Suppose that an annuity will provide for 20 annual payments of 1440 dollars, with the first payment coming 7 years from now. If the nominal rate of interest is 8.2 percent convertible monthly, what is the present value of the annuity?
My attempt:
$1440\left(\frac{\left(1-v^{20}\right)}{j}\right)v^7$ where $j=0.082/12$ and $v=1/1+j$
I have tried many different methods but can't seem to get the right answer
The cash flow is $$PV = 1440 v^7 + 1440 v^8 + \cdots + 1440 v^{26},$$ where $v = 1/(1+i)$ is the effective annual present value discount factor. We must solve for $i$, the effective annual interest rate: $$1 + i = \left(1 + \frac{i^{(12)}}{12}\right)^{12},$$ where $i^{(12)} = 0.082$ is the nominal monthly rate of interest. Hence $i \approx 0.0851531$, and $$\begin{align} \require{enclose}PV &= 1440 v^6 (v + v^2 + \cdots + v^{20}) \\ &= 1440 v^6 a_{\enclose{actuarial}{20} i} \\ &= 1440 v^6 \frac{1 - v^{20}}{i} \\ &\approx 1440 (1.0851531)^{-6} \frac{1 - (1.0851531)^{-20}}{0.0851531} \\ &\approx 8336.36. \end{align}$$
You made a few errors in your solution. First, your discounting schedule is incorrect. You used the formula for an annuity-immediate, which has its first payment occurring one period after the initiation of the annuity. So either you should have written $$v^6 a_{\enclose{actuarial}{20}}$$ with a deferral of $6$ years plus the one year from the annuity-immediate, or $$v^7 \ddot a_{\enclose{actuarial}{20}}$$ which defers $7$ years and then uses an annuity-due.
Second, you did not compute the correct effective annual interest rate. This is necessary because although the interest is specified nominally as a monthly rate, what matters for discounting is the equivalent annual rate.
