Net Present Value

Definitions

• Opportunity Cost of Capital– Expected rate of return forgone by investing in

a project

• Net Present Value (NPV)– Present value of cashflows minus investment

BM&M p225

Example

• Mr Sharp is considering a property investment

• He can lease a flat for 1 year for an upfront payment of £10,000

• Mr Dim, his friend, has agreed to rent the flat from him for a year and will pay £12,000 in rent at the end of the year

• Is this a good investment for Mr Sharp?

12,000

-10,000

t=0 t=1

Opportunity Cost of CapitalThe scheme is attractive to Mr Sharp, only if the payoff exceeds the rate of return offered by comparable investment alternatives

•Assume that the £12,000 payment from Mr Dim is guaranteed

•Assume that the rate of return on 1 year government bonds is 5%, this is the rate of return which Mr Sharp will forgo by investing in the project

•How much would you have to pay to receive £12,000 at the end of the year?

Initial investment = £12,000 = £11,429

1.05

•Mr Sharp’s required initial investment = £10,000

•NPV = PV of cashflows minus investment = £11,429 - £10,000 = £1,429

•NPV Rule – Managers increase shareholder value by accepting projects that are worth more than they cost and should accept projects with a positive NPV

Will Mr Dim pay up?

• Key assumption – Mr Dim’s £12,000 future payment is guaranteed

• Risks– Mr Dim may lose his job– Mr Dim may be run over by a bus– Mr Dim may be dishonest

• Not all investments are equally risky• A risky investment requires a greater return than

a safe one to compensate for potential losses• Opportunity Cost of Capital should reflect the

riskiness of the project

Risk and Opportunity Cost of Capital

The scheme is attractive to Mr Sharp, only if the payoff exceeds the rate of return offered by comparable investment alternatives

•The rate of return on 1 year government bonds is 5%, however, the rate of return on projects which offer a similar level of risk as Mr Dim is 15% - this is the rate of return which Mr Sharp will forgo by investing in the project

•How much would you have to pay to receive £12,000 at the end of the year?

Initial investment = £12,000 = £10,435

1.15

•Mr Sharp’s required initial investment = £10,000

•NPV = PV of cashflows minus investment = £11,429 - £10,435 = £435

•Mr Sharp should still accept the project, but the NPV is much lower

What if Mr Dim wants to rent the flat for 5 years?

• Mr Sharp is required to pay £40,000 for a 5 year lease

• He offers the flat to Mr Dim for £12,000 for the first year, with an increase of £1000p.a. thereafter

• Is this a good investment?

12,000 13,000 14,000 15,000 16,000

-40,000

t=0 t=1 t=2 t=3 t=4 t=5

Multiple Cashflows

Discount the future cashflows at the 15% Opportunity Cost of Capital

Cashflow NPV @ 15%t=0 -40,000 -40,000t=1 12,000 10,435t=2 13,000 9,830t=3 14,000 9,205t=4 15,000 8,576t=5 16,000 7,955

Total 6,001

PV rental income= C1 + C2 +…+ C5 = £46,001

(1+r) (1+r)^2 (1+r)^5

NPV = PV rental Income – Initial Investment = £46,001 - £40,000 = £6,001

Mr Sharp should go ahead

Mr Sharp needs some Cash

• Mr Sharp goes ahead with the project but immediately suffers a cashflow crisis

• Mr Rich offers to buy his investment in the flat

• How much should Mr Rich pay?

Mr Rich will be entitled to the future cashflows from the project

Value of the future cashflows = PV = £46,001

Mr Rich should pay Mr Sharp £46,001

Mr Sharp has outlayed £40,000 so his net gain is £6,001 which is exactly equal to the NPV of the project

Does a project have to generate positive cashflows to be attractive?

• Not necessarily– Projects which offer cost reductions to the business will also

positively impact upon profits in exactly the same way as cash generating opportunities

– Discounting the cost reductions using the opportunity cost of capital will identify the increase in value to the firm

– Some projects may generate negative net cashflows for the first few years and it is necessary to forecast associated cashflows over the long term

– It should be noted that long term forecasting is itself subject to risk

Pentagon Law of Large Projects

Anything big takes longer and costs more than you are originally led to believe

Choosing between projects

• Mutually exclusive projects are two or more projects which cannot be pursued simultaneously

• Calculate the cashflows for each project, compare NPV’s and choose the most attractive

Example• Mr Sharp has

purchased a 5 year lease for £40,000

• Mr Dim offers to pay rent for £12,000 for the first year, with an increase of £1000p.a. thereafter

• Mr Bold offers to pay £25,000 rent for the first year, £20,000 for the second year and £5,000 p.a. thereafter

• Which offer should Mr Sharp accept?

