122132991 Financial Management the Time Value of Money (1)

7/29/2019 122132991 Financial Management the Time Value of Money (1)

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LECTURE TWO: THE TIME VALUE OF MONEY

2.0 INTRODUCTIONIn this lecture you will be provided with compounding and discounting skills required in

making many financial decisions. Because you can earn interest on money you have, a

shilling today would grow to more than a shilling later. Our goal in this lesson is to

evaluate the trade-off between money held today and money promised at some future

point in time.

Objectives

At the end of this lecture you should be able to:

1. Discuss the role of time value in finance2. Explain the concept of future value and

perform compounding calculations.

3. Explain the concept of present value andperform discounting calculations.

4. Apply the mathematics of finance toaccumulate a future sum, preparing loan

amortization schedules, and determining

interest or growth rates.

2.1 THE TIME VALUE OF MONEY

Time value of money is one of the most important concepts in finance. The techniques

and skills you learn in this lesson will be essential in handling many of the financial

decision situations in latter lessons, so you should pay particular care to the material

presented. Time value computations will be necessary in many and diverse decision

situations, such as financial structure decisions, lease versus purchase of capital assets

choices, bond refinancing decisions, valuation of securities , cost of capital, among

others.


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The concept of time value of money is based on the fact that, in general, a shilling

received today is more valuable than a shilling to be received at future point in time (a

bird in hand is worth two in the bush). This concept recognizes that the passage of time

affects the value of money: A shilling in ones possession today is more valuable than a

shilling to be received in future because, first, the shilling in hand can be put to

immediate productive use, and, secondly, a shilling in hand is free from the uncertainties

of future expectations (It is asure shilling).

2.1.1 Time value Concepts

We begin the lesson by considering two views of time value:

Future value (FV) Present value (PV)

Financial values and decisions can be assessed by using either future value (FV) or

present value (PV) techniques. These techniques result in the same decisions, but adopt

different approaches to the decision.

Future value techniques measure cash flow at the some future point in timetypically

at the end of a projects life. The Future Value (FV), or terminal value, is the value at

some time in future of a present sum of money, or a series of payments or receipts. In

other words the FV refers to the amount of money an investment will grow to over some

period of time at some given interest rate.

Present value techniques measure each cash flows at the start of a projects life (time

zero).The Present Value (PV) is the current value of a future amount of money, or a

series of future payments or receipts. Present value is just like cash in hand today.

To illustrate the two concepts of time value, a time line could be employed. A time line

is a horizontal line on which time zero appears at the left most end and future periods are

marked from left to right. Figure 2.1 depicts cash flows occurring over time and the

associated future values and present values.

Because money has time value, all of the cash flows associated with an investment must

be measures at the same point in time. Typically the point is chosen is either at the end or

the beginning of the investments life.

FV techniques use compounding to find the future value of each cash flow at the given

future date and the sums those values to find the value of cash flows. Alternatively the


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PV techniques use discounting to find the PV of each cash flow at time zero and then

sums these values to find the total value of the cash flows.

Although FV and PV techniques result in the same decisions, since financial managers

make decisions in the present, they tend to rely primarily on PV techniques.

FIGURE 2.1Time line depicts an investment's cash flows covering five periods, and

FV and PV concepts. The timeline represents the case where an initial investment of

Sh. 100,000 at time zero is followed by cash inflows for the next five years of

Sh.30,000, Sh. 50,000, Sh.40,000, Sh030,000, and Sh.20,000 respectively

-100,000 30,000 50,000 40,000 30,000 20,000

0 1 2 3 4 5

2.2 FUTURE VALUE OF A SINGLE AMOUNT

The question of whether or not interest is earned on the preceding years interest earningsneeds to be addressed first. Two forms of treatment of interest are possible. In the case of

Simple interest, interest is paid (earned) only on the original amount (principal)borrowed. In the case ofCompound interest, interest is paid (earned) on any previousinterest earned as well as on the principal borrowed (lent). Compound interest is crucial

to the understanding of the mathematics of finance. In most situations involving the time

value of money compounding of interest is assumed.

FV

PV


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The future value of present amount is found by applying compound interest over a

specified period of time

The Equation for finding future values of a single amount is derived as follows:

Let FVn = future value at the end of period n

PV (Po) =Initial principal, or present value

k= annual rate of interest

n = number of periods the money is left on deposit.

The future value (FV), or compound value, of a present amount, Po, is found as follows.

At end of Year 1, FV1 =Po (1+k) = Po (1+k)1

At end of Year 2, FV2=FV1 (1+k) = Po(1+k) (1+k) = Po ( 1+k)2

At end of Year 3, FV3 = FV2 (1+k) = Po ( 1+k) ( 1+k) (1+k) = Po (1+k)3

A general equation for the future value at end ofn periods can therefore be formulated as,

FVn = Po ( 1+k)n

(2-1)

.

