Independent solution

How to solve this Annuities and Perpetuities question

Setup

Setup

The estate principal cancels because both beneficiaries receive the same annual interest amount at different dates.

PVJ=PaX‾∣ 0.05,PVK=PvX/0.05PV_J=P a_{\overline X|\,0.05},\qquad PV_K=Pv^X/0.05

Model

Model

Value the first beneficiary's finite annuity and the second beneficiary's deferred perpetuity at the estate date.

vX1−vX=1.59\frac{v^X}{1-v^X}=1.59

Compute

Compute

Taking their ratio isolates the discount factor raised to the unknown number of years; logarithms then recover X.

vX=1.592.59,X=ln⁡(1.59/2.59)ln⁡(1/1.05)=10.00v^X=\frac{1.59}{2.59},\qquad X=\frac{\ln(1.59/2.59)}{\ln(1/1.05)}=10.00

Answer

Answer

The switch occurs after ten annual payments to the first beneficiary, corresponding to choice E.

X=10(E)\boxed{X=10\quad\text{(E)}}

Calculator reproduction

BA II Plus keystrokes

Check END/BGN, period, sign, TVM, and cash-flow setup

  1. 1.59 ÷ 2.59 =; LN; 1 ÷ 1.05 =; LN; ÷10.00The numerator and denominator logarithms are both negative, so the quotient is positive.