I am trying to solve the following increasing (decreasing?) annuity problem from exam FM.
A perpetuity costs $77.1$ and makes annual payments at the end of the year. The perpetuity pays $1$ at the end of year $2$, $2$ at the end of year $3$, ..., $n$ at the end of year $n+1$. After this year, the payments remain constant at $n$. The effective annual interest rate is $10.5\%$. Calculate $n$.
This looks like such a simple question and yet I need assistance. It would be great if someone could show me steps to solve this question explaining in detail, but the following is what I've tried.
1), I am a bit puzzled by the wording, "pay 1 at the end of year 2". These wordings really throw me off and I really want to learn a good way to know if...
this means a), pay 1 at the end of the first year b), pay 1 at the beginning of the second year c), pay 1 at the end of the second year d), pay 1 at the beginning of the third year ?
I am assuming that it is the end of the second year.
2), I think that one can split this into
a), an increasing annuity immediate looking at $t=1$ for $n$ years plus an $n$ year deferred perpetuity immediate.
$$77.1=(Ia)_{\overline{n}\rceil 10.5\%} + (1.105)^{-n}(a_{\overline{\infty}\rceil 10.5\%})$$
or
b), an perpetuity immediate that starts right away at $t=1$ (is this a perpetuity due?) minus $n$ years of decreasing annuity.
$$77.1=a_{\overline{\infty}\rceil 10.5\%}-(Da)_{\overline{n}\rceil 10.5\%}$$
Since this is a problem from Exam FM, I am assuming that the answer is solvable by hand and honestly, I have so much uncertainties in this problem that I have no confidence that I am doing anything right. (not to mention I don't get the right answer)
I would appreciate any help.
I always write out the cash flow explicitly, then express it in terms of actuarial notation, for complex situations with non-level payments.
We have $$77.1 = (v^2 + 2v^3 + 3v^4 + \cdots + nv^{n+1}) + (nv^{n+2} + nv^{n+3} + \cdots),$$ where $v = (1+i)^{-1} = (1.105)^{-1}$ is the present value discount factor. Now that we can see the cash flow written as a series, I prefer your second approach: $$\begin{align*} 77.1 &= n(v + v^2 + \cdots ) - (nv + (n-1)v^2 + (n-2)v^3 + \cdots + v^n) \\ &= na_{\overline{\infty}\rceil} - (Da)_{\overline{n}\rceil}.\end{align*}$$ Since $$na_{\overline{\infty}\rceil} = \frac{n}{i}$$ and $$(Da)_{\overline{n}\rceil} = \frac{n - a_{\overline{n}\rceil}}{i}$$ it follows that $$77.1 = \frac{a_{\overline{n}\rceil}}{i} = \frac{1-v^n}{i^2}.$$ Consequently, $$n = \frac{\log (1 - 77.1 i^2)}{\log v} \approx 19.$$ You can do it the other way and the formulas are different, but the result is algebraically equivalent: $$\begin{align*} 77.1 &= v (Ia)_{\overline{n}\rceil} + nv^{n+1} a_{\overline{\infty}\rceil} \\ &= v \frac{\ddot a_{\overline{n}\rceil} - nv^n}{i} + \frac{nv^{n+1}}{i} \\ &= \frac{v\ddot a_{\overline{n}\rceil}}{i} = \frac{a_{\overline{n}\rceil}}{i}. \end{align*}$$ I think you will agree that the use of the decreasing annuity was a little bit simpler than the increasing annuity, although both give the same result. Your error in your first solution method (a) is that you did not take into account that the payments are deferred for a year; in both your solutions, you also did not multiply the perpetuity portion by $n$.