Take note

Equation (2-1) is the basic relationship in the time-value-of-money formulations. All other equations

and relationships we shall discuss later derive

from this equation

Example 1

Jane deposited Sh.800,000 in a saving account paying 6% interest compounded annually

and would like to know the money the account after 3, 6,7, and 10 years.

Solution

At the end of 3 years, FV3 = 800,000 ( 1+0.06)3 = Sh.952,813

At the end of 6 years, FV6 = 800,000 ( 1+0.06)6

= Sh. 103,9534


= Sh. 1,202,904


= Sh. 1,432,678


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2.2.1 Using Tables to Find Future Values

Unless you have financial calculator at hand, solving for future values using Equation

2.1 can be quite time consuming because you will have to raise (1+k) to the nth power.

Luckily, there are published tables available giving values of (1+k)n

for various values of

kand n. Appendix A at the back of this manual contains a set of these interest rate tables.

Appendix A table A-1 gives the future value interest factors. These factors are the

multipliers used to calculate at a specified interest rate the future values of a present

amount as of a given date. The future value interest factor for an initial investment of

Sh.1 compounded at kpercent fornperiods is referred to as FVIFk n.

Future value interest factors = FVIFk n. = (1+k)n

. (2.2)

We can, consequently, rewrite Equation 2.1 as follows.

FVn = Po * FVIFk,n (2.3)

The FVIF for an initial principal of Sh.1 compounded at k percent for n periods

can be found in Appendix Table A-1 by looking for the intersection of the nth row

an the k% column. A future value interest factor is the multiplier used to calculate

at the specified rate the future value of a present amount as of a given date.

Example 2 We refer to Example 1 where Jane wanted to find out the amount in

her account at future dates given prevailing interest rate of 6%.

Solution Using Table A-1 we find the relevant factors by looking at the

intersection ofk-column, and n-row, and multiply by initial deposit.

FV after 3 years will be, 800,000*FVIF 6%,3yrs = 800,000*1.191 = Sh.952,800

FV after 6 years will be800,000 * FVIF 6%,6yrs = 800,000*1.419 =Sh.1,135,200

FV after 8 years will be, 800,000 * FVIF 6%,8yrs = 800,000*1.594 =Sh.1,275,200

FV after 10yearswill be, 800,000 * FVIF 6%,10yrs = 800,000*1.791 =Sh.1,432,800

Example

What is the value of Janes deposit after six years assuming the following annual interest

rates; 3%, 5%, 8%, 10%, and 15%.


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FV with 3% rate is 800,000*FVIF3%,6yrs=800,000 x 1.194=Sh.955,200

FV with 5% rate is 800,000*FVIF5%,6yrs=800,000 x 1.340=Sh.1,072,000




A Graphic View of Future Value

Figure 2.1 Influence of time and interest rates on the FVIFs.

30

25

20

15

10

5%

5

1 0%

0 2 4 6 8 10 12 14 16 18 20

Future values are normally stated at the end of the given period. Figure 2.1 graphically

illustrates the relationships between interest rate, the time periods, and the future value

interest factors. The relationships can be summed as follows:

1. The higher the interest rate, the higher the future value of Sh.1.2. The longer the period of time, the higher the future value of Sh.1.3. When the interest rate is zero there is no growth and the future value equals

the present value of Sh.1.

FV of

1/=

20

150

10

5%

Periods


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2.2.2 Compounding More Frequently.

Interest is often compounded more frequently than once a year. Financial institutions

compound interest semi-annually, quarterly, monthly, weekly, daily or even

continuously.

Semi Annual Compounding

This involves the compounding of interest over two periods of six months each within a

year. Instead of stated interest rate being paid once a year one half of the stated interest is

paid twice a year..

Example

Jane decided to invest Sh.100,000 in savings account paying 8% interest compounded

semi annually. If she leaves the money in the account for 2 years how much will she have

at the end of the two years?

She will be paid 4% interest for each 6-months period. Thus her money will amount to.

FV4 = 100,000 ( 1+.08/2)2*2

=100,000(1+.04)4

=Sh.116,990

Or

Using tables = 100,000 x FVIF 4%,4periods =100,000*1.17 = Sh.117,000

Quarterly Compounding

This involves compounding of interest over four periods of three months each at one

fourth of stated annual interest rate.

Example

Suppose Jane found an institution that will pay her 8% interest compounded quarterly.

How much will she have in the account at the end of 2 years?

FV8 = 100,000(1+.08/4)4*2

=100,000(1+.02)8

=100,000 x 1.1716 =

117,160

Or,

Using tables 100,000 x FVIF 2%,8periods = 100,000*1.172 = 117,200

As shown by the calculations in the two preceding examples of semi-annual and quarterly

compounding, the more frequently interest is compounded ,the greater the rate of growth

of an initial deposit. This holds for any interest rate and any period.


