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A 20 year annuity certain provides payments of 200 at time 1 year, 180 at time 2 years , 160 at time 3 years, and so on until the payments have been reduced to $ 60. Payments then continue at 60 per year until the 20th payment has been made. The annual effective interest rate is 4%. Determine the present value of this annuity.

So from what I can gather this is a decreasing annuity. From time t=0 to time t=8, where the decreasing annuity hits the 60 mark. Giving us the first part of our equation 20(Da) angle-8,i=4%. which would give us present value of the first 8 years. However I am not certain as to how to handled the remaining 12 years.

would we discount 8 years and take the present value of the remaining 12 years at 60 per year. i.e. V^8* (A) angle-20, i=4%?

or would it be easier to just take the future 60 for 12 years? i.e. (s) angle 11, i=4%

user294448
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1 Answers1

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Write out the cash flow: $$PV = 200v + 180v^2 + 160v^3 + \cdots + 60v^8 + 60v^9 + \cdots + 60v^{20},$$ where $v = (1+i)^{-1}$ is the present value discount factor and $i = 0.04$ is the effective annual interest rate.

Now we can see that $$\begin{align*} PV &= 60(v + v^2 + \cdots + v^{20}) + 140v + 120v^2 + \cdots + 20v^7 \\ &= 60a_{\overline{20}\rceil i} + 20(Da)_{\overline{7}\rceil i} \\ &= 60 \frac{1 - v^{20}}{i} + 20 \frac{7 - a_{\overline{7}\rceil i}}{i}, \end{align*}$$ and the rest is straightforward.

I would also strongly advise you to compute the decreasing portion term by term, as a way to check your answer, since the number of such terms is relatively small: $$PV = 20(10v + 9v^2 + \cdots + 4v^7) + 60v^8(1 + v + v^2 + \cdots + v^{12}),$$ and the first part, with only $7$ terms, should not take too long to compute by brute force.

heropup
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