FINANCE MANAGEMENT FIN420 chp 5

The Time Value of Money Chapter 5

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Most financial decisions, personal as well as business,

involve the time value of money. We use the rate of

interest to express the time value of money.

Learning objectives

After learning this chapter, you should be able to:

1. Understand the concept of time value of money

2. Understand the time value process

The Time Value of

Money

GOAL

Chapter 5 The Time Value of Money

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5.1 INTRODUCTION

Time value of money is a concept concerning the value of the money we have at different

points of time. As such, the promise or a Ringgit one-year from now is not equal to the value

of a Ringgit to be received today. A Ringgit received today can be invested and it will grow

with time.

1. If RM100 is deposited into a saving account paying 10 percent interest, it will grow to

RM110 a year from now.

2. Similarly, if you deposited RM90.91 in the same account, it will grow to RM100 a year

from now.

This illustrates the concept of what is called time value of money. It is simply to say that a

Ringgit today is not equal to a Ringgit some time in future or for that matter, in the past.

5.2 FUTURE VALUE CONCEPT

When discussing the time value of money (TVM), we will eventually talk about future value

(FV) or compound and present value (PV) or discount value. First, let us look at future value

that deals with the accumulation of today's funds or money that will be increased in the

future to a common point of time. For example, if you deposited your money in a bank's

account today, what will be the accumulated amount in the future?

To illustrate this growth concept, let assume that you deposit RM100 today in a bank that

pays 10% interest, what would be the amount of money that you will have three years from now or the FV at end of year 3? As shown in Table 3-1, future value at end of year 3

equals to RM133.10 that consists of:

1. Original principal value of RM100, and

2. Compound interest of RM33.10.

The RM133.10 is the compound value; that is value after taking considerations that interest

left in an account itself earns interest in the following period. The whole process of

calculating the future value is known as "compounding" using the following equations:

FV3 = PV (1 + k)(1 + k)(1 + k)

= PV (1 + k)3


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FV3 = 100(1.10)(1.10)(1.10)

= 100 (1 + 0.10)3

= 100 (1.3310)

= RM133.10

Table 5-1 TVM Basics and Compound Interest

Interest earned on

Year Beg. Value (PV)

(1 + k) End. Value (FV)

Principal Interest

1 100 1.10 110.00 100(0.1) = 10 None 2 110 1.10 121.00 100(0.1) = 10 10(0.1) = 1.00 3 121 1.10 133.10* 100(0.1) = 10 21(0.1) = 2.10 Totals 30 3.10

*Total interest earned = Interest on principal + Interest on interest

= RM30 + RM3.10

= RM33.10

The above calculation shows that the money will grow to RM133.10 after two years, and

1.3310 is simply the Future Value Interest Factor (FVIF) for 10 percent and 3 periods.

Another way of presenting the above calculations is by using the pre-calculated values of

FVIF as follows:

FVn = PV (FVIFk,n )

FV3 = PV (FVIF10%,3 )

= 100 (1.3310)

= RM133.10

To have a better understanding of the mathematical tables for pre calculated values of the

interest factors, the following symbols are relevant and are important to understand in

solving the time value of money problems:

FV : Future value

PV : Present value

k : Interest rate per period of compounding or discounting

n : Number of period(s); refers to number of times of compounding or discounting

process

PVIFk,n : Present Value Interest Factor at k,n Refer to Appendix B-1

PVIFAk,n : Present Value Interest Annuity Factor at k,n Refer to Appendix B-2

FVIFk,n : Future Value Interest Factor at k,n Refer to Appendix B-3

FVIFAk,n : Future Value Interest Annuity Factor at k,n Refer to Appendix B-4


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The table provides an easy access to pre-calculated values for most of possible k and n.

Refer to future value table for values of future value interest factor (FVIF). Therefore, our

previous problem can be rewritten as:

FVn = PV (FVIFk,n)

FV3 = 100 (FVIF10%, 3)

= 100 (1.3310)

= RM133.10

To have a better view of the problem, time line can be drawn to illustrate the cash flow

involved as follows:

Beginning of 1 Beginning of 2 Beginning of 3 Today Ending of 1 Ending of 2

Year 0 1 2 3

Value 100 133.10 FVIF10%,3

The use of time line to present the cash flow will ease understanding and in determining the

amount and timing of the cash flows involved. The use of time line is important, especially

when it involves complex cash flows with longer periods.