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General Equation for Compounding more Frequently than Once a Year.

Let;

m = the number of times per year interest is compounded

n= number of years deposit is held.

k= annual interest rate.

nm

kn m

kPFV

*

0,1

(2.4)

Continuous Compounding.

This involves compounding of interest an infinite number of times per year, at intervals

of microseconds - the smallest time period imaginable. In this case m approaches infinity

and through calculus the Future Value equation 2.1 would become,

FVn (continuous compounding) = Po * ek x n

(2.5)

Where e is the exponential function, which has a value of 2.7183. The FVIFk,n

(continuous compounding) is therefore ekn

, which can be found on calculators.

Example

If Jane deposited her 100,000/= in an institution that pays 8% compounded continuously,

what would be the amount on the account after 2 years?

Amount = 100,000 x 2.718

.08*2

= 100,000 x 2.718

0.16

=100,000*1.1735 =117,350

Nominal Rate versus Effective Annual Rate (EAR) of Interest.

When interest is compounded more frequently than once annually, then the actual interest

earned or paid would be higher than the nominal or stated interest rate. The nominal

(stated, or quoted) annual rates is the contractual annual rate of interest charged by a

lender or promised by a borrower.

The Effective annual rate (EAR) is the annual rate of interest actually paid or earned.

The EAR is best viewed as the interest rate expressed as if it were compounded once per

year.

The effective annual rate reflects the impact of compounding frequency, whereas the

nominal annual rate does not. At times ,interest rates may be quoted in ways that mislead

borrowers and investors regarding the real interest burden or return they are contracting.


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To correctly assess the interest involved one should go beyond the stated rate and work

out the effective annual rate and compare with the stated annual rates.

We can calculate the EAR by the following equation

11

m

m

kEAR (2.6)

Where m is the frequency of compounding per year, and kis the stated annualrate.

Example

Jane wishes to find the effective annual rate associated with 8% nominal annual rate

when interest is compounded (1) annually, (2) semi annually, and (3) quarterly

Solution

Using the above equations;

1. Annual compounding

EAR = (1 + 0.08 )1

-1 = ( 1+0.08)1-1 = 8%

2. Semi annual compounding

EAR = ( 1+ 0.08/2 )

2

1 = ( 1+0.04 )

2

-1 = 8.16%

3. Quarterly

EAR = ( 1 + 0.08/4)41 = ( 1 + 0.02)

41 = 8.24%.

From the foregoing answers, we can note two points:

Take note

1)

The effective annual rate increases withincreasing compounding frequency.

2) The nominal and effective rates are equivalentfor annual compounding.


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2.3 FUTURE VALUE OF AN ANNUITY.

An annuity is a series of payments or receipts of equal amounts (i.e. a pensioner

receiving Sh.100,000 per year for ten years after his retirement). The two basic types of

annuities are the ordinary annuity and the annuity due. An ordinary annuity is anannuity where the cash flow occurs at the end of each period. In an annuity due the cash

flows occur at the beginning of each period. This means that cash flows are sooner

received with an annuity due than for a similar ordinary annuity. Consequently, the future

value of an annuity due is higher than that of an ordinary annuity because the annuity

dues cash flows earn interest for one more year.

2.3.1 Finding the Future Value of an Ordinary Annuity .

The determination of the future value of an ordinary annuity can be illustrated using the

following example.

Example

John wishes to determine how much money he will have at end of 5 years if he deposits

Sh.100,000 annually at the endof the 5 years into an account paying 7% annual interest.

The following time line and Table 2.2 depicts the situation faced by John.

131100

122500

114500

107000

100000

575100

0 1 2 3 4 5

100,000 100,000 100,000 100,000 100,000


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Table 2.2 Future value of a Sh.100,000 5-year ordinary annuity compounded at 7%

End of

year

(1)

Amount

deposited

(2)

Number of

years

companied

(3)

Future value interest

factor (FVIF) from

discount tables

(4)

Future value at

end of year

(5)

(2) *(4)

1 100000 4 1.311 131100

2 100000 3 1.225 122500

3 100000 2 1.145 114500

4 100000 1 1.070 107000

4.751

5 100000 0 1.000 100000

FVIFA and the FV after 5 years 5.751 Sh.575100

Using Tables to Find Future Value of an Ordinary Annuity

Annuity calculations can be simplified by using an interest table. The value of an annuity

is founding by multiplying the annuity with an appropriate multiplier called the future

value interest factor for an annuity (FVIFA) which expresses the value at the end of a

given number of periods of an annuity of Sh.1 per period invested at a stated interest rate.

