CHAPTER 2 THE TIME VALUE OF MONEY OF MONEY 1 ©Correia, Flynn, Uliana & Wormald n Using formulae, tables, financial calculators and spreadsheets to determine

CHAPTER 2

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©Correia, Flynn, Uliana & Wormald

Using formulae, tables, financial calculators and Using formulae, tables, financial calculators and spreadsheets to determine the:spreadsheets to determine the:

Future Value of:Future Value of: a single suma single sum an annuity an annuity

Present Value of:Present Value of: a single suma single sum an annuityan annuity a perpetuitya perpetuity a growing perpetuitya growing perpetuity a cash flow growing at a constant ratea cash flow growing at a constant rate an uneven cash flow streaman uneven cash flow stream

OverviewOverview

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Define and calculate an effective rateDefine and calculate an effective rate Distinguish between nominal and real Distinguish between nominal and real

interest ratesinterest rates Apply compounding and discounting to Apply compounding and discounting to

complex cash flow streamscomplex cash flow streams Apply time value of money principles to real Apply time value of money principles to real

world problems and the valuation of bondsworld problems and the valuation of bonds Establish the factors that determine the term Establish the factors that determine the term

structure of interest ratesstructure of interest rates

Overview (continued)Overview (continued)

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Future ValueFuture Value

Future value is the value in Future value is the value in dollars that an investment or dollars that an investment or series of investments will grow series of investments will grow over a stated time period at a over a stated time period at a specified interest rate.specified interest rate.

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FUTURE VALUES

FV: The amount of cash which will have accrued by a

given date resulting from earlier lump-sum or

periodic investments.

PV: The value of an investment at the beginning of a

period, sometimes referred to as the principal sum.

r: The interest rate, expressed as a decimal fraction.

I: The periodic investments or instalments made,

excluding single lump-sum investments.

n: The number of periods for which the investment is to

receive interest.5


00 1 1

PV = 100PV = 100 FV = ? FV = ?

An amount of R100 is invested for a one year rate of 12% p.a. What is An amount of R100 is invested for a one year rate of 12% p.a. What is the future value of the investment?the future value of the investment?

FV = PV (1 + r)FV = PV (1 + r)

= R100 (1.12)= R100 (1.12)= R112= R112

Future Value - Single SumFuture Value - Single Sum

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Lump Sum: Multiple periods – Annual Interest Compounded

Example 2.2: Calculating the future value: more than one yearAn amount of R100 is invested for 10 years at a rate of 12% p.a. compound interest. What is the future value of this investment at the end of 10 years?

For year one, FV = R100 (1.12) = R112For year two, FV = R112 (1.12) = R125.44For year three, FV = R125.44 (1.12)= R140.50 etc

This can be generalized to:

FV = PV (1 + r)n(Formula 2.2)

FV = R100 (1.12)10

= R100 × 3.1058= R310.58

FUTURE VALUESFUTURE VALUESFUTURE VALUESFUTURE VALUES

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Example 2.3: Calculating the principalAn investor wishes to invest a sum of money which will accumulate to R310.58 in 10 years time. How much must be invested today, if a rate of 12% p.a. is obtained?

Changing the subject of Formula 2.2, it can be stated as:

PV = FV / (1+r)n(Formula

2.3)

= 310.58 / (1.12)10

= 310.58 / 3.1058

= R100

As expected from the results of Example 2.2, the required investment is R100. Table A may clearly be used to determine the calculation of (1.12)10 .



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Example 2.4: Calculating the number of periodsAn investor is informed that an investment of R100 will grow to R310.58. If it is known that the applied rate is 12%, after how long can the R310.58 be collected? The use of logarithms (or Table A) is required:

(1+r)n = FV / PV (Formula 2.4)

= 310.58 / 100

= 3.1058

The number 3.1058 is the factor defined in table A. Because it is known that the interest rate is 12%, it is possible to move down the 12% column until the number nearest to 3.1058 is found. In this example it is found in the 10-period row.

Also: Ln3.1058/Ln1.12 = 10 years



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Example 2.5: Calculating the interest rateAn investor is given the opportunity of investing R100 today with a promised future value of R310.58 in 10 years’ time.

