Loan amortisation and sinking funds

The level-payment loan

A loan is an annuity seen from the other side. The lender advances a sum today and receives a level payment for n periods, so the loan amount is the present value of that annuity and the payment is the loan divided by the annuity factor.

Take a loan of 250,000 at 6 percent effective over fifteen years. The annuity-immediate factor at 6 percent for fifteen years is 9.712249, so the annual payment is 250,000 divided by that, which is 25,740.69. Over the full term the borrower pays 386,110.36, of which 136,110.36 is interest - a figure worth computing once, because it makes the shape of the schedule concrete in a way the formula does not.

Every later quantity in this topic is derived from those two numbers. If the payment is wrong, everything after it is wrong, so it is worth carrying more decimal places through the schedule than you intend to report.

One caution on rounding. Real lenders round the payment to the cent and absorb the difference in a slightly adjusted final payment, but exam questions almost never do. Unless a question says otherwise, carry the unrounded payment through the whole schedule and round only the final answer; rounding the payment first will move a fifteen-year balance by several units and produce an answer that matches no option on the paper.

Two routes to the outstanding balance

The prospective method values what is still owed: the balance after t payments is the payment times the annuity factor for the remaining n minus t periods. The retrospective method values what has happened: the original loan accumulated to time t, less the accumulated value of the payments made.

On the loan above, the balance immediately after the fifth payment is 25,740.69 times the annuity factor for ten years at 6 percent, which is 189,453.73. The retrospective route - 250,000 accumulated for five years less the accumulated value of five payments - gives the same 189,453.73, as it must.

Prefer the prospective method when the remaining term is short, the retrospective when the elapsed term is short, and use whichever is not asked for as a check when the number matters. They are algebraically identical, so any disagreement is arithmetic rather than method.

The interest and principal split

Each payment pays the interest that accrued on the outstanding balance, and whatever is left reduces the principal. That sentence is the whole of amortisation, and every formula in the topic is a shortcut for it.

The first payment on the loan above meets interest of 6 percent on 250,000, which is 15,000, leaving 10,740.69 of principal repaid and a balance of 239,259.31. By the sixth payment the interest has fallen to 11,367.22 and the principal portion has risen to 14,373.47, the two still summing to the same 25,740.69. By the final payment the interest is 1,457.02 and the principal 24,283.67, which closes the loan exactly.

The interest portion is falling geometrically and the principal portion rising geometrically, each by a factor of one plus the interest rate per period. That relationship is the fastest way to move between two rows of a schedule without rebuilding it, and it is the fact most amortisation questions are built around.

The sinking-fund method

Under the sinking-fund method the borrower does not repay principal gradually. Instead the borrower pays the lender interest on the full loan every period and separately deposits into a fund that accumulates to the principal by maturity, at which point the fund discharges the debt in one payment.

With the same 250,000 loan at 6 percent, and a sinking fund earning 4 percent over fifteen years, the interest to the lender is a flat 15,000 a year. The accumulated-value factor at 4 percent for fifteen years is 20.023588, so the deposit is 250,000 divided by that, which is 12,485.28. The total annual outlay is 27,485.28, against 25,740.69 under amortisation.

The sinking-fund method is more expensive here for one reason: the fund earns 4 percent while the loan costs 6 percent, so money set aside grows more slowly than the debt it is set against. That gap, not the mechanism, is what makes the difference.

The identity that connects them

When the sinking fund earns exactly the loan rate, the two methods cost the same, and that is not a coincidence. One over the annuity-immediate factor equals the interest rate plus one over the accumulated-value factor, at the same rate and term.

At 6 percent over fifteen years both sides come to 0.10296276, and multiplying through by 250,000 turns the identity into the statement that the amortisation payment equals the interest plus the sinking-fund deposit. The methods differ only when the two rates differ.

Examiners like this identity because it can be asked from either end: given a sinking-fund arrangement, find the equivalent amortisation rate, or given both rates, find the extra cost. Recognising it saves rebuilding both schedules.

What gets asked

Loan questions on Exam FM are concentrated on a short list, and working the list deliberately is more efficient than working randomly through a problem bank.

• The payment, given the loan, the rate and the term - or any one of those, given the other three.
• The outstanding balance at a stated time, by either method.
• The interest or principal portion of a specific payment, which the geometric relationship gives without a schedule.
• The total interest paid over the life of the loan, which is the payment times the term less the loan.
• The sinking-fund deposit, the total outlay, and the comparison with amortisation at a different fund rate.
• A loan with a payment that changes partway through, which is two equations of value stitched together at the change date.