Time Value Of Money - Detailed Discussion.doc

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Upon completion of this chapter, students should be able to : 

Understand and use the concept of time value of money.

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Represent the cash flows occurred in different time period using the cash flow time line.

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Calculate the present value and future value of given streams of cash flows.

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Identify the impact of time period and required rate of return on present value and future value.

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Prepare amortization schedule for amortized term loan.

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Compare various types of interest rates.

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The concept of time value of money suggests that the money received at different point of time has different value. The financial manager must appreciate this fact and understand why they are different and how they are made comparable. Therefore, the basic objective of this chapter is to enable the student to calculate present and future value of cash flows and apply these concepts in addressing real life problems. This chapter begins with fundamental concepts of present value and future value and explains how they are calculated. Then it presents how the pattern of cash flows and required rate of return impact the present value and future value. Finally, different concepts related to interest rates have also been deat on for their proper uses by the students.

CONCEPT Time value of money A concept to understand the value of cash flow occurred at different period

ime value of money is a concept to understand the value of cash flows occurred at different point in time. If we are given the alternatives whether to accept Rs 100 today or one year from now, then we certainly accept Rs 100 today. It is because there is a time value to money. Every sum of money received earlier has reinvestment opportunity. For example, if we deposit Rs 100 today in saving account at 5 percent annual rate of interest, it will increase to Rs 105 at the end of year one. Money received at present is preferred even if we do not have reinvestment opportunity. The reason is that the money that we receive at future has less purchasing power than the money that we have at present due to the inflation. What happens if there is no inflation? Still, many received at present is preferred. It is because most of us have a fundamental behaviour to prefer current consumption to future consumption; money at hand allows current consumption. Thus, (i) the reinvestment opportunity or earning power of the money, (ii) the (risk of) inflation and (iii) an individual's preference for current consumption to future consumption are the reasons for the time value of money. The concept of time value of money is useful in addressing our real life problems relating to planning for future family expenditure. For instance, if we need Rs 500,000 after the retirement from job in 15 years, the amount we need to deposit at an interest rate every year from now until the retirement is conveniently determined by using the time value of money concept. Many financial decisions of a firm require a consideration regarding time value of money. In chapter one, we argued that a corporate manager must always concentrate on maximizing shareholders wealth. Maximizing shareholders wealth, to

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a larger extent, depends on the timing of cash flows from investment alternatives. In this regard, time value of money concept deserves serious considerations on all financial decisions. In the following sections, we present some concepts and techniques to understand time value of money and apply them in financial decision.

SIGNIFICANCE OF THE CONCEPT OF TIME VALUE OF MONEY Time value of money is a widely used concept in literature of finance. Financial decision models based on finance theories basically deal with maximization of economic welfare of shareholders. The concept of time value of money contributes to this aspect to a greater extent. The significance of the concept of time value of money could be stated as below:

Investment Decision Investment decision is concerned with the allocation of capital into long-term investment projects. The cash flows from long-term investment occur at different point in time in the future. They are not comparable to each other and against the cost of the project spent at present. To make them comparable, the future cash flows are discounted back to present value. The concept of time value of money is useful to securities investors. They use valuation models while making investment in securities such as stocks and bonds. These security valuation models consider time value of cash flows from securities.

Financing Decision Financing decision is concerned with designing optimum capital structure and raising funds from least cost sources. The concept of time value of money is equally useful in financing decision, specially when we deal with comparing the cost of different sources of financing. The effective rate of interest of each source of financing is calculated based on time value of money concept. Similarly, in leasing versus buying decision, we calculate the present value of cost of leasing and cost of buying. The present value of costs of these two alternatives are compared against each other to decide on appropriate source of financing. Besides, the concept of time value of money is also used in evaluating proposed credit policies and the firm's efficiency in managing cash collections under current assets management.

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CASH FLOW TIME LINE Cash flow time line A graphical presentation of cash flows at different time period

Cash flow time line is an important tool used to understand the timing of cash flows. It is a graphical presentation of cash flows occurring in different time periods, and is helpful for analyzing the time value of cash flows. To gain an idea about timing of cash flows, let us consider the following time line:

Time

3

2

1

0

5

4

The time line represents the time period stated above the vertical scale. Time zero represents today or just now or at the beginning of period 1. Zero states that the time period just begins from this point. Time 1 denotes the end of period one; time 2 denotes the end of period two, and so on. However, it should be noted that the end of any period also means the beginning of the succeeding period. For example, time 1 states that the period one has just been ended and period two has just began. Time period denoted in the scale has generally a length of one year from 0 to 1, from 1 to 2, from 2 to 3 and so on. However, it could be for six months or three months or one month depending on the period for compounding or discounting used. The corresponding cash flows are placed below the scale as shown in the following time line of cash flows:

1

2

3

4

5

10

50

70

100

90

0

Time

Cash Flows -100

Note that Rs 100 in time zero has negative sign. The negative sign represents the cash outflows, which means that Rs 100 is deposited or paid or cost incurred at time zero. All other cash flows in time 1, 2, 3, 4 and 5 have positive signs. Positive sign is used to denote the cash inflows, which means a cash receipt in the given time periods correspondingly. The time line of cash flow is also used to denote the interest rate that each cash flow earns. Let us consider the following time line.

Time

0

8%

1

2

3

4

5

Cash flow -100 The interest rate is placed in between two corresponding time periods. The interest rate 8 percent placed in between the time zero and one denotes that Rs 100 invested today will earn 8 percent interest in year 1 so that it grows to Rs 108 at the end of year one. Similarly, Rs 108 at the beginning of year two earns 8 percent interest during the year two so that it grows to Rs 116.64 at the end of year two and so on. If the interest rate for every period is similar, it is not necessary to show in between of every time period in the scale. However, if the interest rates differ from year to year, it should be stated in between every time period.

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FUTURE VALUE AND COMPOUNDING Future value Future value of a sum of money is defined as the total of the sum of the money plus A present sum of money plus stream of interest the stream of interest amount received for the period, the money was invested. The amount received during process of finding future value is called ‘compounding’. Compounding is the investment period

process of determining the future value of a cash flow or series of cash flows when compound interest is used. For instance, if we invest Rs 1,000 today in a security at Compounding 10 percent annual interest rate for two years, we receive Rs 100 (that is 10 percent of A process of finding future Rs 1,000 original investment) interest during year one so that we will end up with Rs value 1,100 at the end of year one. Again, we receive Rs 110 on our investment of Rs 1,100 in year 2 (that is 10 percent of Rs 1,100) interest at the end of year two plus Rs 1,100 investment during the year two, so that our original investment Rs 1,000 grows to a total of Rs 1,210 at the termination of year two. Here, Rs 1,210 at the end of year two is regarded as the future value of Rs 1,000 today compounded at 10 percent annual rate for two years. The following time line shows it: Time Cash flow Interest income Year-end amount

0

10%

1

-1,000 100 1,100

2

110 1,210

The future value of a sum of money compounded at 'i' percentage annual rate of interest for ‘n’ year is given by the equation (3.1): 1 FVn = PV (1 + i)n (3.1) Where, FVn = future value of a sum of money at the end of period n. PV = present value or the sum of money today. i = the annual rate of interest at which the sum of money is invested. n = the number of years for which the sum of money is compounded.

Using equation (3.1), the future value of the sum of Rs 1,000 compounded at 10 percent annual rate for 2 years is given by: FV2 = PV (1 + i)2 = Rs 1,000 (1 + 0.1)2 = Rs 1,000 x 1.21 = Rs 1,210. Tabular Solution

Besides equation (3.1), the future value of a sum of money also could be calculated by using future value interest factor (FVIF) table (in the appendix). It is given by the equation (3.2) as follows: FVn = PV (FVIF i, n)

1

This equation is derived as follows FV of PV at i percent for 1 period is FV1 = PV + INT (where INT is the amount of interest) = PV + PV (i) = PV (1 + i) FV of PV at i percent for 2 periods is FV2 = PV1 (1 + i) = PV (1 + i) (1 + i) = PV (1 + i)2 Accordingly, FV of PV at i percent for n periods is FVn = PV(1 + i)n

(3.2)

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In our example, looking at FVIF table at 10 percent for 2 years the FVIF factor is 1.21 so that future value of the sum of Rs 1,000 compounded at 10 percent annual rate of interest for 2 years is given by: FV2 = PV (FVIF 10%, 2) = Rs 1,000 x 1.21 = Rs 1,210

Graphic View of Compounding Process Future value of a sum of money has positive relationship with the interest rate and the time period. This means, larger the interest rate larger will be the future value of a present sum of money. This relationship also holds with time period, that is, longer the time period larger will be the future values. This relationship is shown in Figure 3.1. FIGURE 3.1 Relationship between future values and interest rates over different time periods

Future value of Rs 1 4.0

3.0

i = 15%

2.0

i = 10% i = 5%

1.0 i = 0%

Time periods 0

2

4

6

8

10

The Figure 3.1 shows how a sum of rupee one will grow at different interest rates to different time periods in the future. It is observed from the upward sloping curves that value grows over the time in future. Similarly, the growth in value is larger at higher rate of interest. The interest rate itself is the rate of growth in value. For example, if we invest rupee one at 10 percent annual rate of interest, the value of investment grows at the rate of 10 percent every year. The growth in value is larger at later years because of the effect of compounding.

PRESENT VALUE AND DISCOUNTING We already mentioned that Rs 100 that we have at present has more value than Rs 100 received at future dates. It means the same amount received at two different dates are not comparable. To make them comparable, we need to discount the future Present value value. The discounted value of the future sum is the present value. In other words, The value of future sum of money the present value is the value today of a future cash flow or a series of cash flows. The present value of a future sum of money is the amount of current money that is equally desirable to a decision maker today against a specified amount of money to be received or paid at a future date, given the certain rate of interest. In our future value calculation, we recognized that Rs 1,000 invested at 10 percent annual

Discounting A process of finding present value

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rate of return would grow up to Rs 1,210 in two years from now. In this example, Rs 1,000 today is called the present value of a future sum of Rs 1,210 after two years discounted back at 10 percent rate of interest. The process of finding present value of future cash flows or series of cash flows is called ‘discounting’. Discounting is just reverse of compounding. Let us consider the following time line to understand the discounting process.

