposal the total interest payment is 2,000,000 x 0.12 x 3 = Rs. 720,000
The interest payment of Rs. 720,000 is allocated over the 3 years period using the sum of the years digits method as follows:
Year Interest allocation
366 1 x Rs.720,000 = Rs.395,676
666
222
2 x Rs.720,000 = Rs.240,000 666
783 x Rs.720,000 = Rs.84,324
666
The annual hire purchase instalments will be :
Rs.2,000,000 + Rs.720,000 = Rs.906,667
3
The annual hire purchase instalments would be split as follows
Year Hire purchase instalment Interest Principal repayment 1 Rs.906,667 Rs.395,676 Rs. 510,991 2 Rs.906,667 Rs.240,000 Rs. 666,667
3 Rs.906,667 Rs. 84,324 Rs. 822,343
The lease rental will be as follows :Rs. 560,000 per year for the first 5 yearsRs. 20,000 per year for the next 5 years
The cash flows of the leasing and hire purchse options are shown below
Year Leasing Hire Purchase -It(1-tc)-PRt+ - LRt (1-tc) -It(1-tc) -PRt Dt(tc) NSVt Dt(tc)+NSVt
1 -560,000(1-.4)=-336,000 -395,676(1-.4) -510,991 500,000(0.4) -548,397 2 -560,000(1-.4)=-336,000 -240,000(1-.4) -666,667 375,000(0.4) -660,667 3 -560,000(1-.4)=-336,000 - 84,324(1-.4) -822,343 281,250(0.4) -760,437 4 -560,000(1-.4)=-336,000 210,938(0.4) 84,375 5 -560,000(1-.4)=-336,000 158,203(0.4) 63,281 6 - 20,000(1-.4)= - 12,000 118,652(0.4) 47,461 7 - 20,000(1-.4)= - 12,000 88,989(0.4) 35,596 8 - 20,000(1-.4)= - 12,000 66,742(0.4) 26,697 9 - 20,000(1-.4)= - 12,000 50,056(0.4) 20,02310 - 20,000(1-.4)= - 12,000 37,542(0.4) 200,000 215,017
Present value of the leasing option
5 336,000 10 12,000 = - = - 1,302,207 t=1 (1.10)t t=6 (1.10)t
Present value of the hire purchase option
548,397 660,667 760,437 84,375= - - - - (1.10) (1.10)2 (1.10)3 (1.10)4
63,281 47,461 35,596 26,697 + + +
(1.10)5 (1.10)6 (1.10)7 (1.10)8
20,023 215,017 +
(1.10)9 (1.10)10
= - 1,369,383
Since the leasing option costs less than the hire purchase option , Apex should choose the leasing option.
MINICASE
(a)
Year 1 2 3 4 5 6 7 8 9 10Principal repayment -6 -6 -6 -6 -6 Interest payment -3.6 -2.88 -2.16 -1.44 -0.72 Depreciation 12 7.20 4.32 2.59 1.56 0.93 0.56 0.34 0.20 0.12Tax shield on depn. 4.00 2.40 1.44 0.86 0.52 0.31 0.19 0.11 0.07 0.04Post tax interest payment -2.4 -1.92 -1.44 -0.96 -0.48 Net salvage value 6Net cash flow -4.40 -5.52 -6.00 -6.10 -5.96 0.31 0.19 0.11 0.07 6.04
Present value of the cash ‘ borrowing cum buying option’ is4.40 5.52 6.00 6.10 5.96 0.31 0.19 0.11 0.07 6.04
= - ----- - ------ - ------ - ----- - ----- + ----- + ------ + ------ + ------- + ------ (1.08) (1.08)2 (1.08)3 (1.08)4 (1.08)5 (1.08)6 (1.08)7 (1.08)8 (1.08)9 (1.08)10
= - 4.07 – 4.73 – 4.76 – 4.48 – 4.06 + 0.20 + 0.11 + 0.06 + 0.04 + 2.80= - 18.89 million
(b)Present value of lease cash flows =-7(1-0.3333)PVIFA8%, 5years –0.5(1- 0.3333)PVIFA8%, 5years
PVIF8% , 5years
= -7 x 0.6667 x 3.993 – 0.5 x 0.6667 x 3.993 x 0.681 = -19.54million
(c) Total interest =30,000,000 x 0.08 x 3 = Rs.720,000Monthly HP instalment = (30,000,000 + 720,000) / 36 = Rs.853,333Annual instalment = (30,000,000 + 720,000) / 3 = Rs.10,240,000Proportions for interest allocation:
36 +35+-----------------+ 25 366 I year = =
36 +35+-----------------+ 1 666
24 +23 +-----------------+13 222 II year = =
36 +35+-----------------+ 1 666
12 +35+-----------------+ 1 78III year = =
36 +35+-----------------+ 1 666Interest allocations for the three years:
I year = 720,000 x 366/666 = Rs.395,676II year = 720,000 x 222/666 = Rs.240,000IIIyear = 720,000 x 78/666 = Rs .84,324The cash flows under the HP option:
Year -It(1-Tc) -Pr Dt(Tc) NSV Total CF PVIF PV1 -263,797 -9,844,324 3,999,600 -6,108,521 0.9259 -5,656,0382 -160,008 -10,000,000 2,399,760 -7,760,248 0.8573 -6,653,1623 -56,219 -10,155,676 1,439,856 -8,772,039 0.7938 -6,963,5274 863,914 863,914 0.7350 635,0025 518,348 518,348 0.6806 352,7796 311,009 311,009 0.6302 195,9887 186,605 186,605 0.5835 108,8828 111,963 111,963 0.5403 60,4909 67,178 67,178 0.5002 33,606
10 40,307 6,000,000 6,040,307 0.4632 2,797,831
Total=
---15,088,148
Present value of the cash flows under the HP option = - Rs.15.09 million
Chapter 25HYBRID FINANCING
1.l (S/E) + (r + σ2 /2) t
d1 = t
= ln (40 / 25) + [0.16 + (0.35) 2 /2]2 0.35(2)1/2
= 0.4700 + 0.4425 0.4950
= 1.8434
d2 = d1 - t= 1.8434 – 0.35= 1.3484
N(d1) = N (1.8434). From the tables we have N(1.80) = 1- 0.0359 = 0.9641
and N(1.85)= 1- 0.0322= 0.9678By linear extrapolation, we get N(1.8434) = 0.9641 + (1.8434 – 1.8000)(0.9678-0.9641)/0.05
= 0.9641 + 0.003212 = 0.9673N(d2) = N(1.3484) From the tables we have N(1.30) = 1- 0.0968 = 0.9032
N(1.35) = 1- 0.0885 = 0.9115By linear extrapolation, we getN(1.3484) = 0.9032 + (1.3484 – 1.3000)(0.9115 – 0.9032)/0.05
= 0.9032 + 0.008034 = 0.9112E/ert = 25/1.3771 = 18.1541C = So N(d1) – E. e-rt. N(d2) = 40 x 0.9673 – 18.1541 x 0.9112= 22.15Value of the warrant is Rs.22.15.