The formula for the future value interest factor for an annuity when interest is

compounded annually at kpercent forn periods (years) is ;

k

kkFVIFA

nn

t

t

nk

]1[ )1()1(

1

1

,

(2.7)

The future value interest factor for an n-year, k%, ordinary annuity (FVIFA) can also befound by adding the sum of the first n-1 FVIFs to 1.000,as follows;

FVIFAk,n = 1.000 +

1

1,

n

ttkFVIF


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Applying the future value interest factor of an annuity, the future value of an annuity is

determined using the following expression. The future value of an annuity (PMT) can be

found by,

FVAn = PMT x (FVIFAk, n) (2.8)

Where FVAnis the future value of an n-period annuity, PMT is the periodic payment or

cash flow, and FVIFAk,n is the future value interest factor of an annuity. The value

FVIFAk,n can be accessed in appropriate annuity tables using kand n. The Table A-2

gives the PVIFA for an ordinary annuity given the appropriate kpercent and n-periods.

Example

Using Table A-2, the appropriate FVIFA at 7%, for 5 years is 5.751

Thus future value of Johns Sh.100,000 5-year annuity is

FVA = 100,000 x 5.751 = Sh.575,100

Activity

Find the future value of an annuity of Sh.100,000,

a) at 12% for 7 yearsb) at 18% for 4 yearsc)

at 9% for 15 years

d) at 20% for 50 yearse) at 35% for 2 years

2.3.2 Finding Value of an Annuity Due.

We can demonstrate the calculation of the future value of an annuity due by using the

preceding example, only that now John makes the deposits at the beginning of the year

rather than at the end.

John wants to find now much money he will have at the end of five years if he deposits

Sh.100,000 annually at the beginning of each of the next 5 years.


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Solution

The Time line and Table below shows the future value of a Sh.100,000 5-year annuity

due compounded at 7%.

Timeline

140,300

131,100

122,500

114,500

107,000

615,400

0 1 2 3 4

100,000 100,000 100,000 100,000 100,000

Table

Beginning of year

deposit

No of years interest

compounded

FVIF FV at the end of

the year 5

1 5 1.403 140300

2 4 1.311 131100

3 3 1.225 122500

4 2 1.145 114500

5 1 1.070 107000

Sh.615,400

Using Tables to Find Future Value of an Annuity Due

A simple conversion can be applied to use the FVIFA (ordinary annuity)in Table A-2

with annuities due.

The Conversion is represented by Equation 2.6 below.

FVIFk,n(annuity due) = FVIFk,n(ordinary annuity) x ( 1 +k) (2.6)

The conversion is necessary because each cash flow of the annuity due earns interest for

1 more year than an ordinary annuity.


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Example

Using the expression above, we can calculate the future value of the preceding annuity

due as follows.

FVIFA7%,5yrs (annuity due) = FVIFA7%,5yrs(ordinary) x ( 1 + k)

= 5.751 x (1+.07)

= 5.751 x 1.07

=6 .154

Therefore future value of the annuity due = 100, 000 x 6.154 = Sh.615, 400.

Because the annuity dues cash flows occur at the beginning of the period rather than at

the end, its future value is greater than that of a similar ordinary annuity..

2.3.3 Finding unknown Interest Rate

A situation may arise in which we know the future value of a present sum as well as the

number of time periods involved but do not know the compound interest rate implicit in

the situation. The following example illustrates how the interest rate can be determined.

Example

Suppose you are offered an opportunity to invest Sh.100000 today with an assurance of

receiving exactly Sh.300,000 in eight years. The interest rate implicit in this question can

be found by rearranging Equation 2.1 as follows.

FV8 = P0 (FVIF k, 8 )

300,000 = 100,000 (FVIFKk,8)

FVIFk, 8 = 300,000 / 100,000 = 3.000

Readingacross the 8-period row in the FVIFs table (Table A-1) we find the factor that

comes closest to our value of 3 is 3.059 and is found in the 15% column. Because 3.059

is slightly larger than 3 we conclude that the implicit interest rate is slightly less than 15

percent.

To be more accurate, recognize that

FVIFk,8 = (1+k)8

(1+k)8

= 3

(1+k) = 31/8

= 30.125


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1+k = 1.1472

k = 0.1472 = 14.72%

2.3.4 Finding the Number of Compounding Periods.

We may need to know how long it will take for a shilling invested today to grow to a

certain future value given a particular compound rate of interest.

Example

How long will it take Sh.1000 to grow to Sh.1900 if invested at compound annual

interest of 10%.

Using Equation 2.1 we rewrite it as:

FVn = P0 (FVIFk%, n )

1900 = 1000 ( FVIF10%, n)

FVIF10%,n = 1.900

Reading down the 10% column in the FVIF table we look for the FVIF in that column

that is closest to our calculated value 1.900. We see that 1.949 comes closest to 1.900 and

that the number is in the 7-period row. Because 1.949 is a bit larger than 1.900 we

conclude that the implicit compounding periods are slightly less than 7 periods.