0

Time Cash Flow

10%

-1,000

1

2

1,100

1,210 1.10

1.10

The above time line of cash flows shows that Rs 1,210 at the end of year two divided by 1.10 two times produce Rs 1,000 present value. The present value of a future sum of money due in n years is calculated by using the equation (3.3) as follows:2 PV =

(3.3)

Let us suppose that we are offered the alternative of either Rs 1,331 after three years or a specified sum of money today. Assuming that any sum of money today could be invested at 10 percent rate of return (i.e required rate of return is 10 percent). The present value of the future sum of Rs 1,331 after three years discounted back at 10 percent required rate of return is given by: PV = = Rs 1,000 Tabular Solution

Besides the above equation (3.3), the present value of a sum of money also could be calculated by using present value interest factor (PVIF) table (given in the appendix). The relationship is given in equation (3.4) as follows. PV = FVn (PVIF, i, n)

(3.4)

Looking at the present value interest factor table at 10 percent for 3 years the PVIF factor is 0.7513, so that the present value of the sum of Rs 1,331 to be realized at the end of year 3 discounted at 10 percent rate of interest is given by: PV = Rs 1,331 (PVIF 10%, 3) = Rs 1,331 x 0.7513 = Rs 1,000

Graphic View of Discounting Process The present value of a sum of money has inverse relation with the time period and interest rate. As the time period to receive a sum of money increases, the present value of the sum of money will decline. This relationship also holds with interest rate, that is, larger the interest rate lower will be the present value of a future sum. This relationship is depicted in Figure 3.2 FIGURE 3.2 Relationship between present values and interest rates over different time periods

Present Values of Rs 1 1.00 i = 0% 2

0.75

This equation is derived as follows: FVn = PV(1 + i)n PV =

..... (3.1)

i = 5%

0.50

i = 10% 0.25

i = 15% Periods 0

2

4

6

8

10

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The Figure 3.2 shows how the present value of rupee one declines with longer period and larger interest rate. For example, the present value of a future sum is lower at 15 percent than at 10 percent interest rate. The present value also depends on time period. Given the interest rate the present value diminishes with longer period. For example, at 10 percent discount rate the present value of a sum of money (say Rs 10,000) will be low (Rs 3,855) in ten years as compare to Rs 6,209 in five years.

FINDING DISCOUNT RATE AND NUMBER OF PERIODS To this point we explored some fundamental aspects relating to the calculation of present value and future value. It can be observed from equation (3.1) and (3.3) that the present value and future value are the reversal of each other. In the equation, we see four basic variables associated to present and future value calculation– present value (PV), future value (FV), time (n), and interest rate (i). Given the values of any three variables in the equation, we can reformulate both present value and future value equations to calculate the value of fourth variable, which is unknown.

Finding Interest Rate Suppose you deposit Rs 3,500 in a saving account today that will grow to a future value of Rs 5,000 at the end of five years. What rate of return will you earn annually from this investment? In this example Rs 3,500 today is the present value (PV), and Rs 5,000 to be received five years from now is the future value (FV 5). Given the information, the appropriate annual rate of interest can be found by reformulating equation (3.1) and (3.3) and is stated in equation (3.5) as follows: 3 i= –1 For our example, the appropriate annual interest rate is calculated below: i = – 1 = 1.0739 – 1 = 0.074 or 7.4% Tabular Solution

3

This equation is derived as follows: PV = (1 + i)n = (1 + i) = i = –1

... (3.3)

(3.5)

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The same solution could be obtained by using present value interest factor (PVIF) or future value interest factor (FVIF) table. Let us refer to the equation (3.4) that gives the tabular solution of present value as follows: PV = FVn (PVIFi, n) ... (3.4) Substituting the respective values of our example in equation (3.4): Rs 3,500 = Rs 5,000 (PVIF i, 5) = (PVIF i, 5) PVIFi, 5 = 0.7

Looking at PVIF table at five year row the factor 0.7 lies between 7 percent (lower rate = LR) and 8 percent (higher rate = HR). Therefore, to obtain the exact interest rate we interpolate the result as follows: Interest rate (i)

= LR + [(Factor at LR-Exact factor)/(Factor at LR –Factor at HR)] = 7% + [(0.7130- 0.7)/(0.7130 – 0.6806)] = 7% + (0.0130/0.0324) = 7.4%

The above result shows that if Rs 3,500 is deposited today, it will grow to Rs 5,000 at the end of year five. This will yield an annual 7.4 percent return to the investor.

Finding the Number of Periods (Time) Suppose Rs 2,000 is deposited today at 8 percent annual interest rate, then in how many years this will double? This solution could be found out by solving for number of periods or time (n) in equation 3.1. In this example Rs 2,000 today is the present value (PV), at annual interest rate (i) of 8 percent it will double to Rs 4,000 future value (FV) at the end of certain years.

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 BUSINESS FINANCE Using equation (3.1) FVn = PV (1+i)n .... (3.1) Rs 4,000 = Rs 2,000 (1.08)n (1.08)n = 2 Taking logarithm in both sides n log 1.08 = log 2 n = log2/log1.08 n = 0.3010/0.0334 = 9.01 years.

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The above solution shows that Rs 2,000 deposited today at 8 percent annual interest rate will double in nearly 9 years. Tabular Solution

The same solution could be again obtained by using the present value interest factor (PVIF) table. Let us refer to the equation (3.4): PV = FVn (PVIFi, n) ... (3.4) Rs 2,000 = Rs 4,000 (PVIF, 8,% n) PVIF8%, n = 0.5

Looking into the PVIF table at 8 percent interest rate, the factor 0.5 is close to 9 year's factor (that is 0.5002). Therefore, the respective time period that Rs 2,000 doubles at 8 percent annual interest rate is 9 years (that is n = 9 years).

ANNUITY Annuity An annuity is defined as a series of payment of fixed amount at each specified A series of equal payment at equal interval of time for interval of time for a given number of periods. An annuity can be an ordinary a given number of periods annuity or annuity due. In case of an ordinary annuity, each equal payment is made

at the end of each interval of time throughout the period. Whereas in case of annuity due equal payments are made at the beginning of each interval throughout the Ordinary annuity periods. Series of equal payments at the end of each period

For example, if an individual promises to pay Rs 1,000 at the end of each of three years for amortization of a loan, then it is called an ordinary annuity. If it were Annuity due the annuity due, each payment would be made at the beginning of each of the three Series of equal payments at years. They are illustrated in the following time line of cash flows: the beginning of each period

Time

0

8%

Ordinary annuity

Time Annuity due

0 1,000

8%

1

2

1,000

1,000

1

2

1,000

1,000

3 1,000

3

The above time lines show that each cash flow occurs one period earlier in annuity due than in the ordinary annuity.

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Future Value of an Ordinary Annuity Suppose we invest Rs 1,000 at the end of each year for there years in a security paying 8 percent annual interest, how much would we have at the end of three years? This is a problem concerning to the future value of an ordinary annuity. In this example, the first year-end payment of Rs 1,000 is compounded at 8 percent for the rest of two years, second Rs 1,000 year-end payment is compounded for one year and the last Rs 1,000 year-end payment is not compounded at all, since it is only made at the end of year 3, the end of compounding period. Such problems of ordinary annuity are solved by using the equation (3.6) presented below: 4 FVAn = Where,

(3.6)

FVAn = future value of an ordinary annuity for ‘n’ years PMT = annual amount of equal payment. = Rs 1,000 n = number of compounding periods = 3 years i = annual rate of interest at which each payment is compounded= 8 percent or 0.08

Substituting the respective values in equation (3.6), the future value of Rs 1000 ordinary annuity for three years compounded at 8 percent annual rate is given by: FVA3 = = Rs 3,246.4

The following time line gives an idea about compounding of each annual payment and their future value at the end of year three.

Time

0

Ordinary annuity

8%

1

2

3

1,000

1,000

1,000

1,000 × 1.08 1,000 × 1.082

1,080 1,166.4

FVA3 = Rs 3,246.4

Tabular Solution

The future value of annuity stated in above example also could be found by using the future vale interest factors table. It is shown in Table 3.1. TABLE 3.1

End of Year

Future value of a 3 years ordinary annuity of Rs 1,000 compounded at 8 percent per year

1 2 3

Payment (PMT) Rs 1,000 1,000 1,000 Future value of annuity

FVIF at 8%

FV

1.1664 1.0800 1.0000

Rs 1,166.4 1,080.0 1,000.0 Rs 3,246.4

Note that in above calculation in table 3.1, the first year’s payment of Rs 1,000 occurs at the end of year 1 so that it is compounded for two years. Similarly, the second year-end payment of Rs 1,000 is compounded for one year. And the last payment is not compounded at all as it occurs at the end of year three. By using future value interest factor of annuity table, we can calculate the future value of an ordinary annuity as follows. 4

This equation is derived as below: FVAn = PMT (1 + i)n–1 + PMT (1 + i)n–2 + ... ... + PMT (1 + i)1 + PMT (1 + i)0 = PMT = … (3.6)

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FINANCE

(3.7)

Looking at FVIFA table at 8 percent for three years the factor is 3.2464. Now substituting the respective values in equation (3.7): FVA3

= PMT [FVIFA i,n] = Rs 1,000 [FVIFA8%, 3] = Rs 1,000 x 3.2464 = Rs 3,246.4

Future Value of an Annuity Due If Rs 1,000 annuity is the annuity due such that each payment occurs at the beginning of each of the three year, the future value of annuity due is given by equation (3.8) as follows: FVAn (due) = (1+i)

(3.8)

Substituting the respective values in equation (3.8), the future value of threeyear annuity due of Rs 1,000 compounded at 8 percent per year is given by: FVA3 (due) = (1 + 0.08) = Rs 3,246.4 (1.08) = Rs 3,506.11 This calculation process of future value of annuity due is easily understood with the help of cash flow time line:

Time Annuity Due

0

8%

1,000

3

1

2

1,000

1,000 1,000 × 1.08

1,000 × 1.08

2

1,000 × 1.083

1,080 1,166.4 1,259.71

FVA3 = 3,506.11 Tabular Solution

The future value of annuity due stated in above example also could be found by using the future value interest factors table. It is shown in the Table 3.2. TABLE 3.2 Future value of a 3 years annuity due of Rs 1,000 compounded at 8 percent per year

Beginning of Year 1 2 3

Payment (PMT) Rs 1,000 1,000 1,000 Future value of annuity due

FVIF 8% 1.2597 1.1664 1.0800

FV 1,259.7 1,166.4 1,080.0 Rs 3,506.1

In Table 3.2, the first year’s payment of Rs 1,000 occurs at the beginning of year 1, hence, it is compounded for three years. The second year payment of Rs 1,000 is compounded for two years. And the last payment is compounded for one year only. By using future value interest factor of annuity table, we can calculate the future value of an annuity due as follows: FVAn (due) = PMT [FVIFAi,n] (1+i)