2l (S/E) + (r + σ2 /2) t
d1 = t
= ln (50 / 30) + [0.12 + (0.4) 2 /2]2 0.4(2)1/2
= 0.5108 + 0.4 0.5657
= 1.6100
d2 = d1 - t= 1.6100 – 0.40= 1.0443
N(d1) = N (1.6100). From the tables we have N(1.60) = 1- 0.0548 = 0.9452
and N(1.65)= 1- 0.0495= 0.9505By linear extrapolation, we get N(1.6100) = 0.9452 + (1.6100 – 1.6000)(0.9505-0.9452)/0.05
= 0.9452 + 0.00106 = 0.9463N(d2) = N(1.0443) From the tables we have N(1.00) = 1- 0.1587 = 0.8413
N(1.05) = 1- 0.1469 = 0.8531By linear extrapolation, we getN(1.0443) = 0.8413 + (1.0443 – 1.0000)(0.8531 – 0.8413)/0.05
= 0.8413 + 0.01045 = 0.8517E/ert = 30/1.2712 = 23.60C = So N(d1) – E. e-rt. N(d2) = 50 x 0.9463 – 23.60 x 0.8517= 27.21Value of the warrant = Rs.27.21
3. (a) No.of shares after conversion in one year = 2
Value of the shares at the price of Rs.150 = 2 x 150 = Rs.300PV of the convertible portion at the required rate of 15% = 300/1.15 = Rs.260.87 Payments that would be received from the debenture portion:
Value of the convertible debenture = 260.87 + 418.18 = Rs. 679.05
Year Payments PVIF10%,t PV1 60 0.909 54.552 40 0.826 33.063 40 0.751 30.054 40 0.683 27.325 240 0.621 149.026 220 0.564 124.18
Total= 418.18
(b)The cash flow for Shiva is worked out as under:
Year Cash flow0 600
1 =-240-60*(1-0.3)-
2822 =-40*(1-0.3) -283 =-40*(1-0.3) -284 =-40*(1-0.3) -28
5 =-40*(1-0.3)-200-
228
6 =-20*(1-0.3)-200-
214
The post-tax cost of the convertible debenture to Shiva is the IRR of the abovecash flow stream.Let us try a discount rate of 9%. The PV of the cash flow will then be = 600 – 282/(1.09) -28/(1.09)2 - 28/(1.09)3 -28/(1.09)4-228/(1.09)5-214/(1.09)6
= 600 – 258.72 – 23.57 – 21.62 – 19.84 – 148.18 – 127.60 = 0.47 which is very near to zero.So the post –tax cost of the convertible debenture to Shiva is 9%
Chapter 26 WORKING CAPITAL POLICY
Average inventory1 Inventory period =
Annual cost of goods sold/365
(60+64)/2 = = 62.9 days
360/365
Average accounts receivableAccounts receivable = period Annual sales/365
(80+88)/2 = = 61.3 days
500/365
Average accounts payableAccounts payable =
period Annual cost of goods sold/365
(40+46)/2= = 43.43 days
360/365
Operating cycle = 62.9 + 61.3 = 124.2 daysCash cycle = 124.2 – 43.43 = 80.77 days
(110+120)/22. Inventory period = = 56.0 days
750/365
(140+150)/2Accounts receivable = = 52.9 days
period 1000/365
(60+66)/2Accounts payable = = 30.7 days
period 750/365
Operating cycle = 56.0 + 52.9 = 108.9 daysCash cycle = 108.9 – 30.7 = 78.2 days
3. This is a repetition of the solved problem 26.1 .
Chapter 27CASH AND LIQUIDITY MANAGEMENT
1 The projected cash inflows and outflows for the quarter, January through March, is shown below .
Month December January February March (Rs.) (Rs.) (Rs.) (Rs.)
Inflows : Sales collection 50,000 55,000 60,000
Outflows :Purchases 22,000 20,000 22,000 25,000Payment to sundry creditors 22,000 20,000 22,000Rent 5,000 5,000 5,000Drawings 5,000 5,000 5,000Salaries & other expenses 15,000 18,000 20,000Purchase of furniture - 25,000 -
Total outflows(2to6) 47,000 73,000 52,000
Given an opening cash balance of Rs.5000 and a target cash balance of Rs.8000, the surplus/deficit in relation to the target cash balance is worked out below :
January February March (Rs.) (Rs.) (Rs.)
1. Opening balance 5,0002. Inflows 50,000 55,000 60,0003. Outflows 47,000 73,000 52,0004. Net cash flow (2 - 3) 3,000 (18,000) 8,0005. Cumulative net cash flow 3,000 (15,000) (7,000)6. Opening balance + Cumulative net cash flow 8,000 (10,000) (2,000)7. Minimum cash balance required 8,000 8,000 8,0008. Surplus/(Deficit) - (18,000) (10,000)
2 The balances in the books of Datta co and the books of the bank are shown below:
(Rs.)
1 2 3 4 5 6 7 8 9 10Books of Datta Co:
Opening Balance
30,000 46,000 62,000 78,000 94,000 1,10,000 1,26,000 1,42,000 1,58,000 1,74,000
Add: Cheque received
20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000
Less: Cheque issued
4,000 4,000 4,000 4,000 4,000 4,000 4,000 4,000 4,000 4,000
Closing Balance
46,000 62,000 78,000 94,000 1,10,000 1,26,000 1,42,000 1,58,000 1,74,000 1,90,000
Books of the Bank:
Opening Balance
30,000 30,000 30,000 30,000 30,000 30,000 50,000 70,000 90,000 1,06,000
Add: Cheques realised
- - - - - 20,000 20,000 20,000 20,000 20,000
Less: Cheques debited
- - - - - - - - 4,000 4,000
Closing Balance
30,000 30,000 30,000 30,000 30,000 50,000 70,000 90,000 1,06,000 1,22,000
From day 9 we find that the balance as per the bank’s books is less than the balance as per Datta Company’s books by a constant sum of Rs.68,000. Hence in the steady situation Datta Company has a negative net float of Rs.68,000.
3. Optimal conversion size is2bT
C = I
b = Rs.1200, T= Rs.2,500,000, I = 5% (10% dividend by two)
So, 2 x 1200 x 2,500,000
C = = Rs.346,4100.05
4. 3 3 b2
RP = + LL 4I
UL = 3 RP – 2 LL
I = 0.12/360 = .00033, b = Rs.1,500, = Rs.6,000, LL = Rs.100,000
3 3 x 1500 x 6,000 x 6,000RP = + 100,000
4 x .00033
= 49,695 + 100,000 = Rs.149,695
UL = 3RP – 2LL = 3 x 149,695 – 2 x 100,000 = Rs.249,085
5. Optimal conversion size is
2bTC =
Ib = Rs.2800, T= Rs.35,000,000, I = 5% (10% dividend by two)
So, 2 x 2800 x 35,000,000
C = = Rs.1,979,8990.05
6 3 3 b2
RP = + LL 4I
UL = 3 RP – 2 LL
I = 0.12/360 = .00033, b = Rs.3,200, = Rs.22,000, LL = Rs.800,000
3 3 x 3200 x 22,000 x 22,000RP = + 800,000
4 x .00033
= 152,118 + 800,000 = Rs.952,118
UL = 3RP – 2LL = 3 x 952,118 – 2 x 800,000 = Rs.1,256,354
Chapter 28CREDIT MANAGEMENT
1. Δ RI = [ΔS(1-V)- ΔSbn](1-t)- k ΔIΔ S
Δ I = x ACP x V360
Δ S = Rs.10 million, V=0.85, bn =0.08, ACP= 60 days, k=0.15, t = 0.40
Hence, ΔRI = [ 10,000,000(1-0.85)- 10,000,000 x 0.08 ] (1-0.4)
-0.15 x 10,000,000 x 60 x 0.85
360 = Rs. 207,500
2. Δ RI = [ΔS(1-V)- ΔSbn] (1-t) – k Δ I
So ΔSΔ I = (ACPN – ACPo) +V(ACPN)
360 360
ΔS=Rs.1.5 million, V=0.80, bn=0.05, t=0.45, k=0.15, ACPN=60, ACPo=45, So=Rs.15 millionHence ΔRI = [1,500,000(1-0.8) – 1,500,000 x 0.05] (1-.45)
-0.15 (60-45) 15,000,000 + 0.8 x 60 x 1,500,000
360 360 = 123750 – 123750 = Rs. 0
3. Δ RI = [ΔS(1-V) –Δ DIS ] (1-t) + k Δ I Δ DIS = pn(So+ΔS)dn – poSodo
So ΔSΔ I = (ACPo-ACPN) - x ACPN x V
360 360
So =Rs.12 million, ACPo=24, V=0.80, t= 0.50, r=0.15, po=0.3, pn=0.7,ACPN=16, ΔS=Rs.1.2 million, do=.01, dn= .02Hence
ΔRI = [ 1,200,000(1-0.80)-{0.7(12,000,000+1,200,000).02- 0.3(12,000,000).01}](1-0.5)
12,000,000 1,200,000 + 0.15 (24-16) - x 16 x 0.80
360 360
= Rs.79,200
