Independent solution

How to solve this Arithmetic Increasing Perpetuity question

Setup

Setup

Let P be the payment on January 1, 2000. Payments at times t = 0, 1, 2 and so on are P plus 1000t.

1,000,000=Pa¨∞+1,000∑t=0∞tvt1{,}000{,}000=P\ddot a_{\infty}+1{,}000\sum_{t=0}^{\infty}tv^t

Model

Model

At 10%, the due level factor is 11 and the increasing component has value 110000.

a¨∞=1.100.10=11\ddot a_{\infty}=\frac{1.10}{0.10}=11
∑t=0∞tvt=1.100.102=110\sum_{t=0}^{\infty}tv^t=\frac{1.10}{0.10^2}=110

Compute

Compute

Thus P = 80909.09. Immediately after the time-20 payment, the next payment is P plus 21000.

P=80,909.09P=80{,}909.09
V20+=P+21,0000.10+1,0000.102=1,119,090.91V_{20+}=\frac{P+21{,}000}{0.10}+\frac{1{,}000}{0.10^2}=1{,}119{,}090.91

Answer

Answer

The prospective value rounds to 1119091, choice B.

V20+≈1,119,091(B)\boxed{V_{20+}\approx1{,}119{,}091\quad\text{(B)}}