# solution verification – Annuity & Perpetuity problem

Suppose Joe has been paying $$600$$ from his monthly salary at the end of every month for the past $$n$$ years. After $$n$$ years of payments, he retires having purchased a perpetuity-due plan that begins right away, which gets him a continuous supply of annual salary payments at the start of each year. If $$i>0$$ is the underlying annual effective interest rate for all growth, and we know that $$(1 + i)^n = 10.894481$$, $$a_{n, i}^{(12)} = 7.568285$$, and $$ddot{a}_{infty, i} = 8.884259$$, what is the yearly pension Joe receives?

My attempt:

Let $$X$$ be the yearly pension payments.

First, notice the fact that $$n$$ is a yearly amount whereas Joe makes monthly payments. Moreover, these monthly payments are at the end of each month so we have an annuity-immediate.

Using the information we are given we can extract actual values:

$$ddot{a}_{infty, i} = frac{1}{d} implies d = frac{1}{8.884259}$$

$$i = frac{1}{1-d} – 1 = frac{1}{1-frac{1}{8.884259}} – 1 = 0.127$$

$$(1 + 0.127)^n = 10.894481 implies n = 20 (years)$$

$$a_{n, i}^{(12)} = a_{n, i} cdot frac{i}{i^{(12)}} = frac{1-v^n}{i} cdot frac{i}{i^{(12)}} = frac{1-v^n}{i^{(12)}} implies i^{(12)} = 0.12$$

, where $$d$$ is the discount rate and $$i^{(12)}$$ is the nominal yearly rate.

Now the future value of the annuity immediate with $$240$$ monthly payments is

$$F = 600 S_{240, frac{i^{(12)}}{12}} = 600frac{(1 + 0.12/12)^240 – 1}{0.12/12} = 593624.68$$

The present value of the perpetuity due is

$$P = Xcdot ddot{a}_{infty, i} = Xcdot 8.884259$$

The future value and present value should be equal

$$8.884259X = 593624.68 implies X = 66817.58$$

Is this correct? I did a lot of research but I am not sure if this is how to approach this problem. Any assistance is much appreciated.