To further illustrate the use of time line and FVIF table, let us look at another example of

multiple cash flow problems. If you deposited RM1,000 today and RM2,000 at the beginning

of next year, how much will you have at the end of the third year? The bank gives you 5%

interest on your savings. The future value at year 3:

Year 0 1 2 3 RM1,000 RM2,000 FVIF5%,2 RM2,205.00 FVIF5%,3 RM1,157.70 Total future value RM3,362.70

FV3 = 1,000 (FVIF5%,3) + 2,000 (FVIF5%,2)

= 1,000 (1.1577) + 2,000 (1.1025)

= 1,157.70 + 2,205.00

= RM3,362.70


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It shows that the cash flows will grow to RM3,362.70 after 3 years. The key is to determine

the periods in compounding are to analyze the number of times each cash flow needs to be

compounded to reach the target time frame. For example, RM1,000 deposited today will

have to go through 3 compounding process (1) from year 0 to year 1; (2) from year 1 to year

2; and (3) from year 2 to target year 3. This method of determining the relevant periods is

also applicable for discounting process in determining the present value in the following

sections.

5.3 PRESENT VALUE CONCEPT

The present value and the future value are in reverse and reciprocals to each other. It

involves determining the current or today's value of expected cash flows to be received in

the future. This value is called present value (PV) and is determined by "discounting"

process. Discount usually means cheaper but in this situation what will happen is the present

value that we will get should be smaller than the future value in question. The present value

approach is widely used in financial decision-making model because it provides a mean of

measuring the value of future cash flows in terms of present or immediate value.

To illustrate, let assume that you are going to receive RM10,000 at the end of year 5.

1. How much is it worth today?

2. If you plan to have RM1,100 a year later, what is the value now?

Under each of the circumstances, how much must you deposit in the bank today respectively

at a given interest of 10%? To handle the above present value situations, the future value

equation can be rearranged to obtain present value equation. This is possible as PVIF and

FVIF are reciprocals.

FV = PV (FVIFk,n)

PV = FV / (FVIFk,n)

= 1 / (FVIFk,n) Where 1 is the Present Value Interest Factor at n=0.

Therefore, the present value: PV = FV(PVIFk,n). Refer to Appendix B-1 for pre-calculated

values of present value interest factor (PVIF). To illustrate let assume that you are going

to receive RM10,000 five years from today. What is the value of the money today, if the

interest rate is 10%?


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Year 0 1 2 3 4 5 10,000 6,209.00 PVIF10%,5

PV0 = 10,000 (PVIF10%,5)

= 10,000 (0.6209)

= RM6,209.00 This shows that the value of RM10,000 that you are going to receive five years from now is

only worth RM6,209.00 today if discounted at the rate of 10%. The present value will decrease as the discount rate and frequency of discounting increases. For a multiple

cash flow, let assume the following example. You are expected to receive RM10,000 five

years from now and another RM5,000 four years later. What is the present value of the cash

flows if the interest rate is 10%?

Year 0 1 2 3 4 5 6 7 8 9 10,000 5,000 6,209.00 PVIF10%,5 2,120.50 PVIF10%,9 8,329.50 Total present value

PV0 = 10,000 (PVIF10%,5) + 5,000 (PVIF10%,9)

= 10,000 (0.6209) + 5,000 (0.4241)

= 6,209.00 + 2,120.50

= RM8,329.50

The above calculation shows that the value is only RM8,329.50 out of total RM15,000

received in year 5 and year 9.

5.4 ANNUITIES Annuity is a unique form of multiple payments stream or cash flows with a special

characteristic. It involves cash flows with the same amount or value paid or received in each

period occurring over a specified period. The annuity cash flows are quite common in

financial transactions such as payments for mortgages, hire purchase, leasing, insurance,

and loans. For example, the hire purchase payments may look like this, RM300 a month for

60 months. In order to be an annuity the stream of cash flow must involve:

1. The same amount of payment per period and

2. At least for two periods or more.


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To accommodate the annuity cash flows, the basic time value of money equations changed

as follows:

• Future Value Annuity

FVA n, = CF1 (FVIFk,n) + CF 2 (FVIF k,n-1) + ... + CFn-1 (FVIF k,1) + CFn (FVIF k,0)

= ACF (FVIFA k,n)

• Present Value Annuity

PVA n = CF1 (PVIFk,1) + CF2 (PVIF k,2) + ... + CFn (PVIF k,n)

= ACF (PVIFA k,n)

Where, FVA : Future value annuity

PVA : Present value annuity

CF : Cash flow

ACF : Annuity cash flows

FVIFA : Future Value Interest Factor of Annuity

PVIFA : Present Value Interest Factor of Annuity

There are two types of annuities. There are annuity due and ordinary annuity.

