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I am working on the following question (on SOA FM exam)

The present value of a perpetuity paying 1 every two years with first payment due immediately is 7.21 at an annual effective rate of $i$. Another perpetuity paying $R$ every three years with the first payment due at the beginning of year two has the same present value at an annual effective rate of $i + 0.01$. Calculate $R$.

To solve this problem I first found $i$, specifically, if $v=1/(1+i)$, then

\begin{equation} 7.21=1+v^2+v^4+\cdots=\dfrac{1}{1-v^2} \end{equation}

Solving for $v$, I get that $i = .0775$. According to the solutions, this is the correct value.

To find $R$, I know the annual effective interest rate is $i+.01=.08775$. Setting, $w=1/(1.08775)$ I set up the following equation of value:

\begin{equation} 7.21=Rv^2+Rv^5+Rv^8+\cdots=Rw^2(1+w^3+w^6+\cdots)=R\cdot\dfrac{w^2}{1-w^3} \end{equation}

Thus, $7.21=R\cdot\dfrac{w^2}{1-w^3}\implies 7.21=3.8R \implies R=1.89$. However, the actual answer is $1.74$.

Where did my work go wrong?

Mike
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  • In your calculations, should $i + .01$ be equal to .0875 instead of .08775? This has a minor effect, of course. – svavil May 04 '25 at 09:12

1 Answers1

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In the second perpetuity, since the first payment is due at the beginning of year two this is equivalent to the end of year 1, so your sum should be

$$\begin{equation} 7.21=Rw+Rw^4+Rw^7+\cdots=Rw(1+w^3+w^6+\cdots)=R\cdot\dfrac{w}{1-w^3} \end{equation}$$

That should actually fix things.

DMcMor
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