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During the first four years interest is credited using a simple interest of 5% per year. After 4 years, interest is credited at a force of interest $\delta_t = \frac{0.2}{1+.02t}$ for $t \ge 4$. Find the current value at time $t=4$ of payments of $1000$ at $t=2$ and $400$ at time $t=7$

The accumulation function for $0\le t < 4$ is $a(t) = 1+.05t$ and the accumulation function for $t \ge 4$ is $a(t) = \exp(\int_4^t \frac{0.2}{1+.02r}dr)= \frac{1+ .2t}{1.8}$.

The solution manual says the current value at $t=4$ is the sum of the accumulated value of $1000$ from $t=2$ to $t=4$ added to the present value of $400$ at $t=7$ to $t=4$. Doing this gives us $1000(1+(2\cdot.05)) + 400(\frac{1.8}{1+(.2 \cdot 7)}) = 1400$.

I dont understand why we are adding the accumulated value of $1000$ from $t=2$ to $t=4$ because if we are making a payment of $1000$ at $t=2$ shouldn't we be adding the accumulated value of $A(2)-1000$ from $t=2$ to $t=4$ where $A(2)$ is the value of the account at time $t=2$, ie $A(t) = ka(t)$ where $k$ is the initial amount in the account at time $t=0$ but we are not given the initial amount of the account.

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  • $\begingroup$ $1000(1+2\cdot 0.05) + 400\left( \frac{1.8}{1+(0.2 \cdot 7)} \right)=\color{red}{1400}$ ? $\endgroup$ – callculus May 19 '17 at 18:13
  • $\begingroup$ @callculus sorry youre right, I changed it but my question still stands $\endgroup$ – alpastor May 19 '17 at 18:18
  • $\begingroup$ You can treat the payment at $t=2$ as the intital payment. To get the future value at $t=4$ you have to compound the payment two years. $\endgroup$ – callculus May 19 '17 at 18:21
  • $\begingroup$ @callculus payment means we are taking money out of the account right? So if we take 1000 out of the account why are we compounding that 1000 for another 2 years? $\endgroup$ – alpastor May 19 '17 at 18:24
  • $\begingroup$ @callculus This is how I am trying to justify it. In order to have 1000 in the account at time $t=2$ then we must have $1000(1+ 2 \cdot .05)$ in the account at time $t=4$. But if we make a payment of 1000 at $t=2$ then there will be no money left in the account to compound for another two years. $\endgroup$ – alpastor May 19 '17 at 18:32
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The two calculations are equivalent.

$\texttt{1. Calculation in your question}$:

You compound the 1000 at $t=2$ for two years with the simple iterest of $0.05$. Thus you have the 1000 at $t=4$ $(X)$. Then you discount the $400$ for $3$ years using the reciprocal of $a(t)$ $(Y)$. Now $X$ and $Y$ are both represents the values at $t=4$. Finally you can add them: $C_4=X+Y$

$\texttt{2. Calculation in your comment}$:

You calculate the future value of 1000 at $t=7$. Since $400$ are paid at $t=7$ you can just add them to $1000\cdot 1.1\cdot \frac{1+0.2\cdot 7}{1.8}$ To get the value at $t=4$ you compound the sum with the force of interest $a(t)$.

$$C_4=\underbrace{(1000\cdot 1.1\cdot \frac{1+0.2\cdot 7}{1.8}+400)}_{t=7}\cdot \frac{1.8}{1+0.2\cdot 7} =1400$$

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  • $\begingroup$ I thought I understood this but now im not sure what the word payment or paid means in this context. What does it mean when you say 400 is paid at $t=7$. $\endgroup$ – alpastor May 19 '17 at 23:15
  • $\begingroup$ Also what does it mean that there was a payment of 1000 at $t=2$ when we are at time $t=4$? Does that mean that 1000 was the future value at $t=2$ for some initial investment, say $K$, at $t=0$? $\endgroup$ – alpastor May 19 '17 at 23:29
  • $\begingroup$ $\texttt{"What does it mean when you say 400 is paid at t=7"}$ In my understanding it means that you get 400 on your account at t=7. $\texttt{"Also what does it mean that there was a payment of $1000$ at . $t=2$ when we are at time $t=4$"}$ It means that you calculate the future value of 1000. The money is 2 years on your account. Imagine your are at $t=2$ then the amount of $1000$ is the present value. The 1000 has to be compounded for two years to get the future value at $t=4$. @n.e $\endgroup$ – callculus May 20 '17 at 1:32
  • $\begingroup$ $\texttt{"Does that mean that 1000 was the future value at $t=2$ for some initial investment, say K, at $t=0$" }$? You can say that at if you would discount 1000 two times then you would get the present value of at $t=0$: $\frac{1000}{1.1}=909.09...$. This amount you have to get on your account to have 1000 at $t=2$. @n.e. $\endgroup$ – callculus May 20 '17 at 1:42

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