4. Δ RI = [ΔS(1-V)- ΔBD](1-t) –kΔ IΔBD=bn(So+ΔS) –boSo
So ΔS ΔI = (ACPN –ACPo) + x ACPN x V
360 360
So=Rs.50 million, ACPo=25, V=0.75, k=0.15, bo=0.04, ΔS=Rs.6 million,ACPN=40 , bn= 0.06 , t = 0.3
ΔRI = [ Rs.6,000,000(1-.75) –{.06(Rs.56,000,000)-.04(Rs.50,000,000)](1-0.3)
Rs.50,000,000 Rs.6,000,000 - 0.15 (40-25) + x 40 x 0.75
360 360
= - Rs.289,495
5. 30% of sales will be collected on the 10th day 70% of sales will be collected on the 50th day
ACP = 0.3 x 10 + 0.7 x 50 = 38 days
Rs.40,000,000 Value of receivables = x 38
360
= Rs.4,222,222 Assuming that V is the proportion of variable costs to sales, the investment in receivables is :
Rs.4,222,222 x V
6. 30% of sales are collected on the 5th day and 70% of sales are collected on the25th day. So, (a) ACP = 0.3 x 5 + 0.7 x 25 = 19 days
Rs.10,000,000
Value of receivables = x 19 360
= Rs.527,778 (b) Investment in receivables = 0.7 x 527,778
= Rs.395,833
7. Since the change in credit terms increases the investment in receivables,ΔRI = [ΔS(1-V)- ΔDIS](1-t) – kΔI
So=Rs.50 million, ΔS=Rs.10 million, do=0.02, po=0.70, dn=0.03,pn=0.60, ACPo=20 days, ACPN=24 days, V=0.85, k=0.12 , and t = 0.40 ΔDIS = 0.60 x 60 x 0.03 – 0.70 x 50 x 0.2
= Rs.0.38 million
50 10 Δ I = (24-20) + x 24 x 0.85
360 360
= Rs.1.2222 million Δ RI = [ 10,000,000 (1-.85) – 380,000 ] (1-.4) – 0.12 x 1,222,222
= Rs.525,333
8. The decision tree for granting credit is as follows :
Customer pays(0.95)
Grant credit Profit 1500 Customer pays(0.85)
Grant credit Customer defaults(0.05)Profit 1500 Refuse credit
Loss 8500 Customer defaults(0.15)
Loss 8500 Refuse credit
The expected profit from granting credit, ignoring the time value of money, is :
Expected profit on + Probability of payment x Expected profit onInitial order and repeat order repeat order
{ 0.85(1500)-0.15(8500)} + 0.85 {0.95(1500)-.05(8500)} = 0 + 850 = Rs.850
9. Profit when the customer pays = Rs.10,000 - Rs.8,000 = Rs.2000Loss when the customer does not pay = Rs.8000
Expected profit = p1 x 2000 –(1-p1)8000 Setting expected profit equal to zero and solving for p1 gives : p1 x 2000 – (1- p1)8000 = 0 p1 = 0.80 Hence the minimum probability that the customer must pay is 0.80
MINICASE Solution:
Present Data Sales : Rs.800 million Credit period : 30 days to those deemed eligible Cash discount : 1/10, net 30 Proportion of credit sales and cash sales are 0.7 and 0.3. 50 percent of the credit customers
avail of cash discount Contribution margin ratio : 0.20 Tax rate : 30 percent Post-tax cost of capital : 12 percent ACP on credit sales : 20 days
Effect of Relaxing the Credit Standards on Residual Income
Incremental sales : Rs.50 million Bad debt losses on incremental sales: 12 percent ACP remains unchanged at 20 days
∆RI = [∆S(1 – V) - ∆Sbn] (1 – t) – R ∆ I
∆Swhere ∆ I = x ACP x V
360
∆ RI = [50,000,000 (1-0.8) – 50,000,000 x 0.12] (1 – 0.3)
50,000,000- 0.12 x x 20 x 0.8
360
= 2,800,000 – 266,667 = 2,533,333
Effect of Extending the Credit Period on Residual Income
∆ RI = [∆S(1 – V) - ∆Sbn] (1 – t) – R ∆ I
So ∆Swhere ∆I = (ACPn – ACPo) + V (ACPn)
360 360
∆RI = [50,000,000 (1 – 0.8) – 50,000,000 x 0] (1 – 0.3)
800,000,000 50,000,000 - 0.12 (50 – 20) x + 0.8 x 50 x
360 360
= 7,000,000 – 8,666,667= - Rs.1,666,667
Effect of Relaxing the Cash Discount Policy on Residual Income
∆RI = [∆S (1 – V) - ∆ DIS] (1 – t) + R ∆ Iwhere ∆ I = savings in receivables investment
So ∆S = (ACPo – ACPn) – V x ACPn 360 360
800,000,000 20,000,000 = (20 – 16) – 0.8 x x 16
360 360
= 8,888,889 – 711,111 = 8,177,778
∆ DIS = increase in discount cost = pn (So + ∆S) dn – po So do
= 0.7 (800,000,000 + 20,000,000) x 0.02 – 0.5 x 800,000,000 x 0.01 = 11,480,000 – 4,000,000 = 7,480,000
So, ∆RI = [20,000,000 (1 – 0.8) – 7,480,000] (1 – 0.3) + 0.12 x 8,177,778 = - 2,436,000 + 981,333 = - 1,454,667
Chapter 29 INVENTORY MANAGEMENT
1.a. No. of Order Ordering Cost Carrying Cost Total Cost Orders Per Quantity (U/Q x F) Q/2xPxC of Ordering Year (Q) (where and Carrying (U/Q) PxC=Rs.30)
Units Rs. Rs. Rs.
1 250 200 3,750 3,950 2 125 400 1,875 2,275 5 50 1,000 750 1,750
10 25 2,000 375 2,375
2 UF 2x250x200b. Economic Order Quantity (EOQ) = =
PC 30 2UF = 58 units (approx)
2. a EOQ = PC
U=10,000 , F=Rs.300, PC= Rs.25 x 0.25 =Rs.6.25
2 x 10,000 x 300 EOQ = = 980
6.25 10000
b. Number of orders that will be placed is = 10.20 980
Note that though fractional orders cannot be placed, the number of orders relevant for the year will be 10.2 . In practice 11 orders will be placed during the year. However, the 11th order will serve partly(to the extent of 20 percent) the present year and partly(to the extent of 80 per cent) the following year. So only 20 per cent of the ordering cost of the 11th order relates to the present year. Hence the ordering cost for the present year will be 10.2 x Rs.300
c. Total cost of carrying and ordering inventories 980
= [ 10.2 x 300 + x 6.25 ] = Rs.6122.5 2
3. U=6,000, F=Rs.400 , PC =Rs.100 x 0.2 =Rs.20
2 x 6,000 x 400EOQ = = 490 units
20
U U Q’(P-D)C Q* PC Δπ = UD + - F- -
Q* Q’ 2 2
6,000 6,000 = 6000 x .5 + - x 400
490 1,000
1,000 (95)0.2 490 x 100 x 0.2- -
2 2
= 30,000 + 2498 – 4600 = Rs.27898
4. U=5000 , F= Rs.300 , PC= Rs.30 x 0.2 = Rs.6
2 x 5000 x 300EOQ = = 707 units
6 If 1000 units are ordered the discount is : .05 x Rs.30 = Rs.1.5 Change in profit when 1,000 units are ordered is :
5,000 5,000 Δπ = 5000 x 1.5 + - x 300
707 1,000
1000 x 28.5 x 0.2 707 x 30 x 0.2 - - = 7500 + 622-729 =Rs.7393
2 2
If 2000 units are ordered the discount is : .10 x Rs.30 = Rs.3 Change in profit when 2,000 units are ordered is :
5000 5000 2000x27x0.2 707x30x0.2 Δπ = 5000 x 3.0 + - x 300- -
707 2000 2 2
= 15,000 +1372 – 3279 = Rs.13,093
5. The quantities required for different combinations of daily usage rate(DUR) and lead times(LT) along with their probabilities are given in the following table
LT (Days)DUR 5(0.6) 10(0.2) 15(0.2)
(Units)
4(0.3) 20*(0.18) 40(0.06) 60(0.06)
6(0.5) 30 (0.30) 60(0.10) 90(0.10) 8(0.2) 40 (0.12) 80(0.04) 120(0.04)
* Note that if the DUR is 4 units with a probability of 0.3 and the LT is 5 days with a probability of 0.6, the requirement for the combination DUR = 4 units and LT = 5 days is 20 units with a probability of 0.3x0.6 = 0.18. We have assumed that the probability distributions of DUR and LT are independent. All other entries in the table are derived similarly.
