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FINANCIAL CALCULATIONS

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� Main function is to calculate payments, determine interest rates and to solve for the present or future value of a loan or an annuity

� 5 common keys on financial calculators:◦ N – number of periods◦ I – periodic interest rate◦ PV – present value◦ PMT – payment◦ FV – Future value

� These are all at the top row of the HB10bII and HB10bII+ calculator

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� Determine which variable you would like to solve for. You will need four out of the five variables to start, keeping in mind that PMT, PV or FV may be zero

� Know when to use zero◦ PMT – invest a lump sum or owe all of your money at the end of the term

◦ PV – when you are receiving or making payments

◦ FV – when a loan or annuity is paid off finished paying out

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� Keep in mind that the number of periods is usually listed in the same increment as the interest rate, i.e a 30 year mortgage bond would have 360 periods (30 years)

� If you are uncertain, always use

“Shift N” function.

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�Annually 1 NACA�Semi Annually 2 NACSA�Quarterly 4 NACQ�Monthly 12 NACM�Daily 365 NACD

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� Number of periods per year

� Number of decimals used

� Begin/end mode

� Always clear the calculator (Shift C All) before you start any calculation

� Always practice your calculations well

� Watch these YouTube training videos:◦ https://www.youtube.com/watch?v=LYsF3rY0qmA

◦ https://www.youtube.com/watch?v=LYsF3rY0qmA

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Mrs. Basson wants to take her family to Disney World 10 years from now.The travel agent has estimated that such a trip would currently costR150,000. Inflation is expected to average 8% per year over the next 10years. Mrs. Basson has the following investments available to fund this trip:

•R20 000 in a savings account, invested at an effective rate of 7% p.a.•An endowment policy with a current value of R10 000. Mrs. Bassoninvests a level amount of R1 000 per month into this policy. The policywill mature one year before the planned trip and she has indicatedthat she will merely reinvest the maturity value for another yearwithout making any further payments. The expected growth rate inthis portfolio is 9% per annum, compounded monthly.

Mrs. Basson would like to know how much she has to invest per annum inorder to ensure that she would have enough money for the holiday, takinginto account her current investments earmarked for this purpose. Sheindicates that she can escalate her annual investment by 5% per year, andthat she believes that a new investment can grow at an effective rate of9% per year. Calculate the annual investment required to make up theshortfall.

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Brackets

Others (indices and integers)

Division

Multiplication

Addition

Subtraction

Multiplication and division are performed whichever comes first from left to right

Addition and subtraction are performed whichever comes first from left to right

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� A number that is anything from “0” to “1”

� Therefore a fraction is less than 1, but greater than 0

� ¼

� 1 divide by 4 (Long Division)

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� 2/10� 6/10� 55/100� 73/80� 60/120� 176/200

�0.2�0.6�0.55�0.912 or 0.913 rdg

�0.5�0.88

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� Converting a fraction to a percentage Converting a fraction to a percentage Converting a fraction to a percentage Converting a fraction to a percentage requires multiplying by 100/1requires multiplying by 100/1requires multiplying by 100/1requires multiplying by 100/1

� ThereforeThereforeThereforeTherefore:

��

�x ���

�

� = ���

�

� = 25%

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� Examples:Examples:Examples:Examples:

� 32/50� 75/100� 12/71� 25/100� 120/200

� 64%

� 75%

� 16.91%

� 25%

� 60%

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Amortisation table

Year Start

Interest Year End

2000

Bob invests R2,000 at 10%p.a. simple interest for 4 years

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Year Start

Interest

Year End

2000 200 2200

2200 200 2400

Bob invests R2,000 at 10%p.a. simple

interest for 4 years

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Year Start

Interest Year End

2000 200 2200

2200 200 2400

2400 200 2600

2600 200 2800

Bob invests R2,000 at 10%p.a. simple interest

for 4 years

Therefore:2000 X 10% = 200 200 x 4 yrs = 800

2000 + 800 = 2800

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� For how many years does one have to invest a lump sum of R10 000 at 15% pa simple interest in order to receive R26 600 at the end of the term?

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� 26,600 – 10,000 = 16,600

� 15% of 10,000 = 1,500 simple interest per year.

� 16,600 / 1,500 = 11.066

� (26,600 – 10,000) / 1,500 = 11.066 years 10 000 @ 15% pa annum = R 1 500 Capital invested = 10 000 Growth = 1 500 x 11.066 years = R16 599 R 16 599 + 10 000 = R 26 60017

Year Start Int End

1 2000 200 2200

2 2200 220 2420

3 2420 242 2662

4 2662 266.2 2928.2

� Bob Invests R2,000 at 10% p.a. compounding annually for 4 Years� 2000 x (1+0.1) =� 2200 x 1.1 =� 2420 x 1.1 =� 2662 x 1.1 =

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� Bob Invests R2,000 at 10% p.a. compounding Monthly for 1 Year.

