“Money doesn’t grow on trees.” While this is true, money does make more money, if it collects interest. Interest is money that one pays for the use of someone else’s money.
Most people know that if you put money in a bank, you will get more money over time. The bank is paying you to use your money for investments and loans.
The longer the money is in the bank, the more money you will get. And it makes sense that if you get put your money in a bank with high interest, you will get more money than if you put it in a lower interest bank. But how does it all work? Let’s investigate.
To begin, we need to define some terms. When a person deposits money in a bank, that initial amount is called the principal and is typically denoted by the letter P. The rate at which that the principal collects interest is called the interest rate and is denoted by the letter r. (The rate is typically expressed as a decimal and not as a percentage.) While all of this is going on, time elapses, and is denoted by t.
There are two types of interest: basic interest and compound interest. Basic interest is paid only once. For example, you may lend $100 to a friend and ask for 10% (0.1) interest every year until you are repaid. If your friend pays you back in two years, your friend will owe you $120, but why?
Your friend is responsible for paying back the initial amount, the principal, as well as paying 10% on that principal for two years. 10% of $100 is $10, and times two years is $20. So your friend will pay you $100 + $20 = $120. This reasoning is summarized in the following formula.
Basic Interest Formula
The interest charged according to formula (1) is called basic interest.
Compound interest, on the other hand, is interest paid on previously earned interest. Suppose, instead of paying 10% interest when the money is given back, you have your friend pay you 10% on the money he still has every year. If he doesn’t pay you back for two years, then you will get 10% on $100 the first year, which is $110 and you will get 10% on $110, which is $121 at the end of the second year. Instead of getting $20, you will get $21. Another example might help explain this concept.
Suppose a bank offers an interest rate of 6% per annum (yearly) on a savings account that is compounded quarterly (four times a year). A person deposits $4,000 in the bank. How much money will be in his account at the end of one year?
Since the interest is compounded quarterly, we can use the formula for basic interest, , four times. The principal amount P is $4,000, the interest rate r is 6% (r = 0.06), and t = of a year. (Recall that t is measured in years.)
So, the amount of interest earned on the savings account is:
Now, as we start the second quarter, the principal is no longer $4,000. Since we have accumulated interest, the new principal is $4,000 + $60 = $4,060. Using that new principal, the interest gained in the second quarter is:
The interest paid at the end of the third quarter on the principal of $6,060 + $60.90 = $6,120.90 is:
The final amount of interest paid, at the end of the fourth quarter, will be equal to:
Adding that interest amount to the principal going into that period ($4,182.71), we get the balance in the account after one year. It will be $4,182.71 + $62.74 = $4,245.45.
We can use the process above to obtain a general formula for computing compound interest. To do this, notice that we started with a principal, P, and an interest rate r per annum. Let n be the number of times the principal is compounded. Then the interest earned at the end of each compounding period will be the principal amount times the interest rate times the fraction , ie . The amount A in the account after one compounding period will be:
After two compounding periods, the amount A, based on the new principal of will be:
After three compounding periods, the amount will be:
Following this pattern, after n compounding periods, the amount A will be:
However, the above formula only works for interest compounded n times in one year. In the more general case with t years, the principal will compound nt times. So, we have the following formula.
Compound Interest Formula
If we rework Example 1 using formula (3), we would use P = $4,000, r = 0.06, n = 4, and t = 1. Plugging that in yields:
, which is what we found it to be above using the longer, more tedious method of computing the amount after each period.
Now let’s examine the effect that using different compounding periods has on the amount of a principal investment after 1 year.
Let P = $1,000 be an initial deposit into a savings account with a per annum interest rate of r = 0.05 (5%). What is the amount in the account after 1 year if the account (i) compounds annually, (ii) compounds quarterly, (iii) compounds monthly, (iv) compounds weekly, (v) compounds daily?
(i) Compounded annually:
(ii) Compounded quarterly:
(iii) Compounded monthly:
(iv) Compounded weekly:
(v) Compounded daily:
Notice that as the number of compounding periods increases, so does the ending amount. With only one compounding period, the amount is $1,050. If the account compounds daily, the ending amount is $1,051.27. While not significantly higher (it is only $1.27 more), it is better than compounding only once. But what happens as ?
To find the answer, we need to rewrite formula (2).
Where the substitution was used. Notice that as , as well.
Recall that the . So, Eq. (4) tells us that as , .
If the amount A after 1 year of a principal P and a per annum interest rate r is , then we say that the interest is compounded continuously. Again, the above formula only holds if we are dealing with 1 year. Below is the general case for t years:
Continuous Compounding Interest Formula
The following table compares different values of n and r with a principal of $1. That means that we are using the formula: . Notice that as n gets larger, the values in the table get closer to .
Suppose that a savings account is compounded continuously at a rate of 8% per annum. After three years, the amount in the account is $2,500. How much was deposited at the beginning, assuming that nothing was added or withdrawn during that period?
We are told that account is compounded continuously, so we are going to use . Also, we know that the interest rate is 8% per annum, so r = 0.08. The amount A is $2,500 when t = 3. All that is left is to find P. We can do that if we plug in everything that we know.
Suppose that a savings account is compounded continuously with a principal of $1,500. After 20 years, the amount increased to $5,000. (a) What was the per annum interest rate, assuming that nothing was added or withdrawn during that period? (b) If the account sits for another 20 years, how much will be in the account?
To answer (a) we need to find the value of r. We are told that P = $1,500 and that when t =20, A = $5,000. Using , we can solve for r.
Now, (b) asks us to find out how much money will be in the account after another 20 years. Notice that means that t = 40, not 20. Using the same formula, keeping P the same, but replacing r with 0.06 and t with 40 gives us the following:
When dealing with interest, the terms nominal interest rate and effective interest rate arise. Don’t let the terms confuse you. The nominal interest rate is just the per annum interest rate, r, as it is used in the basic interest formula.
The effective interest rate, on the other hand, is the interest rate when compounding more often than once a year is taken into consideration. Another way to think of the effective interest rate is as the equivalent nominal interest rate that would yield the same amount as compounding after 1 year.
In Example 1, $4,000 compounded quarterly at 6% per annum resulted in $4,245.45 at the end of the year. Subtracting off the principal, we have $4,245.45 - $4,000 = $245.45. Calculations reveal that $245.45 is 6.14% of $4,000 ($245.45/$4,000 = 0.0613625). So, it would take a basic interest rate of 6.14% to equal 6% compounded quarterly.
We examined the effects that frequent compounding has on interest rates in Example 2 and in Table 1, but it was somewhat disguised. Another example is warranted.
Suppose a principal of $10,000 is compounded (a) annually, (b) quarterly, (c) monthly, (d) weekly, (e) daily, and (f) continuously at a per annum interest rate of 5%. Write out the corresponding effective interest rates.
A basic interest rate of 5.095% would be needed to match 5% compounded quarterly. A basic interest rate of 5.116% would be needed to equal 5% compounded monthly, and so forth. The more often we compound, the higher the basic interest rate needs to be in order to make the two quantities equal.