I am trying to solve the following problem which I am having a bit of trouble with.
Olga buys a 5-yr increasing annuity for $X$. Olga will receive $2$ at the end of the first month, $4$ at the end of the second month, and for each month thereafter the payment increases by $2$. The nominal interest rate is $9\%$ convertible quarterly. Calculate $X$.
The following is what I have tried which could have a problem with.
A), I tried to calculate the effective interest rate per conversion period, which is per 3 months. So,
$$1+i = 1+\frac{0.9}{4} = 1.0225$$
B), Although Olga receives $2$ per month plus the $2$ increase every month, her money is convertible only quarterly, so she receives $2+4+6$ the first quarter, $8+10+12$ the second quarter, etc… so that we can see that her money that she receives is
$$12+18t \quad (t \ge 0)$$
for $t$ in number of quarters.
This means that the money she receives consists of annuity-present with each payments $2$ and the other part that is an increasing annuity-present with each payment increasing by $18$. This keeps going for $5$ years which is $20$ quarters… so
$$X=12a_{\overline {20} \rceil .0225}+18(Ia)_{\overline {20} \rceil .0225}$$
C), The calculation would be
$$\begin{align}
X &=12a_{\overline {20} \rceil .0225}+18(Ia)_{\overline {20} \rceil .0225} \\
&= 12(\frac{1-1.0225^{-20}}{.0225}) + 18\frac{(1.0225(\frac{1-1.0225^{-20}}{.0225})-20(1.0225)^{-20})}{.0225} \\
& \approx 191.56 + 2805.25 = 2996.84
\end{align}$$
But the answer is supposedly $2729$. Can I have some advice, please?
Thank you.
Best Answer
I think you do not quite understand what this is asking. We are given a nominal rate compounded quarterly, but our payments occur on a monthly basis so we must convert to an effective monthly interest rate. $j = \left(1+\tfrac{.09}{4}\right)^{1/3}-1 \approx 0.007444$. Note we have $12 \cdot 5 = 60 $ payments.
The present value of all these payments is $X$. Thus;
$$X = 2(Ia)_{\overline{60}|j} \approx 2729$$
Note, you made a fatal error when you summed the face value of the first 3 payments. Those payments occurred at different times and it is not meaningful to sum them unless $i=0$.