Most of us borrow money when we purchase a car, a house, education, etc. Most businesses borrow money when they purchase new equipment or just to keep up the daily operation of their businesses. We call the amount of money we borrow the (original) principal.
When people borrow money, they have to agree to repay this amount – usually plus some extra – in the future. This extra amount is called the interest. These terms are also used when we invest money and earn interest, and our story indeed involves investing money.
Consider the following situation. Suppose you have invested $100 at an interest rate of 10 per cent per annum. Suppose also that interest is compounded annually, i.e. the interest earned by the principal is reinvested so that it, too, earns interest. How much will you obtain by the end of the tenth year?
After one year, the value of investment will be the original principal ($100), plus the interest on the principal ($100× 0.1):
100+ 100 × 0.1 = $110.
So the interest for the second year is earned for $110, not just $100. At the end of the second year, the value of investment will be the principal at the end of the first year ($110), plus the interest on it ($110× 0.1):
110+ 110 × 0.1 = $121.
It means that the principal increases each year by 10 per cent. The $121 represents the original principal, plus all accrued interest, and is called the compound amount.
The difference between the compound amount and the original principal is called the
compound interest. In this example, the compound interest (at the end of the second year) is $121− $100 = $21.
In general, the compound amount St of the principal P at the end of t years at the rate of i (expressed as a fraction or decimal) compounded annually can be expressed as follows.
(1) After the first year:
S1= P + P i
= P (1 + i).
(2) After the second year:
S2 = S1+ S1i
= S1(1+ i)
= P (1 + i)(1 + i)
= P (1 + i)2. (3) After the third year:
S3 = S2+ S2i
= S2(1+ i)
= P (1 + i)2(1+ i)
= P (1 + i)3. ... ... (t) At the end of year t:
St = P (1 + i)t. (3.24)
Now try the following question.
Question What is the compound amount of $1000 invested at an annual rate of 6 per cent for 6 years? What are the compound amount and the compound interest? You may round your answer to two decimal places at the end of calculation.
Solution:
S6= 1000 · (1 + 0.06)6≈ 1418.52.
S6− P = 1000 · (1 + 0.06)6− 1000 ≈ 418.52.
Exercise 3.5 Compound interest.
67 3.4 Compound interest
3.4.1 What do we need to do if the compounding period is not annual?
Compounding may not necessarily take place annually. It may take place daily, monthly, quarterly, etc. If compounding takes place quarterly (every three months), we say that there are four interest periods or conversion periods per year. However, regardless of how often compounding occurs, an interest rate is usually quoted as an annual rate, which is called the nominal rate.
The periodic rate (of interest) is obtained by dividing the nominal rate by the num-ber of conversion periods per year. For example, if the nominal rate is 8 per cent and compounding occurs quarterly, the periodic rate is 8%
4 = 2%. In this chapter, unless oth-erwise stated, all interest rates will be assumed to be nominal rates. We now can generalise Equation (3.24).
The compound amount Sn of the principal P at the end of n interest periods when the periodic rate is r (expressed as a fraction or decimal) can be expressed as follows.
(1) After the first interest period:
S1 = P + P r
= P (1 + r).
(2) After the second interest period:
S2= S1+ S1r
= S1(1+ r)
= P (1 + r)(1 + r)
= P (1 + r)2. (3) After the third interest period:
S3= S2+ S2r
= S2(1+ r)
= P (1 + r)2(1+ r)
= P (1 + r)3 ... ... (n) At the end of nth interest period:
Sn = P (1 + r)n. (3.25)
Now try this one.
Question What is the compound amount of $1000 invested at an annual rate of 6 per cent compounded semi-annually for 6 years? What is the compound interest? You may round your answer to two decimal places at the end of calculation.
Solution Observe that there are 12 interest periods (twice per year for 6 years) and the periodic rate is 3 per cent.
S12= 1000 ·
1+0.06 2
6×2
= 1000 · (1 + 0.03)12= 1425.76.
S12− P = 1000 · (1 + 0.03)12− 1000 = 425.76.
Exercise 3.6 Compound interest (but not annually).
We have seen that for a principal of $1000 at a nominal rate of 6 per cent over a period of 6 years, annual compounding results in a compound interest of $418.52, and with semi-annual compounding the compound interest is $425.76. So the important lesson here is:
for a given positive nominal rate, the more frequent the compounding, the greater is the compound interest.
3.4.2 The effective rate of interest
If we invest $1000 (principal) at a nominal rate of 6 per cent compounded semi-annually for one year, it will earn more than 6 per cent that year.
Question How much will you earn in the above situation?
Solution
S− P = P
1+0.06 2
2
− P
= P (1.03)2− P
=
(1.03)2− 1 P
= (1.0609 − 1) P
= 0.0609P.
Exercise 3.7 The effective rate of interest.
As we show above the compound interest is 0.0609P , which is 6.09 per cent of P . This means that 6.09 per cent is the rate of interest compounded annually that is actually earned. We call this rate the effective rate of interest, or the yield. In other words, the effective rate is just the rate of change in the principal over a period of one year. Hence we have shown that the nominal rate of 6 per cent compounded semi-annually is equivalent to an effective rate of 6.09 per cent.
69 3.4 Compound interest
In fact, we can formally state the relationship between the effective rate and the nominal rate in general. Suppose the principal is P and the nominal rate of i is compounded n times a year. Then the compound amount after a year can be denoted by P
Solving this for rewe get:
re=
Again, don’t just try remembering the result per se. You can always derive Equation (3.26) by yourselves if you understand the notion of the effective rate. Under-standing the idea is far more important. The effective rate is quite useful in comparing different compounding methods with different nominal rates. If you convert those into the effective rates, comparison becomes possible and we can see which one of the methods will yield more interest in one year. Now you should be able to answer the following question.
Question If you have a choice of investing money at 6.3 per cent compounded annually or 6.125 per cent compounded quarterly, which one should you prefer? You may round your answer (expressed as a percentage) to two decimal places at the end of calculation.
Solution
The effective rate for the first option:
re=
By definition, it is 6.3 per cent.
The effective rate for the second option:
re =
It is 6.27 per cent. Hence the first option is preferred.
Exercise 3.8 Comparing the effective rates.