(3.9)

Looking at FVIFA table at 8 percent for three years the factor is 3.2464. Now substituting the respective values in equation (3.9):

FVA3 (due)

T I M E VA L U E O F M O N E Y  = PMT [FVIFAi, n] (1+i) = Rs 1,000 [FVIFA8%, 3] (1+0.08) = Rs 1,000 x 3.2464 x 1.08 = Rs 3,506.11

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Present Value of an Ordinary Annuity If you are given the alternatives of accepting either a three-year annuity with a payment of Rs 1,000 at the end of each year or a lump sum payment of Rs 2,500 today, which alternative would you prefer? This problem could be solved by calculating present value of ordinary annuity. If the lump sum payment today is equal to the present value of three-years annuity of Rs 1,000 each year discounted at a given rate of return (that is, the required rate of return or the opportunity cost of funds), we would be indifferent in choosing either of the alternatives, otherwise, we would prefer the higher value. The present value of an annuity is calculated either by using formula or by using present value interest factor of annuity table. The present value of an annuity for n years discounted at 'i' percent required rate of return is given by equation (3.10) as follows:5 PVAn = PMT

(3.10)

Let us suppose that opportunity cost of funds is 10 percent, so that present value of three-year annuity of Rs 1,000 each year is given by: PVA3

= Rs 1,000 = Rs 1,000 x 2.4868 = Rs 2,486.8

That is Rs 1,000, 3 years annuity discounted each year at 10 percent interest rate will have a present value of Rs 2,486.8. Thus, we would be indifferent between a lump sum of Rs 2,486.8 received at present and an ordinary annuity of Rs 1,000 for three years. In the above example, if the choice were between an annuity of Rs 1,000 for three years and a lump sum of Rs 2,500, at present, we would choose higher value i.e. Rs 2,500. Tabular Solution

Another method of calculating the present value of an annuity is to use the present value interest factor for an annuity (PVIFA) table. In this method, the annuity is multiplied by the factor at i percent for n years by referring the PVIFA table as given in the equation (3.11). PVAn = PMT [PVIFAi, n]

(3.11)

Looking at PVIFA table at 10 percent for three years, the present value interest factor of an annuity is 2.4868. By substituting the respective values in equation (3.11) we get Rs 2,486.8 the present value of annuity: PVA3

5

= Rs 1,000 [PVIFA 10, 3] = Rs 1,000 x 2.4868 = Rs 2,486.8

This equation is derived as below: PVAn = PMT [1/(1 + i)1] + PMT[1/(1 + i)2] + ... ... + PMT [1/(1 + i)n] = PMT = PMT … (3.10)

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Present Value of an Annuity Due If each payment were made at the beginning of each year instead of at the end, what would be the present value of the annuity? The simple consideration required is that each of three payments would shift one year earlier so that each of them is discounted for one year less. The formula for present value of an annuity is: PVAn (due) = PMT (1+i)

( 3.12)

Let us refer the same example. If each Rs 1,000 annual payment is made at the beginning of each of three years, the present value of annuity due is given by: PVAn (due)

= Rs 1,000 (1+0.1) = Rs 1,000 x 2.4868 x 1.10 = Rs 2,735.48

Note that present value of an annuity due is greater than the present value of an ordinary annuity because each payment occurs one period earlier. Tabular Solution

Another method of calculating the present value of an annuity due is to use the present value interest factor for an annuity (PVIFA) table. The annuity is multiplied by the factor at 'i' percent for 'n' years by referring to the PVIFA table and the product is further multiplied by (1 + i) as given in the equation (3.13). PVAn (due) = PMT [PVIFA i, n] (1+i)

(3.13)

Looking at PVIFA table at 10 percent for three years the present value interest factor of an annuity is 2.4868. Substituting the respective values in equation (3.11), the present value of annuity due of Rs 1,000 for 3 years at 10 percent is: PVA3

= Rs 1,000 [PVIFA 10%, 3] (1+0.1) = Rs 1,000 x 2.4868 x 1.10 = Rs 2,735.48

SOLVING FOR INTEREST RATES IN ANNUITIES If we were told to invest a lump sum of Rs 10,000 today in a security that pays Rs 4,021.23 at the end of each of next 3 years, what rate of return would we generate from our investment? In this problem we have one unknown variable ‘k’- the interest rate. Rs 10,000 today represents a present value of Rs 4,021.23 three years ordinary annuity, so that the interest rate (k) could be found by using the relationship of present value of an ordinary annuity as given by equations (3.10 or 3.11). Here we are using equation (3.11) PVAn = PMT [PVIFA i, n] .... (3.11) Rs 10,000 = Rs 4,021.23 [PVIFAi, 3] [PVIFAi, 3] = Rs 10,000/Rs 4,021.23 [PVIFAi, 3] = 2.4868

Looking at PVIFA table for three years, the above factor 2.4868 is exactly equal to PVIFA at 10 percent for three years. It means that the rate of return we would earn is 10 percent.

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PRESENT VALUE OF PERPETUITY In the previous sections, we dealt with annuity payments of fixed maturity period, say a three-year annuity, a five-year annuity, a ten-year annuity and so on. However, some annuities may go for paying equal installment at each equal interval of time, Perpetuity indefinitely. Such annuities are called perpetuities. In other words, perpetuity is a An infinite stream of equal payment stream of equal payment made at the end of equal interval of time to indefinite period. The present value of perpetuity is calculated as follows: PVPERPETUITY = =

(3. 14)

For example, the present value of a perpetuity of Rs 1,000 each year at 10 percent interest rate is: PV PERPETUITY = = = = Rs 10,000.

UNEVEN CASH FLOW STREAMS In our previous section we noted that annuities call for a stream of equal payment over the time. However in many cases, in a stream of cash flow, the cash flow in each period may differ from period to period. Such cash flows are called uneven stream of cash flows. In case of uneven stream of cash flows the calculation process of present value and future value is discussed below.

Present Value of Uneven Cash Flow Stream Present value of an uneven cash flow stream is simply the sum of the present value of each of the cash flow occurred throughout the stated period to maturity. The present value equation for uneven stream of cash flows could be stated as follows: PV = + +

+... ... +

(3.15)

For instance, let us suppose a security provides the following stream of cash flows till its maturity in five years. End of Year Cash Flow (Rs )

1 100

2 150

3 200

4 250

5 400

If appropriate discount rate is 10 percent, the present value of this uneven stream of cash flows is calculated as follows: TABLE 3.3 Present value of an uneven cash flow stream

Year

Cash Flows

PVIF 10%

PV

1

Rs 100

0.9091

Rs 90.91

2

150

0.8264

123.96

3

200

0.7513

150.26

4

250

0.6830

170.75

5

400

0.6209

248.36

PV of uneven CF stream

Rs 784.24

The present value of the above uneven cash flow stream is Rs 784.24.

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Future Value of Uneven Cash Flow Streams Future value of uneven cash flow stream is the sum of the future value of each cash flow compounded to the end of the stream at required rate of return. It is calculated by using the following relationship: FVn = CF1(1+i)n-1 + CF2(1+i)n-2 + CF3(1+i)n-3 + . . . . . . + CFn(1+i)0

(3.16)

Let us consider the same cash flow stream as above and a 10 percent rate of compounding. The future value is computed as follows: TABLE 3.4

Year

Future value of an uneven cash flow stream

Cash Flows

10%FVIF 4

FV

1

Rs 100

(1.1) = 1.4641

Rs 146.41

2

150

(1.1)3 = 1.3310

199.65

2

3

200

(1.1) = 1.2100

242.00

4

250

(1.1)1 = 1.1000

275.00

5

400

0

(1.1) = 1.0000

FV of uneven CF stream

400.00 Rs 1,263.06

Note that in above future value calculation, the first year-end cash flow is compounded for four years, the second year-end cash flow is compounded for three years and so on.

SEMIANNUAL AND OTHER COMPOUNDING PERIODS In all preceding discussions, we have considered interest rate compounded annually. It is called simple or quoted rate. However in many cases, the interests are paid monthly or quarterly or semiannually or any other periods which is less than one year. It is called periodic rate. In such cases, interests are compounded more than once during the year. For instance, if interest is paid quarterly, it is compounded four times during the year. If it is paid semiannually, it is compounded two times during the year. The number of compounding periods during the year is denoted as ‘m’. For any compounding period less than one year, we make following two changes in all our present and future value calculation: first, the rate of interest (i) is divided by the number of compounding periods (m) during the year; and second, the number of years that cash flow occurs (n) is multiplied by the number of compounding periods during the year. Let us illustrate a future value calculation for semiannual compounding. Assume that we place a Rs 10,000 today for 5 years in a security paying interest rate of 10 percent. If interest is compounded semiannually, we receive 5 percent interest for each of the six-monthly period through a total of 10 six-monthly periods. The FV of this semiannual compounding appears as: FVmn = PV [1+i/m]mn

Substituting the respective values in equation (3.17): FVmn

= Rs 10,000 [1+0.10/2]2 x 5 = Rs 10,000 (1.05)10 = Rs 10,000 x 1.6289 = Rs 16,289

(3.17)

T I M E VA L U E O F M O N E Y

Effective annual rate An annual equivalent rate of some semiannual or other rates less than one year

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75

In this calculation, we have simply changed a 10 percent annual interest directly to a 5 percent semiannual interest. However, it should be noted that each 5 percent interest payment occurs earlier so that they could be reinvested for rest of the periods during the year. Because of this reason, effective annual rate (EAR) is somewhat greater than annual percentage rate of 10 percent. Effective annual rate is the rate that would produce same terminal value if the annual compounding was used. It is calculated as follows: EAR = – 1

(3.18)

With reference to our example for semiannual compounding the effective annual rate (EAR) is: EAR

=–1 = – 1 = [1.05]2 – 1 = 1.1025 – 1 = 0.1025 or 10.25%.

That is, if the initial deposit of Rs 10,000 were compounded annually at 10.25 percent for 5 years, this would result into a terminal value of Rs 16,289 at the end of 5 years. Note that this is equal to the amount which we calculated by using 5 percent semiannual rate for 10 semiannual periods. The future value calculated using semiannual compounding is always greater than that of annual compounding. It occurs because of more compounding periods during the year.

