Find the accumulated value at the end of ten years of an annuity in
which payments are made at the beginning of each half-year for five
years. The first payment is 2,000, and each of the other payments is
98% of the previous payment. Interest is credited at 10% convertible
quarterly.
correct answer: 40,042
My work:
I found equivalent semi-annual rate $i^{(2)}$, which gives us the interest rate per half-year $i=0.0506$, and using this the first five years is just a geometrically decreasing annuity-due with conversion period = payment period, which we can use the equation $2000(1+i)\frac{1-\left(\frac{1-k}{1+i}\right)^{10}}{i+k}$ with $n=10$ conversion periods (using $i$) and $k=0.02$ = common ratio of geometric progression. This is equal to the accumulated value at $t=5$, which is 14916.70. Then we can simply compound this up to year 10, which is straightforward. I get the answer 24,442.75, which is incorrect. What am I doing wrong?
Edit: It appears that I calculated the PV of geometric progression at t=0, instead of the accumulated value at t=5 like I thought I was doing
Best Answer
The cash flow looks like this:
$$AV = 2000\left(\left(1+\frac{i^{(4)}}{4}\right)^{\!40} \!\!\!\! + (0.98)\left(1+\frac{i^{(4)}}{4}\right)^{\!38} \!\!\!\! + (0.98)^2 \left(1+\frac{i^{(4)}}{4}\right)^{\!36} \!\!\!\!+ \cdots + (0.98)^9 \left(1 + \frac{i^{(4)}}{4}\right)^{\!22}\right)$$ where $i^{(4)} = 0.10$ is the nominal rate of interest compounded quarterly.
Explanation: the effective rate of interest per quarter period is simply $i^{(4)}/4$. To account for the payments occurring every other compounding period, we just skip those periods. Because payments are made at the beginning of each half-year, the first payment of $2000$ has had the full $10$ years, or $40$ quarters, to accumulate. To ensure that we have $5$ years of semiannual payments, or a total of $10$ payments, we require that the last payment be reduced by $(0.98)^{10 - 1}$, and that $40 - 2(9) = 22$ is the number of periods that the last payment accumulates interest.
Once you see how this is all put together, the meaning should become plainly obvious. This is why I recommend writing out the cash flow. Actuarial notation comes next. We note that we can write the above as
$$\begin{align} AV &= 2000(1+j)^{22} \left( (1 + j)^{18} + (0.98) (1+j)^{16} + \cdots + (0.98)^9 (1+j)^0 \right) \\ &= 2000(0.98)^9 (1+j)^{22} \left( \left(\frac{(1+j)^2}{0.98}\right)^{\!9} + \left(\frac{(1+j)^2}{0.98}\right)^{\!8} + \cdots + 1 \right) \\ &= 2000(0.98)^9 (1+j)^{22} \require{enclose}s_{\enclose{actuarial}{10} j'} \\ &= 2000(0.98)^9 (1+j)^{22} \frac{(1+j')^{10} - 1}{j'}, \end{align}$$ where $j = i^{(4)}/4 = 0.025$ is the effective quarterly interest rate, and $$j' = \frac{(1+j)^2}{0.98} - 1 = \frac{113}{1568} \approx 0.072066$$ is the equivalent semiannual effective rate after adjusting for the geometric decrease in payments. It follows that $$AV \approx 40052.28.$$ The claimed answer $40042$ is inaccurate.
Alternatively, using your approach and converting the rate to a semiannual frequency, we have $j = i^{(2)}/2 = 0.050625$ as you stated, and the cash flow is then written $$AV = 2000 \left((1 + j)^{20} + (0.98)(1 + j)^{19} + \cdots + (0.98)^9(1 + j)^{11}\right) = 2000 (0.98)^9 (1 + j)^{11} \require{enclose}s_{\enclose{actuarial}{10} j'}$$ where now $$j' = \frac{1+j}{0.98} - 1.$$ Either way gives the same result.