� 2000 x (1+(0.1/12)) = 2000 x (1+(0.0083)) =

� 2016.6 x 1.0083 =�

� 2033.33 x 1.0083 =

� 2050.21 x 1.0083 = 2067.23

� Therefore:◦ 2000 x (1.0083)4 = 2067.23◦ 2000 x (1.0083)12 = 2208.5498

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� PV x (1 + (I/PY / P/YR)N = FV

� Future Value = Present Value x (1 + (Int rate / Comp Periods per year))Power of Total Compounding

periods

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� NNNN = TOTAL number of compounding periods in the calculation.

� I/YRI/YRI/YRI/YR = Nominal per annum interest / growth rate applicable

� PVPVPVPV = Present Value

� PMTPMTPMTPMT = Any Regular payment (same as periods per year)

� 2nd F P/YRP/YRP/YRP/YR= Number of compounding periods in 1111 Year .

� FVFVFVFV = Future Value

� PV x (1 + (I/PY / P/YR)N = FV

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� Nominal interest rate◦ Simply the stated interest rate of a given bond, loan or investment

◦ E.g. if nominal rate on a loan is 5%, then borrowers can expect to pay R5 on every R100 loaned to them

� Effective interest rate◦ Take the power of compounding into consideration

� The difference between nominal and effective rates increases with the number of compounding periods within a specific time period

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� A Bank advises you can earn 15% pa on a one year fixed deposit – interest accumulates once a yearonce a yearonce a yearonce a year:

Capital Invested R 100Plus interest R 15Nominal and effective the same as there is no interest on interest.

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� A Bank advises you can earn 15% pa on a one year fixed deposit – interest accumulates monthlymonthlymonthlymonthly:

� Capital invested R100� Plus interest R 16.08

� 12 P/YR 15 Shift Nom %Shift EFF% = 16.08%

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� R1000 invested over a 5 year period with growth at 10%. What will I get at the end?

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� Billy wins the lotto and invest the money for 10 years at an interest rate of 11% per annum. After the 10 years he receives R940 515.61. What is the capital sum that Billy invested?

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� If I invest R1000 a month for the next 5 years at 10%, what will the maturity value be?

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� Mr P invested R10 000 5 years ago and receives R25 000 now. What is the rate of return on the investment? Show all steps

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� Elizabeth invests a lump sum amount of R230 000 for a period of 5 years. The interest rate is 6.5% per annum which will be credited to the account on a monthly basis. Ignore income tax. At the end of the term she will receive an amount of?

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� Mr A invests R550 at the beginning of each month for 10 years. What is the maturity value if the investment has an 8% effective interest rate? [3]

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� Tish signed an investment contract where she will invest R15000 + a further R1000 pa at beginning of each year for next 5 years at 12% interest: Maturity value will be?

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� Alma buys a house & takes a bond of R600 000 at an interest rate of 12% for 20 years –calculate monthly instalment:

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� Mr N’s investment receives an annuity income of R20 000 pa in advance for 15 years as well as R100 000 at the end of the term. The interest rate is 10% calculated in advance. How much did Mr N invest initially?

[4]

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Bond of R500 000 at an interest rate of 11% and term is 20 yrs. Calculate the monthly instalment:

End Mode12 Shift P/YR500 000 PV11 I/YR20 Shift NPMT? -5 160.94

DO NOT CLEAR THE CALCULATORDO NOT CLEAR THE CALCULATORDO NOT CLEAR THE CALCULATORDO NOT CLEAR THE CALCULATOR!!

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� SHIFT ‘AMORT’

� PRESS ‘=‘ prin - capital amount paid ( 7291.66)

‘=‘ int - interest portion paid (54639.65)

‘=‘ bal – balance ( 4927080

13 INPUT 2413 INPUT 2413 INPUT 2413 INPUT 24Shift Amort

Capital = R 8135.44Interest = R 53 795.86Balance = R 484 572.90

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� At beg of year 3 rates increase to 12%

a) what is new monthly instalments?

b) what is the balance at end of year 2?

End Mode

12 Shift P/YR

484 572.90 PV

12 I/YR

18 Shift N (remaining years)

PMT? -R 5485.12

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� What if after 2 yrs bond holder wants to reduce term by 2 yrs?

Use same example as above

Answer?

