Time Value of Money Dr. Himanshu Joshi FORE School of Management New Delhi

Learning Outcomes.. Interpret interest rates as required rates of return, discount rates, or opportunity costs; Explain an interest rate as the sum of real risk free rate, and premiums that compensate investors for bearing distinct type of risk; Calculate and interpret the effective annual rate, given the stated annual interest rate and frequency of compounding; Solve time value of money problems for different frequencies of compounding; Demonstrate the use of a time line in modeling and solving time value of money problems.

Introduction As individuals, we often face decisions that involve saving money for a future use, or borrowing money for current consumption. We then need to determine the amount we need to invest, if we are saving, or the cost of borrowing if we are shopping for a loan. As an analyst, much of our work also involves evaluating transactions with present or future cash flows. (Security Valuation)

Introduction… To carry out all the above tasks accurately, we must understand the mathematics of time value of money problems. Money has time value in that individuals value a given amount of money more highly the earlier it is received. Therefore a smaller amount of money today may be equal to a large amount of money received in future. The time value of money as a topic in corporate finance deals with equivalence relationship between cash flows with different dates.

Interest Rates: Interpretation

The Time Value of Money Conceptually ‘time value of money’ means that the value of a unit of money is different in different time periods. The value of a sum of money received today is more than its value received after some time. Few methods of dealing time value of money are : - future value of a single amount - present value of a single amount - future value of an annuity - present value of an annuity - intra-year compounding and discounting In general, Compounding means calculating future value of present amount and Discounting means calculating present value of future amount.

I. Future Value Of A Single Amount In this case we calculate the future value of a single amount. The process of investing money as well as reinvesting the interest earned thereon is called compounding. The future value or compounded value of an investment after “n” years when the interest rate is “r” percent is – FV n = PV (1+r) n In this case, the equation (1+r) n is called the “future value interest factor (FVIF)” or simply the future value factor. Either you calculate the value of FVIF or there is a table which gives you the value of FVIF.

Compound and Simple interest : So far we have assumed that money is invested at compound interest which means that each interest payment received is reinvested in future periods. But in case of simple interest, the investment will grow as – FV n = PV [ 1 + No of years × Interest rate] Doubling Period : There is a rule of thumb in FVIF, although not very accurate, but very useful. This is called as rule of 72. According to this rule, the doubling period is obtained by dividing 72 by the interest rate.

Another rule of thumb is rule of 69. It is more calculative but gives more accurate result. According to this rule, the doubling period is equal to – Interest Rate The application of future value of a single amount is to calculate – future value, rate of interest or growth rate, or time period; if other things are given. II. Present Value Of A Single Amount In this case we calculate the present value of future amount. The present value can be calculated by discounting the amount to the present point of time.

The process of discounting is simply the inverse of compounding. The formula for calculating present value is – PV = FV n [ 1 / (1+r) n ] The factor 1 / (1 + r) n is called as the discounting factor or present value interest factor (PVIF). There is a table of PVIF which gives the value of PVIF at different combination of r and n. Present Value of an Uneven Series : If cash flow stream is uneven i.e., the same amount of cash is not flowing every year, then the formula will be different.

The formula for calculating present value of cash flow stream – uneven or even – is PV n =. A1. +. A2. +. A3. + …. +. An. (1+r) 1 (1+r) 2 (1+r) 3 (1+r) n = where, PV n = present value of cash flow stream A t = cash flow occuring at the end of year t r = discount rate n = duration of the cash flow.

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III. Future Value Of An Annuity : An annuity is a stream of cash flow (payment or receipt) occurring at regular interval of time. When cash flows occur at the end of each period the annuity is called an ordinary annuity or a deferred annuity. When cash flow occur at the beginning each period, the annuity is called an annuity due. The formula for calculating the future value of an annuity is FVA n = A [(1+r) n – 1] / r where – FVA n = future value of an annuity for duration of n years A = constant periodic flow r = interest rate per period n = duration of the annuity

The term [(1+r) n – 1] / r is referred to as the future value interest factor for an annuity (FVIFA r,n ). Applications : The future annuity formula can be applied in different cases. IV. Present Value Of An Annuity : In this case we calculate the present value of stream of cash inflows. The present value of an annuity is actually the sum of the present value of all the inflows of this annuity.

The formula for calculating the present value of an annuity is – PVA n = PVA n = A {[(1+r) n – 1] / r (1+r) n } where – PVA n = present value of an annuity for n years A = constant periodic flow r = discount rate The factor - [(1+r) n – 1] / r (1+r) n is referred as present value interest factor for an annuity (PVIFA r,n ).

Present Value of a Growing Annuity : A cash flow that grows at a rate for a specified period of time is a growing annuity. The formula for present value of a growing annuity is – PVGA = A (1+g) The above formula can be used when the growth rate is either less than or more than discount rate (g r). However it does not work when growth rate and discount rate are equal. In this case, the present value is simply equal to “n A”.

Annuities Due : When cash flow occur at the beginning of each period, such an annuity is called as annuity due. Since the cash flows of an annuity due occur one period earlier in comparison to the cash flows of an ordinary annuity, the formula for annuity due will be – Annuity due value = Ordinary annuity value × (1+r) Present Value of a Perpetuity : A perpetuity is an annuity of infinite duration. It may be expressed as – P ∞ = A × PVIFA r, ∞ where P ∞ = present value of a perpetuity A = constant annual payment

and PVIFA r,∞ = present value interest factor for a perpetuity and – PVIFA r, ∞ = ∞ Thus it means that the present value interest factor of a perpetuity is simply 1 divided by the interest rate expressed in decimal form. Intra-Year Compounding And Discounting : Here we will discuss the case where compounding / discounting is done more frequently, i.e., more than once a year (like semi-annually, quarterly, monthly etc)

The general formula for the future value of a single cash flow after n years when compounding is done m times a year – FV n = PV If compounding is done more than once a year, effective interest rate will be different than stated interest rate. The general relationship between the effective interest rate and the stated annual interest rate is as follows – Effective interest rate = where m is the frequency of compounding per year.

Sometimes cash flows have to be discounted more frequently than once a year. The general formula for calculating the present value in the case of shorter discounting period is – PV = FV n where – PV = present value FVn = cash flow after n years m = number of times per year discounting is done r=annual discount rate When compounding becomes continuous, the effective interest is expressed as – e r – 1, where e = base of natural logarithm and r = stated interest rate.