For accuracy, rewrite FVIF10%,n as ( 1+0.1)n

and solve for n

( 1 + 0.1)n

= 1.9

9.1)1.1( n

Using natural logarithms,

n ln1.1 = ln 1.9

n= (ln1.9)/ (ln 1.1) = 6.73 years

Rule of 72

Occasionally you may want to estimate how long a given sum must earn at a given

annual rate to double the amount (or, what interest the principal must earn for a given


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number of periods for the principal to double); the rule of 72 can be used to make this

estimate.

1. Numbers of years to double amount =rateinterestAnnual

72

2. The interest rate to double amount =deposited.isamountyearsofNo

72

Examples

At 2% interest it would take ~ 36 years (72/2)

At 9% interest it would take ~ 8 years (72/9)

If you keep money for 4 years you need to earn approximately 18% (72/4) to double it.

If you keep money for 6 years you need to earn approximately 12% (72/6) to double it.

Activity

Using discount tables, test the accuracy of Rule 72

2.4 PRESENT VALUES

We may be interested in determining how much a given sum of money to be received at a

future date is worth at present. To do this we use the concept of thepresent value. The

present value (PV) is the current shilling value of a future amount i.e. the amount of

money that would have to be invested today at a given interest rate to equal the future

amount.

The process of finding present values is referred to as discounting. It is the inverse of

compounding and seeks to answer the question. If I can earn k% on my money, what is

the most I will be willing to pay now for an opportunity to receive FV(a future value)

shillings n periods from now? The annual rate of return k% is referred to as the discount

rate, required rate of return, cost of capital, or opportunity cost.

Example


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Paul has an opportunity to receive Sh.30,000 one year from now. If he can earn 6% on his

investments what is the most he should pay for this opportunity now?

We are required to determine how much he needs to invest now at 6% in order to

accumulate Sh.30,000 after one year.

Let PV (or Po) equal this amount.

PV (1+0.06) = 30,000

PV = 302,2806.1

000,30

The PV of Sh.30,000 received one year from now is Sh.28,302.

Equation for Present Value

The PV of some future amount FVn to be received n periods from now assuming an

opportunity cost k% is calculated by rearranging our basic future value equation 2.1,

makingP0 (PV) the subject as follows.

)1()1(

1*

,

kFV

k

FVPV nnn

n

nk

(2.8)

Example

Paul wishes to find PV of Sh.170,000 that he will receive 8 years from now. Pauls

opportunity cost is 8%

Substituting in Equation 2.7,

PV8%,8yrs = 170,000/( 1 + 0.8)8

=170000/1.851 = Sh.91,842

Time line depicting the present value of a sum to be received 8 years from now.

0 1 2 3 4 5 6 7 8

PV=Sh.91,842 FV8 = Sh.170,000

2.4.1 Using Present Value Interest Factor (PVIF) Tables


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The factor in Equation 2.8 denoted by )1()1(

,1

Kk

n

nor

is called the present

value interest factor (PVIF). The PVIF is the multiplier used to calculate at a specified

discount rate the present value of an amount to be received at a future date. The PVIFk,n

is the present value of one shilling discounted at k% for n-periods.

Therefore the present value (PV)of a future sum ( FVn) can be found by

PV = FVn x (PVIFk, n). (2.9)

Example

In the preceding example the PV could be found by multiplying Sh.170,000 by the

relevant PVIF. Table A-2 gives a factor of 0.540 for 8% and 8 years.

PV = 170,000 x 0.540 = Sh.91,800.

Present Value Relationships

PVIFs

1.00 0% 0%

0.75 5%

0.5 15% 20%

0.25

0 2 4 6 8 10 12 14 16 18 20 22 24

Periods

Remember that PV calculations assume that FV are measured at the end of the given

period.


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Take note

The figure above shows that

1. The higher the discount rate the lower thePV.

2. The longer the period of time, the lowerthe PV

3.PVIFs cannot be greater than 1 nor less than

zero.

Note that PVIFK, n = 1/FVIFk,n, thus given a table for FVIFs one can find

corresponding PVIFs.

2.4.2 Present Value of Cash Flow Streams

Two basic types of cash flow streams are possible: the mixed stream (unequal cash

flows) and the annuity (uniform cash flows). A mixed stream cash flow reflects no

particular pattern while an annuity is a pattern of equal periodic cash flows.

Present Value of a Mixed Stream

We determine the PV of each future amount and then add together all the individual PVs

Example

The following is a mixed stream of cash flows occurring at the end of year

Year Cash flow

Sh.000

1 4002 8003 5004 4005 300

If a firm has been offered the opportunity to receive the above amounts and if its

required rate of return is 9% what is the most it should pay for this opportunities?

Table 2.4 PV of mixed stream of cash flows.