At what rate is the investment accruing interest? Developing from equation 2.4, it is possible to make r the subject of the formula as follows:

(1+r)n = FV / PV (Formula 2.5)

r = (FV/PV)(1/n) - 1

= (310.58/100)(1/10) – 1

= 0.12 OR 12%

Table A can again be used. This time one would search for a number close to 3.106 by

looking along the 10 period row. Once the closest number to 3.106 is located, the column

in which it is situated is the required interest rate.



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Investing with Warren Buffett

Warren Buffett earned 20.3% per year from 1965 to 2009 Warren Buffett earned 20.3% per year from 1965 to 2009 (45 yrs). If your grandfather had invested $1000 on your (45 yrs). If your grandfather had invested $1000 on your behalf then, how much would you have at the end of behalf then, how much would you have at the end of 2009?2009?

FV = 1000 (1+0.203)FV = 1000 (1+0.203)4545

= $4 092 166= $4 092 166How much would you have accumulated if he had rather How much would you have accumulated if he had rather

invested in the general share market (S&P500)? [See invested in the general share market (S&P500)? [See page 2-7]page 2-7]

If you invested with Allan Gray, R10 000 in 1974, how If you invested with Allan Gray, R10 000 in 1974, how much would you have in 2010? [See page 2-8]much would you have in 2010? [See page 2-8] 11


Example 2.6: Future value, interest compounded monthly

An investor deposits R100 into an account which offers 12% p.a. interest compounded monthly. Find the value of the investment at the end of one year.12% over 12 months = 1% interest added every month. At the end of the twelfth month it would be:

FV = R100 (1.01)12

= R100 × 1.1268= R112.68

The effective rate in this example is 12.68%. The formula required to generalize this calculation is as follows:

FV = PV (1+ r/m)mn (Formula 2.6)



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Single SumSingle SumMultiple periods and;Multiple periods and;Non - Annual Non - Annual CompoundingCompounding

Example: Example:

An investor deposits R100 into an account An investor deposits R100 into an account which offers 12% p.a. interest compounded which offers 12% p.a. interest compounded monthly. What is the value of the investment monthly. What is the value of the investment at the end of 10 years?at the end of 10 years?

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Future Value Calculation

Answer:Answer:

FV = PV (1 + r/m)FV = PV (1 + r/m)mnmn

= R100 (1 + 0.12/12)= R100 (1 + 0.12/12)12x1012x10

= R100 x 3.300= R100 x 3.300

= R 330= R 330

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Annual Effective RateAnnual Effective Rate

1 - m

Rn 1 Rate Effective

m

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Interest rates quoted by three banks:Interest rates quoted by three banks: Bank X:Bank X: 15%, compounded daily15%, compounded daily Bank Y:Bank Y: 15.5%, compounded quarterly15.5%, compounded quarterly Bank Z:Bank Z: 16%, compounded annually16%, compounded annually

Which bank would you borrow from?Which bank would you borrow from?

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16% 1 - 1

0.16 1 Rate Effective

16.42% 1 - 4


16.18% 1 - 365


1

zBank

4

YBank

365

XBank

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You would borrow from the bank quoting the “highest” interest rate!


Year end

Investment

(1.12)2

R 112.00

R 125.44

R 337.44

(1.12)1

2 3

R 100.00

1

R 100.00 R 100.00

0

Future value of an annuity (FVA) at 12%

What is the future value of an Ordinary Annuity?

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Future Value using Tables & Formula

FormulaFormula

100 x 3.3744 = 337.44100 x 3.3744 = 337.44

Go to Table B - select factor of for 3 years and Go to Table B - select factor of for 3 years and 12%12%

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Ordinary Annuity or Annuity Due?

Annuity DueAnnuity Due

Ordinary AnnuityOrdinary Annuity

0 1 2 3

0 1 2 3

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Year end

Investment

(1.12)3

R 140.49

R 377.93

2

R 100.00

R 112.00

Future value of an annuity due

(1.12)2

(1.12)1

R 100.00 R 100.00

R 125.44

Future value of an annuity due at 12%

0 1 3

Future Value of an Annuity Due

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(1 + r)n+1

- 1

r

(1 + 0.12)4 - 1

0.12

=

-1

R377.93

= R100 x

-1 (Formula 2.9)FVAdue = I x

What is the Future Value if the annuity is payable in advance? Using the FormulaUsing the Formula

Table B - select 4 periods (3+1) and 12% = Table B - select 4 periods (3+1) and 12% = 4.7793 and then minus 1 = 3.77934.7793 and then minus 1 = 3.7793

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Example 2.11: An investment offers the opportunity to receive R100 one year from now if R90 is paid immediately. Should an investor who applies a 12% discount rate make the investment?