CONTINUOUS COMPOUNDING In addition to semiannual and other compounding, sometimes interest is compound continuously. Let us recall the Equation (3.19) for semiannual and other compounding. FVmn = PV [1+i/m]mn

(3.19)

In Equation (3.19), when number of compounding period ‘m’ approaches to infinity, we get continuous compounding and the compounding factor ‘[ 1 + i/m] mn’ approaches ‘ei x n’. Therefore, when interest is compounded continuously, the Equation (3.19) could be re-written as follows,: FVn = PV(e)i x n

(3.20)

In Equation (3.20), ‘e’ is called an exponential terms, whose numerical value approximates to 2.71828. To illustrate, let us suppose, if we deposit Rs 1,000 today at an annual interest rate of 12 percent for five years. The future value of this present sum Rs 1,000 at the end of year five, when interest is compounded continuously, is given by: FVmn = PV(e)i x n = Rs 1,000 x (e).12 x 5 = Rs 1,000 x 1.822112 = Rs 1,822.12 Note that the future value of any given amount of present sum is comparatively larger when continuous compounding is used than any other compounding periods. It happens because number of compounding period is significantly larger under continuous compounding.

76

 BUSINESS FINANCE The Equation (3.20), in case of continuous compounding, also can be restated as follows to calculate the present value of a future sum:

Chapter 3

PV = FVn x

(3.21)

The present value calculated by using Equation (3.20) in case of continuous compounding is lower than in any other compounding periods because of significantly larger number of discounting periods.

FRACTIONAL TIME PERIODS In our previous section we noted how present values and future values are calculated using semi-annual and other compounding periods such as three monthly compounding, monthly compounding and so on. This section deals with compounding and discounting when payment occurs in some fractional period. Let us suppose, for example, you deposit Rs 5,000 in Himalayan Bank account that pays annual interest of 12 percent. The interest is compounded daily by the bank. If you hold your deposit for 226 days, what will be the future value of this deposit at the termination of deposit period? If bank compounds interest daily, the periodic rate, that is daily rate, is 0.032877 percent assuming 365 days in a year. Therefore future value is given by: FV = Rs 5,000 (1 + 0.00032877)226 = Rs5,385.59. That is sum of Rs 5,000 deposited today will grow to Rs 5,385.59 at the end of 226days at a 12 percent interest rate, assuming the interest is compounded daily.

AMORTIZED LOANS Amortized loan Amortized loan refers to the loan that is to be repaid in equal periodic installments A loan to be repaid in equal installments including both principal and interest. The concepts of present value and compound throughout the given interest rate are used to amortize a loan over the time in equal installments. periods

Let us suppose a loan of Rs 10,000 is to be repaid in 4 equal installments including principal and 10 percent interest per annum. We apply the following steps to determine the annual payment and set up an amortization schedule of the loan. Determining annual payment

The annual amount of installments to be paid off that includes both principal and interest amount is calculated as follows: PMT = = =

(3.22) = Rs 3,154.67

It means an installment of Rs 3,154.67 paid annually for four years will pay off both principal and interest of the loan. Setting loan amortization schedule

Once the annual amount of installment is determined, the loan amortization schedule could be set up as follows: TABLE 3.5 Loan amortization schedule

T I M E VA L U E O F M O N E Y

Year (1) 1 2 3 4 

Beginning Amount (2)

Payment

Rs 10,000.00 7,845.30 5,475.16 2,868.01

Rs 3,154.67 3,154.67 3,154.67 3,154.67

(3)

Interest (4) = (2) x 0.10 Rs 1,000.00 784.53 547.52 *286.66

 Chapter 3 77 Repayment of Ending Balance Principal (5) = (3) – (4) (6) = (2) – (5) Rs 2,154.67 2,370.14 2,607.15 2,868.01

Rs 7,845.33 5,475.16 2,868.01 -

Interest payment in the 4th year has been rounded to make the sum of principal plus interest equal to the annual payment of Rs 3,154.67.

COMPARISON OF DIFFERENT TYPES OF INTEREST RATES To this point we used three types of interest rates: simple or quoted rate, periodic interest rate and effective annual rate. This section compares them and explains their uses. Simple or quoted interest rate

The rate of interest, which is quoted by borrowers and lenders, is known as simple

Simple rate The interest rate quoted by or quoted interest rate. The practitioners in the stock, bond, commercial loan, lender banking and finance company’s loan express all financial contracts in terms of simple

loan. Simple rate of interest is the general rate that we use in practice while talking about borrowing and lending. However, the quotation of simple rate must also include the number of periods used in compounding per year. For example, a bank may offer 10 percent simple interest loan compounded monthly or quarterly or semiannually or annually. It should be noted that the simple rate of one instrument could be compared with other only when they have same number of compounding periods during the year. This means if is a bank offers 8 percent simple interest loan with quarterly compounding where as another bank offers 8.5 percent simple interest loan compounded semiannually then they can not be compared on the basis of simple interest rate because of difference in compounding periods used in a year. Periodic rate Periodic rate The interest rate for each interest period such as monthly, quarterly, semiannually, annually

The rate of interest charged by lender or paid by borrower at each interest period is known as periodic rate of interest. It can be stated as interest rate per year or interest rate per six month, or per quarter or per month and so on. Periodic rate is calculated as simple interest rate divided by number of period in a year as given in equation (3.23). Periodic Rate (iPER) =

(3.23)

Above equation shows that if periodic rate is multiplied by number of compounding period during the year then the periodic rate is stated on approximate annual rate. This approximate annual rate of periodic rate is known as annual percentage rate (APR). It is to be noted that the APR never is used in actual calculations; it is simply reported to borrowers. The periodic interest rate is equal to simple interest rate only if there is only one interest payment in the year, that is, once in a year. But when interests are paid

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BUSINESS FINANCE

more frequently than once in a year and the payment is made on each compounding date then periodic interest rate is different from simple rate. Effective annual rate

Effective annual rate (EAR) is the annual equivalent interest rate of a given periodic rate. However, it is not the APR. The APR does not consider the compounding effect of periodic rate, whereas the EAR considers it. For example, if we use 3 percent periodic rate per quarter, its APR is 12 percent, but EAR is more than 12 percent because of compounding effect. The basic use of EAR is that it facilitates the comparison of different interest rates with different number of compounding period during the year. For example, if we are going to compare a 8 percent simple interest loan paying interest every six month against a 7.5 percent simple interest loan paying interest every three months, both the simple rates must be converted into effective annual rate.

ILLUSTRATIVE PROBLEMS Illustration

1

Find the following present and future values: An initial Rs 500 compounded for 1 year at 6 percent. b. An initial Rs 500 compounded for 2 years at 6 percent. c. The present value of Rs 500 due in 1 year at a discount rate of 6 percent. d. The present value of Rs 500 due in 2 years at a discount rate of 6 percent.

Present and future a. values

S OL U T I O N a.

Given, Present value (PV) = Rs. 500 Interest rate (i) = 6% 0

6%

1

- 500

FV = ?

FVn FV1 b.

= PV(1 + i)n = PV (1 + i)1 = Rs. 500 (1 + 0.06)1 = Rs. 530 Present value (PV) = Rs. 500 Interest rate (i) = 6% 0

6%

1

2

- 500

FV = ?

FVn FV2 c.

= PV(1 + i)n = PV (1 + i)2 = Rs. 500 (1 + 0.06)2 = Rs. 561.80 Future value (FV) = Rs. 500 Interest rate (i) = 6% No. of periods (n) = 1 0 PV = ?

6%

1 FV = 500



T I M E VA L U E O F M O N E Y

Chapter 3

79

PV d.

= = = = = Rs. 471.70 Future value (FV) = Rs. 500 Interest rate (i) = 6% No. of periods (n) = 2 Present value (PV) = ? 0

6%

1

2

PV = ?

PV

Illustration

FV = 500

= = = = = Rs. 445

Suppose Mr. Sharma deposits Rs 10,000 in a bank account that pays 10 percent interest

Future value 2 annually. How much money will be in his account after 5 years?

SOLUTION

Here, Present value (P) = Rs 10,000, Interest rate (k) = 10% Number of years (n) = 5 years, Future value (FV5) = ? 0

10%

1

2

3

5 FV = ?

Rs10000

We have, FV5 = PV × (1 + k)n = Rs 10,000 × (1.10)5 = Rs 10,000 × 1.6105 = Rs 16,105.10 Mr. Sharma will have Rs16,105.10 at the end of year 5 in his account.

Illustration

3

Present value

What is the present value of a security that promises to pay you Rs 5,000 in 20 years? Assume that you can earn 7 percent if you were to invest in other securities of equal risk?

SOLUTION

Here, Future value (FV) = Rs 5,000 Number of years (n) = 20 years Interest rate (k) = 7% Present value (PV) = ? 0

7%

1

2

3

PV = ?

20 Rs5,000

We have, PV = = = = Rs 1,292.09

Illustration

4

If you deposit money today into an account that pays 6.5 percent interest, how long will it

Time for a lump sum to take for you to double your money? double SOLUTION

Here, Interest rate (i) = 6.5% Number of period (n) = ? Present value (PV) = Rs 1000 (assume)

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Chapter 3

 BUSINESS FINANCE Future value (FV) = Rs 2000 0

6.5%

1

2

3

PV = Rs1000

n=?

FV = Rs2,000

We have, Present value (PV) = or, Rs 1000 = or, (1 + 0.065)n = or, (1.065)n = 2 .... (i) Trying at n = 11 We get, If n = 11, the left hand side in above equation (i) is approximately equal to 2. Hence the required no. of years to double the sum of money is 11 years.

Illustration 5

Find the future values of the following ordinary annuities: FV of Rs 400 each 6 months for 5 years at a simple rate of 12 percent, compounded semiannually. b. FV of Rs 200 each 3 months for 5 years at a simple rate of 12 percent, compounded quarterly. c. The annuities described in parts a and b have the same amount of money paid into them during the 5-year period and both earn interest at the same simple rate, yet the annuity in part b earns Rs 101.60 more than the one in part a over the 5 years. Why does this occur?

Future value of annuity a. at different compounding periods

S OL U T I O N a.

Here, Periodic equal payment (PMT) = Rs. 400 Number of period (n) = 5 years Interest rate (i) = 12% semiannual compounding Future value of an annuity (FVA) = ? Now, FVA = PMT = Rs. 400 = Rs. 400 × 13.1808 = Rs. 5272.32 b. Here, Periodic equal payment (PMT) = Rs. 200 Number of periods (n) = 5 years Interest rate (i) = 12% compounded quarterly Future value of an ordinary annuity (FVA) = ? Now, FVA = PMT = Rs. 200 = Rs. 200 × 26.8704 = Rs. 5374.08 c. It is because, other things held constant, higher the number of compounding higher will be the FV and vice versa.