The NPV of Mr Dim’s proposal is £6,001

The NPV of Mr Bold’s proposal is £5,494

Mr Sharp should accept Mr Dim’s offer

Note – it is assumed that both offers carry equal risk and so the same opportunity cost of capital is employed

Mr Dim Mr Bold

Cashflow NPV @ 15% Cashflow NPV @ 15%t=0 -40,000 -40,000 -40,000 -40,000

t=1 12,000 10,435 25,000 21,739t=2 13,000 9,830 20,000 15,123t=3 14,000 9,205 5,000 3,288t=4 15,000 8,576 5,000 2,859t=5 16,000 7,955 5,000 2,486

Total 6,001 Total 5,494

PaybackPayback period is the time until cashflows recover the initial investment in the project

Project A requires an initial investment of £50,000 and will generate income of £25,000 at the end of year 1, year 3 and year 5. The payback period is 3 years because total income of £50,000 will have been received by the end of year 3.

Project B also requires an initial investment of £50,000 and will generate income of £30,000 at the end of year 1, £20,000 at the end of year 3 and £10,000 at the end of year 5. The payback period is also 3 years because total income of £50,000 will have been received by the end of year 3.

The payback rule states that a project should be accepted is its payback period is less than a specified cutoff period

t=0 t=1 t=2 t=3 t=4 t=5Project A Cashflow -50,000 25,000 0 25,000 0 25,000Project B Cashflow -50,000 20,000 0 30,000 0 10,000

Ref:BM&M p230

Limitations of Payback

• Payback is intuitive and answers the question ‘How long will it take to cover its costs?

• However, there are limitations with payback– Payback ignores cashflows after the cutoff period, tends to favour short

term projects and rejects long term investments– Payback gives equal weight to all cashflows arriving before the cutoff

period– Payback does not consider whether the project increases shareholder

wealth

•Project B has a negative NPV and should be rejected,• Project A has a positive NPV and should be accepted•This is overlooked by the Payback rule.

t=0 t=1 t=2 t=3 t=4 t=5 NPV @15%Cashflow -50,000 25,000 0 25,000 0 25,000 606Cashflow -50,000 20,000 0 30,000 0 10,000 -7,911

Discounted Payback

• The discounted payback measure answers the question ‘How long must the project last until it offers a positive NPV?’

• The rule will never accept a project with a negative NPV

• However, cashflows after the cutoff point remain ignored

• Nevertheless may be useful for highlighting projects which have a long payback and the associated risks with long term forecasting

Internal Rate of ReturnWhat is the actual return which the project will deliver?

12,000

-10,000

t=0 t=1

•PV income @ opportunity cost of capital of 15% = £12,000 = £10,435 1.15

•NPV = PV of cashflows minus PV income = £11,429 - £10,435 = £435

•Internal Rate of Return= Profit Investment= 12,000 10,000= 20%

Rate of Return Rule

Invest in any project offering a rate of return higher than the opportunity cost of capital

Internal Rate of Return and NPV

• If the project is discounted at 15%, the NPV is £435

• If the project is discounted at 20%, the NPV is zero

• The rate of return on the project is the discount rate which gives an NPV of zero

-1,000

-500

0

500

1,000

1,500

2,000

2% 6% 10% 14% 18% 22% 26% 30%

Discount rate %

Net

Pre

sen

t V

alu

e

Rate of return =20%

NPV Profile

Ref:BM&M p232

Calculating IRR for multiple cashflows

•The project rate of return is the discount rate which gives the project an NPV of zero

•Sometimes called the discounted cashflow (DCF) rate of return

Time CashflowDiscount

factorDiscount

factorDiscounted

cashflow

formula 20.85%Cashflow/Discount

factor

t=0 -40,000 1 1 -40,000t=1 12,000 1+r 1.209 9,929t=2 13,000 (1+r)^2 1.461 8,901t=3 14,000 (1+r)^3 1.765 7,932t=4 15,000 (1+r)^4 2.133 7,032t=5 16,000 (1+r)^5 2.578 6,206

Sum 0

•Method of calculation

•Linear interpolation

•Spreadsheet Goal Seek function

•Spreadsheet IRR function

•Financial Calculator

Linear Interpolation

Technique793 + 135 = 928135/928 = 0.14521.00 – 0.145 = 20.85IRR = 20.85%

NPV @ 20% = 793 NPV @ 21% = -135

Time CashflowDiscount

factorDiscount

factorDiscounted

cashflow

formula 20.00%Cashflow/Discount

factor

0 -40,000 1 1.000 -40,0001 12,000 1+r 1.200 10,0002 13,000 (1+r)^2 1.440 9,0283 14,000 (1+r)^3 1.728 8,1024 15,000 (1+r)^4 2.074 7,2345 16,000 (1+r)^5 2.488 6,430

