Question

Course: Theory of Interest

A perpetuity-immediate has an initial payment of $5,000 at the end of the first year. Payments increase by $500 each year. A level payment annuity-due provides 25 payments of $X per year. If the interest rate is 6.5% what is the value of X if these two annuities are of equal value to an investor?

Answer: $15,031.19

Answer 1

The formula to calculate present value of a perpetuity that is growing by a fixed amount every year is:

$PV=\frac{C}{r}+\frac{I}{r^{2}}$

Where,

• C is the first payment
• I is the constant growth amount
• r is the discount rate

Substituting the values, we get

$PV=\frac{5,000}{0.065}+\frac{500}{0.065^{2}}=195,266.27$

Now, the formula to calculate present value of an annuity due is:

$PV=C+C\left [ \frac{1-(1+r)^{-(n-1)}}{r} \right ]$

Substituting the given values, we get

$195,266.27=C+C\left [ \frac{1-(1+0.065)^{-(25-1)}}{0.065} \right ]$

$195,266.27=12.99C$

or,$C=15,031.19$