I am currently faced with a practice problem for a financial mathematics course, and I would like to verify that I am approaching, the problem correctly (and ultimately that my solution is indeed correct).
The problem is as follows:
A loan of $\$250000$ charging a nominal annual interest rate of $6\%$ convertible monthly is to be repaid via the sinking fund method. If payments are made at the end of each
month for ten years and the sinking fund earns a nominal annual rate of $4\%$ convertible monthly, find the total monthly expense for the repayment of this loan.
Here are my thoughts:
- We know the loan size $L = 250,000$ and that there are $10 * 12 = 120$ payment periods.
- We know that the loan charges a nominal rate of $6\%$. Since this is a nominal rate, we divide this by $12$ to get the monthly effective rate of $0.06/12 = 0.005 = 0.5\% $
- The sinking fund nominal interest rate is $4\%$, so the sinking fund’s monthly effective rate is $0.04/12 = 0.00333333… = 0.333333…\%$.
- Thus, the interest paid per period is $iL = (0.005) * (250,000) = \$1250$
- And, the sinking fund deposit per period is $250,000 / s_{n=120, j = 0.3\%} = \$1697.80$.
Thus, the total monthly expense = $Interest Paid + Sinking Fund Deposit = 1250 + 1697.80 = 2947.80$
If anyone would be willing to verify the correctness of this (or suggest an alternative approach), it would be greatly appreciated!
Thanks!
Best Answer
Your work is correct as written. The idea is to observe that the principal on the loan is the accumulated value of the sinking fund, which in turn is equal to an annuity-immediate over the term of the loan.
That is to say, if the level monthly payment into the fund is $K$, then $$\require{enclose} L = Ks_{\enclose{actuarial}{120} j},$$ where $j = i^{(12)}/12 = 0.04/12$ is the effective monthly interest rate for the fund, and $L = 250000$ is the principal. Thus the total monthly outlay is the sum of the interest-only monthly payment, plus the level payment into the sinking fund: $$\left(\frac{0.06}{12} + \frac{1}{s_{\enclose{actuarial}{120} j}} \right)L = \left(\frac{1}{200} + \frac{\frac{1}{300}}{(1 + \frac{1}{300})^{120} - 1}\right)(250000) \approx 2947.79512.$$
I would not interpret "total monthly expense for the repayment of this loan" in the manner described in the other answer, since that quantity would be better described as a monthly borrowing cost of the loan net of the principal received. Alternatively, it is in a sense the monthly excess interest paid on the loan. The expense of taking out a loan is not net of the value of the money received from the loan itself, because the purpose of securing that loan is to receive a lump sum up front to pay for goods and services at that time, in exchange for a series of payments in the future. In other words, if you plan to use the loan to pay back the loan, why take it out in the first place? The only situation where that would be to the borrower's benefit is if there is an arbitrage opportunity.