I was helping my comrade answer some questions when we found this question. It goes like this:
A loan is to be repaid quarterly for five years that will start at the end of two years. If interest rate is $6$% converted quarterly,
how much is the loan if the quarterly payment is $10000$?
My work
I recognize that the problem above is a deferred annuity problem. The payment will start at the end of two years (The payment is deferred
by two years) and the payment will last five years.
The term "$6$% converted quarterly", I believe, would mean that the interest rate is $6$ percent per year divided by 4, giving $\frac{0.06}{4}$ or
$0.015$. In short, the interest rate $6$% 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, and $k$ 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 five years, so
the number of payment periods is now $\left(\frac{4}{year}\right)(5 \space years) = 20 $. The number of deferred periods is $\left( \frac{4}{year} \right)(2 \space years) = 8$
because the interest rate already took effect even if there is no payment within the deferred period.
Now, we have…
$$k|P = A(P/A,i\%,n)(P/F,i\%,k)$$
$$k|P = A \left( \frac{(1+i)^n-1}{i(1+i)^n}\right) \left(\frac{1}{(1+i)^k} \right)$$
$$k|P = A \left( \frac{\left(1+\left(\frac{0.06}{4}\right)\right)^{20}-1}{\left(\frac{0.06}{4}\right)\left(1+\left(\frac{0.06}{4}\right)\right)^{20}}\right) \left(\frac{1}{(1+\left(\frac{0.06}{4}\right))^8} \right)$$
$$k|P = 152407.91$$
Therefore, the present value of the loan after five years would be $\color{green}{152407.91}$
Is my answer correct?
Best Answer
Yes, your answer is correct. Using actuarial notation, we have $$PV = 10000 \; {}_{8|} a_{\overline{20}| 0.015} = 10000 v^8 \frac{1 - v^{20}}{i^{(4)}/4},$$ where $i^{(4)} = 0.06$ is the nominal interest rate compounded quarterly, and $v = 1/(1+i^{(4)}/4)$ is the quarterly present value discount factor.