Continuously payable annuities with force of interest

[Eclair_de_XII]

Homework Statement

An account pays interest at a continuously compounded rate of 0.05 per year. Continuous deposits are made to the account at a rate of 1000 per year for 6 years and then at a rate of 2000 per year for the next 4 years. What is the account balance at the end of 10 years?

Relevant Equations

##\bar{a}_n=\frac{1-v^n}{\delta}##
##v=\frac{1}{1+i}##
##\delta=\ln(1+i)##

Answer: ##17402.48##

I split the continuous payment stream into two separate streams:

1. one 10-year payment stream with a rate of 1000 per year
2. one 4-year payment stream with a rate of 1000 pewr year

I expressed the present value of this payment stream as:

##\begin{align}
1000a_{10}+1000a_4&=&1000(a_{10}+a_4)\\
&=&\frac{1000}{\delta}(2-(v^{10}+v^4))
\end{align}
##

The problem gave:

##i=0.05##
##\delta=\ln(1.05)\approx 0.04879##
##v=\frac{1}{1.05}\approx 0.95238##

I evaluated the equation with these values and got: ##11547.09##. Where did I go wrong?

 

[andrewkirk]

You have calculated a present value, whereas the question asks for a final value.
Also, you have not deferred the second stream. It runs for four years, commencing in six years time.

 

[Eclair_de_XII]

Okay, I got it. I seem to have also mixed up the interest rate and the force of interest values:

##\delta=0.05##
##i=\exp(\delta)-1\approx 0.0513##
##v=\frac{1}{1+i}\approx 0.9512##

##\begin{align*}
(1+i)^{-10}FV&=&PV\\
&=&1000a_{10}+v^{6}1000a_{4}\\
&=&1000\left(\frac{1-v^{10}}{\delta}+v^{6}\frac{1-v^4}{\delta}\right)\\
&=&\frac{1000}{\delta}(1-v^{10}+v^6-v^{10})\\
&=&\frac{1000}{\delta}(1-2v^{10}+v^6)
\end{align*}##

I multiply the thing on the right by ##(1+i)^{10}## and then I get the desired value.

Thank you for pointing out my errors.