PV = FV / (1+r)n (using Formula 2.3)

= 100 / (1.12)1

= R89.29

Calculation: The Present Value of a future amount due one year from today

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An investment offers the opportunity to receive R100 in ten years’ time. If an investor applies an interest factor of 12%, what is the highest price which will be offered for the investment?

PV = FV / (1+r)n (using Formula 2.3)

= 100 / (1.12)10

= R32.20

Calculation:The Present Value of a future amount due more than one year from today

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Present Value of an Annuity – using the Formula & Table D

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The formula is;The formula is;

R100 x 2.4018 = R240.18R100 x 2.4018 = R240.18

Also, use Table D – 3 periods, 12% = 2.4018Also, use Table D – 3 periods, 12% = 2.4018

TheThe


Year end

Investment

R269.01 Present value of the annuity DUE

R79.72 1/(1.12)2

R 100.00 R 100.00R 100.00

R89.29 1/(1.12)1

Present value of an annuity DUE at 12%

0 1 2 3

Present Value of an Annuity Due

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1(1 + r)n-1

1(1.12)2

= R100 x=

+ 1

R269.00

1 -

0.12

2.69005

= R100 x

PVAdue = I x 1 -

r+ 1

PV of an Annuity Due – using a Formula

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Year end

Investment

Present value of the DEFERRED annuity

1/(1.12)3

1/(1.12)4

1/(1.12)5

R 100.00 R 100.00

R191.47

4 5

R 100.00

R63.55

R56.74

R71.18

Present value of a DEFERRED ANNUITY at 12%

0 1 2 3

Deferred Annuity A deferred annuity commences a number of A deferred annuity commences a number of

years in the future. An important example is years in the future. An important example is a Pension.a Pension.

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An investor wants to buy 1000 non-redeemable 9% preference shares of R1 each. If the interest rate which he applies is 12%, what is the present value of the investment?The investor is buying a future cash flow in perpetuity amounting to 9% of R1000, that is R90.

Because a 12% return on the investment is expected, this problem requires the principal sum to be determined.

PV = CF / r

= 90 / 0.12

= R750

Present Value of a Perpetuity

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Growing PerpetuitiesGrowing Perpetuities

PV = PMTPV = PMT00 (1+ (1+gg))

((r – g)r – g)

If a company has just made a payment of R10 000 If a company has just made a payment of R10 000 and this is expected to grow at the expected inflation and this is expected to grow at the expected inflation rate of 3% per year and the discount rate is 8%, then rate of 3% per year and the discount rate is 8%, then the present value of this perpetual payment stream the present value of this perpetual payment stream is?is?

PV = PV = 10000 (1+0.03)10000 (1+0.03)

(0.08 – 0.03)(0.08 – 0.03)

= R206000= R20600030


Growing annuityGrowing annuity

PV of a Growing AnnuityPV of a Growing Annuity= PMT (1+= PMT (1+gg) 1 – (1+g)) 1 – (1+g)nn

(1+(1+rr))nn

r – g r – g What is the PV of a future salary growing at 6% per year What is the PV of a future salary growing at 6% per year

for 40 years, if the required return is 9% per year.for 40 years, if the required return is 9% per year.

= R180 000 (1+0.06) 1 – (1.06)= R180 000 (1+0.06) 1 – (1.06)4040

(1.09) (1.09) 4040

0.09 – 0.06 0.09 – 0.06 = R4 277 276= R4 277 276

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An investor wishes to:• Retire in 25 year's time . Although, the investor is due a pension from

her employment, she wishes to augment this by acquiring a further pension of R6000 per month by contributing to a retirement fund.

• Receive a monthly pension income of R6000 per month for 12 years• Receive a lump sum payment of one third of the accumulated sum on

retirement.

Additional informationThe fund is currently earning a return of 12% per annum, interest compounded monthly. The return is expected to remain unchanged and to be sustainable over the next 37 years

RequiredDetermine the monthly contribution that the investor is required to make to the retirement annuity fund over the next 25 years.

