how to find delta when the force of interest is unknown

Question

An investment of 1000 accumulates to 1360.86 at the end of five years. If the force of interest is δ during the first year and 1.5δ in each subsequent year. How to find δ?

Answer 1

You missed the integral at the exponent.

$$1000\cdot e^{\int_0^1 \delta \ ds +4\cdot \int_0^1 1.5\cdot \delta \ ds}=1360.86$$

Dividing the equation by 1000 the equation becomes

$$e^{\int_0^1 \delta \ ds +4\cdot \int_0^1 1.5\cdot \delta \ ds}=1.36086$$

$$e^{7\cdot \delta}=1.36086$$ $$7\cdot \delta=\ln(1.36086)$$ $$\delta=\frac{1}{7}\cdot \ln(1.36086)\approx 0.04401669$$

It seems that you haven´t omitted the integrals at your calculation if I compare our results. So I can confirm your result. Can you now find the equivalent annual effective interest rate in the first year?

Answer 2

You need to solve the equation

$$1000 (1+\delta)(1+1.5\delta)^4=1360.80$$

(I get $4.54225\%$)