I need help with the following questions. In order to answer these questions, I am only allowed to use two of the tables of values for $\ddot a_n , \bar a_n$ (continuous annuity) and $(I \ddot a)_n$ from n=1,...,30 and i=3%.
(h) Given an effective annual rate of 3%, calculate the present value at time 0 of a 15-year arithmetically increasing annuity immediate whereby the first annual payment is 2,452 and subsequent annual increment is 450.
$PV_0 = 2002 \ddot a_{15} + 450 (I \ddot a)_{15}$
$ = 2002*12.29607 + 450*91.60627$
=$65,839.55
(i) Consider a stream of 30-year continuous cash flows with a payment rate of 404t during year t. For instance, the payment rate during year 5 is $2,020. Calculate its present value at time 0 given an effective annual rate of 3%.
$PV_0 = 404(I \bar a)_{30} + 404 \bar a_{30}$
$= 404(I \ddot a)_{30}* \frac{d}{\delta} + 404 \bar a_{30}$
$=404*268.7906* \frac {1-(1.03)^{-1}}{ln(1.03)} +404*19.893$
=$115,038.96
Any help is appreciated, especially for i) as I'm not too sure about that question.
Thanks
You need to write out the cash flows and set up the equation of value.
For (h), an increasing annuity-immediate has the first payment of $2452$ at time $t = 1$, so the equation of value is $$PV = 2452 v + 2902 v^2 + \cdots + 8752 v^{15}$$ where $v = 1/(1+i)$ and $i = 0.03$. Using your line of reasoning, this would be $$PV = 2002(v + v^2 + \cdots + v^{15}) + 450(v + 2v^2 + 3v^3 + \cdots 15v^{15}).$$ Therefore, the correct equation in actuarial notation would need to be $$PV = v\left(2002 \ddot a_{\overline{15}\rceil i} + 450 (I \ddot a)_{\overline{15}\rceil i}\right).$$
For (i), the equation of value is more difficult to express. Consider the cash flow in year $t$ alone. This corresponds to $$404t {\bar a}_{\overline{1} \rceil i}$$ before discounting for the year in which the payment is made. In other words, we are only taking the present value of the payments in year $t$ to time $t-1$. Now if we discount for the year, this is an extra factor of $v^{t-1}$. So the cash flow is $$PV = 404 {\bar a}_{\overline{1}\rceil i} + 808 v {\bar a}_{\overline{1}\rceil i} + \cdots + 12120 v^{29} {\bar a}_{\overline{1}\rceil i}.$$ Factoring, we get $$PV = 404 {\bar a}_{\overline{1}\rceil i} \left( 1 + 2v + 3v^2 + \cdots + 30v^{29} \right) = 404 {\bar a}_{\overline{1}\rceil i} (I\ddot a)_{\overline{30}\rceil i}.$$