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Year (n) Cash flow PVIF9%, n PV

1 400,000 0.917 366,800

2 800,000 0.842 673,600

3 500,000 0.775 386, 000

4 400,000 0.708 283,200

5 300,000 0.650 195,000

PV 1,904,600

Present Value of an Annuity

The method for finding the PV of an annuity is similar for that of a mixed stream but can

be simplified using present value interest factor of an annuity (PVIFA) tables.

The present value interest factor of an annuity with endof-year cash flows that are

discounted at k per cent for n periods is

PVIFAK,n =

n

t

n

k1 1

1=

k

n

k 1

11

1(2.9)

Table A-3 provides the PVIFAk,n, which can be used in calculating the present value of

an annuity (PVA) as follows:

PVA = PMT* PVIFAk n (2.10)

Example

Mark wants to find the PV of a 5- year annuity of Sh.700,000, assuming opportunity cost

of 8%.

The PVIFA at 8% for 5 years (PVIFA8%,5yrs) from Table A-3 is 3.993.

Therefore, PVA = 3.993 X 700,000 = Sh.2,795,100

2.4.3 Present Value of Perpetuity

A perpetuity is an annuity with an infinite life never stops producing a cash flow at the

end of each year forever.

The PVIF for a perpetuity discounted at the rate k is


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PVIFAk, = 1/k (2.11)

Example

Rose wishes to determine the PV of a Sh.1000 perpetuity discounted at 10%.

The present value of the perpetuity is 1000 x PVIFAk, = 1000 x 1/0.1= Sh.10,000.

This implies that the receipt of Sh.1,000 for an indefinite period is worth only Sh .10,000

today if Rose can earn 10% on her investments (If she had Sh.10,000 and earned 10%

interest on it each year, she could withdraw Sh.1000 annually without touching the

initial Sh.10,000).

2.4.4 Deposits to Accumulate a Future Sum.

It may be necessary to find out the periodic deposits that should lead to the built of a

needed sum of money in future.We can use the expression below, which is a rewriting of Equation 2.3.

PMT = FVAn/FVIFAk n (2.10)

Where PMT is the periodic deposit, FVAn is the future sum to be accumulated, and

FVIFAk n is the future value interest factor of an n-year annuity discounted at k%.

Example

Jane needs to accumulate Sh. 5 million at the end of 5 years to make special purchase.

She can make deposits in an account that pays 10% interest compounded annually. How

much should she deposit in her account annually to accumulate this sum?

Solution

PMT = FVAn/FVIFAk n = 5,000,000/6.105 = Sh.819,000

Example

Suppose you want to buy a house in 5 years from now and estimate that the initial down

payment of Sh. 2 million will be required at that time. You wish to make equal annual

end of year deposits in an account paying annual interest of 6%. Determine the size of the

annual deposit.

FVAN = PMT X FVIFAK, N


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PMT = FVAn/FVIFAk n

PMT = 2,000,000/5.637= Sh.354,799

2.4.5. Amortizing a Loan

An important application of discounting and compounding concepts is in determining the

payments required for an installmenttype loan. The distinguishing features of this loan

is that it is repaid in equal periodic (monthly, quarterly, semiannually or annually)

payments that include both interest and principal. Such arrangements are prevalent in

mortgage loans, auto loans, consumer loans etc.

Amortization Schedule. An amortization schedule is a table showing the timing ofpayment of interest and principal necessary to pay off a loan by maturity.

Example

Determine the equal end of the year payment necessary to amortize fully a Sh.600,000,

10% loan over 4 years. Assume payment is to be rendered (i) annually, (ii) semi-

annually.

Solution

(i) Annual repayments

First compute the periodic payment

PMT = PVAn /PVIFAk,n.

Using tables we find the PVIFA10%,4yrs = 3.170, and we know that PVAn = Sh.600,000

PMT = 600,000/3.170 = Sh.189, 274 per year.

Table 2.4 Loan Amortization schedule

Payments

End of year Loan

payment

Beg. Of

year

principal

Interest Principal End of year

principal.

[10%x (2)] [ (1)(3) [ (2)(4)]

(1) (2) (3) (4) (5)


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1 189,274 600,000 60,000 129,274 470,726

2 189,274 470726 47,073 142,201 328,525

3 189,274 328525 32,853 156,421 172,104

4 189,274 172104 17,210 172,064 -

(ii) Semi-annual repayments

For semi-annual repayments the number of periods, n, is 8 and the discount rate is 5%.

Lets compute the periodic payment.

PMT = PVAn /PVIFAk,n.

Using tables we find the PVIFA5%,8periods =6.4632, and we know that PVAn = Sh.600,000

PMT = 600,000/6.4632 = Sh.92,833 per year.

Table 2.4 Loan Amortization schedule

Payments

End of

period

(6months)

Loan

payment

Beg. Of

year

principal

Interest Principal End of

period

principal.