The normal (expected) consumption during the lead time is :20x0.18 + 30x0.30 + 40x0.12 + 40x0.06 + 60x0.10 + 80x0.04 + 60x0.06 + 90x0.10 + 120x0.04 = 46.4 tonnes
a. Costs associated with various levels of safety stock are given below :
Safety Stock Stock out Probability Expected Carrying Total CostStock* outs(in Cost Stock out Cost
tonnes)
1 2 3 4 5 6 7[3x4] [(1)x1,000] [5+6]
Tonnes Rs. Rs. Rs. 73.6 0 0 0 0 73,600 73,600 43.6 30 120,000 0.04 4,800 43,600 48,400
33.6 10 40,000 0.10 40 160,000 0.04 10,400 33,600 44,000
13.6 20 80,000 0.04 30 120,000 0.10 24,800 13,600 38,400 60 240,000 0.04
0 13.6 54,400 0.16 33.6 134,400 0.04 43,296 0 43,296 43.6 174,400 0.10
73.6 294,400 0.04
* Safety stock = Maximum consumption during lead time – Normal consumption during lead time
So the optimal safety stock= 13.6 tonnes Reorder level = Normal consumption during lead time + safety stock
K= 46.4 + 13.6 = 60 tonnes
b. Probability of stock out at the optimal level of safety stock = Probability(consumption being 80 or 90 or 120 tonnes)
Probability (consumption = 80 tonnes) + Probability (consumption = 90 tonnes) + Probability (consumption = 120 tonnes) = 0.04 +0.10+0.04 = 0.18
6.
Item Annual Usage Price per Annual Ranking(in Units) Unit Usage Value
Rs. Rs.
1 400 20.00 8,000 6 2 15 150.00 2,250 10 3 6,000 2.00 12,000 5 4 750 18.00 13,500 4 5 1,200 25.00 30,000 1 6 25 160.00 4,000 9
7 300 2.00 600 14 8 450 1.00 450 15 9 1,500 4.00 6,000 7 10 1,300 20.00 26,000 2 11 900 2.00 1,800 11 12 1,600 15.00 24,000 3 13 600 7.50 4,500 8 14 30 40.00 1,200 12 15 45 20.00 900 13
1,35,200
Cumulative Value of Items & Usage
Item Rank Annual Cumulative Cumulative Cumulative No. UsageValue Annual Usage % of Usage % of Items
(Rs.) Value (Rs.) Value
5 1 30,000 30,000 22.2 6.7 10 2 26,000 56,000 41.4 13.3 12 3 24,000 80,000 59.2 20.0 4 4 13,500 93,500 69.2 26.7 3 5 12,000 105,500 78.0 33.3 1 6 8,000 113,500 83.9 40.0 9 7 6,000 119,500 88.4 46.7 13 8 4,500 124,000 91.7 53.3 6 9 4,000 128,000 94.7 60.0 2 10 2,250 130,250 96.3 66.7 11 11 1,800 132,050 97.7 73.3 14 12 1,200 133,250 98.6 80.0 15 13 900 134,150 99.2 86.7 7 14 600 134,750 99.7 93.3 8 15 450 135,200 100.0 100.0
Class No. of Items % to the Total Annual Usage % to Total Value Value Rs.
A 4 26.7 93,500 69.2 B 3 20.0 26,000 19.2 C 18 53.3 15,700 11.6
15 135,200
7. The quantities required for different combinations of daily usage rate(DUR) and lead times(LT) along with their probabilities are given in the following table
LT (Days)DUR 5(0.4) 8(0.4) 12(0.2)
(Units)
2(0.2) 10 (0.08) 16(0.08) 24(0.04)
3(0.6) 15 (0.24) 24(0.24) 36(0.12) 4(0.2) 20 (0.08) 32(0.08) 48(0.04)
The normal (expected) consumption during the lead time is :10x0.08 + 15 x0.24 + 20x0.08 + 16x0.08 + 24x0.24 + 32 x0.08 + 24x0.04 + 36 x0.12 + 48 x0.04 = 22.8 tonnes
c. Costs associated with various levels of safety stock are given below :
Safety Stock Stock out Probability Expected Carrying Total CostStock* outs(in Cost Stock out Cost
tonnes)
1 2 3 4 5 6 7[3x4] [(1)x1,500] [5+6]
Tonnes Rs. Rs. Rs. 25.2 0 0 0 0 37,800 37,800 13.2 12 60,000 0.04 2,400 19,800 22,200
9.2 4 20,000 0.12 16 80,000 0.04 5,600 13,800 19,400
1.2 8 40,000 0.08 12 60,000 0.12 15,200 1,800 17,000 24 120,000 0.04
0 1.2 6,000 0.28 9.2 46,000 0.08 18,320 0 18,320 13.2 66,000 0.12
25.2 126,000 0.04
* Safety stock = Maximum consumption during lead time – Normal consumption during lead time
a) So the optimal safety stock= 1.2 tonnes Reorder level = Normal consumption during lead time + safety stock
K= 22.8 + 1.2 = 24 tonnes
b) Probability of stock out at the optimal level of safety stock = Probability(consumption being 32 or 36 or 48 tonnes)
= 0.08 +0.12+0.04 = 0.24
Chapter 30WORKING CAPITAL FINANCING
1. Annual interest cost is given by , Discount % 360
x 1- Discount % Credit period – Discount period
Therefore, the annual per cent interest cost for the given credit terms will be as follows:
a. 0.01 360 x = 0.182 = 18.2%
0.99 20
b. 0.02 360 x = 0.367 = 36.7%
0.98 20
c. 0.03 360 x = 0.318 = 31.8%
0.97 35
d. 0.01 360 x = 0.364 = 36.4%
0.99 10
2.a. 0.01 360
x = 0.104 = 10.4%0.99 35
b. 0.02 360 x = 0.21 = 21%
0.98 35
c. 0.03 360 x = 0.223 = 22.3%
0.97 50
d. 0.01 360 x = 0.145 = 14.5%
0.99 253. The maximum permissible bank finance under the three methods suggested by
The Tandon Committee are :
Method 1 : 0.75(CA-CL) = 0.75(36-12) = Rs.18 millionMethod 2 : 0.75(CA)-CL = 0.75(36-12 = Rs.15 million
Method 3 : 0.75(CA-CCA)-CL = 0.75(36-18)-12 = Rs.1.5 million
4. Raw material and stores and spares consumed (RMC)= Opening stock of raw materials and stores and spares + purchases – closing stock of raw materials and stores and spares
= 524 + 1821 – 540 = 1805Cost of production = RMC + Other operating expenses(including depreciation) +
Opening stock of work-in-process – Closing stock of work-in-process= 1805 + 674 + 218 – 226 = 2471
Cost of sales = Cost of production + Opening stock of finished goods – Closing stock of finished goods
= 2471 + 485 – 588 = 2368Holding level of raw material and stores and spares(months consumption)
=( 540 x 12) / 1805 = 3.59 monthsHolding level of work-in-process ( months cost of production)
= ( 226 x 12) / 2471 = 1.10 monthsHolding level of finished goods(months cost of sales)
= (588 x 12)/ 2368 = 2.98 months
Chapter 31WORKING CAPITAL MANAGEMENT :EXTENSIONS
1.(a) The discriminant function is :
Zi = aXi + bYi
where Zi = discriminant score for the ith account Xi = quick ratio for the ith accountYi = EBDIT/Sales ratio for the ith account
The estimates of a and b are : y
2. dx - xy . dy a =
x 2. y 2 - xy . xy
x 2. dy xy . dx
b = x
2 y 2 xy xy
The basic calculations for deriving the estimates of a and b are giventhe accompanying table.