Sum 793

Time CashflowDiscount

factorDiscount

factorDiscounted

cashflow

formula 21.00%Cashflow/Discount

factor

0 -40,000 1 1.000 -40,0001 12,000 1+r 1.210 9,9172 13,000 (1+r)^2 1.464 8,8793 14,000 (1+r)^3 1.772 7,9034 15,000 (1+r)^4 2.144 6,9985 16,000 (1+r)^5 2.594 6,169

Sum -135

Goal Seek Function

Time CashflowDiscount

factorDiscount

factorDiscounted

cashflow

formula 0.208527094643138Cashflow/Discount

factor

0 -40000 1 =(1+$D$2)^A3 =B3/D31 12000 1+r =(1+$D$2)^A4 =B4/D42 13000 (1+r)^2 =(1+$D$2)^A5 =B5/D53 14000 (1+r)^3 =(1+$D$2)^A6 =B6/D64 15000 (1+r)^4 =(1+$D$2)^A7 =B7/D75 16000 (1+r)^5 =(1+$D$2)^A8 =B8/D8

Sum =SUM(E3:E8)

Time CashflowDiscount

factorDiscount

factorDiscounted

cashflow

formula 20.85%Cashflow/Discount

factor

0 -40,000 1 1.000 -40,0001 12,000 1+r 1.209 9,9292 13,000 (1+r) 2̂ 1.461 8,9013 14,000 (1+r) 3̂ 1.765 7,9324 15,000 (1+r) 4̂ 2.133 7,0325 16,000 (1+r) 5̂ 2.578 6,206

Sum 0

On the Tools menu, click Goal Seek. In the Set cell box, enter the reference for the cell that contains the sum of the

discounted cashflows, appears above as=SUM(E3:E8)In the To value box, type 0 In the By changing cell box, enter the reference for the cell that contains the

discount rate (appears above as 0.20852709)Click OK.

Values Formulae

IRR Function

IRR is calculated using the Excel IRR function on the cashflows

Values Formulae

Time CashflowDiscount

factorDiscounted

cashflow

20.85%Cashflow/Discount

factor

0 -40,000 1.000 -40,0001 12,000 1.209 9,9292 13,000 1.461 8,9013 14,000 1.765 7,9324 15,000 2.133 7,0325 16,000 2.578 6,206

Sum 0

Time Cashflow Discount factorDiscounted

cashflow

=IRR(B3:B8)Cashflow/Discount

factor

0 -40000 =(1+$C$2)^A3 =B3/C31 12000 =(1+$C$2)^A4 =B4/C42 13000 =(1+$C$2)^A5 =B5/C53 14000 =(1+$C$2)^A6 =B6/C64 15000 =(1+$C$2)^A7 =B7/C75 16000 =(1+$C$2)^A8 =B8/C8

Sum =SUM(D3:D8)

Pitfalls of IRR

• IRR does not distinguish between lending and borrowing

• Where cashflows consist of multiple receipts and payments, it is possible to have more than one IRR figure

• Mutually Exclusive Projects• High IRR does not always mean high NPV

which is the objective for increasing shareholder wealth

Ref:BM&M p234

Lending and Borrowing

Time Project A Project B

Cashflow Cashflow

0 -40,000 40,0001 12,000 -12,0002 13,000 -13,0003 14,000 -14,0004 15,000 -15,0005 16,000 -16,000

IRR 21% 21%NPV @10% £12,351.24 -£12,351.24

•Project A and Project B have the same IRR of 21%

•Does this mean that the projects are equally attractive?

•Project A involves lending £40,000 and receiving cashflows equivalent to interest of 21% p.a.

•Project B involves borrowing £40,000, then paying interest at a rate equivalent to 21% p.a.

•Project A is attractive, Project B is not

•Calculate the NPV to determine which is a good proposition

Rule

When NPV increases as the discount rate increases, a project is acceptable only if its IRR is less than the Opportunity Cost of Capital

Multiple Rates of Return

•Project C has an IRR of 8%

•The NPV of project C using a discount rate of zero is 0 so Project C has an IRR of 0%

•Is this possible?

•Note – cashflow sign changes twice

•There can be as many different IRR’s as there are changes in the sign of the cashflow stream

•Opportunity Cost of Capital is 5% and NPV of the project is £148.43

Time Project CDiscount

factorDiscounted

cashflow

Cashflow 0.00%Cashflow/Discount

factor

0 -24,000 1.000 -24,0001 12,000 1.000 12,0002 13,000 1.000 13,0003 14,000 1.000 14,0004 15,000 1.000 15,0005 -30,000 1.000 -30,000

IRR 8% Sum 0NPV @10%£148.43

NPV of Project C

-£6,000

-£5,000

-£4,000

-£3,000

-£2,000

-£1,000

£0

£1,000

-20% -10% 0% 10% 20% 30% 40% 50%

Discount Rate

NP

V

IRR = 0% IRR = 8%

Which IRR is correct?