[5%x (2)] [ (1)(3) [ (2)(4)]

(1) (2) (3) (4) (5)

1 92833 600,000 30,000 62,833 537,167

2 92,833 537,167 26,858 65,975 471,192

3 92,833 471,192 23,559 69,274 401918

4 92,833 401, 918 20,096 72,737 329,181

5 92,833 329,181 16,459 76,374 252,807

6 92,833 252,807 12,481 80,192 172,615

7 92,833 172,615 8,631 84,202 88,413

8 92,833 88,413 4,421 88,412 -0-


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2.4.6 Determining Interest or Growth Rate

It is often necessary to calculate the compound annual interest or growth rate implicit in a

series of cash flows. We can use either PVIFs or FVIFs tables. Lets proceed by way of

the following illustration.

Example

Roy wishes to find the rate of interest or growth rate of the following series of cash flows

Year Cash flow (Sh.)

2004 1,520,000

2003 1,440,000

2002 1,370,000

2001 1,300,000

2000 1,250,000

Solution

Using 2000 as base year, and noting that interest has been earned for 4 years, we proceed

as follows:

Divide amounts received in the earliest year by amount received in the latest year.

1,250,000/1,520,000 = 0.822. This is thePVIF yrsk 4, . We read across row for 4

years for the interest rate corresponding to factor 0.822. In the row for 4 year in table of

Table A-3 of PVIFs, the factor for 5% is .823, almost equal to 0.822. Therefore interest

or growth rate is approximately 5%.

Note that the FVIF yrsk 4, (1,520,000/1,250,000) is 1.216. . In the row for 4 year in table

of Table A-1 of FVIFs, the factor for 5% is 1.2155 almost equal to 1.216.We estimate the

growth rate to be 5% as before.


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REVIEW QUESTIONS

1. Why does money have time value?2. List five different applications of time value of money.3. If, as an investor, you had a choice of daily, monthly, or quarterly

compounding, which would you choose? Why?

4. Describe the procedure used to amortize a loan into a series of equal annualpayments. What is a loan amortization schedule?

5. What is perpetuity?6. What is continuous compounding? What effect does frequency of

compounding have on the effective annual rate (EAR)?

7. Explain the difference between an ordinary annuity and an annuity due

PROBLEMS

2-1 Tom Mapesa has Sh.500,000 that he can deposit in any of three savings accounts

for a 3 years period. Bank I compounds interest on an annual basis, Bank II

compounds interest twice each year, and Bank III compounds interest each

quarters. All the three Banks have a stated annual interest rate of 4%.

a) Compute the amount in each bank account at the end of 3 years.

b) What is the effective annual rate (EAR) that is earned from each bank?

c) In which bank should Tom deposit his money?

d) Bank IV is different from the others only in the sense that compounding is done

continuously. What would be the amount at end of 3 years if it was deposited in

Bank IV.

2.2. Flo Waridi wishes to accumulate Sh.800,000 by the end of 5 years by making

equal annual end-of-year deposits over the next 5 years. If Flo can earn 7% on

her investments, how much must she deposit at the end of each year to meet this

goal?

2-3 The East African Railways Ltd has outstanding an issue of 4 percent perpetual

bonds and promises to pay Sh.2000 semiannually forever. If the market rate of

interest for similar bonds is 9%,at what price should be perpetual bond sell

currently.


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2-4 Gina Watseka wants to take a holiday in Las Vegas 5 years from today. It is

estimated that the trip will cost Sh. 600,000.

a) Determine the periodic annual deposits in the account necessary to

accumulate the required sh.600,000. Assume first deposit first deposit is

made 1 year from today.

b) How much will Gina have to deposit annually if the final deposit is made

1 year before the holiday.

c) Suppose the cost of living increases 4 percent per year over the next 5

years. What would be the effective purchasing power in year 5 of your

deposits calculated in part (a) above?

2-5 A Bank offers a loan of Sh.1,500,000. The terms of the loan require you to

payback the loan in five equal annual installments of sh.416,100. The first

payment will be made a year from today. What it the effective annual rate of

interest on this loan.

2-6 a) Determine the equal annual end-off-year payments required each year over the

life of the loans shown in the following table to repay them fully during the stated

term in the loan.

Loan Principal Interest rate Term of loan

A Sh. 120,000 8% 3 years

B 600,000 12 10

C 750,000 10 30

D 40,000 15 5

b) Prepare a loan amortization schedule for each of the loans in (a).c) Explain why interest on the loan declines with the passage of time.

2-7 An investment promises the following end-of-year cash flows


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Year Cashflow

1 Sh. 100,000

2 300,000

3 200,000

Assuming an interest rate of 10 per cent calculate the single amount that is equivalent in

value to the cash flow stream.

a) If received today

b) If received at the end of year 1

c) If received at the end of year 2

d) If received at the end of year 3

Problems

2-8. If you invest Sh.12,000 today, how much will you have :a. In 6 years at 7%.

b. In 15 years at 12 %.

c. In 25 years at 10 %.

d. In 25 years at 10 % compounded semi-annually.