Example: Calculating the required contribution to a retirement fund

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Retirement Funding [Three step calculation]

1. Calculate the present value required at retirement date to generate an annuity of R6000 per month for 12 years.

2. R456 823 is 2/3 of required amount. Find the full amount.

Full amount required = R456823 / 0,6667 = R685 200

3. The monthly contribution required over the next 25 years to generate an accumulated sum of R685 200 is;

6000 x

6000 x 76.14= 456,823

01.

)01.1/(11 144

685,200 = I x

685,200 = I x 1878.8466I = 364.69 per month

01.0

1)01.01( 300

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Using Financial Calculators

What do these keys mean?What do these keys mean? NN = number of periods = number of periods I/YI/Y = interest rate as a percentage. Enter a number, so if the = interest rate as a percentage. Enter a number, so if the

rate is 10%, enter 10, not 0.10. Other calculators may rate is 10%, enter 10, not 0.10. Other calculators may reflect the interest rate per period as reflect the interest rate per period as [i][i] and the number of and the number of periods as periods as [n].[n].

PV PV = present value= present value PMTPMT = annuity payment. Specify this as a zero when = annuity payment. Specify this as a zero when

working with single sums onlyworking with single sums only In most cases, three or four inputs will be specified, and the In most cases, three or four inputs will be specified, and the

financial calculator will solve for the remaining variable. financial calculator will solve for the remaining variable. On some calculators you will need to first press the On some calculators you will need to first press the COMPUTE key prior to pressing the missing input key.COMPUTE key prior to pressing the missing input key.

N I/YR PV PMT FV

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Let’s do some of the previous examples by using an HP-Let’s do some of the previous examples by using an HP-10B10BIIII financial calculator. financial calculator.

An investment of R100 invested for 10 years earning 12% An investment of R100 invested for 10 years earning 12% per year compound interest will result in a future value of per year compound interest will result in a future value of R310.6. Key in the following input values and press the FV R310.6. Key in the following input values and press the FV key for the solution.key for the solution.

First enter the present value as a negative number, -100 or First enter the present value as a negative number, -100 or press 100 followed by (-), depending on the calculator being press 100 followed by (-), depending on the calculator being used, and then press the PV key,. Then enter 10 and press N, used, and then press the PV key,. Then enter 10 and press N, enter 12 and press I/Y, enter 0 and press PMT and then press enter 12 and press I/Y, enter 0 and press PMT and then press FV (or Comp FV) to find the answer.FV (or Comp FV) to find the answer.

What is the interest rate that will achieve a present value of What is the interest rate that will achieve a present value of R100 growing to R310.60 within 10 years? Enter the inputs R100 growing to R310.60 within 10 years? Enter the inputs in the following sequence, then press the [I/YR] key to in the following sequence, then press the [I/YR] key to determine the interest rate of 12%.determine the interest rate of 12%.

Seq. 2 3 1 4

N I/YR PV PMT FV

10 12 -100 0 310.6

Seq. 2 1 3 4

N I/YR PV PMT FV

10 12 -100 0 310.635


The role of Interest Rates

Interest is a payment for the Interest is a payment for the use of moneyuse of money

The The supply and demand supply and demand for money (loans) for money (loans) is determined by is determined by 3 main factors:3 main factors:

The time value of money (preferring it The time value of money (preferring it now rather than later)now rather than later)

The risk of capital repaymentThe risk of capital repayment

Expected inflation Expected inflation 36


Interest Rate TheoryInterest Rate Theory

The expectation theory The expectation theory The slope of the term structure of interest rate The slope of the term structure of interest rate

depends on the expected future spot rates of depends on the expected future spot rates of interestinterest

The liquidity preference theoryThe liquidity preference theory The interest rate risk is greater, the longer the The interest rate risk is greater, the longer the

time to maturitytime to maturity The market segmentation theoryThe market segmentation theory

Different investors have different investment Different investors have different investment preferences as to timing due to legal, regulatory, preferences as to timing due to legal, regulatory, business and personal motives.business and personal motives.

