Independent solution

How to solve this Annuities question

Setup

Setup

Deposit 2(t + 1) at each year-end t + 1 and accumulate every deposit to time 60.

F60=2∑t=059s60−t‾∣iF_{60}=2\sum_{t=0}^{59}s_{\overline{60-t}|i}

Model

Model

At time 60, value the five withdrawals beginning one year later with 5% annual growth.

PV60=X(v+1.05v2+1.052v3+1.053v4+1.054v5)PV_{60}=X\left(v+1.05v^2+1.05^2v^3+1.05^3v^4+1.05^4v^5\right)

Compute

Compute

Equating these two quantities gives exactly statement III; the other expressions use an annuity-due timing on one side.

2∑t=059s60−t‾∣i=X∑k=151.05k−1vk2\sum_{t=0}^{59}s_{\overline{60-t}|i}=X\sum_{k=1}^{5}1.05^{k-1}v^k

Answer

Answer

Only statement III is correct, so choice C is the answer.

III only(C)\boxed{\text{III only}\quad\text{(C)}}