Time Value of Money 51
In general,
Effective rate = [1 + (r/m)]m – 1 (5)
where r = nominal annual rate and
m= number of times compounding is done in a year.
Try calculating the effective rate in the above example when compounding is done daily, quarterly, and
semi-annually.
Let us extend the idea a little bit. Following data is available for a fixed deposit scheme:
Investment = Rs 5000
Rate of interest = 13 percent
Compounding = Semi-annual
Term = 5 years
The future value would be:
FV = A[1 + (r/m)]m × n (6)
where n = number of years
FV = 5000 [1 + (0.13/2)]2 × 5 = Rs 9385
Had the compounding been annual,
FV = 5000 (1 + 0.13)^5 = Rs 9212
The difference of Rs (9385 – 9212) = Rs 173 is due to semi-annual compounding.
Perpetuities
Some investments make a regular income for ever. Suppose Company-X declares a dividend of Rs 2 per
share. It is expected to remain at this level for ever (the firm is a going concern). If the appropriate discount
rate is 13 percent, the present value of this perpetual stream = (Rs 2/0.13) = Rs 15.40.
In general,
PV of a perpetuity = Per period amount /Discount rate (7)
PV = A/K
If an amount is to grow at a constant rate for ever it is called a growing perpetuity. The present value of a
growing perpetuity can be estimated using the following formula.
PV =
gk
A
–
where k is the discount rate and g is the growth rate.