Modified IRR• Combine the cashflows until only

one change of sign remains• Use Opportunity Cost of Capital (5%)

to discount the cashflows • The cashflows for Years 3,4 and 5

are discounted to year 3 to become a single cashflow of value 1075

Time Project CDiscount

factorDiscounted

cashflow

Cashflow 5.00%Cashflow/Discount

factor

3 14,000 1.000 14,0004 15,000 1.050 14,2865 -30,000 1.103 -27,211

Sum 1,075

Time Project ADiscount

factorDiscounted

cashflow

Cashflow 5.00%Cashflow/Discount

factor

0 -24,000 1.000 -24,0001 12,000 1.050 11,4292 13,000 1.103 11,7913 1,075 1.158 9294 0 1.216 05 0 1.276 0

IRR 5.42% Sum 149NPV @10%£148.43

•Replace the cashflows for Years 3,4,5 with 1075 in year 3

•Calculate IRR

•Modified IRR = 5.42%

•Note – NPV of £148.43 remains unchanged

Mutually Exclusive ProjectsMr Sharp, Mr Dim and Mr Bold

• Mr Sharp accepted Mr Dim’s offer because it had a higher NPV - £6001 v £5494

• The IRR from Mr Dim is 21%

• The IRR from Mr Bold is 23%

• Is this a contradiction?

Mr Dim Mr Bold

Time CashflowNPV @ 15% Cashflow

NPV @ 15%

0 -40,000 -40,000 -40,000 -40,0001 12,000 10,435 25,000 21,7392 13,000 9,830 20,000 15,1233 14,000 9,205 5,000 3,2884 15,000 8,576 5,000 2,8595 16,000 7,955 5,000 2,486

NPV 6,001 NPV 5,494IRR 21% IRR 23%

Mutually Exclusive ProjectsMr Sharp, Mr Dim and Mr Bold

Mutually Exclusive ProjectsMr Sharp, Mr Dim and Mr Bold

• The NPV profiles cross at 16.31%

• Mr Dim’s offer has a higher NPV if the discount rate is greater than 16.31%

• Mr Bold’s offer has a higher NPV if the discount rate is lower than 16.31%

-10,000

-5,000

0

5,000

10,000

15,000

10% 12% 14% 16% 18% 20% 22% 24% 26% 28% 30%

Discount Rate

NP

V

Mr Dim

Mr Bold16.31%

Higher IRR does not always mean higher NPV – shareholder wealth is paramount

Comparing NPV’s

• Investment Timing Decision

• Quality versus Price

• Replacement Decision

Ref:BM&M p240

Investment Timing

• Mr Careful is considering purchasing a second hand car

• The car save him £2000 p.a. in bus fares and will cost £500 to run

• If he buys the car now, it will cost £10,000 and will last for 10 years

• Depreciation in the second hand market runs at 15% p.a.

• His savings account pays 5% p.a.• Should he buy the car now or wait ?

Mr Careful

Year of purchase Cost PV cost

PV Savings

PV net savings

0 10,000 10,000 11,583 1,5831 8,696 8,282 10,154 1,8722 7,561 6,858 8,793 1,9353 6,575 5,680 7,498 1,8184 5,718 4,704 6,264 1,5605 4,972 3,896 5,088 1,193

•If Mr Careful purchases the car this year, he will save £1500 p.a. for the next 10 years

•This saving must be discounted at Mr Careful’s opportunity cost of capital – his deposit account rate,5%

•The car becomes 15% cheaper each year, this costs should also be discounted at Mr Careful’s opportunity cost of capital rate, 5%

•Net Saving = PV Saving - PV Cost

The PV of the net savings are maximised if Mr Careful waits for 2 years and buys the car at a reduced price

The decision for investment timing is to choose the investment date which results in the highest NPV today

Quality versus Value

• Mr Careful decides he needs the car now

• However, he could choose another car

• This car provides net savings of £1500 p.a. but costs only £8000, however, it will only last for 8 years

• Which car is best?

Comparison

At first sight, the longer life seems better since the savings are greater. But the longer life car provides a service for 2 more years.

Is the annual saving for the longer life car as good?

The savings are converted into annual equivalent figures

Equivalent annual annuity = present value of savings annuity factor @5%

Select the car that has the highest equivalent annuity – Cheaper

Cost PV costPV Savings

PV net savings

Equivalent Annuity

Cheaper 8,000 8,000 9,695 1,372 £212.28Longer Life 10,000 10,000 11,583 1,583 £204.95

Capital Rationing

• Capital Rationing refers to the limit set on the amount of funds available for investment

• Soft Rationing – capital rationing is not limited by investors but by management

• Hard Rationing – firm has insufficient resources to invest in attractive projects

• Profitability Index = NPV Initial Investment