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The present value of $2x$ paid at the end of $k$ years and the present value of $x$ paid at the end of $2k$ years sum up to $2x$. Show that the annual rate of interest is $(\frac{\sqrt{3}+1}{2})-1$

I tried summing the present values of each case.

Let $i$ be the annual effective rate of interest.

$$\begin{align} 2x(1+i)^k +x(1+i)^{2k} & =2x\\ 2(1+i)^k+(i+i)^{2k} & =2 \\ \end{align}$$

Let $(1+i)^k=a$

$2a+a^2-2=0$

Using the quadratic formulae, I get $a=-1+\sqrt{3}$

Then I get $i=(-1+\sqrt{3})^{\frac{1}{k}}-1$.

But that is not the answer.

Tosh
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1 Answers1

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There is a typo in the solution. First of all the present value of $2x$ paid in $k$ years is $\frac{2x}{(1+i)^k}$. And the present value of $x$ paid in $2k$ years is $\frac{x}{(1+i)^{2k}}$. Therefore the equation is

$\frac{2x}{(1+i)^k}+\frac{x}{(1+i)^{2k}}=2x$

Let $\frac{1}{(1+i)^k}=z$

The equation becomes $2z+z^2=2$.

You will see that the solution is $i=\left(\frac{\sqrt 3+1}{2} \right)^{1/k}-1$.

callculus42
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