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I'm having trouble understanding the solution of this problem.

Find the present value of a ten-year annuity which pays $400$ at the beginning of each quarter for the first 5 years, increasing to $600$ per quarter thereafter. The annual effective rate of interest is $12%$. Answer to the nearest dollar.

My attempt was to find the quarterly rate of interest $j$ which I found to be $.02874$, then find

$400\ddot{a}_{\overline{40|}j} + 600\ddot{a}_{\overline{20|}j} = 15484$

However, the solution says the correct answer is

$600\ddot{a}_{\overline{40|}j} - 200\ddot{a}_{\overline{20|}j} = 11466$,

Can someone tell me why we subtract $200\ddot{a}_{\overline{20|}j}$ and why we started with and why we started with $600\ddot{a}_{\overline{40|}j}$ when the question is $600$ for only $20$ quarters?

user270494
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1 Answers1

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You have a $10$-years annuity-due that pays $400$ for the first $5$ years and $600$ for the second $5$ years. This is equivalent to have a $10$-years annuity that pays $600$ for $10$ years and then subtract the excess of $200$ from the first $5$ years, that is $$ 600\ddot{a}_{\overline{40|}j} - 200\ddot{a}_{\overline{20|}j} = 600\times 24.27-200\times 15.49=14563-3097=11466 $$ where $j=(1+i)^{1/4}-1=2.87\%$ is quarterly interest rate and $i=12\%$.

Another method is consider an annuity-due that pays $400$ for the first $5$ years and then $5$-years annuity-due deferred of $5$ years (or $20$ quarters) that pays $600$, that is $$ 400\ddot{a}_{\overline{20|}j} + 600\,v^{20}\ddot{a}_{\overline{20|}j} = 600\times 15.49+400\times 0.5674\times 15.49=6194+5272=11466 $$ where $v=\frac{1}{1+j}$ and, as you can see, we've found the same result. Infact $$ 400\ddot{a}_{\overline{20|}j} + 600\,v^{20}\ddot{a}_{\overline{20|}j} =400\ddot{a}_{\overline{20|}j} + 600(\ddot{a}_{\overline{40|}j}-\ddot{a}_{\overline{20|}j})=600\ddot{a}_{\overline{40|}j}-200\ddot{a}_{\overline{20|}j} $$ observing that $$ v^m\ddot{a}_{\overline{n|}i}= \ddot{a}_{\overline{n+m|}i}-\ddot{a}_{\overline{m|}i} $$

alexjo
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