Sinking Funds: How They Work and Exam FM Calculations

The Sinking Fund Method

Under the sinking fund method, the borrower pays interest-only on the loan each period (interest at rate i on the original loan L) and simultaneously deposits into a sinking fund that earns rate j to accumulate to L at the end of n periods. The sinking fund deposit is D = L / s-angle-n at rate j. The total periodic outlay is L*i + L/s-angle-n(j).

This method separates the two components of loan repayment: servicing the debt (interest payments) and repaying the principal (sinking fund accumulation). The borrower effectively maintains two financial relationships: one with the lender (paying interest on L) and one with the sinking fund (making deposits that grow to L).

When i = j, the total outlay equals the amortization payment: L*i + L/s-angle-n(i) = L/a-angle-n(i). This identity, which follows from the relationship 1/a-angle-n = 1/s-angle-n + i, connects the sinking fund and amortization methods. It is worth memorizing because it explains why the two methods are equivalent when the rates are equal.

Sinking Fund Schedule

The sinking fund balance after t deposits is D * s-angle-t at rate j. The net amount owed (book value of the loan) is L - D * s-angle-t. The increase in the sinking fund during year t is D*(1+j)^(t-1), which represents the deposit plus the interest earned on the accumulated balance.

Example: A $100,000 loan at i = 8% is repaid using a sinking fund earning j = 6% over 10 years. The annual interest payment is 100,000 * 0.08 = $8,000. The sinking fund deposit is 100,000 / s-angle-10 at 6% = 100,000 / 13.1808 = $7,586.80. The total annual outlay is $15,586.80. Compare this to the amortization payment at 8%: 100,000 / a-angle-10 at 8% = 100,000 / 6.7101 = $14,902.95. Since i > j, the sinking fund method costs more.

After 5 years, the sinking fund balance is 7,586.80 * s-angle-5 at 6% = 7,586.80 * 5.6371 = $42,772.68. The net amount owed is 100,000 - 42,772.68 = $57,227.32.

Comparing Sinking Fund and Amortization

When i > j, the total periodic outlay under the sinking fund method exceeds the amortization payment. When i < j, the sinking fund method costs less. The intuition is straightforward: if the sinking fund earns less than the loan rate, the borrower loses on the rate differential, making the arrangement more expensive than direct amortization.

The difference in total annual outlay is L*(i + 1/s-angle-n(j)) - L/a-angle-n(i). This quantity is positive when i > j and negative when i < j. On Exam FM, you may be asked to find the rate j that makes the two methods equally costly (the answer is j = i) or to compute the present value of the difference over the loan term.

Net Interest and Book Value

The net interest paid in year t is the loan interest minus the sinking fund interest earned: Net I_t = L*i - j*D*s-angle-(t-1)(j). The net principal repaid in year t is the increase in the sinking fund: Net P_t = D*(1+j)^(t-1). These quantities parallel the interest and principal portions of an amortization schedule.

The net loan balance (book value) decreases over time as the sinking fund grows. At time t, book value = L - D*s-angle-t(j). At time 0, book value = L. At time n, book value = L - D*s-angle-n(j) = L - L = 0, confirming the loan is fully repaid.

Sinking Fund with Different Payment Structures

Some problems modify the basic setup. For instance, the sinking fund deposits may not be level. If deposits increase by a fixed amount each period, use an increasing annuity formula: total accumulated = Q*(Is)-angle-n(j) + P*s-angle-n(j), where P is the first deposit and Q is the annual increase. The deposits must accumulate to L.

Another variation has the borrower making sinking fund deposits at a different frequency than the interest payments. In this case, convert the deposit frequency to match the interest payment frequency, or work with the appropriate periodic rate. Always ensure that the accumulated value of all deposits at time n equals the loan amount L.

Study Tips and Common Pitfalls

The most important thing to remember is that the sinking fund has its own interest rate j, which is usually different from the loan rate i. Many students mistakenly use the same rate for both. Always read the problem carefully to identify both rates.

Another common mistake is confusing the sinking fund balance with the net amount owed. The sinking fund balance is what has been saved; the net amount owed is the loan minus the sinking fund balance. Exam FM may ask for either quantity, so be clear about which one the question requests.

Practice at least 5 problems that compare sinking fund and amortization methods, and 3 problems that involve finding the sinking fund balance or net loan balance at a specific point in time. These are the most frequently tested problem types.