Illustration 6

Effective rate of interest

Your broker offers to sell a note for Rs 13250 that will pay Rs 2345.05 per year for 10 years. If you buy the note, what rate of interest will you be earning? Calculate to the closest percentage.

S OL U T I O N Here, Present value of annuity (PVA) = Rs. 13,250 Periodic equal payment (PMT) = Rs. 2345.05 No. of periods (n) = 10 years Interest rate (i) = ?



T I M E VA L U E O F M O N E Y

Time Line 1

0 PVA = 13250

2345.05

2

3

4

2345.05

2345.05

5

2345.05 2345.05

6

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Chapter 3

7

8

2345.05 2345.05

9

10

2345.05 2345.05 2345.05

We have, PVA = PMT × PVIFA i × n yrs. or, Rs. 13,250 = Rs. 2345.05 × PVIFAi% 10 yrs or, PVIFAi%, 10 yrs = 5.6502 From the PVIFA table, the value of 5.6502 in 10 years lies at 12%.  The required interest rate is 12%.

Illustration

7

Effective rate of interest

Your parents are planning to retire in 18 years. They currently have Rs 250,000, and they would like to have Rs 1,000,000 when they retire. What annual rate of interest would they have to earn on their Rs 250,000 in order to reach their goal, assuming they save no more money?

SOLUTION

Here, Future value (FV) = Rs 1,000,000 Present value (PV) = Rs 250,000 Time period (n) = 18 years Interest rate (i) = ? 0

i=?

1

2

18

3

Rs250,000

Rs1,000,000

We have, FV = PV (1 + i)n or, Rs 1,000,000 = Rs 250,000 (1 + i)18 or, (1 + i)18 = or, (1 + i)18 = 4 or, 1 + i = (4)1/18 or, i = 1.08 - 1 = 0.08 or 8%  The required rate of interest to reach the goal is 8%.

Illustration

8

What is the future value of a 5-year ordinary annuity that promises to pay you Rs 300

Future value of an each year? The rate of interest is 7 percent. annuity

SOLUTION

Here, Future value of annuity (FVA) = ? Payment (PMT) = Rs 300 Number of period (n) = 5 years Interest rate (i) = 7% 0

7%

1

Rs300

2 Rs300

3 Rs300

We have, FVA = PMT = Rs 300 = Rs 300 × 5.7507 = Rs 1,725.21

5 Rs300 FVA = ?

82



Chapter 3

Illustration

9

Future value of an annuity due SOLUTION

BUSINESS FINANCE

What is the future value of a 5-year annuity due that promises to pay out Rs 300 each year? Assume that all payments are reinvested at 7% a year, until year 5. Here, Future value of annuity due (FVAdue) = ? Payment (PMT) = Rs 300 Number of period (n) = 5 years Interest rate (i) = 7% 0 Rs300

7%

1

Rs300

2 Rs300

5

3 Rs300

FVA (due) = ?

We have, FVAdue = PMT (1 + i) = Rs 300 (1 + 0.07) = Rs 300 × 5.7507 × 1.07 = Rs 1,845.97

Illustration 10

An investment pays you Rs 100 at the end of each of the next 3 years. The investment will then pay you Rs 200 at the end of year 4, Rs 300 at the end of year 5, and Rs 500 at the end Present and future value of year 6. If the rate of interest earned on the investment is 8 percent, what is its present of a cash flow stream value? What is its future value?

SOLUTION Year 1 2 3 4 5 6

Illustration

Cash flow Rs 100 100 100 200 300 500

Present value PVIF at 8% 0.9259 0.8573 0.7938 0.7350 0.6806 0.6302 TPV

PV Rs 92.59 85.73 79.38 147 204.18 315.10 Rs 923.98

Future value FIVE at 8% FV 1.4693 146.93 1.3605 136.05 1.2597 125.97 1.1664 233.28 1.0800 324.00 1.0000 500.00 TVF Rs 1,466.23

11 You are thinking about buying a car, and a local bank is willing to lend you Rs 20,000 to

Loan amortization and buy the car. Under the terms of the loan, it will be fully amortized over 5 years (60 effective interest rate months), and the nominal rate of interest will be 12 percent, with interest paid monthly.

What would be the monthly payment on the loan? What would be the effective rate of interest on the loan?

SOLUTION

Here, Price of the car (PVA) = Rs 20,000 n = 5 years Interest rate (i) = 12% annually (i.e monthly interest rate is 1%) Monthly installment (PMT) = ? Effective interest rate (EIR) = ? We have, PMT = = = = Rs 444.889 EIR = - 1.0 = - 1.0 = 1.1268 - 1 = 0.1268 or 12.68%

Illustration

12 Nepal Horticulture Ltd. invests Rs 4 million to clear a tract of land and to set out some

Expected rate of return young pine trees. The trees will mature in 10 years, at which time the company plans to

sell the forest at an expected price of Rs 8 million. What is company's expected rate of return?

SOLUTION

T I M E VA L U E O F M O N E Y

Here, Future value (FV) = Rs 8,000,000 Present value (PV) = Rs 4,000,000 Time period (n) = 10 years Expected rate of return (i) = ? First set up time line as follows: 0

i=?

1

2

3

Rs4 million We have, FV or, or, or, or, 

Illustration



Chapter 3

83

10 Rs8 million

= PV (1 + i)n Rs 8,000,000 (1 + i)10 (1 + i)10 1+i i

= Rs 4,000,000 (1 + i)10 = =2 = (2)1/10 = 1.0718 - 1 = 0.0718 or 7.18%

13 You need to accumulate Rs 10,000. To do so, you plan to make deposits of Rs 1,250 per

Reaching a financial goal

year, with the first payment being made a year from today, in a bank account which pays 12 percent annual interest compounded annually. Your last deposit will be less than Rs 1,250 if less is needed to round out to Rs 10,000. How many years will it take you to reach your Rs 10,000 goal, and how large will the last deposit be?

SOLUTION

Here, Annual payment (PMT) = Rs 1,250 Future value of annuity (FVAn) = Rs 10,000 Interest rate (i) = 12% Time to maturity (n) = ? Last deposit = ? 0

12%

1

2

3

n =?

Rs1,250 Rs1,250 Rs1,250

Last deposit = ? FVA = Rs10,000 First, we determine the number of periods of the financial goal. This is calculated using future value of annuity formula as follows: We have, FVAn = PMT × FVIFAi, n Rs 10,000 = Rs 1,250 × PViFA12, n FVIFA12, n = =8 Looking FVIFA table the value 8 at 12 percent interest rate lies approximately in 6 years. Therefore the number of years to reach the financial goal is 6 years. Now we calculate the future value of Rs 1,250 for 5 years at 12%, it is Rs 7,941.06 FV = Rs 1,250 × FVIFA12, 5 = Rs 1,250 × 6.3528 = Rs 7,941 Compounding this value after 6 years and before the last payment is made, it is Rs 7,941 (1.12) = Rs 8,893.92. Thus, we will have to make a payment of Rs 10,000 - Rs 8,893.92 = Rs 1,106.08 at year 6, therefore it will take 6 years, and Rs 1,106.08 must be paid in the last installment.

Illustration 14

Mr. Dhakal is in the process of negotiating his first contract. A Company has offered him

Present value of a cash three possible contracts. Each of the contracts lasts for 4 years. All of the money is flow stream

 BUSINESS FINANCE guaranteed and is paid at the end of each year. The terms of each of the contracts are listed below:

84

Chapter 3

Year 1 Year 2 Year 3 Year 4

Contract 1 Payment Rs 3 million 3 million 3 million 3 million

Contract 2 Payment Rs 2 million 3 million 4 million 5 million

Contract 3 Payment Rs 7 million 1 million 1 million 1 million

The Mr. Dhakal discounts all cash flows at 10 percent. Which of the three contracts offers him the most value?

SOLUTION Year PVIF at 10%

1 2 3 4

0.9091 0.8264 0.7513 0.6830

Contract 1 (In Million) CF PV 3 2.7273 3 2.4792 3 2.2539 3 2.0490 TPV 9.5094

Contract 2 Contract 3 (In Million) (In Million) CF PV CF PV 2 1.8182 7 6.3637 3 2.4792 1 0.8264 4 3.0052 1 0.7513 5 3.4150 1 0.6830 TPV 10.7176 TPV 8.6244

The total present value of the Contract 2 is the largest so that it offers him the most value.

Illustration 15

Assume that you inherited some money. A friend of yours is working as an unpaid intern PV and effective at a local brokerage firm, and her boss is selling some securities which call for 4 annual rate payments, Rs 50 at the end of each of the next 3 years, plus a payment of Rs 1,050 at the end of year 4. Your friend says she can get you some of these securities at a cost of Rs 900 each. Your money is now invested in a bank that pays an 8 percent nominal (quoted) interest rate, but with quarterly compounding. You regard the securities as being just as safe, and as liquid, as your bank deposit, so your required effective annual rate of return on the securities is the same as that on your bank deposit. You must calculate the value of the securities to decide whether they are a good investment. What is their present value to you?

SOLUTION

Here, Payment = Rs 50 Fourth year payment = Rs 1,050 Cost of the securities = Rs 900 Interest rate = 8% Compounding = Quarterly compounding, Effective interest rate (EAR) = ? PV = ? First we calculate the effective annual rate: Effective interest rate (EAR) = (1 + 0.08/4)4 - 1 = 8.24% Calculation of the present value of cash flow stream at 8.24% effective rate Year 1 2 3 4

Cash flow 50 50 50 1,050

PVIF @ 8.24% 0.9239 0.8535 0.7886 0.7285 PV

PV 46.195 42.675 39.43 764.925 Rs 893.225

The present value of this cash flow stream is Rs 893.225, which is less than their current selling price so that they are not a good investment

Illustration 16

Loan amortization

SOLUTION

Your Company is planning to borrow Rs 1,000,000 on a 5-year, 15 percent, annual payments, fully amortized term loan. What fraction of the payment made at the end of the second year will represent repayment of principal?

T I M E VA L U E O F M O N E Y  Chapter Here, Loan amount (PVA) = Rs 1,000,000 Number of years (n) = 5 years Interest rate (i) = 15% First we determine the annual installment or payment (PMT) We have, PMT = = = = Rs 293,311.55 Preparation of Amortization Schedule, Amortization schedule

Year 1 2

Payments Rs 298,311.5566 298,311.5566

InterestPayment of Principal Rs 150,000 Rs 148,311.5566 127,753.2665 170,558.2901

85

3

Ending Balance Rs 851,688.4434 681,130.1533

% principal in 2nd year = = = 57.17% That is 57.17% of the payment in second year represents the principal.

