How to find the present value of an annuity-due after a certain period p while interest also varies for each year t Given:
i) X is the current value at the end of year two of a 20-year annuity-due of 1 per annum.
ii) The annual effective rate for year t is: $$i_t = \frac {1}{8+t}$$
Calculate X.

*

*$$ a(t) = (1+i_t) = \frac {9+t}{8+t}$$
From this point, I honestly have no idea how to evaluate the annuity...
 A: At the end of year $2$, there are $18$ payments of $1$ still to come. The first one come immediately, so its present value is $1$.  The second one comes at the end of year $3$, in which the interest rate is $1/11$ so the present value is $$\left(1+\frac1{11}\right)^{-1}=\frac{11}{12}$$  The third comes at the end of year $4$, so the present value is
$$\left(1+\frac1{11}\right)^{-1}\left(1+\frac1{12}\right)^{-1}=\frac{11}{12}\frac{12}{13}=\frac{11}{13}$$
I'm sure you see how to continue.  We have to discount each payment by the rate in every year from the end of yer $2$ to the payment date, so we get for the present value of the future payments
$$1+11\left(\frac1{12}+\frac1{13}+\cdots+\frac1{28}\right)$$
We must add to this the accumulated value of the past payments, of which there have been $2$.  The payment at the beginning of year $1$ has grown by $1+\frac1{10}=\frac{11}{10}$ and the payment at the beginning of year $1$ has grown by $\left(1+\frac19\right)\frac{11}{10}=\frac{11}9$ so the current value is $$11\left(\frac19+\frac1{10}+\cdots+\frac1{28}\right)=\sum_{t=9}^{28}\frac{11}t$$
