Independent solution

How to solve this Annuities and Perpetuities question

Setup

Setup

Write both continuous present-value factors in terms of the same force of interest.

1−e−20δδ=1.51−e−10δδ\frac{1-e^{-20\delta}}{\delta}=1.5\frac{1-e^{-10\delta}}{\delta}

Model

Model

After cancelling the common force denominator, substituting x for the ten-year discount factor reduces the ratio to a quadratic.

x=e−10δ:x2−1.5x+0.5=0,x=0.5x=e^{-10\delta}:\quad x^2-1.5x+0.5=0,\quad x=0.5

Compute

Compute

The nontrivial root determines the force; the seven-year continuous accumulation factor then evaluates to 9.01.

δ=0.069315,s‾7‾∣=e7δ−1δ=9.01\delta=0.069315,\qquad \overline s_{\overline7|}=\frac{e^{7\delta}-1}{\delta}=9.01

Answer

Answer

The accumulated value of the seven-year unit stream is 9.01, choice D.

9.01(D)\boxed{9.01\quad\text{(D)}}