Illustration 17

Loan amortization T.U. BBA 2004

You are branch manager of Nepal Bank Limited, Balaju. A borrower approaches you for a term loan of Rs500,000. You agreed to give loan to be fully amortized in a period of 5 year at 10 percent, annual payment. What will be the size of each installment? What fraction of the payment made at the end of second year represents repayment of interest?

SOLUTION

Here, Loan amount (PVA) = Rs 500,000 Number of years (n) = 5 years Interest rate (i) = 10% First we determine the annual installment or payment (PMT) We have, PMT = = = = Rs 131898.28 Preparation of Amortization Schedule, Amortization schedule Year

Beginning balance

1 2

Rs500,000 418,101.72

PMT Rs131,898.28 131,898.28

Interest Rs50,000 41,810.17

Repayment of principal Rs81898.28 90,088.11

Ending balance Rs418,101.72 328,013.61

% interest in 2nd year = = = 31.7% That is 31.7% of the payment in second year represents the interest.

Illustration 18

a.

Non annual compounding

b.

It is now January 1, 2007. You plan to make 5 deposits of Rs 100 each, on every 6 months, with the first payment being made today. If the bank pays a nominal interest rate of 12 percent, but uses semiannual compounding, how much will be in your account after 10 years? Ten years from today you must make a payment of Rs 1,432.02. To prepare for this payment, you will make 5 equal deposits, beginning today and for the next 4 quarters, in a bank that pays a nominal interest rate of 12 percent, quarterly compounding. How large must each of the 5 payments be?

SOLUTION

a.

Here,

Number of deposits (n) = 5 deposits; Semiannual (every 6 months), payment = Rs 100; Nominal interest rate (i) = 12%, Present value of annuity (PVA) = ? 0

Rs100

6% Rs100

1

2

Rs100 Rs100 Rs100

10 FV = ?

 BUSINESS FINANCE We have, FVA = PMT × (1 + i) = Rs 100 (1 + 0.06) = Rs 100 × 5.6371 × 1.06 = Rs 597.5326 Now remaining period is 15 periods (20 periods - 5 periods), so we calculate the future value of this Rs 597.5326 for remaining periods. We have, FV = PV (1 + i)n = Rs 597.5326 (1 + 0.06)15 = Rs 1,432.02 b. Here, Future value at the end of 10 years = Rs 1,432.02; n = 35 periods because quarterly compounding (in 10 years there are 40 quarters); Quarterly interest rate = 3%, PMT = ?, PV = ?

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0

3%

1

10

PMT = ? PMT = ? PMT = ? PMT = ? PMT = ? FV = Rs1,432.02 We have, PV = = = = Rs 508.91 Now we calculate the payment (PMP) Here, n = 5 periods, i = 3%, PV = ?; FV = Rs 508.91, FVA = Rs 508.91 PMT = ? We have, FVA = PMT (1 + i) or, Rs 508.91 = PMT (1 + 0.03) or, Rs 508.91 = PMT × 5.3091 × 1.03  PMT = = Rs 93.06

Illustration 19 19

The prize in last week's Himalayan Lottery was estimated to be worth Rs 35 million. If you were lucky enough to win, the Himalayan will pay you Rs 1.75 million per year over Value of an annuity the next 20 years. Assume that the first installment is received immediately. a. If interest rates are 8 percent, what is the present value of the prize? b. If interest rates are 8 percent, what is the future value after 20 years? c. How would your answers change if the payments were received at the end of each year?

SOLUTION

Here, Payment (MPT) = Rs 1.75 million Number of periods (n) = 20 years, a. Present value of annuity (PVA) = ? interest rate (i) = 8% PVA = PMT × (1 + i) = Rs 1.75 (1 + 0.08) = Rs 1.75 × 9.8181 × 1.08 = Rs 18.56 million b. Future value of annuity (FVA) = ?, Interest rate (i) = 8% FVA = PMT (1 + i) = Rs 1.75 (1 + 0.08) = Rs 1.75 × 45.7620 × 1.08 = Rs 86.49 million c. PVA and FVA assuming payments received at the end of year. Present value of annuity (PVA) = ?, Interest rate (i) = 8% We have, PVA = PMT × = Rs 1.75 = Rs 1.75 × 9.8181 = Rs 17.18 million Future value of annuity (FV) = ?, Interest rate (i) 8% FVA = PMT = Rs 1.75 = Rs 1.75 × 45.7620 = Rs 80.08 million

T I M E VA L U E O F M O N E Y

Illustration 20

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Your client is 40 years old and wants to begin saving for retirement. You advise the client

Future value of an to put Rs 5,000 a year into the stock market. You estimate that the market's return will be, annuity on average, 12 percent a year. Assume the investment will be made at the end of the year.

a. b. SOLUTION

If the client follows your advice, how much money will she have by age 65? How much will she have by age 70?

Here, Your client is 40 years old, Payment (PMT) = Rs 5,000, Interest rate (i) = 12% Investment will be made at the end of the year a. Future value of annuity (FVA) at the age of 65? Number of periods (n) = 65 - 40 = 25 years FVA = PMT = Rs 5,000 = Rs 5,000 × 133.3338 = Rs 666,669 b. Future value of annuity (FVA) at the age of 70? Number of periods (n) = 70 - 40 = 30 years FVA = PMT = Rs , 5,000 = Rs 5,000 × 241.3327 = Rs 1,206,66

Illustration 21

Mr. Lamsal has inherited Rs 25,000 and wishes to purchase an annuity that will provide him with a steady income over the next 12 years. He has heard that the local savings and Solving for payment loan association is currently paying 6 percent compound interest on an annual basis. If he were to deposit his funds, what year-end equal rupee amount (to the nearest rupee) would he be able to withdraw annually such that he would have a zero balance after his last withdrawal 12 years from now?

SOLUTION

Here, Present value of annuity (PVA) = Rs 25,000 Number of years (n) = 12 years Interest rate (i) = 6% Equal annual withdraw (PMT) = ? PVA = PMT × or, Rs 25,000 = PMT or, Rs 25,000 = PMT × 8.3838  PMT = Rs 2,981.9414

Illustration 22

You need to have Rs 50,000 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 Solving for payment the bank. The bank pays 8 percent interest compounded annually for long term deposits. How much will you have to save each year (to the nearest rupee)?

SOLUTION

Illustration 23 Annual interest rate

FVA or, Rs 50,000 or, Rs 50,000  PMT

= PMT = PMT = PTM × 14.4866 = = Rs 3,451.46

Mrs. Karki wishes to borrow Rs 10,000 for three years. A group of individuals agrees to lend her this amount if she contracts to pay them Rs 16,000 at the end of the three years. What is the implicit compound annual interest rate you receive (to the nearest whole percent)?

SOLUTION

Here, Present value (PV) = Rs 10,000 Number of year (n) = 3 years Future value (FV) = Rs 16,000 End payment, interest rate (i) = ? We have,

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 BUSINESS FINANCE FV = PV (1 + i)n or, Rs 16,000 = Rs 10,000 (1 + i)3 or, 1.6 = (1 + i)3 or, (1.6)1/3 - 1 = 1 or, i = 0.1695 or 16.95%

Chapter 3

Illustration 24

You have been offered a note with four years to maturity, which will pay Rs 3,000 at the

Solving for interest rate end of each of the four years. The price of the note to you is Rs 10,200. What is the implicit

compound annual interest rate implied by this contract (to the nearest whole percent)?

SOLUTION

Here, Payment (PMT) = Rs 3,000 per year End payment, years (n) = 4 years Present value (PV) = Rs 10,200 Interest rate (i) = ? We have, PVA = PMT × or, Rs 10,200 = Rs 3,000 × PVIFAi%, 4 years or, Rs 10,200/Rs 3,000 = PVIFAi%, 4 years or, 3.4 = PVIFAi%, 5 years According to PVIFA table the value of 3.4 at 4-year lies between 6% and 7%. For the actual expected return interpolate between these two rates. Actual expected return = Low rate + = 6% + = 6.835%

Illustration 25

The Sriram Brick Company is considering the purchase of a debarking machine this is

PV of uneven cash flow expected to provide cash flows as follows: stream End of Year Year

1

Cash flow Rs 1,200

2

3

4

5

6

7

8

9

10

2,000

2,400

1,900

1,600

1,400

1,400

1,400

1,400

1,400

If the appropriate annual discount rate is 14 percent, what is the present value of this cash flow stream? SOLUTION

Calculation of present value Year 1 2 3 4 5 6-10

Cash flow

*PVIFA for 6-10

Illustration 26

PVIF at 14%

Rs 1,200 2,000 2,400 1,900 1,600 1,400

PV

0.8772 0.7695 0.6750 0.5921 0.5194 1.7830*

Rs 1,052.64 1,539 1,620 1,124.99 831.04 2,496.2

Total present value

Rs 8,663.87

= PVIFA for 10 years - PVIFA for 5 years = 5.2161 - 3.4331 = 1.7830

The following cash flow stream needs to be analyzed

PV and FV of uneven cash flow stream T.U. BBA 2004

Cash flow stream

a. b.

End of year 1

2

3

X

Rs100

Rs200

Rs200

Rs300

4

Rs300

5

Y

200

0

500

0

300

Calculate the present value of each X cash flow at 14 percent discount rate. Calculate the future value of each Y cash flow at 10 percent discount rate.

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S OL U T I O N a.

Calculation of present value of cash flow stream X at 14 percent discount rate Year

Cash flow ‘X’

1 2 3 4 5

14% PVIF

Rs100 200 200 300 300

0.8772 0.7695 0.6750 0.5921 0.5194

Total present value

b.

