Pricing an Increasing Annuity with Quarterly-Convertible Interest

Summary

The note values a five-year monthly annuity whose payment increases by two units each month, starting with a payment of two at the first month-end. It expresses the present value as the sum of each payment discounted over its payment period, using an effective monthly interest rate converted from a nominal rate convertible quarterly. A closed-form formula for the discounted sum of payments growing linearly with the month index is also supplied.

The calculation gives a present value of about 2,729.21, close to the study manual’s stated value of 2,730. The answer cautions that interest-rate conventions can affect the exact result. The excerpt does not discuss alternative payment timing or derive the conversion convention in detail, so those assumptions should be checked when applying the method to a differently specified annuity.

Key ideas

• Represent each month’s payment as twice its payment number over the 60-month term.
• Convert the nominal quarterly-convertible rate to an effective monthly rate before discounting.
• Discount each payment to the valuation date and sum the resulting present values.
• A closed-form expression can evaluate a linearly increasing discounted payment stream.
• Payment timing and the chosen interest convention affect the valuation.

Tags

# Increasing Annuities


# Increasing Annuities












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 by David Addison (score 2, accepted)

https://quant.stackexchange.com/a/38194

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:

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.