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