2-9How much would you have to invest today to get:a. Sh.12,000 in 6 years at 12%.b. Sh.8,000 in 5 years at 20%c. Sh.15,000 in 15 years at 8%.d. Sh.30,000 each year for 20 years at 6%.e. Sh.40, 000 each year for 40 years at 5%.

2-10 Brian Makamu borrows Sh.7,000,000 at 12% interest rate toward the purchase ofa new house. His mortgage is for 30 years.

a. How much will his annual payments be?b. How much interest will he pay over the life of the loan?c. How much should he be willing to pay to get out of a 12% mortgage and

into a 10% mortgage with 30 years remaining on the mortgage?

2-11 Your younger sister, Susie, will start college in one years time. The college feeswill amount to Sh.80,000 per year for four years payable at the beginning of eachyear. Anticipating Susies ambition, yourparents started investing Sh.10,000 per year

five years ago and will continue to do so for five more years. How much more will

your parents have to invest for the next five years to have the necessary funds forSusies education.. Use 10% as the appropriate interest rate throughout the question.


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2-12 You are the chairperson of the investment retirement fund for Actors Fund. Youare asked to set up a fund of semi-annual payments to be compounded semi-annually

to accumulate a sum of Sh.5,000,000 after 10 years at 8% annual rate (20 payments).The first payment into the fund is to take place 6 months from now, and the last

payment is to take place at the end of the tenth year.

a.

Determine how much the semi-annual payment should be.On the day after the sixth payment is made ( the beginning of the fourth year), theinterest rate goes up to 10% annual rate, and you can earn 10% on the funds that

have accumulated as well as on all future payments into the fund. Interest is to be

compounded semi-annually on all funds.

b. Determine how much the revised semi-annual payments should beafter this rate change (there are 14 semi-annual payments remaining).

The next payment will be in the middle of the fourth year.

2-13 You can deposit Sh.100,000 into an account paying 9% annual interest eithertoday or exactly 10 years from today. How much better off would you be at the end of

40 years if you decide to make the initial deposit today rather than 10 years fromtoday.

2-14 Lumumba needs to have Sh.1,500,000 at the end of 5 years in order to fulfill hisgoal of purchasing a sports car. He is willing to invest the amount as a lump sum

today but wonders what type of investment return he will need to earn. Figure out theannual compound rate of return needed in each of the following cases.

a. Lumumba can invest Sh.1,020,000 today.b. Lumumba can invest Sh.815,000 todayc. Lumumba can invest Sh.715,000 today

2-15 Lucas wishes to determine the future value at the end of two years 0f a

Sh.150,000 deposit made today into an account paying a nominal interest rate of

12%.

a. Find the future value of Lucas deposit assuming that interest rate is

compounded:

1) Annually

2) Quarterly

3) Monthly

4) Continuouslyb. Using your findings in a demonstrate the relationship between

compounding frequency and future value

c. What is the maximum future value obtainable given the Sh.150,000

deposit, 2-year time period, and 12% nominal rate? Using your findings in

a to explain.

2-16 Rita Wanda wishes to select the better of two ten-year annuities.

Annuity X is an ordinary annuity of Sh.250,000 per for 10 years.

Annuity Y is an annuity due of Sh.220,000 per for ten years.

a. Find the future value of both annuities assuming that Rita can earn (1) 10%annual interest, (2) 20% annual interest.

b. Briefly explain the differences between the values


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2-17 You plan to retire in exactly 20 years. Your goal is to create a fund that will allowyou to draw Sh.200,000 per for 30 years between retirement and probable death. You

will be able to earn 11% during the retirement period..

a. How large a fund will you need when you retire in 20 years to provide the30-year Sh.200,000 annuity.

b.

How much would you need today as a lump sum to provide the amountcalculated in a above if you earn only 9% during the 20 years precedingretirement?

2-18 While vacationing in Lamu Island Chris Musundi saw the vacation home of hisdreams. It was listed with a sale price of Sh.20,000,000. The only catch is that Chrisis 40and plans to continue working till he is 65. Chris expects property values to

increase at the general rate of inflation. Chris can earn 9% annually and is willing to

invest a fixed amount for the 25 years to fund the cash purchase of such a house

when he retires.

a. If inflation is expected to average 5% for the next 25 years what will Chrisdream house cost when he retires/

b.

How much must Chris invest at the end of each of the 25 years in order tohave the cash purchase price of the house when he retires?

c. If Chris invests at the beginning instead of at the end of teach of the 25years, how much must he invest each year?

2-19 Rodgers Masengo just closed a Sh.2,000,000 business loan that is to be repaid in3 equal end-of year repayments . The interest rate on the loan is 13%.

Prepare an amortization schedule for the loan.


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Documents

122132991 Financial Management the Time Value of Money (1)