1. Ordinary Annuity. Ordinary annuity is most prominently used in practice and refers

to annuity where their cash flows occur at the end of the period; that is for example at

the end of the year on 31 December, at the end of each month and so on.

2. Annuity Due. Annuity due refers to the cash flows which occur at the beginning of

the period, rather than at the end of the period; that is for example on 1st January of

the year or the first day of the month and so on.

Before going through any examples, it is necessary to understand and able to differentiate

between these two types of annuity payment; annuity due versus ordinary annuity. It

requires different approach to solve for the values under different annuity format.

To illustrate the differences between the two types of annuities, consider the following

examples. Assume that Anita is offered payments of RM500 per year for 3 years at 7%

compounded annually. and under different annuity format, the relevant cash flows are shown

in Figure 5-1 followed by sample calculations of its associated values.


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Figure 5-1 Annuity Due versus Ordinary Annuity

Year 0 1 2 3

500 500 500 Ordinary annuity

500 500 500 Annuity due

As shown, the annuity due cash flows will occurs earlier, thus its value is higher than the

ordinary annuity. Due to the differences in timing of cash flows, the equation to compute the

annuity due has to be adjusted appropriately. This adjustment will be shown in the following

segment.

Future Value Ordinary Annuity

Year 0 1 2 3 500 500 500 FVIF7%,0 500.00 FVIF7%,1 535.00 FVIF7%,2 572.50 Total future value 1,607.50

FV3 = 500(FVIF 7%,2) + 500 (FVIF 7%,1) + 500(FVIF 7%,0,)

= 500(1.1450) + 500(1.0700) + 500(1.000)

= RM1,607.50

Alternatively, under annuity format FV3 = 500(FVIFA7%,3)

= 500(3.215)

= RM1,607.50

Present Value of Ordinary Annuity

Year 0 1 2 3 500 500 500

467.50 PVIF 7%,1 436.50 PVIF 7%,2 408.00 FVIF 7%,3 1,312.00 Total present value


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PV 0 = 500(PVIF 7%,1) + 500 (PVIF 7%,2) + 500(PVIF 7%,3)

= 500(0.9346) + 500(0.8734) + 500(0.8163)

= RM1,312.15

Alternatively, under annuity format PV 0 = 500 (PVIFA 7%,3)

= 500(2.6243)

= RM1,312.15 Future Value of an Annuity Due

Year 0 1 2 3 500 500 500 FVIF7%,1 535.00 FVIF7%,2 572.45 FVIF7%,3 612.50 Total future value 1,719.95

FV3 = 500(FVIF7%,3) + 500 (FVIF7%,2) + 500 (FVIF7%,1)

= 500(1.2250) + 500(1.1449) + 500(1.0700)

= RM1,719.95

Alternatively, under annuity format FV3 = 500 (FVIFA7%,4 – 1)

= 500(4.4399 – 1)

= RM1,719.95

The “minus 1” (–1) factor in the above equation represents the value for FVIF7%,0; which is

deducted from FVIFA7%,4 value because there is no cash flow in year 3. The deduction is

essential to determine the real value as FVIFA7%,4 consists of FVIF7%,3 plus FVIF7%,2 plus

FVIF7%,1 plus FVIF7%,0. Another method to solve for FV3 is as follows:

FV3 = 500 (FVIFA7%,3)(FVIF7%,1)

= 500(3.2149)(1.0700)

= RM1,719.95

Alternatively = RM500 (FVIFA7%,3)(1 + k)


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Present Value of an Annuity Due

Year 0 1 2 3 500 500 500 PVIF7%,0

500.00 467.50 PVIF7%,1 436.50 PVIF7%,2 1,404.00 Total present value

PV0 = 500(PVIF7%,0) + 500 (PVIF7%,1) + 500(PVIF7%,2)

= 500(1.000) + 500(0.9346) + 500(0.8734)

= RM1,404.00

Alternatively, under annuity format PV0 = 500 (PVIFA7%,2 + 1)

= 500(1.8080 + 1)

= RM1,404.00

The “plus 1” (+1) factor in the above equation represents the value for PVIF7%,0; which is not

counted in the sum of PVIFA7%,2, which covers only PVIF7%,1 plus PVIF7%,2.

