Question

1. Find the PV of an annuity consisting of 60 quarterly payments of Rs. 200, the first being made at the end of 4 years 9 months. Assume that the rate of interest is 12.3% p.a. compounded monthly.

Answer 1

We have to calculate the PV of annuity that starts at time t = 4 years 9 months and then discount this back to t=0 to get the value as of today.

$PV of annuity = PMT \times\frac{1 − (1 + i)^{−n}}{i}$

PV=present value

PMT=amount in each annuity payment

I=interest

n=number of payment

Step 1

Effective Annual Rats of the rate of interest:

12.3% compounded monthly

$= (1+\frac{r}{m})^{m}-1$

$= (1+\frac{0.123}{12})^{12}-1$

$=0.130176595 = 13.02$ %

Value of annuity at 4 years 9 months from today:

$PV = 200\times[\frac{1−(1+\frac{0.1302}{4})^{-60}}{\frac{0.1302}{4}}]$

$PV=5245.91901$

Step 2

We need to discount this value of $5,245.92 to today's value,

4 years and 9 months is $4\times4 +3 = 19$ quarters

$PV0=\frac{ PV1}{(1+r)^{19}}$

$=\frac{ 5,245.92 }{(1+\frac{0..1302}{4})^{19}}$

$=2,854.68$

The present value of the annuity today is $2,854.68