I'm having trouble solving this FM problem.
For $3000$, Nick purchases an perpetuity-immediate paying $100$ at the end of each $6$ months period. For the same amount and for the same effective annual rate Paul purchase an annuity-immediate with $80$ quarterly payments that begin at amount $P$ and decreases by $1.1$ each quarter. Find $P$.
I honestly have no clue where to start.
So far I have:
Nick:
$3000 = 200*/i$
$i = 0.067$
For Paul:
Paul's effective rate is $j = (1.067)^{(1/4)} - 1 = 0.0163$
Is it $3000 = P*a_{80|0.0163}-(1.1*79)$
Write out the cash flows.
Let $i$ be the effective annual interest rate; then Nick's cash flow is $$3000 = 100(v^{1/2} + v + v^{3/2} + v^2 + \cdots) = 200 a_{\overline{\infty}\rceil i}^{(2)} = 100 \cdot \frac{1}{v^{-1/2}-1}$$ where $v = 1/(1+i)$ is the effective annual present value discount factor. Therefore, $v = 900/961 \approx 0.936524$ and $i = 61/900 \approx 0.0677778$.
For Paul, the present value of his cash flow is $$\begin{align*}3000 &= Pv^{1/4} + (P-1.1)v^{2/4} + (P-2.2)v^{3/4} + \cdots + (P-86.9)v^{20} \\ &= (P+1.1)a_{\overline{80}\rceil j} - 1.1(Ia)_{\overline{80} \rceil j}, \end{align*}$$ where $$j = (1+i)^{1/4} - 1 \approx 0.01653$$ is the effective quarterly interest rate. Since $$a_{\overline{n}\rceil i} = \frac{1 - v^n}{i}, \quad (Ia)_{\overline{n}\rceil i} = \frac{\ddot a_{\overline{n}\rceil i} - nv^n}{i},$$ we obtain $$\begin{align*} P &= \frac{3000 + 1.1 (Ia)_{\overline{80}\rceil j}}{a_{\overline{80}\rceil j}} - 1.1 \\ &\approx \frac{3000 + (1.1)(1414.29)}{44.1989} - 1.1 \\ &\approx 101.973. \end{align*}$$