So I have this homework question for actuarial mathematics:
Two perpetuities have the same annual effective interest rate. Perpetuity A pays \$4 at the end of each year for the first 20 years and then $ 2 at the end of each year thereafter. Perpetuity B is a perpetuity due which has a level annual payment of \$ 3. At time t=0, the present value of Perpetuity A is equal to that of Perpetuity B. What is the effective annual interest rate, i?
So what I've done is I calculated is let the present value of A equal to $$\sum_{n=1}^{20}(4/(1+i)^n +\sum_{n=20}^{\infty}(2/(1+i))^n$$ And the present value of B equal to $$\sum_{n=1}^{\infty}(3/(1+i)^n)$$ Letting these equal, I got the interest rate $i$ to be equal to 3.35% which is wrong. Can anyone tell me where I've gone wrong? Thanks.
There are two issues with your present value formulas.
For the first perpetuity, the second summation should start at $n=21$. That is because the payments of 2 don't start until the end of the 21st year.
For the second perpetuity, the summation should start at $n=0$. That is because it is a perpetuity due, meaning the first payment occurs immediately at time 0.
So the correct equation is: $$\sum_{n=1}^{20}\frac{4}{(1+i)^n} + \sum_{n=21}^\infty \frac{2}{(1+i)^n} = \sum_{n=0}^\infty \frac{3}{(1+i)^n}.$$ Solving this, I find two real solutions $i \approx 4.23\%$ and $i \approx 33.11\%$.