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Mr Nkosi has a mortgage bond of R400 000

repayable over 20 years at an interest rate of 13%

1.1 Calculate monthly repayments [2]

1.2 The interest rate drops from 13% to 11% at the beg of the 2nd year. Mr Nkosi elects not to reduce his monthly payments. Calculate how long it will now take Mr Nkosi to repay his bond? [4]

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� Frans buys a new car for R550 000. He pays a deposit of R 50 000 and takes a loan from the bank for the balance for a period of 3 years. His monthly instalment is R 15 668.18. What is the percentage interest that he pays on the loan?

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� Scenario – Annual premium pd to an investment must increase each year by 7% and the investment’s growth is 10%

� Can we account for 2 “growth factors”?

� Only 1 I/YR key – no escalation key.

� Can’t add them together! ie can’t do a FVcalc.

� Can deduct and get a “net effect” and thereby do an “equivalent” PV calc

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� Brad wants to invest 100 per year, escalating at 7% p.a. for 3 years. Growth on the investment is 10%

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100 107 114.49

FV

133.10

129.47

125.939

97.2727

100.00

94.6198

PV

291.8925 388.5090

1 2 3

10% for 3 yrs

10% for 2 yrs

10% for 1 yr

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Incorporates both interest and escalation rate:

���

���� 1100

I = interest or growth rate

E = escalation

Alternative method: Interest 12%

Escalation rate 10%

12 – 10 / 1,10 = 1,81818% = resultant rate

NB: NB: NB: NB: I AM BEFORE E ISI AM BEFORE E ISI AM BEFORE E ISI AM BEFORE E IS

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PMT PV

Resultant Interest

Rate Rate

PV FV

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� Brad invests R100 pa at the beginning of each year, escalating at 7% pa for 5 years at an interest of 9%. What is the FV?

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12mth pmt PMT PV

Nominal (Eff Rate) Nominal

Interest Resultant Interest

Rate Rate Rate

12 P/YR

Or 1/py with eff rate

PV of annual PV of FV of Payment escalating invest

annuity itself!

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� Mr J wants to invest R2 500 pm for the next 5 years at an interest rate of 7.5% and wants to increase his premiums by 6% every year. How much will he receive after 5 years if he invests the R2 500 at the beg of each month?

[6]

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� Step 1: Calculate PV of the annual contributions

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Step 2 calculate resultant rate

Step 3 calculate PV of the escalating annual investment

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� Step 4 Calculate the FV of the investment using the interest rate

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Ann wants to invest R100 per month for 5 years. This monthly investment must increase by 6% per annum. The investment will earn 8%. What will the

future value of this monthly investment be?

We want to know the FV of an investment if we are investing MONTHLY.

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� Step 1 – determine the annual equivalent irothe monthly instalments – discounted at the nominal rate.

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� Step 2 – change to effective rate and calculate resultant rate.

� Resultant rate

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Step 3 – discount the annual equivalent to PV using resultant rate.

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Step 4 – calculate the FV of the instalment using the nominal rate.

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Tom pays out an amount of R50 000 and receives monthly payments of R3000, R6000, R6000, R22000 and R15000

� Calculate internal rate of return

� If discounted at 12% what will the net present value be?

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� Calculate NPV if discounted at 12%

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� Your client has been making uneven adhoc contributions into her investment for the past year. Contributions made as follows:

� March R1000

� April R2000

� June R1750

� September R350

� October R900

� December R175

� January R1000

� February R 250

� The current value is R8 587

Calculate the annual rate of return, assuming the fund compounds monthly:

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Mr Nel has just taken cession of a life assurance contract. The policy is due to mature in 4 years time. Premiums of R750 pa are payable towards the policy. The estimated maturity value of the policy in 4 years time is R28 250. The growth rate is assumed to be 10%. What is the PV of this policy?

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Mrs Waterman invests R5 000. The nominal rate of interest is 10% and the interest is compounded half-annually. What is her FV after 2 years?

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Mrs Van Wyk wants to invest R500 at the beginning of each year for 10 years. The interest payable on this

investment will be 15%

• What will the future value of this investment be?

• Still using the same figures above, how much capital would she need to buy an annuity of R 500 per

annum – payable at beg of each year- for 10 years,

if the life assurer pays 15% on her investment?

• What is the FV if we take the result from the

previous calculation if she invests a lump sum of R2 885.79 for 10 years at 15%

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� FV of the investment will be:

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Capital needed to buy an annuity?

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FV OF R2886?

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Mr Verwey wants to invest R800 pm for the next 5 years at an interest rate of 8% and wants to increase his premiums by 5% every year.

How much will he receive after 5 years if he invests the R800 at the beginning of each month?