Drawing on the information in the accompanying table we find that
Xi = 19.81 Yi= 391 (Xi-X)2 Yi-Y)2 Xi-X)(Yi-Y)
X = 0.7924 Y = 15.64 = 0.8311 =1661.76 = 10.007
Account Xi Yi (Xi-X) (Yi-Y) (Xi-X)2 (Yi-Y)2 (Xi-X)(Yi-Y) Number
1 0.90 15 0.1076 -0.64 0.0116 0.4096 -0.06892 0.75 20 -0.0424 4.36 0.0018 19.0096 -0.18493 1.05 10 -0.2576 -5.64 0.0664 31.8096 -1.45294 0.85 14 0.0576 -1.64 0.0033 2.6896 -0.0945
G 5 0.65 16 -0.1424 0.36 0.0203 0.1296 -0.513R 6 1.20 20 0.4076 4.36 0.1661 19.0096 1.7771O 7 0.90 24 0.1076 8.36 0.0116 69.8896 0.8995U 8 0.84 26 0.0476 10.36 0.0023 107.3296 0.4931P 9 0.93 11 0.1376 -4.64 0.0189 21.5296 -0.6385 10 0.78 18 -0.0124 2.36 0.0002 5.5696 -0.0293I 11 0.96 12 0.1676 -3.64 0.0281 13.2496 -0.6101 12 1.02 25 0.2276 9.36 0.0518 87.6096 2.1303
13 0.81 26 0.0176 10.36 0.0003 107.3296 0.1823 14 0.76 30 -0.0324 14.36 0.0010 206.2096 -0.4653 15 1.02 28 0.2276 12.36 0.0518 152.7696 2.8131
16 0.76 10 -0.0324 -5.64 0.0010 31.8069 0.1827 17 0.68 12 -0.1124 -3.64 0.0126 13.2496 0.4091G 18 0.56 4 -0.2324 -11.64 0.0540 135.4896 2.7051R 19 0.62 18 -0.1724 2.36 0.0297 5.5696 -0.4069O 20 0.92 -4 0.1276 -19.64 0.0163 385.7296 -2.5061U 21 0.58 20 -0.2124 4.36 0.0451 19.0096 -0.9261P 22 0.70 8 -0.0924 - 7.64 0.0085 58.3696 0.7059 23 0.52 15 –0.2724 -0.64 0.0742 0.4096 0.1743II 24 0.45 6 –0.3424 -9.64 0.1172 92.9296 3.3007 25 0.60 7 –0.1924 -8.64 0.0370 74.6496 1.6623
19.81 391 0.8311 1661.76 9.539
Sum of Xi for group 1 13.42X1 = = = 0.8947
15 15
Sum of Xi for group 2 6.39X2 = = = 0.6390
10 10
Sum of Yi for group 1 295Y1 = = = 19.67
15 15
Sum of Yi for group 2 96Y2 = = = 9.60
10 10
1 0.8311x 2 = Xi –X)2 = = 0.0346
n-1 25-1
1 1661.76 y
2 = Yi – Y)2 = = 69.24n-1 25-1
1 10.0007xy = Xi-X)(Yi-Y) = = 0.4167
n-1 25-1
dx = X1 - X2 = 0.8947 – 0.6390 = 0.2557
dy = Y1 – Y2 = 19.67 – 9.60 = 10.07
Substituting these values in the equations for a and b we get :
69.24 x 0.2557 – 0.4167 x 10.07a = = 6.079 0.0346 x 69.24 – 0.4167 x 0.4167
0.0346 x 10.07 – 0.4167 x 0.2557b = = 0.1089
0.0346 x 69.24 – 0.4167 x 0.4167
Hence , the discriminant function is :Zi = 6.079 Xi + 0.1089 Yi
(b) Choice of the cutoff pointThe Zi score for various accounts are shown below
Zi scores for various accounts
Account No. Zi Score
1 7.10462 6.73733 7.47204 6.69185 5.69386 9.47287 8.08478 7.93789 6.851410 6.701811 7.142612 8.923113 7.755414 7.887015 9.2498
16 5.709017 5.440518 3.839819 5.729220 5.157121 5.703822 5.126523 4.794624 3.389025 4.4097
The Zi scores arranged in an ascending order are shown below
Good(G)Account Number Zi Score or
Bad (B)
24 3.3890 B18 3.8398 B25 4.4097 B23 4.7946 B22 5.1265 B20 5.1571 B17 5.4405 B 5 5.6938 G21 5.7038 B16 5.7090 B19 5.7292 B 4 6.6918 G10 6.7018 G 2 6.7373 G 9 6.8514 G 1 7.1046 G11 7.1426 G 3 7.4720 G13 7.7554 G14 7.8870 G 8 7.9378 G 7 8.0847 G12 8.9231 G15 9.2498 G 6 9.4728 G
From the above table, it is evident that a Zi score which represents the mid-point between the Zi scores of account numbers 19 and 4 results in the minimum number of misclassifications . This Zi score is :
5.7292 + 6.6918= 6.2105
2Given this cut-off Zi score, there is just one misclassification (Account number 5)
CA
2 WCL = (CA + NFA)– 0.2 CA
Dividing both the numerator and denominator by CA, we get
1= 1
1 +( NFA/CA) -0.2
0.8 +NFA/CA = 1 or NFA/CA =0.2
31.3
From the above table we get the followingXi = 139 Yi= 18 (Xi-X)2 Yi-Y)2 Xi-X)(Yi-Y)
X = 8.6875 Y = 1.125 = 1519.438 = 3.69 = -55.975 Sum of Xi for Good Accts. 133X1 = = = 16.625
8 8
Sum of Xi for Bad Accts. 6X2 = = = 0.75
8 8
Sum of Yi for Good Accts. 6.3Y1 = = = 0.7875
8 8
Sum of Yi for Bad Accts. 11.7Y2 = = = 1.4625
8 8 1 1519.438
x 2 = Xi –X)2 = = 101.30 n-1 16-1
1 3.69 y
2 = Yi – Y)2 = = 0.246n-1 16-1
1 -55.975xy = Xi-X)(Yi-Y) = = - 3.73
n-1 16 -1
dx = X1 - X2 = 16.625 – 0.75 = 15.875
dy = Y1 – Y2 = 0.7875 – 1.4625 = - 0.675
Substituting these values in the equations for a and b we get :
0.246 x 15.875 - 3.73 x 0.675a = = 0.1261 101.30 x 0.246 – 3.73 x 3.73
-101.30 x 0.675 + 3.73 x 15.875b = = - 0.8325
101.30 x 0.246 – 3.73 x 3.73
Hence , the discriminant function is :Zi = 0.1261 Xi - 0.8325 Yi
Chapter 32CORPORATE VALUATION
1. (a) The calculations for Hitech Limited are shown below :Year 2 Year3
EBIT PBT 86 102+ Interest expense 24 28- Interest income (10) (15)- Non-operating income (5) (10) EBIT 95 105
Tax on EBIT Tax provision on income statement 26 32+ Tax shield on interest expense 9.6 11.2- Tax on interest income (4) (6)- Tax on non-operating income (2) (4) Tax on EBIT 29.6 33.2
NOPLAT 65.4 71.8Net investment (50) (50)Non-operating cash flow (post-tax) 3 6FCFF 18.4 27.8
(b) The financing flow for years 2 and 3 is as follows :Year 2 Year 3
After-tax interest expense 14.4 16.8 Cash dividend 30 40- Net borrowings (30) (30)+ Excess marketable securities 30 10- After-tax income on excess (6) (9) marketable securities- Share issue (20) -
18.4 27.8
(c) Year 2 Year 3Invested capital (Beginning) 310 360Invested capital (Ending) 360 410NOPLAT 65.4 71.8Turnover 400 460
Net investment 50 50
Post-tax operating margin 16.35% 15.61%Capital turnover 1.29 1.28ROIC 21.1% 19.9%Growth rate 16.1% 13.9%FCF 15.4 21.8
2. Televista Corporation
0 1 2 3 4 5 Base year
1. Revenues 1600 1920 2304 2765 3318 36502. EBIT 240 288 346 415 498 5473. EBIT (1-t) 156 187 225 270 323 3564. Cap. exp. 200 240 288 346 415 -
- Depreciation 120 144 173 207 2495. Working capital 400 480 576 691 829 9126. Working capital 80 96 115 138 83 7. FCFF 11 13 16 19 273
(3-4-6)
Discount factor 0.876 0.767 0.672 .589Present value 9.64 9.97 10.76 11.19
Cost of capital for the high growth period
0.4 [12% + 1.25 x 7%] + 0.6 [15% (1 - .35)]8.3% + 5.85%
= 14.15%
Cost of capital for the stable growth period0.5 [12% + 1.00 x 6%] + 0.5 [14% (1 - .35)]
9% + 4.55% = 13.55%
Present value of FCFF during the explicit forecast period= 9.64 + 9.97 + 10.76 + 11.19 = 41.56
273 273Horizon value = = = 7690
0.1355 – 0.10 0.0355
Present value of horizon value = 4529.5
Value of the firm = 41.56 + 4529.50 = Rs.4571.06 million
3. The WACC for different periods may be calculated :
WACC in the high growth period
Year kd(1-t) = 15% (1-t) ke = Rf + x Market risk premium ka = wd kd (1-t)+ we ke
1 15 (0.94) = 14.1% 12 + 1.3 x 7 = 21.1% 0.5 x 14.1 + 0.5 x 21.1 = 17.6%2 15 (0.88) = 13.2% 21.1% 0.5 x 13.2 + 0.5 x 21.1 = 17.2%3 15 (0.82) = 12.3% 21.1% 0.5 x 12.3 + 0.5 x 21.1 = 16.7%4 15 (0.76) = 11.4% 21.1% 0.5 x 11.4 + 0.5 x 21.1 = 16.3%5 15 (0.70) = 10.5% 21.1% 0.5 x 10.5 + 0.5 x 21.1 = 15.8%
WACC in the transition periodkd(1-t) = 14 (1 – 0.3) = 9.8%ke = 11 + 1.1 x 6 = 17.6%ka = 0.44 x 9.8 + 0.56 x 17.6 = 14.2%
WACC for the stable growth periodkd(1-t) = 13 (1 – 0.3) = 9.1%ke = 11 + 1.0 x 5 = 16%ka = 1/3 x 9.1 + 2/3 x 16 = 13.7%
The FCFF for years 1 to 11 is calculated below. The present value of the FCFF for the years 1 to 10 is also calculated below.
