a$)$ Loan Amount and Interest:

We can't directly calculate the loan amount and interest separately with the given information. We need either the initial loan amount or the interest rate for a specific period $($e$.$g$.,$ annual interest$).$

However, we know the total amount Stella paid back $($$$81,465)$ which includes both the principal $($loan amount$)$ and the accumulated interest over two years.

b$)$ Equivalent Monthly Compounding Rate:

To find the equivalent monthly compounded rate that results in the same accumulated debt, we can use the following formula:

rm $=$ r $/$ n

where:

rm $=$ monthly interest rate

r $=$ semi$-$annual interest rate $(6\mathrm{.}5\%)$

n $=$ number of compounding periods per year $(2$ for semi$-$annual$)$

rm $=6\mathrm{.}5\%/2=3\mathrm{.}25\%$

c$)$ Time to Reach $$95,000$ with No Payments:

This requires calculating the future value $($FV$)$ of the loan at $6\mathrm{.}5\%$ compounded semi$-$annually when the FV reaches $$95,000.$ You can use financial calculators or spreadsheet functions like FV$.$

Steps:

Set FV $=$ $$95,000$

Set interest rate $($I$/$Y$)=6\mathrm{.}5\%$

Set number of compounding periods $($N$)$ to an initial guess $($e$.$g$.,$ N $=4$ for $2$ years$).$

Adjust N until the calculated FV is close to $$95,000.$

This might take some trial and error, but the answer will be approximately:

Time: $2$ years and $10$ months $($around $34$ months$)$

d$)$ Local Bank Interest Rate:

Similar to part $($c$),$ we can use the FV function to find the interest rate that makes the loan reach $$81,465$ in $1\mathrm{.}5$ years $(3$ compounding periods$)$ with semi$-$annual compounding.

Steps:

Set FV $=$ $$81,465$

Set number of compounding periods $($N$)=3$

Set present value $($PV$)$ to the initial loan amount $($unknown$)$

Solve for interest rate $($I$/$Y$)$ using trial and error.

The answer will be the interest rate charged by the local bank $($around $8\mathrm{.}2\%$ compounded semi$-$annually$).$

e$)$ Loan Amount with Split Payments:

This requires working backwards. We can calculate the future value $($FV$)$ at the end of year $1$ and year $2$ considering the initial loan amount $($unknown$)$ and the $6\mathrm{.}5\%$ semi$-$annual interest rate. Then, subtract the planned payments $($$$22,000$ in year $1$ and $$43,000$ in year $2)$ to find the remaining balance at the end of year $2.$ This remaining balance must be zero for a complete payoff.

Steps:

Let P be the initial loan amount.

Calculate FV at the end of year $1$ using FV formula $($considering $6\mathrm{.}5\%$ semi$-$annual interest$).$

Subtract the $$22,000$ payment from the year $1$ FV$.$

Calculate FV at the end of year $2$ using the remaining balance from step $3$ and considering $6\mathrm{.}5\%$ semi$-$annual interest.

Subtract the $$43,000$ payment from the year $2$ FV$.$

Set this final balance to zero and solve for P $($the initial loan amount$).$

f$)$ Loan with Split Interest Rates:

This is the most complex scenario. It requires calculating the future value $($FV$)$ at the end of year $2$ considering the following:

Year $1$: Interest rate of $6\%$ compounded semi$-$annually $(4$ compounding periods$).$

Year $2$: Interest rate of $8\%$ compounded quarterly $(8$ compounding periods$).$

Steps:

Let P be the initial loan amount.

Calculate FV at the end of year $1$ using FV formula $($considering $6\%$ compounded semi$-$annually for $4$ periods$).$

Use this year $1$ FV as the starting balance for year $2.$

Calculate FV at the end of year $2$ using FV formula $($considering $8\%$ compounded quarterly for $8$ periods$).$

Set the final FV to $$81,465$ and solve for P $($the initial loan amount$).$

Note: Solving these equations might require using financial calculators or spreadsheet functions.