Question

A perpetuity-immediate pays 100 per year. Immediately after the fifth payment, the perpetuity is exchanged for a 25-year annuity-immediate that will pay X at the end of the first year. Each subsequent annual payment will be 5% greater than the preceding payment. The annual effective rate of interest is 8%. Calculate X.

Answer 1

Answer:

X = 74.18

Step-by-step explanation:

Given that:

A perpetuity-immediate pays 100 per year and the annual effective rate of interest is 8%.

The PV of Perpetuity = $\frac{100}{0.8}$ = 1250

As this will be equivalent to the PV of the geometrically increase in annuity-immediate. Thus, the npv at time point S will be:

$1250 = X[v+(1.05)v^2+(1.05^2)v^3+...(1.025^{24})v^{25}]$

where

i = annual payment = 5% = 0.05

v = 1/1+ annual effective rate

= 1/1+8%

= 1/1+0.08

= 1/1.08

Our equation can be re-written as:

$PV_5=\frac{X}{1.08}(\frac{1-(\frac{1.05}{1.08})^{25} }{1- \frac{1.05}{1.08} } )$

$PV_5 = 16.35 X$

To determine the value of X, we equate the above value with 1250.

So, 16.35X = 1250

X = $\frac{1250}{16.35}$

X = 74.18