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101214161820

Yield %

Years to Maturity5 10 15 20 25

A Typical Yield Curve[The term structure of interest rates]

The interest rate set by the market

The length of timeof the loan

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Applying Time Value of Money Applying Time Value of Money Principles to BondsPrinciples to Bonds A bond is a A bond is a financial instrument financial instrument issued by issued by

the government and companies to raise the government and companies to raise fundsfunds

The bond will stipulate that the issuer is The bond will stipulate that the issuer is obliged to pay the bond holder obliged to pay the bond holder a fixed a fixed interest or coupon rate interest or coupon rate until the maturity until the maturity of the bond is repaid to bondholders.of the bond is repaid to bondholders.

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Applying Time value of Money Applying Time value of Money principles to bondsprinciples to bonds

ExampleExample

A Government security which has a A Government security which has a fixed coupon fixed coupon rate of 10% per yearrate of 10% per year, coupon interest rate , coupon interest rate payable payable semi-annuallysemi-annually and maturity date is the and maturity date is the end of Oct 2015. Assume the current date is 1 end of Oct 2015. Assume the current date is 1 November 2011. The current market yield is 12% November 2011. The current market yield is 12% p.a. p.a. What is the value of the bond? What is the value of the bond?

The interest rate per half-year is 6% and there are 8 The interest rate per half-year is 6% and there are 8 half-years until maturity.half-years until maturity.

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Example: Valuing bonds

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7

891011

12131415161718192021222324252627

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A B C D E F G H I JTreasury Fixed Coupon Apr-12 Oct-12 Apr-13 Oct-13 Apr-14 Oct-14 Apr-15 Oct-15

10%, Oct 2015Cash flow from Coupon 5 5 5 5 5 5 5 5Cash flow at Maturity 100

5 5 5 5 5 5 5 105

PV factor 1/(1+r)^n 0.9434 0.8900 0.8396 0.7921 0.7473 0.7050 0.6651 0.6274Present Value 4.7170 4.4500 4.1981 3.9605 3.7363 3.5248 3.3253 65.8783Value (sum of PVs) 93.79

Net Present Value (Excel) 93.79 [Using the =NPV function =npv(0.05,B11..I11)]

Valuation of Annuity & Single Sum

Coupon payment 5PV factor for an annuity 6.2098 31.05

Par Value 100PV factor for a single sum 0.6274 62.74

Value 93.79

Here, we use the formula;

[(1-(1/(1+r)n)]/r

Current date 1 November 2011


Volatility of values based on the term Example: Two bonds pay a coupon rate of 7.5% but one is redeemable Example: Two bonds pay a coupon rate of 7.5% but one is redeemable

in a year’s time and the other bond is redeemable in 6 years time. How in a year’s time and the other bond is redeemable in 6 years time. How will values change if interest rates change?will values change if interest rates change?

420.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14%

1-year bond

6-year bond


Self-study example: Transfer of Ronaldo to Real Madrid

Cristiano Ronaldo was transferred to Real Madrid for €93 million at the Cristiano Ronaldo was transferred to Real Madrid for €93 million at the beginning of July 2009. Ronaldo, who is also the captain of Portugal, is beginning of July 2009. Ronaldo, who is also the captain of Portugal, is reported to have signed a 6-year contract and agreed to be paid €11.11 reported to have signed a 6-year contract and agreed to be paid €11.11 million in his first year (2009-2010) with this amount rising by 25% million in his first year (2009-2010) with this amount rising by 25% annually for the remaining 5 years of his contract.annually for the remaining 5 years of his contract.

Required:Required: What will Ronaldo’s salary be in the final year of his contract (2014-What will Ronaldo’s salary be in the final year of his contract (2014-

2015)?2015)? At a discount rate of 4% per year, what is the present value to Ronaldo of At a discount rate of 4% per year, what is the present value to Ronaldo of

his future earnings with Real Madrid? Assume his annual earnings are his future earnings with Real Madrid? Assume his annual earnings are paid on 31 December of each year with his first annual salary due on 31 paid on 31 December of each year with his first annual salary due on 31 December 2009 for the 2009-2010 season. Assume the current date is also December 2009 for the 2009-2010 season. Assume the current date is also 31 December 2009 so that the first payment is due right away.31 December 2009 so that the first payment is due right away.

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PV of Ronaldo’s contract

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Documents

CHAPTER 2 THE TIME VALUE OF MONEY OF MONEY 1 ©Correia, Flynn, Uliana & Wormald n Using formulae, tables, financial calculators and spreadsheets to determine