3.2 Interest

Schmid, B., Vanderby, S.

3.2 Interest

Two services typically offered by banks are savings accounts and loans. If you deposit money into a savings account, the amount of money in the account gradually increases. At a later date, you can withdraw more money than you deposited. In contrast, when you borrow money in the form of a loan, the total amount you will have to repay will be more than the amount you originally borrowed. In both of these scenarios, an increase in the amount of money is due to interest.

Interest is the cost of borrowing money. It is the amount of money that is paid in addition to the amount borrowed, loaned or invested. Just like you pay to rent a car or rent an apartment, you pay to “rent” money. The original amount borrowed, loaned or invested is called the principal. Interest is charged on the principal due to the following factors:

Inflation. As noted in the section 3.1, due to inflation money’s purchasing power slowly decreases. Without charging interest, the lender would be left with less purchasing power than they started with.
Risk. There is a chance that the borrower will default on (not pay back) their loan. To account for this risk, lenders charge interest. The interest they earn helps cover losses from borrowers who are unable to make their payments.

An interest rate is a percentage rate applied on the principal, which is used to calculate the amount of interest generated in an interest period (the frequency that interest is calculated, e.g. monthly, bi-weekly, yearly).

There are two types of interest: simple interest and compound interest.

3.2.1 Simple Interest

Simple interest is calculated on the amount that was originally borrowed, loaned or invested – the principal. Interest accumulated in previous periods does not earn additional interest.

Example 3.1

Suppose you make a deposit of $100 in a bank account that pays 5% interest per year. After one year, you earn 5% interest, or $5, bringing your total balance to $105. After one more year, since simple interest is paid only on your principal, you again earn 5% of the original $100. That means you earn another $5 in the second year, and will earn $5 for every year of the investment. (Boundless Finance, 2016). The diagram below shows the interest the $100 deposit earns each year and how that affects the total value of your deposit. The next table shows account balances for this scenario for the first 5 years.

Year	Beginning Balance	Interest Earned	Ending Balance
0	–	–	$100.00
1	$100.00	$5.00	$105.00
2	$105.00	$5.00	$110.00
3	$110.00	$5.00	$115.00
4	$115.00	$5.00	$120.00
5	$120.00	$5.00	$125.00

Table 3.1 Simple Interest Earned on a $100 Deposit

Typically, the current account balance is called the present value ($100 in period 0) and the account balance at some point in the future is termed the future value ($125 in period 5).

The future value consists of the present value (principal) and the total interest:

$F = P + I$ (3.3)

Where F = Future Value

P = Present Value

I =Total Interest

Total interest I is the total simple interest paid on the principal in all periods. It is calculated as follows:

$I = (iP)N$ (3.4)

Where P = Present value

i = interest rate

N = number of periods

Thus, the future value formula becomes:

$F = P + (iP)N$

Rearranging, we get the future value with simple interest formula:

$F = P(1 + iN)$ (3.5)

Applying this formula to example 3.1 we get:

$F = \$ 100(1 + 0.05 \ast 5) = \$ 125$

So, the future value of the investment in 5 years is $125. Total interest paid is $25.

While simple interest is a useful theoretical concept, it is rarely used in the real world. Loans you get from a bank, money you invest in savings accounts, credit card debt etc. accrue what is called compound interest.

3.2.2 Compound Interest

Compound interest is calculated on the total amount – the principal and previously earned interest – in a given period. In other words, compound interest includes interest earned on interest, not just interest earned on the principal, like simple interest does. The interest period for compound interest is typically termed the compounding period. For the same interest rate, compound interest will always result in more total interest than simple interest.

Example 3.2

Suppose you make the same $100 deposit into a bank account that pays 5% interest, but this time, the interest is compounded annually. After the first year, you will again have $105. At the end of the second year you also earn 5% interest, but this time it is calculated based on your $105 balance. Thus, you earn $5.25 in interest in the second year, bringing your balance to $110.25. In the third year, you earn 5% interest on your $110.25 balance, or $5.51. Table 3.2 shows how interest accumulates in the account for the first 5 years.

Year	Beginning Balance	Interest Earned	Ending Balance
0	–	–	$100.00
1	$100.00	$5.00	$105.00
2	$105.00	$5.25	$110.25
3	$110.25	$5.51	$115.76
4	$115.76	$5.79	$121.55
5	$121.55	$6.08	$127.63

Table 3.2 Compound Interest Earned on a $100 Deposit

While you may use a table to calculate the future value, it may be convenient to develop a formula for this calculation.

In year 1, interest is applied to the principal (present value in period 0):

$I_1 = Pi$

Thus, the future value in year 1 is:

$F_1 = P + I_1 = P + Pi + P(1 + i)$

In year 2, interest is now calculated on the ending balance in year 1, which is $F_1$ :

$I_2 = F_1i = P(1 + i)i$

The Future value in year 2 is therefore:

$F_2 = F_1 + I_2 = P(1 + i) + P(1 + i)i = P(1 + i)(1 +i) = P(1 +i)^2$

Similarly, in year 3 interest is calculated on the ending balance in year 2 – $F_2$ :

$1_3 = F_2i = P(1 +i)_2i$

So, the future value in year 3 is:

$F_3 = F_2 + I_3 = P(1 +i)^2 + P(1 + i)^2i = P(1 +i)^2(1 + i) = P(1 + i)^3$

Continuing for N periods, we get the general future value with compound interest formula for single cash flows:

$F = P(1 + i)^N$ (3.6)

Remembering that , the formula for total compound interest is:

$I = P[(1 + i)^N - 1]$ (3.7)

To summarize, the formulas for simple and compound interest are in Table 3.3 below.

	Simple Interest	Compound Interest
Total Interest	$I = (iP)N$	$I = P[(1 + i)^N - 1]$
Future Value	$F = P(1 + iN)$	$F = P(1 + i)^N$
Interest earned	Only on principal	Both principal and accumulated interest

Table 3.3 Simple and Compound Interest

Let’s see an example problem comparing simple interest to compound interest.

Simple and Compound Interest Examples

Suppose we want to make one deposit into a bank account today in order to withdraw $2000 five years from now. How much would we have to deposit

If the account paid 5% simple interest annually?
If the account paid 5% compound interest, compounded annually?

Solution

a. Simple Interest

From the problem statement, we know that the future value of the account should be $2000, so F = $2000 We are given the interest rate, i = 5% = 0.05 , and the number of years, N = 5. Given all this information, we first use the simple interest formula 3.5 to solve for P – the value of the initial required deposit:

$F = P(1 + iN) \rightarrow P = \frac{F}{1 + iN} = \frac{\$ 2000}{1 + (0.05)(5)} = \$ 1600.00$

b. Compound Interest

For the compound interest account, we use compound interest formula 3.6 to solve for P:

$F = P(1 + i)^N \rightarrow P = \frac{F}{(1 + i)^N} = \frac{\$ 2000}{(1 + 0.05)^5} = \$ 1567.05$

Thus, comparing the two values, we can see that you would have to deposit a smaller amount into a bank account with compound interest. This is due to the fact that compound interest yields higher total interest amounts compared to simple interest, as noted before. By putting money in a compounding account rather than a simple-interest account, even at the same interest rate, you would earn an extra $33. Even though $33 does not seem like much money, it can be significant when considering larger amounts invested over longer time periods. If you were to withdraw $2 million after five years under the same terms, the difference would be $33 000!

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