Second Moment of an Increasing Term Annuity

Question

I have an arithmetically increasing term annuity-due payable to a life aged $x$ for at most $n$ years under which the payment at time $t$ is $t+1$ for $t=0,1,\dots,n-1$. The actuarial present value is $$ \require{enclose} (I\ddot{a})_{x:\enclose{actuarial}{n}}= \sum_{t=0}^{n-1}(t+1)v^t \hspace0.1cm _tp_x \hspace0.2cm , $$ where $v=(1+i)^{-1}$, $i$ is the interest rate, and $_tp_x$ is the probability that a life aged $x$ survives at least $t$ years. I cannot seem to derive the second moment from this definition. My guess was $$ \sum_{t=0}^{n-1}(t+1)^2v^{2t} \hspace0.1cm _tp_x \hspace0.2cm. $$ However, when I evaluate this for specific $x$ and $n$ values, the second moment is too small for the variance to be positive. Could someone please help me figure out is wrong with my formula?

Answer 1

$\require{enclose}$The present value random variable is $$Y = \begin{cases} (I\ddot a)_{\enclose{actuarial}{K+1}}, & 0 \le K < n, \\ (I \ddot a)_{\enclose{actuarial}{n}}, & K \ge n \end{cases}$$ where $K$ is the curtate lifetime random variable.

Consequently the actuarial present value of the annuity may then be expressed as $$(I\ddot a)_{x:\enclose{actuarial}{n}} = \operatorname{E}\left[Y\right] = \sum_{t=0}^{n-1} (I \ddot a)_{\enclose{actuarial}{t+1}} \; {}_t p_x \; q_{x+t} + (I\ddot a)_{\enclose{actuarial}{n}} \; {}_n p_x. \tag{1}$$ Note that this formula differs from yours in that we have expressed it in terms of the total present value payment conditional upon death in the interval $[t, t+1)$. This allows us to write the second moment as

$$\operatorname{E}[Y^2] = \sum_{t=0}^{n-1} \left((I\ddot a)_{\enclose{actuarial}{t+1}}\right)^2 \; {}_t p_x \; q_{x+t} + \left((I\ddot a)_{\enclose{actuarial}{n}}\right)^2 \; {}_n p_x. \tag{2}$$

Whether such an expression can be simplified further (I suspect it can but have not tried), I leave to you as an exercise. One idea is to set up a table of present values of payments at times $t = 0, 1, 2, \ldots$ for each outcome of $K \in \{0, 1, 2, \ldots \}$, and then look at the square of the sums to see if you can simplify.