The preceding computations of the future value and present value of the ordinary annuity

and an annuity due prove that the later value is higher. As mentioned earlier, this is because

of cash flows occur earlier in an annuity due compared to ordinary annuity. In example, the

last cash flow in ordinary annuity is not compounded in calculating future value and therefore

does not earned any interest.

5.5 MULTIPLE PERIODS OF DISCOUNTING AND COMPOUNDING

Up to this point, all of the examples presented deal with yearly discounting or compounding.

In practice, however, it is common that discounting and/or compounding is done more than

once a year; such as semi-annually, quarterly, monthly, or even daily depending on the

situations involved. With multiple compounding or discounting, the effects of compound rate

or discount rate and time on the future value or the present value, respectively will, increase.

This relationship is shown in Table 5-2 with 12 percent interest for 2 years.

Table 5-2 shows that with higher multiples, the present value of future cash flows will

decrease, but inversely the future value will increase. Which multiples are better? The

answer will differ from the perspectives of a borrower or an investor. For a borrower, lesser

multiples are better as compound interest payment will be lower. As for an investor, higher


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multiples are preferred since interests earned on investment are compounded more often

which results in higher return.

Table 5.2 (k) and (n) in Multiple Periods

Frequency (m)

Interest = k / m Periods = n x m

Annually 12% / 1 = 12% 2 x 1 = 2 Semi-annually 12% / 2 = 6% 2 x 2 = 4 Quarterly 12% / 4 = 3% 2 x 4 = 8 Monthly 12% / 12 = 1% 2 x 12 = 24 Daily 12% /360 = 0.033% 2 x 360 = 720

Frequency

Compounding Discounting

Annually FV2 = 1,000(FVIF12%,2) = RM1,254.50

PV0 = 1,000(PVIF12%,2) = RM797.20

Semi-annually FV2 = 1,000 (FVIF6%,4) = RM1,262.50

PV0 = 1,000 (PVIF6%,4) = RM792.10

Quarterly FV2 = 1,000(FVIF3%,8) = RM1,266.80

PV0 = 1,000 (PVIF3%,8) = RM789.40

Monthly FV2 = 1,000(FVIF1%,24) = RM1,269.70

PV0 = 1,000(PVIF1%,24) = RM787.57

Values not found in the mathematical tables are calculated using financial calculator

With the present of multiple periods of compounding and discounting, the effective annual

rate of interest will not equal to the nominal rate of interest used in the basic calculations.

The calculation of effective annual rate of interest takes into account the frequency of

compounding. To illustrate, let assume nominal interest of 12 percent with monthly and

quarterly payments, respectively. The effective annual rate of interest (ER):

Quarterly = [1 + (k / m)] m – 1

Where m : Number of compounding or discounting periods per year

= [1 + (0.12 / 4)]4 – 1

= 12.55%

Monthly = [1 + (0.12/12)]12 – 1

= 12.68%


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The above calculations prove that increase frequency of compounding yearly will lead to an

increase in value or return to investors. On the other hand, increase frequency of discounting

will lead a lower value.

The interest rate used in discounting and compounding process will vary depending on the

situations of time value of money involved. The basic guidelines in choosing appropriate

interest are as follows:

1. Savings and borrowings from the bank. Interest rate set by the bank should be

used in determining the value that involves the savings or other short-term

investment and borrowing facilities offered by the bank.

2. Capital investment by the firm. The firm should use (1) the opportunity cost for the

use of the money, if it uses internal funds for initial capital; or (2) the costs of capital,

if external capital or from other sources are used.

The time value of money is a very important tool in the financial decision process. Normally

one wills take-up the proposal that offers the highest time value of money, either future or

present values. If you intend to use future value in a decision making process, you must use

the same method throughout the whole process. The same rules applied if you plan to use

the present value method.


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1. You place RM25,000 in a saving account paying annual compound interest of

8% three years and then move it into a savings account that pays 10%

interest compounded annually. How much will your money have grown at the

end of six years?

2. Saiful purchase a car for RM35,000 and pay RM5,000 down payment and

agree to pay the rest over the next 10 years in 10 equal annual payments that

include principal payments plus 13% compound interest on the unpaid

balance. What will be the amount of each payment?

3. You need to have RM50,000 at the end of 10 years. To accumulate this sum,

you have decided to save a certain amount at the end of each of the next 10

years and deposit it in the bank. The bank pays 8 percent interest

compounded annually for long-term deposits. How much will you have to

save each year?


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FINANCE MANAGEMENT FIN420 chp 5

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