I was helping my comrade answer some questions when we found this question. It goes like this:
A cash loan is to be repaid by paying $13500$ quarterly for three years starting at the end of four years. If the interest rate is $12$% convertible quarterly, how much is the cash loan?
My work
I recognize that the problem above is a deferred annuity problem. The payment will start at the end of four years (The payment is deferred by four years) and the payment will last three years.
The term "$12$% converted quarterly", I believe, would mean that the interest rate is $12$ percent per year divided by 4, giving $\frac{0.12}{4}$ or $0.03$. In short, the interest rate $12$% is compounded quarterly.
The amount of the loan to be paid for five years would be the present value of the loan at the end of five years. Using the formula
$$k|P = A(P/A,i\%,n)(P/F,i\%,k)$$ $$k|P = A \left( \frac{(1+i)^n-1}{i(1+i)^n}\right) \left(\frac{1}{(1+i)^k} \right)$$
where....
$A$ is the amount of each payment of an ordinary annuity, $i$ is the interest rate, $n$ is the number of payment periods, $k$ is the number of deferred periods.
In this problem, we see that the number of payment periods if we pay quarterly for a year would be $4$. We will pay the amount for three years, so the number of payment periods is now $\left(\frac{4}{year}\right)(3 \space years) = 12 $. The number of deferred periods is $\left(\frac{4}{year} \right)(4 \space years) = 16$ because the interest rate already took effect even if there is no payment within the deferred period.
Now, we have...
$$k|P = A(P/A,i\%,n)(P/F,i\%,k)$$ $$k|P = A \left( \frac{(1+i)^n-1}{i(1+i)^n}\right) \left(\frac{1}{(1+i)^k} \right)$$ $$k|P = A \left( \frac{\left(1+\left(\frac{0.12}{4}\right)\right)^{12}-1}{\left(\frac{0.12}{4}\right)\left(1+\left(\frac{0.12}{4}\right)\right)^{12}}\right) \left(\frac{1}{(1+\left(\frac{0.12}{4}\right))^{16}} \right)$$ $$k|P = 83740.58$$
Therefore, the present value of the loan after three years would be $\color{green}{83740.58}$
Is my answer correct?