Sinking Fund Method - Loan Repayment

Question

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!

Answer 1

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.

Answer 2

I came to a different computation.

Let $i = 0.005$.

Let $\displaystyle j = \frac{1}{3} \times 0.01.$

Let $t = (1 + j)$.

Let $P = 250000$.

Let $R = ~$ the monthly payments.

Then, I am assuming that the intended interpretation for monthly expense is

$$\frac{(120 \times R) - P}{120}.\tag1$$

That is, in (1) above, you take the total paid, subtract the current value of the loan, and divide by the number of payments. This gives you the overage per payment.

All of the computations below are based on the above interpretation.

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At the end of $10$ years, the value of the loan will be $P \times (1+i)^{(120)}.$

At the end of $10$ years, the value of the monthly payments of $R$ will be

$$R \times \left(1 + t + t^2 + \cdots + t^{(119)}\right) = R\times \frac{t^{(120)} - 1}{t - 1}.$$

This implies that $R$ must be computed as

$$R = \frac{t-1}{t^{(120)} - 1} \times P \times (1+i)^{(120)}.\tag2 $$

My computation gives $R \approx 3088.96$.

Plugging this value into (1) above gives a monthly expense of

$$\frac{(120 \times 3088.96) - 250000}{120} \approx 1005.63.$$

Edit-1
An alternative interpretation of monthly expense might plausibly be the value of $R \approx 3088.96$, which seems to be how you are interpreting the phrase.

Your interpretation may well be the intended one. I was simply calculating (in effect) the expense of having to pay interest on the loan, where the interest charged is $i$, which is greater than the sinking fund interest of $j$.

Then, there is the added expense that the loan begins accruing interest immediately, while the sinking fund payments don't begin earning interest until they are paid. This means that the average amount of time that each sinking fund payment will earn interest is about $5$ years.

Edit-2
I am assuming that the first payment begins $1$ month from the date of the loan, and that the final payment is made $120$ months from the date of the loan.