3Multisoft Limited
Period Growth rate (%)
EBIT Tax rate (%)
EBIT (1-t)
Cap. exp.
Dep. WC FCFF D/E Beta WACC %
PV Factor
Present value
0 90 100 601 40 126 6 118 140 84 26 36 1:1 1.3 17.6 .850 30.62 40 176 12 155 196 118 39 38 1:1 1.3 17.2 .726 27.63 40 247 18 203 274 165 50 44 1:1 1.3 16.7 .622 27.44 40 346 24 263 384 230 70 39 1:1 1.3 16.3 .535 20.85 40 484 30 339 538 323 98 26 1:1 1.3 15.8 .462 12.06 34 649 30 454 721 432 132 33 0.8:1 1.1 14.2 .405 13.47 28 830 30 581 922 553 169 43 0.8:1 1.1 14.2 .354 15.48 22 1013 30 709 1125 675 206 53 0.8:1 1.1 14.2 .310 16.7
9 16 1175 30 822 1305 783 239 61 0.8:1 1.1 14.2 .272 16.910 10 1292 30 905 1436 862 263 68 0.8:1 1.1 14.2 .238 16.611 10 1421 30 995 1580 948 289 74 0.5:
1.01.1 13.7 476
673.4The present value of continuing value is :
FCF11 74 x PV factor 10 years = x 0.238 = 476
k – g 0.137 – 0.100
This is shown in the present value cell against year 11.
The value of the firm is equal to :Present value of FCFF during + Present value of continuingThe explicit forecast period of 10 years value
This adds up to Rs.685.4 million as shown below
MINI CASE
Solution:
1 2 3 4 5 61. Revenues 950 1,000 1,200 1,450 1,660 1,7702. PBIT 140 115 130 222 245 2873. NOPAT = PBIT (1 – .35)
91 74.8 84.5 144.3 159.3 186.6
4. Depreciation 55 85 80 83 85 875. Gross cash flow 146 159.8 164.5 227.3 244.3 273.76. Gross investment in fixed assets
100 250 85 100 105 120
7. Investment in net current assets
10 15 70 70 70 54
8. Total investment 110 265 155 170 175 1749. FCFF (5) – (8) 36 (105.2) 9.5 57.3 69.3 99.6
0.4 1.0 WACC = x 12 x (1 – 0.35) + {8 + 1.06 (8)} 1.4 1.4
= 14%
99.6 (1.10)Continuing Value = = 2739.00
0.14 – 0.10
2739Present value of continuing value = = 1249 (1.14)6
PV of the FCFF during the explicit forecast period 3.6 105.2 9.5 57.3 69.3 99.6= – + + + + (1.14) (1.14)2
(1.14)3 (1.14)4 (1.14)5 (1.14)6
= 72.4 Firm value = 72.4 + 1249 = 1321.4
Value of equity = 1321.4 – 200 = 1121.4 million
Chapter 33VALUE BASED MANAGEMENT
1. The value created by the new strategy is calculated below :
Current Income Statement ProjectionValues (Year 0) 1 2 3 4 5
Sales 2000 2240 2509 2810 3147 3147Gross margin (20%) 400 448 502 562 629 629Selling and general 160 179 201 225 252 252 administration (8%)Profit before tax 240 269 301 337 378 378Tax 72 81 90 101 113 113 Profit after tax 168 188 211 236 264 264
Balance Sheet ProjectionsFixed assets 600 672 753 843 944 944Current assets 600 672 753 843 944 944Total assets 1200 1344 1505 1696 1888 1888Equity 1200 1344 1505 1686 1888 1888
Cash Flow ProjectionsProfit after tax 188 211 236 264 264Depreciation 60 67 75 84 94Capital expenditure 132 148 166 185 94Increase in current assets 72 81 90 101 -Operating cash flow 44 49 55 62 264
Present value of the operating cash flow = 147Residual value = 264 / 0.15 = 1760Present value of residual value = 1760 / (1.15)4 = 1007Total shareholder value = 147 + 1007 = 1154Pre-strategy value = 168/0.15 = 1120Value of the strategy = 1154 – 1120 = 34
2. According to the Marakon approachM r – g =
B k – g
r - .102 =
k - .10r - .10 = 2k - .20r = 2k - .10r/k = 2 - (.10/k)
Thus r/k is a function of k. Unless k is specified r/k cannot be determined.
3. NOPAT for 20X1
PBIT (1 – T) = 24 (0.65) = 15.6Cost of equity
6% + 0.9 (6%) = 11.4%
Average cost of capital0.5 x 8% (1 - .35) + 0.5 x 11.4% = 8.3%
EVA for 20X1NOPAT - Average cost of capital x Capital employed15.6 - .083 x 100 = 7.3
4.I = Rs.200 millionr = 0.40c* = 0.20T = 5 years
200 (0.40 – 0.20) 5Value of forward plan =
0.20 (1.20)
= Rs.833.3 million
5. Cost of capital = 0.5 x 0.10 + 0.5 x 0.18 = 0.14 or 14 per cent
1. Revenues 2,000 2,000 2,000 2,000 2,0002. Costs 1,400 1,400 1,400 1,400 1,4003. PBDIT 600 600 600 600 6004. Depreciation 200 200 200 200 2005. PBIT 400 400 400 400 4006. NOPAT 240 240 240 240 2407. Cash flow (4+6) 440 440 440 440 440
8. Capital at charge 1,000 800 600 400 2009. Capital charge (8x0.14) 140 112 84 56 2810. EVA (6-9) 100 128 156 184 212
5 440NPV = - 1000 = 440 x 3.433 – 1000 = 510.5
t=1 (1.14)t
EVAt
NPV = = 100 x 0.877 + 128 x 0.769 + 156 x 0.675 + 184 x 0.592 + (1.14)t 212 x 0.519
= 510.3
6. Equipment cost = 1,000,000 Economic life = 4 years Salvage value = Rs.200,000 Cost of capital = 14 per cent
Present value of salvage value = 200,000 x 0.592 = 118,400
Present value of the annuity = 1,000,000 – 118,400= 881,600
881,600 881,600Annuity amount = =
PVIFA14%, 4yrs 2.914
= Rs.302,540
Depreciation charge under sinking fund method1 2 3 4
Capital 1,000,000 837,460 652,164 440,927Depreciation 162,540 185,296 212,237 240,810Capital charge 140,000 117,244 91,303 61,730Sum 302,540 302,540 302,540 302,540
7. Investment : Rs.2,000,000Life : 10 yearsCost of capital : 15 per centSalvage value : 0
2,000,000Economic depreciation =
FVIFA(10yrs, 15%)
2,000,000 = = 98,503
20.304
8. Investment : Rs.5,000,000Life : 5 yearsCost of capital : 12 per centSalvage value : Nil
PVIFA(5yrs,12%) = 3.605 ; Annuity amount = 5,000,000 / 3.605 = 1,386,963
Depreciation charge under sinking fund method1 2 3 4 5
Capital 5,000,000 4,213,037 3,331,638 2,344,472 1,238,846Depreciation 786,963 881,399 987,166 1,105,626 1,238,301Capital charge 600,000 505,564 399,797 281,336 148,662Sum 1,386,963 1,386,963 1,386,963 1,386,963 1,386,963
5,000,000Economic depreciation =
FVIFA(5yrs, 12%)
5,000,000 = = Rs.787,030
6.353
9. (a) Investment = Rs.100 millionNet working capital = Rs.20 millionLife = 8 yrsSalvage value = Rs.20 million (Net working capital)Annual cash flow = Rs.21.618 millionCost of capital = 15%Straight line depreciation = Rs.10 million per year
80 80Economic depreciation = = = Rs.5.828 million
FVIFA(8, 15%) 13.727
Year 1 Year 4 Profit after tax 11.618 11.618
Depreciation 10.000 10.000 Cash flow 21.618 21.618 Book capital100 70 (Beginning) ROCE 11.62% 16.59% ROGI 21.62% 21.62% CFROI 15.79% 15.79%
(b) Year 1 Year 4
EVA 11.618 – 100 x 0.15 11.618 - 70 x 0.15 = - 3.382 million = - 8.882 million
CVA (11.618 + 10) – 5.828-(100x0.15) (11.618+10)-5.828- (100x0.15) = 0.79 million = 0.79 million
MINICASE
1. Both HLL and Infosys have excluded extraordinary or exceptional items.
2. HLL calculated NOPAT as:PAT (1- T) + INT (1-T)
Remember that NOPAT can be calculated asPBIT (1-T) oras PAT (1-T) + INT (1-T). The two are equivalent.
