__Time Value of Money__
Money has time value. A rupee is less valuable in the future than it is today. Time value of money could be studied under the following heads:

(a) Future value of a single cash flow

(b) Future value of an annuity

(c) Present value of a single cash flow

(d) Present value of an annuity

__a) Future Value of a Single Cash Flow__

For a given present value (PV) of money, future value of money (FV) after a period ‘t’ for which compounding is done at an interest rate of ‘r’, is given by the equation

FV = PV (1+r)t

This assumes that compounding is done at discrete intervals. However, in case of continuous compounding, the future value is determined using the formula

FV = PV * ert

where the compounding factor is calculated by taking natural logarithm (log to base). e=2.71828

*: Calculate the value 5 years hence of a deposit of Rs.1,000 made today if the interest rate is 10%.*

__Example 1__
Solution: By discrete compounding

FV = 1,000 * (1+0.10)5 = 1,000* (1.1)5 = 1,000* 1.61051 = Rs.1,610.51

By continuous compounding:

FV = 1,000 * e(0.10 * 5) = 1,000* 1.648721 = Rs.1,648.72

__Find the value of Rs. 50,000 deposited for a period of 3 years at the end of the period when the interest is 10% and continuous compounding is done.__

*Example 2:*
Solution: Future Value = 50,000* e^(0.01*10*3) = Rs. 67,493.00

The future value (FV) of the present sum (PV) after a period ‘t’ for which compounding is done ‘m’ times a year at an interest rate of ‘r’, is given by the equation

FV = PV (1+(r/m))^mt

*: How much does a deposit of Rs. 5,000 grow to at the end of 3 years, if the nominal rate of interest is 10% and compounding is done quarterly?*

__Example 3__
Solution: Future value = 5,000* ((1 + 0.10/4)^(4*3)) = Rs. 6,724.45

__b) Future Value of an Annuity__

The future value (FV) of a uniform cash flow (CF) made at the end of each period till the time of maturity ‘t’ for which compounding is done at the rate ‘r’ is given by

FV = CF*(1+r)t-1 + CF*(1+r)t-2 + ... + CF*(1+r)1+CF

= CF [{(1+r)t - 1} / r]

__: Suppose, you deposit Rs. 1,000 annually in a bank for 5 years and your deposits earn a compound interest rate of 10 per cent, what will be value of this series of deposits (an annuity) at the end of 5 years? Assume that each deposit occurs at the end of the year.__

*Example 4*
Solution: Future value of this annuity is:

= Rs.1000*(1.10)4 + Rs.1000*(1.10)3 + Rs.1000*(1.10)2 + Rs.1000*(1.10) + Rs.1000

= Rs.1000*(1.4641) + Rs.1000*(1.3310)+Rs.1000*(1.2100) + Rs.1000*(1.10) + Rs.1000

= Rs. 6,105.00

In case of continuous compounding, the future value of annuity is determined using the formula FV = CF * (ert -1)/r

__c) Present Value of a Single Cash Flow__

Present value of (PV) of the future sum (FV) to be received after a period ‘t’ for which discounting is done at an interest rate of ‘r’, is given by the equation

In case of discrete discounting: PV = FV / (1+r)t

__: What is the present value of Rs.1,000 payable 3 years hence, if the interest rate is 12 % p.a.__

*Example 5*
Solution: PV = 1000 / (1.12)3 i.e. = Rs.711.78

In case of continuous discounting: PV = FV * e-rt

__: What is the present value of Rs. 50,000 receivable after 3 years at a discount rate of 10% under continuous discounting?__

*Example 6*
Solution: Present Value = 50,000/(exp^(0.01*10*3)) = Rs. 37,041.00

*d) Present Value of an Annuity*
The present value of annuity is the sum of the present values of all the cash inflows/outflows. Present value of an annuity (in case of discrete discounting)

PV = FV [{(1+r)t - 1 }/ {r * (1+r)t}]

Present value of an annuity (in case of continuous discounting) PVa = FVa * (1-e-rt)/r

## No comments:

## Post a Comment