PV Rs87.72 153.90 135.00 177.63 155.82 Rs710.07

Calculation of future value of cash flow Y at 10 percent compounding rate Year

Cash flow ‘Y’

1 2 3 4 5

Rs200 0 500 0 300

10% FVIF = (1 + i)n - t

FV

(1.1)4 = 1.4641 (1.1)3 = 1.3310 (1.1)2 = 1.2100 (1.1)1 = 1.1000 (1.1)0 = 1.0000

Rs292.82 0 605.00 0 300.00

Total future value

Illustration 27

Calculate the present value of the following cash flow stream. Assume that the stated rate

Uneven cash flow of interest is 14 percent per annum discounted semiannually. stream 1600 1500 Cash flow 1000 T.U. BBA 2005

SOLUTION

Rs1197.82

850

End of year 0 3 1 2 If stated annual rate is 14 percent, discounted semiannually, first we calculate the effective annual rate as follows: Effective interest rate (EAR) = (1 + 0.14/2)2 - 1 = 14.49% Now present value of given cash flow stream discounted at 14.49 percent effective annual rate is calculated as follows: Year 0 1 2 3

Cash flow

14.49% PVIF

Rs1,000 1,600 1,500 850 Total present value

1.0000 0.8734 0.7629 0.6663

PV Rs1,000.00 1,397.44 1,144.35 566.36 Rs4,108.15

SUMMARY Time value of money is a concept to understand the value of cash flow occurred at different point in time. Financial decisions concerned with business firm require a consideration regarding time value of money. Maximizing shareholder wealth, to a larger extent, depends on the timing of cash flows from investment alternatives. Cash flow time line is used to understand the timing of cash flow. Future value of a sum of money is defined as the total of the sum of the money plus the stream of interest amount received for the period of time, the money invested. The process of finding future value is called 'compounding'. Present value of a future sum of money is the amount of current money that is equally desirable today, against a specified amount of

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money to be received or paid at a future date. The process of finding present value is called discounting. Annuity is series of equal payment occurred at equal interval of time throughout a given period. There are two types of annuitiesordinary annuity and annuity due. For ordinary annuity, the series of equal payment occurs at the end, whereas it occurs, at the beginnings of each equal interval of time for annuity due. When a series of equal payment occurs for indefinite period of time it is called a perpetuity.

REVIEW QUESTIONS Indicate whether the following statements are ‘True’ or ‘False’. Support your answer with reason: 1. Rs 100 worth today is equal to Rs 100 worth at the end of year 1. 2. The process of finding future value is called 'compounding'. 3. The present value of a future sum of money is the amount of current money that is equally desirable to a decision maker today against a specified amount of money to be received or paid at a future date. 4. Discounting is just reverse of compounding. 5. An annuity is a series of payment of fixed amount at each specified interval of time for a given number of periods 6. If lump sum payment today is equal to the present value of 3-years annuity of Rs 1000 each year discounted at a given rate of return, we would not be indifferent in choosing either of the alternatives. 7. In semi-annual compounding, the compounding periods are doubled. 8. The loan to be repaid in equal periodic installments is called amortized loan. 9. The present value of a security that promises to pay Rs 5,000 in 20 years at 7 percent discount rate is Rs 1392.10. 10. Given the monthly periodic rate of 1 percent, the annual percentage rate is 12 percent. 11. If monthly periodic rate of interest is 2 percent, the effective annual rate is 24 percent. 12. If your investment doubles in 10 years, you earn approximately 9 percent return in a year. Choose the most appropriate answer for the following: 13. Cash flow time line is a .......... presentation of cash flows associated with different time period. a. tabular b. formula c. graphic d. linear 14. If we deposit Rs 1000 today at an annual interest rate of 10 percent, it is compounded to ........... at the end of year 2. a. Rs 1000 b. Rs 2000 c. Rs 1210 d. Rs 1410 15. The present value of Rs 1100 due in year 1, discounted at 10 percent is .......... a. Rs 1,000 b. Rs 900 c. Rs 1,100 d. Rs 950

T I M E VA L U E O F M O N E Y

16.

17.

18.

19.

20.

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If equal amount of payment occurs at the beginning of each equal interval of time for the given period, the payment is called ........... a. an ordinary annuity b. an annuity due c. present value d. future value A stream of equal payment occur at equal interval of time to infinity is called ......... a. present value of annuity b. future value of annuity c. cash flow d. perpetuity In a stream of cash flow, if the cash flow in each period differs from period to period, it is called ........... a. perpetuity b. payment c. uneven cash flow streams d. amortization schedule The loan to be repaid in equal periodic installment is called .......... a. future value b. present value c. long-term loan d. amortized loan A sum of money due at some future date is called ......... a. time value b. intrinsic value c. present value d. future value

QUESTIONS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

"A rupee in hand today is worth more than a rupee to receive next year". Explain. What is discounting? How it is related to compounding? What do you mean by present value? How it is calculated? What do you mean by future value? How it is calculated? What do you mean by cash flow time lie? What does it show? Illustrate with example. What is the difference between ordinary annuity and annuity due? Illustrate with the help of cash flow time line. What do you mean by perpetuity? How present value of perpetuity is calculated? Illustrate. What do you mean by uneven stream of cash flow? Illustrate how the future value of the uneven stream of cash flow is calculated? What annuity has a greater future value-an ordinary annuity or an annuity due? Why? Explain. What is the difference between annual percentage rate (APR) and effective annual rate (EAR)? Illustrate with suitable example.

PROBLEMS 3–1

Calculate present and future values of the following: An initial Rs 500 compounded for 10 years at 6 percent. b. An initial Rs 500 compounded for 10 years at 12 percent. c. The present value of Rs 500 due in 10 years at 6 percent discount rate.

Present and future a. values

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The present value of Rs 1552.90 due in 10 years at (i) a 12 percent discount rate, and (ii) a 6 percent rate. Give a verbal definition of the term present value, and illustrate it using a cash flow time line with data from this problem. As part of your answer, explain why present values are dependent upon interest rates.

3–2

Which amount is worth more at 14 percent: Rs 1,000 in hand today or Rs 2,000 due in

3–3

To the closest year, how long will it take Rs 200 to double if it is deposited and earns

Present and future 6 years? values

Solving for number of the following rates? periods

a. c.

3–4

7 percent. 18 percent

b. d.

10 percent 100 percent.

Shitalnagar Oil Corporation’s 2006 sales were Rs 12 million. Sales were Rs 6 million 5

Solving for interest years earlier (in 2001). rate or rate of growth

a. b.

To the nearest percentage point, at what rate have sales been growing? Suppose some one calculated the sales growth for MC corporation in part a as follows: “Sales doubled in 5 years. This represents a growth of 100 percent in 5 years, so dividing 100 percent by 5, we find the growth rate to be 20 percent per year.” Explain what is wrong with this calculation.

3–5

Find the future value of the following annuities: Rs 400 per year for 10 years at 10 percent. b. Rs 200 per year for 5 years at 5 percent. c. Rs 400 per year for 5 years at 0 percent. d. Now rework parts a, b and c assuming that payments are made at the beginning of each year; that is, they are annuities due.

3–6

Find the present value of the following ordinary annuities: a. Rs 400 per year for 10 years at 10 percent. b. Rs 200 per year for 5 years at 5 percent. c. Rs 400 per year for 5 years at 0 percent. d. Now rework parts a, b and c assuming that payments are made at the beginning of each year; that is, they are annuities due.

3–7

Find the interest rates, or rates of return, on each of the following: You borrow Rs 700 and promise to pay back Rs 749 at the end of 1 year. b. You lend Rs 700 and receive a promise to be paid Rs 749 at the end of 1 year. c. You borrow Rs 85000 and promise to pay back Rs 201229 at the end of 10 year. d. You borrow Rs 9000 and promise to make payments of Rs 2684.80 per year for 5 years.

Future value of a. annuities

Present value of annuity

Finding interest rates a.

3–8

What is the present value of a perpetuity of Rs 100 per year if the appropriate

Present value of discount rate is 7 percent? If interest rates in general were to double and the perpetuity appropriate discount rate rose to 14 percent, what would happen to the present value

of the perpetuity?

3–9

Following are the cash flow streams associated to two investment proposals under

Uneven cash flow consideration:

Year 1 2 3 4

Cash Flow Streams X Y Rs 100 Rs 300 400 400 400 400 400 400



T I M E VA L U E O F M O N E Y

5

a. b.

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100

If the appropriate interest rate is 8 percent, what is the present value of each cash flow streams? What is the value of each cash flow stream at a 0 percent interest rate?

3–10 Represent the following cash flows in time line and calculate the present value of the

Uneven cash flow stream of cash flows at 11 percent discount rate. The cash flow stream is Rs500,000 in

year zero, -Rs200,000 at the end of year 0ne, Rs200,000 at the end of year two, Rs300,000 at the end of year three and Rs400,000 at the end of year 4.

3–11 Find the amount to which Rs 500 will grow under each of the following conditions: 12 percent compounded annually for 5 years. b. 12 percent compounded semiannually for 5 years. c. 12 percent compounded quarterly for 5 years. d. 12 percent compounded monthly for 5 years.

Future values at different a. compounding periods

3–12 Find the present values of Rs 500 due in the future under each of the following

Present values at different conditions: compounding periods

a. b. c. d.

12 percent simple interest rate, compounded annually. 12 percent simple rate, semiannual compounding, discounted back 5 years. 12 percent simple rate, quarterly compounding, discounted back 5 years. 12 percent simple rate, monthly compounding. Discounted back 1 year.

3–13 You just started your first job, and you want to buy a house within 3 years. You are

Future value currently saving for the down payment. You plan to save Rs 5,000 the first year. You

also anticipate the amount you save each year will rise by 10 percent a year as your salary increases over time. Interest rates are assumed to be 7 percent, and all saving occurs at year end. How much money will you have for a down payment in 3 years?

3–14 Bank A pays 8 percent interest, compounded quarterly, on its money market account. The manager of Bank B want its money market account to equal Bank A’s effective annual rate, but interest is to be compounded on a monthly basis. What simple rate must bank B set? 3–15 The Himalayan Bank pays 7 percent interest, compounded annually, on time

Effective annual rates deposits. The NB bank pays 6.5 percent interest compounded quarterly.

a. b.

Based on effective interest rates, in which bank would you prefer to deposit your money? Could your choice of banks be influenced by the fact that you might want to withdraw your funds during the year as opposed to at the end of the year? In answering this question, assume that funds must be left on deposit during the entire compounding period in order for you to receive any interest.

3–16 Mr. Dhakal invested Rs 150000 eighteen months ago. Currently, the investment is

Effective interest rate worth Rs 168925. Mr. Dhakal knows the investment has paid interest every three

months, but he does not know what the yield on his investment is. Help Mr. Dhakal. Compute both the annual percentage rate (APR) and the effective annual rate of interest.

3–17 A mortgage company offers to lend you Rs 85000; the loan calls for payments of

Effective rate of Rs 8273.59 per year for 30 years. What interest rate is the mortgage company interest

charging you?

3–18 You are thinking to buy a car , and a local bank is willing to lend you Rs20,000 to buy

Amortized loan the car. Under the terms of the loan, it will be fully amortized over 5 years (60

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months), and the nominal rate of interest will be 12 percent, with interest paid monthly. What would be the monthly payment on the loan? What would be the effective rate of interest on the loan? 3–19 Madhyamanchal Inc. just borrowed Rs 25000. Loan is to be repaid in equal

Amortization schedule installments at the end of each of the next 5 years, and the interest rate is 10 percent.

a. b. c.

