Here is the full question:
A level continuously paying annuty pays $\$ 1500$ each month for eight year. The force of interest is $\delta(t)=\frac{2t}{t^2+5}$ where time is measured in years. Find the present value:
What I've got so far:
So using the force of interest we can find the accumulation function. I believe this should be $t^2 +5$. I know that if the payment was being made every year, then I could integrate $(1500)(1/a(t))$ from 0 to 8 to get the answer. However, the problem is that the payment is being made 12 times a year, staggered by a month. How do I account for this?
The correct answer is 52253.29
Thanks!
The force of interest is $\delta(t)=\frac{2t}{t^2+5}$ and the discount function $v(t)$ is $$ v(t)=\mathrm{e}^{-\int_0^t\delta(s)\mathrm d s}=\frac{5}{5+t^2} $$ observing that $$ \int_0^t\delta(s)\mathrm d s=\int_0^t\frac{2s}{s^2+5}\mathrm d s=\log(s^2+5)\Big|_0^t=\log\left(\frac{t^2+5}{5}\right) $$ ant the time $t$ is measured in years.
The rate of payment is $\rho(t)=\rho=1500$ and then the present value of the annuity for $n=8$ years is $$ \int_0^{n\times 12} \rho v\left(\tfrac{t}{12}\right)\mathrm dt= \int_0^{96} 1500\frac{5}{5+\left(\frac{t}{12}\right)^2}\mathrm dt = 18000 \sqrt 5 \tan^{-1}\left(\frac{96}{12\sqrt 5}\right)\approx 52253.29 $$