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.