How to solve Interest Theory — Time Value of Money — accumulation and present value under a time-varying force of interest, and the equivalent level force questions on Exam FM?

Question

A deposit of one is placed in each of two accounts. Account $X$ has force of interest $\delta_t=t^2/k$, while Account $Y$ earns nominal discount $8.0\%$ convertible $2$ times per year. Their values agree after $4$ years. Calculate $k$.

Accepted Answer

A variable force produces accumulation $\exp(\int_0^{4}t^2/k\,dt)=\exp(64/(3k))$. Account $Y$ accumulates by $(1-0.0800/2)^{-8}$, whose logarithm is $0.32657596$. Equating logarithms gives $64/(3k)=0.32657596$, so $k=64/[3(0.32657596)]=65.3243$, which matches the correct option.

Therefore, the result is $\frac{a^{3}}{3 \log{\left({0.96}^{- 2 a} \right)}}\approx 65.3243$, so the correct answer is option (D).

How to Solve Interest Theory — Time Value of Money — accumulation and present value under a time-varying force of interest, and the equivalent level force Questions on Exam FM

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