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STEP 1 – PV OF ANNUAL CONTRIBUTIONS

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� STEP 2 – CALCULATE RESULTANT RATE

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� STEP 3 – PV OF ESCALATING PAYMENTS

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� STEP 4 – CALCULATE FV

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� Susan has R400 000 in a fixed deposit which earns interest of 15%. The inflation rate is 6%. Sue’s marginal tax rate is 40%. What effect will this have on her real rate of return?

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� Mpho owns a house. The interest she pays on the bond is 12%. She won R300 000 from the lotto. Her marginal tax rate is 40%

� What taxable rate of interest must she earn on the R 300 000 to equal the 12% interest rate she is paying on her bond?

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Mr Greedy would like to double his inheritance of R100 000 within 5 years by speculation on the stock market. He is aware that he will have to give up approximately 40% to tax annually. Calculate the annual pre-tax yield rate he will have to achieve in order to reach his goal.

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Mr R needs R58 567.26 in 5 years time.

He will invest by way of annual

installments. He will start with an amount

Of R8 500 and then increase the

Installment by a fixed %. He will earn

interest at 7.5% pa.

Calculate the % by which he has to

increase his installments every year?

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� STEP 1 - WHAT IS THE PV OF WHAT I

WANT?

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STEP 2 – DETERMINE THE RESULTANT RATE

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� STEP 3 – CALCULATE ESCALATION RATE

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� Calculation of Retirement NeedsCalculation of Retirement NeedsCalculation of Retirement NeedsCalculation of Retirement Needs:Mr G who is currently 45 years of agewould like to retire at the age of 65.

His current salary is R 500 000 pa and hewill be happy to receive 75% of his salary.

He believes that his salary will increasewith 8% pa. What is the first year’sincome that he will need at the age of 65?

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Mr G would like this income (75% of his

salary ) for at least until his age of 85 but

the income he receives must be increased

by 6% every year.

The capital will be invested and will attract

8% growth.

How much Capital will Mr G need to have at the age of 65 to address his needs?

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� Mr G is very concerned as he will under no circumstances have this type of money!

He would like to know: - with an investment growth of 12%.

a) How much does he have to invest annually assuming it will be a level premium?

b) What if he decides to increase the premium by 10% every year?

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� Level Annual Investment Amount

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� Step 1 Equivalent Lump Sum (PV)

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� Step 2 Discounted PMT at Esc 10%

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Mrs. Basson wants to take her family to Disney World 10 years from now.The travel agent has estimated that such a trip would currently costR150,000. Inflation is expected to average 8% per year over the next 10years. Mrs. Basson has the following investments available to fund thistrip:

•R20 000 in a savings account, invested at an effective rate of 7%p.a.•An endowment policy with a current value of R10 000. Mrs. Bassoninvests a level amount of R1 000 per month into this policy. The policywill mature one year before the planned trip and she has indicatedthat she will merely reinvest the maturity value for another yearwithout making any further payments. The expected growth rate inthis portfolio is 9% per annum, compounded monthly.

Mrs. Basson would like to know how much she has to invest per annum inorder to ensure that she would have enough money for the holiday, takinginto account her current investments earmarked for this purpose. Sheindicates that she can escalate her annual investment by 5% per year, andthat she believes that a new investment can grow at an effective rate of9% per year. Calculate the annual investment required to make up theshortfall.

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� Need

� Provision

� Surplus / Shortfall

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� Step 2 Equivalent escalating annual cash flow for PV Lump Sum

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Mr Makhensa, born 20 January 1985.

� Wants to receive a monthly income of at least R35,000 (after tax – 40%) in today’s value, when he retires at the age of 60.

� The monthly income should last for 25 years from the date of retirement and escalate by the rate of inflation per annum. (Given as 6% currently)

� His Current retirement savings only consist of a pension preservation fund at R350,000 current value.

� Growth on all investments 10%

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� Q 5.3.1 Calculate the capital required at retirement to meet Makhensa’s retirement goals. (5)

� 5.3.2 Calculate the capital available at retirement (2)

� 5.3.3 Calculate the shortfall at retirement. (2)

� 5.3.4 Calculate the increasing monthly investment that Makhensa should make/save at the beginning of each month, in his retirement annuity fund to make up for the shortfall. He wants to increase the premium at 6% p.a. (3)

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� Calculate the pre-tax income required monthly

� Then grow from now to retirement (30 years)

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� Effective rate for monthly conversion to annual equivalent @10%p.a.

� Resultant rate (use effective)

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Step 1 – convert shortfall to current day equivalent lump sum

(Monthly, so use Nominal rate and 12 P/YR)

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Step 2 – convert current lump sum value needed to an equivalent annual escalating premium

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Step 3 – convert first year’s annual premium to an equivalent monthly premium

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