Time Value of Money · Chapter 3 97
Time Value of Money
Upon completion of this chapter, students should be able to :
Understand and use the concept of time value of money.
Represent the cash flows occurred in different time period using the cash flow time line.
Calculate the present value and future value of given streams of cash flows.
Identify the impact of time period and required rate of return on present value and future value.
Prepare amortization schedule for amortized term loan.
Compare various types of interest rates.
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
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 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.
Cash Flow Time Line
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:
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:
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.
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.
Future Value And Compounding
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, the money was invested. The process of finding future value is called ‘compounding’. Compounding is the 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 10 percent annual interest rate for two years, we receive Rs 100 (that is 10 percent of Rs 1,000 original investment) interest during year one so that we will end up with Rs 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:
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) (3.2)
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.
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 value. The discounted value of the future sum is the present value. In other words, 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 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.
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)