(a) Bullet repayment

The entire loan is paid back at maturity.

The cash flow table would look like this:

Total debt service is the annual sum of interest and principal to be paid back. This is also called debt servicing at each due date.

(b) Constant amortisation

Each year, the borrower pays off a constant proportion of the principal, corresponding to 1/n, where n is the initial maturity of the loan.

The cash flow table would look like this:

(c) Equal instalments

The borrower may want to allocate a fixed sum to the service of debt (capital repayment and interests).

Based on the discounting method described previously, consider a constant annuity A, such that the sum of the five discounted annuities is equal to the present value of the principal, or €1000:

This means that the NPV of the 10% loan is nil; in other words, the 10% nominal rate of interest is also the internal rate of return of the loan.

Using the formula from Section 16.5, paragraph 1, the previous formula can be expressed as follows:

A = €263.80. Hence, the following repayment schedule:

In this case, the interest for each period is indeed equivalent to 10% of the remaining principal (i.e. the nominal rate of return) and the loan is fully paid off in the fifth year. Internal rate of return and nominal rate of interest are identical, as calculation is on an annual basis and the repayment of principal coincides with the payment of interest.

Regardless of which side of the loan you are on, both work the same way. We start with invested (or borrowed) capital, which produces income (or incurs interest costs) at the end of each period. Eventually, the loan is then either paid back (leading to a decline in future revenues or in interest to be paid) or held on to, thus producing a constant flow of income (or a constant cost of interest).

(d) Interest and principal both paid when the loan matures

In this case, the borrower pays nothing until the loan matures. The sum that the borrower will have to pay at maturity is none other than the future value of the sum borrowed, capitalised at the interest rate of the loan:

This is how the repayment schedule would look:

This is a zero-coupon loan.

(a) The concept of effective annual rate

What happens when interest is paid not once but several times per year?

Suppose that somebody lends you money at 10% but says (somewhere in the fine print at the bottom of the page) that interest will have to be paid on a half-yearly basis. For example, suppose you borrowed €100 on 1 January and then had to pay €5 in interest on 1 July and €5 on 1 January of the following year, as well as the €100 in principal at the same date.

This is not the same as borrowing €100 and repaying €110 one year later. The nominal amount of interest may be the same (5 + 5 = 10), but the repayment schedule is not. In the first case, you will have to pay €5 on 1 July (just before leaving on summer holiday), which you could have kept until the following 1 January if using the second case. In the first case you pay €5, instead of investing it for six months as you could have done in the second case.

As a result, the loan in the first case costs more than a loan at 10% with interest due annually. Its effective rate is not 10%, since interest is not being paid on the benchmark annual terms.

To avoid comparing apples and oranges, a financial officer must take into account the effective date of disbursement. We know that one euro today is not the same as one euro tomorrow. Obviously, the financial officer wants to postpone expenditure and accelerate receipts, thereby having the money work for him. So, naturally, the repayment schedule matters when calculating the rate.

Which is the best approach to take? If the interest rate is 10%, with interest payable every six months, then the interest rate is 5% for six months. We then have to calculate an effective annual rate (and not for six months), which is our point of reference and our constant concern.

Two rates referring to two different maturities are said to be equivalent if the future value of the same amount at the same date is the same with the two rates.

In our example, the lender receives €5 on 1 July which, compounded over six months, becomes 5 + (10% × 5) / 2 = €5.25 on the following 1 January, the date on which he receives the second €5 interest payment. So, over one year, he will have received €10.25 in interest on a €100 investment.

Therefore, the effective annual rate is 10.25%. This is the real cost of the loan, since the return for the lender is equal to the cost for the borrower.

If the apparent rate (or nominal rate) (r_a) is to be paid n times per year, then the effective annual rate (t) is obtained by compounding this nominal rate n times , where n is the number of interest payments in the year and r_a/n the proportional rate during one period, or t = (1 + r_a/n)ⁿ − 1.

In our example:

The effective interest rate is thus 10.25%, while the nominal rate is 10%.

It should be common sense that an investment at 10% paying interest every six months produces a higher return at year end than an investment paying interest annually. In the first case, interest is compounded after six months and thus produces interest on interest for the next six months. Obviously a loan on which interest is due every six months will cost more than one on which interest is charged annually.

The table below gives the returns produced by an investment (a loan) at 10% with varying instalment frequencies:

The effective annual rate can be calculated on any timescale. For example, a financial officer might wish to use continuous rates. This might mean, for example, a 10% rate producing €100, paid out evenly throughout the year on principal of €1000. The financial officer will use the annual equivalent rate as his reference rate for this investment.

(b) The concept of proportional rate

In our example of a loan at 10%, we would say that the 5% rate over six months is ­proportional to the 10% rate over one year. More generally, two rates are proportional if they are in the same proportion to each other as the periods to which they apply.

10% per year is proportional to 5% per half-year or 2.5% per quarter, but 5% half-yearly is not equivalent to 10% annually. Effective annual rate and proportional rates are therefore two completely different concepts that should not be confused.

Proportional rates are of interest only when calculating the interest actually paid. In no way can they be evaluated with other proportional rates.

Proportional rates serve only to simplify calculations, but they hide the true cost of a loan. Only the effective annual rate (10.25%/year) gives the true cost, unlike the proportional rate (10%/year).

When the time span between two interest payment dates is less than one year, the proportional rate is lower than the effective annual rate (10% is less than 10.25%). When maturity is more than a year, the proportional rate overestimates the effective annual rate. This is rare, whereas the first case is quite frequent on money markets, where money is lent or borrowed for short periods of time.

As we will see, the bond markets can be misleading since they reason in terms of nominal rate of return: paper is sold above or below par value, the number of days used in calculating interest can vary, there could be original issue discounts, and so on. And, most importantly, on the secondary market, a bond’s present value depends on fluctuations in market interest rates.

In the rest of this book, unless otherwise specified, an interest rate or rate of return is assumed to be an effective annual rate.