I was helping my comrade answer some questions when we found this question. It goes like this:
A car is to be purchased in monthly payments of $19500$ for five years starting at the end of three months. How much is the cash value of the car if the interest rate used is $10$% converted monthly?
My work
I recognize that the problem above is a deferred annuity problem. The payment will start at the end of three months (The payment is deferred by three months) and the payment will last five years.
The term "$10$% converted monthly", I believe, would mean that the interest rate is $10$ percent per year divided by 12, giving $\frac{0.10}{12}$ or $\frac{1}{120}$. In short, the interest rate $10$% is compounded monthly.
The amount of the car to be paid for five years would be the present value of the car 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 monthly for a year would be $12$. We will pay the amount for five years, so the number of payment periods is now $\left(\frac{12}{year}\right)(5 \space years) = 60 $. The number of deferred periods is $3$ because the interest rate already took effect even if there is no payment within the deferred period.
Now, we have...
$$k|P = A(P/A,i\%,n)(P/F,i\%,k)$$ $$k|P = A \left( \frac{(1+i)^n-1}{i(1+i)^n}\right) \left(\frac{1}{(1+i)^k} \right)$$ $$k|P = A \left( \frac{\left(1+\left(\frac{0.10}{12}\right)\right)^{60}-1}{\left(\frac{0.10}{12}\right)\left(1+\left(\frac{0.10}{12}\right)\right)^{60}}\right) \left(\frac{1}{(1+\left(\frac{0.10}{12}\right))^3} \right)$$ $$k|P = 895207.49$$
Therefore, the cash value of the car if the interest rate used is $10$% converted monthly is $\color{green}{895207.49}$
Is my answer correct?