Infosys calculated NOPAT as:OPERATING PROFIT LESS TAXES
Since Infosys is a zero debt company with nil interest, operating profit less taxes is equivalent to PBIT (1-T).
3. For calculating the cost of equity both HLL and Infosys have used the Capital Asset Pricing Model. However, they have used somewhat different inputs for the risk-free rate and the market risk premium.
HLL used a risk-free rate of 6.95% whereas Infosys used a risk-free rate of 7.50%. To some extent this difference may be because HLL’s financial year ended on 31/12/2005 and Infosys’s financial year ended on 31/3/2006.
HLL assumed a market risk premium of 9% whereas Infosys used a market risk premium of 7%. This difference is, however, substantial.
4. Both HLL and Infosys have used a beta variant without explaining how the same has been calculated. HLL has used a beta variant of 0.95 for the year 2005. It seems reasonable for an FMCG major like HLL. Infosys has used a beta variant of 0.78 for 2006. Interestingly, the beta variant of Infosys was 1.41 in 2006 and declined steadily to 0.78 in 2006. This reflects the diminished riskiness of Infosys.
5. On the whole, the procedures used by the two companies seem reasonable. However, one would have liked to know the rationale of their assumptions and the exact method for the calculation of beta variant.
Chapter 34 MERGERS, ACQUISITIONS AND RESTRUCTURING
1. The pre-amalgamation balance sheets of Cox Company and Box Company and the post-amalgamation balance sheet of the combined entity, Cox and Box Company, under the ‘pooling’ method as well as the ‘purchase’ method are shown below :
Before Amalgamation After Amalgamation Cox & Box Company
Cox Box Pooling method Purchase method
Fixed assets 25 10 35 45Current assets Goodwill
20 7.5 27.5 302.5
Total assets 45 17.5 62.5 77.5
Share capital(face value @ Rs.10)
20 5 25 20
Reserves & surplus 10 10 20 10Debt 15 2.5 17.5 17.5
45 17.5 42.5 77.5
2. Post-merger EPS of International Corporation will be
2 x 100,000 + 2 x100,000
100,000 + ER x 100,000
Setting this equal to Rs.2.5 and solving for ER givesER = 0.6
3. PVA = Rs.25 million, PVB = Rs.10 millionBenefit = Rs.4 million, Cash compensation = Rs.11 millionCost = Cash compensation – PVB = Rs.1 millionNPV to Alpha = Benefit – Cost = Rs.3 millionNPV to Beta = Cash Compensation – PVB = Rs.1 million
4. Let A stand for Ajeet and J for Jeet
PVA = Rs.60 x 300,000 = Rs.18 millionPVJ = Rs.25 x 200,000 = Rs.5 millionBenefit = Rs.4 millionPVAJ = 18 + 5 + 4 = Rs.23 millionExchange ratio = 0.5The share of Jeet in the combined entity will be :
100,000= = 0.25
300,000 + 100,000
a) True cost to Ajeet Company for acquiring Jeet CompanyCost = PVAB - PVB
= 0.25 x 27 - 5 = Rs.1.75 million
b) NPV to Ajeet= Benefit - Cost= 4 - 1.75 = Rs.2.25 million
c) NPV to Jeet = Cost = Rs.1.75 million
5. a) PVB = Rs.12 x 2,000,000 = Rs.24 millionThe required return on the equity of Unibex Company is the value of k in the equation.
Rs.1.20 (1.05)Rs.12 =
k - .05
k = 0.155 or 15.5 per cent.
If the growth rate of Unibex rises to 7 per cent as a sequel to merger, the intrinsic value per share would become :
1.20 (1.07)= Rs.15.11
0.155 - .07
Thus the value per share increases by Rs.3.11 Hence the benefit of the acquisition is
2 million x Rs.3.11 = Rs.6.22 million
(b) (i) If Multibex pays Rs.15 per share cash compensation, the cost of the merger is 2 million x (Rs.15 – Rs.12) = Rs.6 million.
(ii) If Multibex offers 1 share for every 3 shares it has to issue 2/3 millionshares to shareholders of Unibex.
So shareholders of Unibex will end up with
0.667 = 0.1177 or 11.77 per cent
5+0.667
shareholding of the combined entity,The present value of the combined entity will be
PVAB = PVA + PVB + Benefit= Rs.225 million + Rs.24 million + Rs.6.2 million = Rs.255.2 million
So the cost of the merger is :Cost = PVAB - PVB
= .1177 x 255.2 - 24 = Rs.6.04 million
6. The expected profile of the combined entity A&B after the merger is shown in the last column below.
A B A&BNumber of shares 5000 2000 6333Aggregate earnings Rs.45000 Rs.4000 Rs.49000Market value Rs.90000 Rs.24000 Rs.114000P/E 2 6 2.33
7. Value of Alpha Limited’s equity as a stand-alone company.
50 55 60 64 70 70 (1.06) 1 + + + + + x(1.12) (1.12)2 (1.12)3 (1.12)4 (1.12)5 0.12 – 0.06 (1.12)5
= Rs. 912.79 million
Value of the equity of the combined company.
80 90 105 120 135 135 (1.05) 1 + + + + + x(1.12) (1.12)2 (1.12)3 (1.12)4 (1.12)5 0.12 – 0.05 (1.12)5
= Rs. 1518.98 million
Let a be the maximum exchange ratio acceptable to the shareholders of Alpha Limited. Since the management of Alpha Limited wants to ensure that the net present value of equity-related cash flows increases by at least 5 percent, the value of a is obtained as follows.
10 x 1518.98 = 1.05 x 912.79 10 + a 8
Solving this for a we get
a = 0.7311
Note that the number of outstanding shares of Alpha Limited and Beta Limited are 10 million and 8 million respectively.
8. (a) The maximum exchange ratio acceptable to shareholders of Vijay Limited is :
S1 (E1+E2) PE12
ER1 = - + S2 P1S2
12 (36+12) 8= - + = 0.1
8 30 x 8
(b) The minimum exchange ratio acceptable to shareholders of Ajay Limited is : P2 S1
ER2 = (PE12) (E1+E2) - P2 S2
9 x 12 = = 0.3
9 (36+12) - 9 x 8
(c) 12 (48) PE12
ER1 = - + 8 240
9 x 12 ER2 =
PE12 (48) - 72
Equating ER1 and ER2 and solving for PE12 gives, PE12 = 9 When PE12 = 9 ER1 = ER2 = 0.3Thus ER1 and ER2 intersect at 0.3
9. The present value of FCF for first seven years is 16.00 14.30 9.7 0
PV(FCF) = - - - + (1.15) (1.15)2 (1.15)3 (1.15)4
0 10.2 16.7 + + +
(1.15)5 (1.15)6 (1.15)7
= - Rs.20.4 millionThe horizon value at the end of seven years, applying the constant growth model is
FCF8 18 V4 = = = Rs.257.1 million
0.15-0.08 0.15 – 0.08
1 PV (VH) = 257.1 x = Rs.96.7 million
(1.15)7
The value of the division is :- 20.4 + 96.7 = Rs.76.3 million
MINICASE
Solution:(a)
Modern Pharma Magnum Drugs Exchange Ratio
Book value per share 2300 650 = Rs.115 = Rs.65 20 10
65
115Earnings per share 450 95
= Rs.22.5 = Rs.9.5 20 10
9.5
22.5Market price per share Rs.320 Rs.102 102
320
Exchange ratio that gives equal weightage to book value per share, earnings per share, and market price per share
65 9.5 102 + + 115 22.5 320 0.57 + 0.42 + 0.32 = = 0.44 3 3
(b) An exchange ratio based on earnings per share fails to take into account the following:
(i) The difference in the growth rate of earnings of the two companies.(ii) The gains in earnings arising out of merger.(iii) The differential risk associated with the earnings of the two companies.
