Increasing Annuities [closed]

Question

Olga buys a 5-year increasing annuity for X. Olga will receive 2 at the end of the first month, 4 at the end of the second month, and for each month thereafter the payment increases by 2. The nominal interest rate is 9% convertible quarterly. Calculate X.

This is from the Study Manual for Exam FM/Exam 2 Eleventh Edition Section 4h and 4i number 2. This whole section has been very confusing for me and I don't quite understand the reasoning. The provided answer is x=2730. If anyone could help me out I would really appreciate it!

Answer 1

Interest rate conversions can be confusing, so an exact answer depends on the convention rate being used. However, I can get you close.

Given a general solution to a series summation:

$$\sum_{n=1}^{N} \frac{xn}{(1+r)^n} = \frac{(1 + r - (1 + r)^{-N} (1 + r + N r)) x}{r^2} $$

We can rewrite the value present of annuity which pays 2n units per period as:

$$V_A = \sum_{n=1}^{n=12*5}\frac{2n}{(1+r)^n}$$

where the effective interest rate per period can be converted as such:

$r = (1+i/4)^{4/12} -1 = .07\bar{4} $

Thus:

$$V_A = \sum_{n=1}^{n=60}\frac{2n}{(1.07\bar{4} )^n} = 2729.21$$