I'm having trouble understanding this solution for a sinking fund question.
A borrower takes out a loan of $2000 for two years. Construct a sinking fund schedule if the lender receives 10% effective on the loan and if the borrower replaces the amount of the loan with semiannual deposits in a sinking fund earning 8% convertible quarterly
The solution lets the deposits of the sinking fund be $D$
$D*(s_{8|.02}/s_{2|.02}) = 2000$
Where does the $s_{8|.02}$ come from?
For a loan $L$, with sinking fund rate $j$, the periodic deposit of the sinking fund satisfies $$ Ds_{\overline{n}|j}=S $$ But in your problem the interest rate $8\%$ is quarterly convertible, i.e. the quarter interest rate is $r=\frac{8\%}{4}=2\%$, and the deposit are semiannual.
It represent the future value of an annuity payable at a different frequency than interest is convertible.
We can follow two ways.
The semiannual interest rate is $j$ such that after two quarters you have $$\left(1+r\right)^2=1+j\quad\Longrightarrow\quad j=\left(1+r\right)^2-1=(1+0.02)^2-1=4.04\% $$ and then after $n=4$ deposits in $2$ years you'll have $$ S=Ds_{\overline{n}|j}=Ds_{\overline{4}|4.04\%} $$
Let $k$ be the number of interest conversion periods in one payment period. Let $r$ be the rate per conversion period and $m$ the total number of conversion periods for the term of the annuity. We will assume that each payment period contains an integral number of interest conversion periods so that $m$ and $k$ are positive integers and also we assume that $m$ is divisible by $k$. The total number of annuity payments made is then $\frac{m}{k}$. The future value an annuity immediate which pays 1 at the end of each $k$ interest conversion periods for a total of $m$ interest conversion periods is $$ \begin{align} FV&=1+(1+r)^k+(1+r)^{2k}+\cdots+(1+r)^{\left(\frac{n}{k}-1\right)k}\\ &=\frac{[(1+r)^k]^{\frac{m}{k}}-1}{(1+r)^k-1}=\frac{(1+r)^m-1}{(1+r)^k-1}\\ &=\frac{\frac{(1+r)^m-1}{r}}{\frac{(1+r)^k-1}{r}}=\frac{s_{\overline{m}|r}}{s_{\overline{k}|r}} \end{align} $$ So in your problem you have: $r=\frac{8\%}{4}=2\%$ is the quarterly convertible rate, $k=2$ is the number of interest conversion periods (two quarters) in one payment period (half year), $m=4\times 2=8$ is the total number of conversion periods. Thus $$ S=D\frac{s_{\overline{m}|r}}{s_{\overline{k}|r}}=D\frac{s_{\overline{8}|0.02}}{s_{\overline{2}|0.02}} $$