Set up an amortization schedule for the loan. How large must each annual payment be if the loan is for Rs 50000? Assume that the interest rate remains at 10 percent and the loan is paid off over 5 years. How large must each payment be if the loan is for Rs 50000, the interest rate is 10 percent, and the loan is paid off in equal installments at the end of each of the next 10 years? This loan is for the same amount as the loan in part b, but the payments are spread out over twice as many periods. Why are these payments not half as large as the payments on the loan in part b?

3–20 The management of Campaign for Peace in Nepal Limited decided to buy a printing

Amortization schedule press by taking a loan of Rs1,500,000 for 4 years from Peace Cooperative Limited.

The loan bears a compound annual interest of 12 percent and calls for equal annual installment payments at the end of each of the 4 years. a. What is the amount of annual payments? b. Prepare a schedule showing the fraction of interest and principal payment for each year. c. What fraction of payment made in year 2 represents the principal? d. What fraction of payment made in year 4 represents the interest?

3–21 You are planning to borrow Rs 1,000,000 on a 5-year, 12 percent annual payment

Amortized loan fully amortized term loan. What fraction of payment made at the end of second year

will represent the payment of interest? What fraction of payment made at the end of third year will represent the repayment of principal? What would be the amount of interest and principal paid in the final year?

3–22 Assume that it is now January 1, 2006. On January 1, 2007, you will deposit Rs 1000 into a savings account that pays 8 percent. a. If the bank compounded interest annually, how much will you have in your account on January 1, 2010? b. What would your January 1, 2010, balance be if the bank used quarterly compounding rather than annual compounding? c. Suppose you deposited the Rs 1000 in 4 payments of Rs 250 each on January 1 of 2007, 2008, 2009, and 2010. How much would you have in your account on January 1, 2010, based on 8 percent annual compounding? d. Suppose you deposited 4 equal installments in your account on January 1 of 2007, 2008, 2009, and 2010. Assuming an 8 percent interest rate, how large would each of your payments have to be for you to obtain the same ending balance as you calculated in part a?

Future value

3–23 Krishna Nepal is 63 years old and recently retired. He wishes to provide retirement

Time value of money income for himself and is considering an annuity contract with the National Life

Insurance Corporation. Such a contract pays him an equal rupee amount each year that he lives. For this cash flow stream, he must put up a specific amount of money at the beginning. According to actuary tables, his life expectancy is 15 years, and that is the duration on which the insurance company bases its calculations regardless of how long he actually lives. a. If the Insurance Company uses a compound annual interest rate of 5 percent in its calculations, what must Mr. Nepal pay at the outset for an annuity to

T I M E VA L U E O F M O N E Y

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provide him with Rs10,000 per year? (Assume that the expected annual payments are at the end of each of the 15 years) What would be the purchase price if the compound annual interest rate is 10 percent? Mr. Nepal had Rs30,000 to put into an annuity. How much would he receive each year if the insurance company uses a 5 percent compound annual interest rate in its calculation?

3–24 You opened an account in Bank of Kathmandu (BOK). The bank pays interest at the

Time value of money rate of 3 percent per annum and compounds quarterly.

a.

If you deposit Rs5,000 now, how much shall it grow at the end of 5 years?

b.

What rate will you earn if the money deposited in the bank account doubles in 5 years?

c.

How long will it take to grow Rs 5,000 to Rs 10,955 if the bank pays interest at 4 percent per annum compounded annually?

d.

Assume that you deposit Rs5,000 at the end of each quarter for 4 years. What will be the balance in your account at the end of fourth year if the bank pays interest at 4 percent per annum compounded quarterly?

3–25 Assume that it is now January 1, 2006, and you will need Rs 1000 on January 1, 2010.

Time value of money Your bank compounds interest at an 8 percent annual rate.

a. b. c. d. e. f.

g.

How much must you deposit on January 1, 2007, to have a balance of Rs 1000 on January 1, 2010? If you want to make equal payments on each January 1 from 2007 through 2010 to accumulate the Rs 1000, how large must each of the 4 payments be? If your father offered either to make the payments calculated in part b or to give you a lump sum of Rs 750 on January 1, 2007, which would you choose? If you have only Rs 750 on January 1, 2007, what interest rate, compounded annually, would you have to earn to have necessary Rs 1000 on January 1, 2010? Suppose you can deposit only Rs 186.29 each January 1 from 2007 through 2010, but you still need Rs 1000 on January 1, 2010. What interest rate, with annual compounding, must you seek out to achieve your goal? To help you reach your Rs 1000 goal, your mother offers to give you Rs 400 on January 1, 2007. You will get a part time job and make 6 additional payments of equal amounts each 6 months thereafter, If all this money is deposited in a bank that pays 8 percent, compounded semiannually, how large must each of the 6 payments be? What is the effective annual rate being paid by the bank in part f?

3–26 To complete your last year in business school and then go through law school, you

Present value of will need Rs 10000 per year for 4 years, starting next years (that is you will need to annuity withdraw the first Rs 10000 one year from today). Your rich uncle offers to put you

through school, and he will deposit in a bank paying 7 percent interest a sum of money that is sufficient to provide the four payments of Rs 10000 each. His deposit will be made today. a. How large must the deposit be? b. How much will be in the account immediately after you make the first withdrawal? After the last withdrawal?

3–27 Mrs. Rita wants a refrigerator that costs Rs 12000. She has arranged to borrow the

Solving for payment total purchase price of refrigerator from a finance company at a simple interest rate

equal to 12 percent. The loan requires quarterly payments for a period of three years.

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If the first payment is due three months after purchasing the refrigerator, what will be the amount of her quarterly payments on the loan? 3–28 You are the manager of Nepal Bangladesh Bank, Butwal Branch. Mr. Lamsal, a

Solving for payment government official visits you for advice. He needs to have Rs 500000 at the end of 3

years for his daughter’s enrolment in MBA. To accumulate this sum, he has decided to deposit certain amount at NB bank at the beginning of every year for 3 years. How much will Mr. Lamsal have to deposit each year at the NB bank that pays interest at 5 percent compounded annually?

3–29 While Mr. A.K Chhetri was a student at the Tribhuvan University, he borrowed

Solving for time Rs 12,000 in student loans at an annual interest rate of 9 percent. If Mr. Chhetri

repays Rs 1500 per year, how long, to the nearest year, will it take him to repay the loan?

3–30 Suppose you had just celebrated your 19th birthday. A rich uncle set up a trust fund

Solving for present value for you that will pay Rs 100,000 when you turn 25 years. If the relevant discount rate

is 11 percent, how much is this fund worth today?

3–31 You have just joined the investment-banking firm of Pandey and Pandey Company.

Evaluation cash flow They have offered you two different salary arrangements. You can have Rs 30,000

per year for next two years or Rs 20,000 for the next two years, along with a Rs 30,000 signing bonus today. If the interest rate is 12% compounded quarterly, which do you prefer?

3–32 You need to accumulate Rs 10,000. To do so, you plan to make deposits of Rs 1750 Solving for time and per year, with the first payment being made a year from to day, in a bank account payments which pays 6 percent annual interest. Your last deposit will be more than Rs 1750 if more is needed to round out to Rs 10,000. How many years will it take you to reach your Rs 10,000 goal, and how large will the last deposit be? 3–33 Mr,. Sharma has Rs 42180.53 in brokerage account, and plans to contribute an additional Rs 5000 every year at an annual interest rate of 12 percent. If Mr. Sharma has to accumulate Rs 250,000, how many years will it take for him to reach his goal?

Solving for time

3–34 A 15-year security has a price of Rs 340.4689. The security pays Rs 50 at the end of each of next 5 years, and then it pays a different cash flow amount at the end of each of the following 10 years. Interest rates are 9 percent. What is the annual cash flow amount between 6 and 15 years?

Solving for payment

3–35 Miss Sabita has just own the National Lottery and has three award options to choose from. She can elect to receive a lump sum payment today of Rs 61 million, 10 annual end-of-year payments of Rs 9.5 million, or 30 annual end-of-year payments of Rs 5.5 million. If she expects to earn an 8 percent annual return on her investment, which option should she choose?

Evaluating cash flow

3–36 Mr. Hari, like many college students, recently filled out a credit card application. Not surprisingly, his application was accepted at a nominal interest rate of 24 percent. Reaching a financial goal Upon receiving the credit card, he purchased a new stereo, which costs him Rs 305.44, with the card. Upon receiving his first bill, he was delighted to learn that the credit card company only requires a minimum payments of Rs 10 per month. a. If Hari makes the minimum payment every months, how many months will it be before the account is completely paid off? b. If Hari continues to make the minimum payment, how much will the final payment be? Assume that the last payment occurs at the end of the month, rather than in the middle of the billing period.

T I M E VA L U E O F M O N E Y

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Chapter 3

97

MINI CASE Case 1:

Mr. Ramesh Maharjan has just completed his Bachelor Degree in Business Administration from Tribhuvan University. Assume that it is now January 1, 2007, and he is planning to accumulate Rs 150,000 in January 2012 for his post graduate study in abroad. Today he is thinking for an investment in a security that pays 12 percent annual interest. His only source of income today is his monthly salary of Rs 10,000 from his job in Standard Chartered Bank, main branch, Kathmandu. Out of his monthly income he spends 60 percent amount for his living. He has also saved Rs 100,000 in his bank account from his job over the years. In adherence to his plan about further study, you are required to answer the following: a. How much must he deposit in lump sum on January 1, 2007, to accumulate a balance of Rs 150,000 on January 1, 2012? b. If he wants to make equal payments on each January 1 from 2008 through 2012 to accumulate Rs 150,000, how large must each annual payments be? c. If he wants to invest his monthly net saving in the security, the first payment being made one month from now, how much he could accumulate in January 1, 2012? Assume interest is compounded monthly. d. If he invests his monthly saving into the security, the first payment being made today, how many months it would take him to accumulate the sum of Rs 150,000? e. What is the effective annual rate of monthly compounded interest rate in part ‘c’? f. Instead, if he could invest Rs 25,000 every year starting one year from now, at what annual rate of interest he could accumulate Rs 150,000 on January 1, 2012? g. If his bank balance of Rs 100,000 today pays 10 percent annual interest compounded quarterly, in how many years he could accumulate required sum of Rs 150,000? h. If his bank balance of Rs 100,000 today pays 10 percent annual interest compounded quarterly, to which value it will grow on January 1, 2012?

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