Have I understood the question properly? Annuities in Actuarial math

Question

I am wondering if I have interpreted the language correctly in the following question

The force of interest at time $t$ is given by $\delta(t) = 0.05-0.005t$ for $\leq t < 5$ and $\delta(t)=0.02+0.001t$ for $5 \leq t \leq 10$. Calculate the present value of an annuity of $\$1000$ payable annually for $10$ years in arrears.

It's the " annuity of $\$1000$ payable annually for $10$ years in arrears" I am not sure about.

1.My interpretation is; $\$1000$ is payable at the "end" of each year, for the next $10$ years. So say, if the annuity payment starts at $t=0$, then $\$1000$ is paid at $t=1$ which is the "end" of $t=0$(i.e. after one unit time from $t=0$). Then the next (second payment is at $t=2$ etc and the final will come at $t=10$ and the total amount paid in this period is $\$1000 \times 10 = \$10000$.

I took "payment in arrears" as "end of the year payment." "$\$1000$ payable annually" as "$\$1000$ is paid each year."

2.I found the discount factor from the force of interest so all I need to do now is discount each $\$1000$ for each year accordingly I think but I'd also like to check if I am right.

I'll denote $u_1(t)$ as my discount factor for $0 \leq t < 5$ and $u_2(t)$ for $5 \leq t \leq 10$. Then, from year $0$ to year $5$, the discounted value for year $0$ is $0$(since no payment if my interpretation is right), $1000 \cdot u_1(1)$ for $t=0$, $1000 \cdot u_1(2)$ for $t=2$... up to $t=4$.Then for $t=5$ onward, $1000 \cdot u_2(5),1000 \cdot u_2(6),...,1000 \cdot u_2(10)$.

Then my final answer would be the sum of all these discounted values of $1000$. Am I correct? If not, lease tell me why and correct me!

Thank you in advance

Answer 1

1. An annuity in arrears is an ordinary annuity or annuity-immediate: the payments are made at the end of the time periods, so that interest is accumulated before the payment.
2. Let be $$ \delta(t)=\begin{cases} \delta_1(t)= 0.05 - 0.005 t& \text{ if } 0 \leq t < 5 \\ \delta_2(t)= 0.02 + 0.001 t & \text{ if } 5 \leq t < 10 \end{cases} $$ The discount function is $v(t)=\mathrm e^{- \int_0^t \delta(s)\mathrm d s}$ that is $$ v(t)=\begin{cases} v_1(t)=\mathrm e^{- \int_0^t \delta_1(s)\mathrm d s}& \text{ if } 0 \leq t < 5 \\ v_2(t)=v_1(5)\mathrm e^{- \int_0^t \delta_(s)\mathrm d s}& \text{ if } 5 \leq t < 10 \end{cases} $$ The present value will be $$ PV=1000\cdot \sum_{n=1}^{10}v(n)=1000\cdot \left(\sum_{n=1}^{4}v_1(n)+\sum_{n=5}^{10}v_2(n)\right) $$