A $20$-year annuity-due makes payments of $100$ each year for the first $10$ years and then each subsequent payment decreases by $5$ for the next $10$ years. The effective annual interest rate is $9\%$. Find the accumulated amount of the annuity at the time of (and including) the last payment.
Hi all! i'm doing practice questions for the FM exam for actuarial science and the solution to this question confused me.
We're saying:
- $n = 20$ years; $i = 0.09$
- First annuity: level payments of $100$ each for $10$ years at $0.09$ interest which is $\ddot s_{\overline{n}\rceil i}$. Then we multiply it by $(1+i)^{10}$ because we have to accumulate this for up to $20$ years.
- Second annuity is decreasing so I split it between:
1) an annuity of level payments of $45$ each between time $11$ and $20$.
So it's $45 \ddot s_{\overline{n}\rceil i}$ for $n = 10$.
2) Then I add a decreasing annuity with factor $5$ for $10$ years; $5(D\ddot s)_{\overline{n}\rceil i}$.
However, the answer given was like mine except no $\ddot s_{\overline{n}\rceil i}$. I understand the questions at the end says 'at the time of and including last payment, which is the definition of $\ddot s_{\overline n \rceil}$'. But what happens in this case when it also says it's an annuity due?
The answer given was:
$$AV = 100s_{\overline{10}\rceil 0.09} (1.09)^{10} + 45 s_{\overline{10}\rceil 0.09} + 5(Ds)_{\overline{10}\rceil 0.09} = 4751.5512.$$
Best Answer
Just write out the cash flow.
$$\begin{align*} AV &= 100(1+i)^{19} + 100(1+i)^{18} + \cdots + 100(1+i)^{10} \\ &\hphantom{=} + 95(1+i)^9 + 90(1+i)^8 + \cdots + 55(1+i)^1 + 50. \end{align*}$$
Note there are $19-10+1 = 10$ payments of $100$, and the remaining $9 - 0 + 1 = 10$ payments follow the arithmetic sequence $95, 90, \ldots, 50$.
Now you can see there are multiple ways of writing out this cash flow in actuarial notation. You could do it this way:
$$\begin{align*} AV &= 100(1+i)^{10}\left((1+i)^9 + \cdots + 1 \right) \\ &\hphantom{=} + 45(1+i)^9 + 45(1+i)^8 + \cdots + 45 \\ &\hphantom{=} + 50(1+i)^9 + 45(1+i)^8 + \cdots + 5 \\ &= 100 (1+i)^{10} s_{\overline{10}\rceil i} + 45 s_{\overline{10}\rceil i} + 5(Ds)_{\overline{10}\rceil i} \end{align*}$$
which is the textbook solution. What they did was take the first $10$ payments as an annuity-immediate of $10$ years on $100$, and accumulated it for an additional $10$ years; then split the decreasing annuity into a level annuity-immediate of $45$, and a decreasing annuity of $5$.
But you don't have to do it this way. You can do it a number of other ways, one of which is to write
$$\begin{align*} AV &= 100\left((1+i)^{19} + \cdots + 1\right) \\ &\hphantom{=} - \left(5(1+i)^9 + 10(1+i)^8 + \cdots + 50\right) \\ &\hphantom{=} = 100s_{\overline{20}\rceil i} - 5(Is)_{\overline{10}\rceil i}. \end{align*}$$
In both cases, the result is the same.
I have given this advice to numerous students of actuarial science: when in doubt, write out the cash flow. It takes practice and experience to be able to directly set up the equation of value in terms of actuarial notation, and when one is still learning, skipping the cash flow step is not only more error-prone, it may actually be more time-consuming than just writing it out because you are wasting time trying to come up with how to write it compactly.