(c) Current EPS of Modern Pharma 450= = Rs.22.5
20
If there is a synergy gain of 5 percent, the post-merger EPS of Modern Pharma is
(450 + 95) (1.05)
20 + ER X 10Equating this with Rs.22.5, we get
(450 + 95) (1.05) = 22.5
20 + 10ERThis gives ER = 0.54
Thus the maximum exchange ratio Modern Pharma should accept to avoid initial dilution of EPS is 0.54
(d) Post-merger EPS of Modern Pharma if the exchange ratio is 1:4, assuming no synergy gain:
450 + 95 = Rs.24.2 20 + 0.25 x 10
(e) The maximum exchange ratio acceptable to the shareholders of Modern Pharma if the P/E ratio of the combined entity is 13 and there is no synergy gain
-S1 (E1 + E2) P/E12
ER1 = + S2 P1 S2
- 20 (450 + 95) 13 = + = 0.21
10 320 x 10
(f) The minimum exchange ratio acceptable to the shareholders of Magnum Drugs if the P/E ratio of the combined entity is 12 and the synergy benefit is 2 percent
P2S1
ER2 = (P/E12) (E1 + E2) (1 + S) – P2S2
102 x 20 =
12 (450 + 95) (1.02) – 102 X 10 = 0.36
(g) The level of P/E ratio where the lines ER1 and ER2 intersect.
To get this, solve the following for P/E12
- S1 (E1 + E2) P/E12 P2S1
+ = S2 P1S2 P/E12 (E1 + E2) – P2S2
- 20 (450 +95) P/E12 102 x 20 + = 10 320 x 10 P/E12 (450 +95) – 1020
- 6400 + 545 P/E12 2040 = 3200 545 P/E12 – 1020
(545 P/E12 – 1020) (545 P/E12 – 6400) = 2040 x 3200
297025 P/E212 – 3488000 P/E12 – 555900 P/E12
+6528000 = 6528000297025 P/E2
12 = 4043900 P/E297025 P/E12 = 4043900
P/E12 = 13.61
Chapter 37INTERNATIONAL FINANCIAL MANAGEMENT
1. The annualised premium is :
Forward rate – Spot rate 12x
Spot rate Forward contract length in months
46.50 – 46.00 12 = x = 4.3%
46.00 3
2. 100100 (1.06) = x 1.07 x F
1.553
106 x 1.553F = = 1.538
107 A forward exchange rate of 1.538 dollars per sterling pound will mean indifference between
investing in the U.S and in the U.K.
3. (a) The annual percentage premium of the dollar on the yen may be calculated with reference to 30-days futures
105.5 – 105 12 x = 5.7%
105 1
(b) The most likely spot rate 6 months hence will be : 107 yen / dollar
(c) Forward rate 1 + domestic interest rate =
Spot rate 1 + foreign interest rate
107 1 + domestic interest rate in Japan=
105 1.03
Domestic interest rate in Japan = .0496 = 4.96 per cent
4. S0 = Rs.46 , rh = 11 per cent , rf = 6 per cent Hence the forecasted spot rates are :Year Forecasted spot exchange rate
1 Rs.46 (1.11 / 1.06)1 = Rs.48.17 2 Rs.46 (1.11 / 1.06)2 = Rs.50.44 3 Rs.46 (1.11 / 1.06)3 = Rs.52.82 4 Rs.46 (1.11 / 1.06)4 = Rs.55.31 5 Rs.46 (1.11 / 1.06)5 = Rs.57.92
The expected rupee cash flows for the project
Year Cash flow in dollars Expected exchange Cash flow in rupees (million) rate (million)
0 -200 46 -9200 1 50 48.17 2408.5 2 70 50.44 3530.8 3 90 52.82 4753.8 4 105 55.31 5807.6 5 80 57.92 4633.6
Given a rupee discount rate of 20 per cent, the NPV in rupees is :
2408.5 3530.8 4753.8NPV = -9200 + + +
(1.18) (1.18)2 (1.18)3
5807.6 4633.6 + +
(1.18)4 (1.18)5
= Rs.3291.06 million
The dollar NPV is : 3291.06 / 46 = 71.54 million dollars
5. Forward rate 1 + domestic interest rate =
Spot rate 1 + foreign interest rate
F 1 + .015 =
1.60 1 + .020F = $ 1.592 / £
6. Expected spot rate a year from now 1 + expected inflation in home country
=Current spot rate 1 + expected inflation in foreign country
Expected spot rate a year from now 1.06 =
Rs.70 1.03
So, the expected spot rate a year from now is : 72 x (1.06 / 1.03) = Rs.72.04
7. (a) The spot exchange rate of one US dollar should be :12000
= Rs.48 250
(b) One year forward rate of one US dollar should be :13000
= Rs.50 260
8. (1 + expected inflation in Japan)2
Expected spot rate = Current spot rate x2 years from now (1 + expected inflation in UK)2
(1.01)2
= 170 x = 163.46 yen / £ (1.03)2
9. (i) Determine the present value of the foreign currency liability (£100,000) by using 90-day money market lending rate applicable to the foreign country. This works out to :
£100,000 = £ 98522
(1.015)(ii) Obtain £98522 on today’s spot market(iii) Invest £98522 in the UK money market. This investment will grow to
£100,000 after 90 days
10. (i) Determine the present value of the foreign currency asset (£100,000) by using the 90-day money market borrowing rate of 2 per cent.
100,000 = £98039
(1.02)
(ii) Borrow £98039 in the UK money market and convert them to dollars in the spot market.
(iii) Repay the borrowing of £98039 which will compound to £100000 after 90 days with the collection of the receivable
11. A lower interest rate in the Swiss market will be offset by the depreciation of the US dollar vis-à-vis the Swiss franc. So Mr.Sehgal’s argument is not tenable.
12INR/CHF = (INR/USD) x (USD/CHF) = 0.0248 x 1.2056 = 0.0299
13As the forward bid in points is more than the offer rate in points the forward rate is at a discount. So we have to subtract the points from the respective spot rate. The outright one month forward quotation for USD/INR is therefore: 41.3524 / 41.3534 ( Note that one swap point = 0.0001)
14 USD/INR Spot midrate = (41.3424 + 41.3435)/2 = 41.34295
USD/INR 1 month forward midrate = ( 41.2050 + 41.2060)/2 = 41.2055As the forward rate indicates lesser rupee for a dollar, the rupee is at a premium.The annual percentage of premium = [(41.34295 – 41.2055)/ 41.34295] x 12 = 0.0399
MINICASE
Outright rates Spot 90 days 180 daysJPY per USD 117.43/117.45 115.83/116.03 113.55/114.73INR per USD 44.86/44.87 45.14/45.17 45.32/45.35Rupee receivable per JPY 44.87/117.43 45.17/115.83 45.35/113.55
=0.3821 0.3900 0.3994Annualised Premium 8.27 % 9.06 % (A)
Required annualised return 12% 12% (B)Interest rate per annum to be quoted ( B - A) 3.73 % 2.94 %Price to be quoted in Yen for immediate payment 5,000/0.3821
=13,085.58
Chapter 40CORPORATE RISK MANAGEMENT
1. (a) The investor must short sell Rs.1.43 million (Rs.1 million / 0.70) of B
(b) His hedge ratio is 0.70(c) To create a zero value hedge he must deposit Rs.0.43 million
2. Futures price Spot price x Dividend yield = Spot price -
(1+Risk-free rate)0.5 (1+Risk-free rate)0.5
4200 4000 x Dividend yield = 4000 -
(1.145) 0.5 (1.145) 0.5
The dividend yield on a six months basis is 2 per cent. On an annual basis it is approximately 4 per cent.
3. Futures price = Spot price + Present value of – Present value
(1+Risk-free rate)1 storagecosts of convenience yield
5400 = 5000 + 250 – Present value of convenience yield
(1.15)1
Hence the present value of convenience yield is Rs.554.3 per ton.
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