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I'm reading Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 and have resolved my difficulty with one of his questions (question 25.2 on page 231). (Thanks a lot, Calvin Lin!!) He asks:

Find the force of interest at which $\overline{s}_{\begin{array}{r|}\hline\!20\end{array}}=3\overline{s}_{\begin{array}{r|}\hline\!10\end{array}}$.

(Notation and terminology: $\overline{s}_{\begin{array}{r|}\hline\!n\end{array}}$ is the accumulated value at the end of the annuity of an annuity of 1 unit of currency per year for $n$ years, paid continuously; the force of interest, when (as here) constant, is the number $\delta$ such that $e^\delta=1+i$ for an interest rate of $i$.)

I've done as follows:

We have $$e^{20\delta}\int_0^{20}\frac{dt}{e^{\delta t}}=3e^{10\delta}\int_0^{10}\frac{dt}{e^{\delta t}}$$ so that $$1-e^{20\delta}=3(1-e^{10\delta})$$ and letting $x=e^{10\delta}$ we obtain $x^2-3x+2=0, x=2.$

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Can you explain why you switched from 10 to 30? – Calvin Lin Jan 7 '13 at 18:21

(Assuming you mean 30 and not 10)

Note that $x=0$ means $\delta$ is undefined (or negative $\infty$). It's a common error of $\log 1 = 0 $ but $\log 0 \neq 1$.

An answer of $x=0$ makes sense mathematically, since it means that in the second year onwards, you have $0 in the bank! This should almost always be a solution in such types of problems if you think about it this way.

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