Simple and Compound Interest

Interest can be thought of as the price a lender charges a borrower for the use of his money. In the case of loans, the lender and borrower are quite clear; in other cases, such as investing in a mutual fund, the "lender" and "borrower" are not as clear. The investor can be thought of as lending his money to the mutual fund, which is in this case the "borrower".
Whether or not interest is re-invested, and at what frequency, impact the accumulated value of the initial investment at the end of each investment period. This topic covers the two most common re-investment strategies, first where interest is not re-invested, and secondly where interest is re-invested at the same rate as the initial investment. Effective and nominal rates of interest are defined, and the relationship between them is derived.

Definition (Interest) The difference between the accumulated value of an investment after a period of time and the initial amount invested (principal) is called the interest earned during that period.

The above definition of interest assumes that no withdrawals or additional deposits are made during the investment period. This requirement will be relaxed when the more general case is developed.

Example (Interest) An initial deposit of 1,000 is made into an investment account. No withdrawals are made and no other principal is invested. At the end of three years, the account is worth 1,300. The interest earned during the three-year period is

Definition (Effective Rate of Interest) The effective rate of interest is the amount of money that one unit of principal will earn during one investment period. Alternatively, it is the ratio of the amount of interest earned during the period to the amount of principal invested at the beginning of the period.

Example (Effective Rate of Interest) (a) In Example 1.3, the effective rate of interest earned over the three-year period can be calculated as
(b) An initial deposit of 500 accumulates to 520 at the end of one year and 550 at the end of the second year. The effective rate of interest for the first year is The effective rate of interest for the second year is The effective rate of interest for the two-year period is
Note that it was assumed that the interest earned in the first year was re-invested during the second year, so that the principal amount at the beginning of the second year was 520. In part (a) no such assumption is needed. Also, note that the effective rate of interest can be calculated for any time period; in part (b) it was calculated separately for the first year, second year, and the two-year period. Note that the and are annual rates while the is a two-year rate. Topic 4, Equations of Value, will discuss methods of converting the two-year rate to an annual rate.

Definition (Simple Interest) A pattern of interest accrual such that the amount of interest earned in each investment period is constant is called simple interest. If is the initial investment and is the constant interest amount, the accumulated value at the end of investment periods is given by

Definition (Compound Interest) A pattern of interest accrual such that the rate of interest earned in each investment period is constant is called compound interest. If is the initial investment and is the constant interest rate, the accumulated value at the end of investment periods is given by

    Under a pattern of simple interest, interest is not reinvested at the end of each period to earn additional interest. Therefore the constant amount of interest earned in each investment period can be thought of as a rate of interest, called the simple interest rate, applied to the principal amount only. In contrast, compound interest assumes that the interest earned in one period is reinvested in the next period (at the same rate as the initial investment) to earn additional interest.
    The formulas given in the definitions of simple and compound interest are easily derived from the definitions. First, suppose that the simple interest amount earned in each period is At the end of the first period, the accumulated amount is At the end of the second period, the accumulated amount is At the end of periods, the accumulated amount is
    Next, if the compound interest rate is then is the amount of interest earned in the first period. At the end of the first period, the accumulated amount is The accumulated amount at the end of the first period can be thought of as the principal amount at the beginning of the second period. The amount of interest earned in the second period is thus and the accumulated amount at the end of the second period is



At the end of periods the accumulated amount is
    Note that with simple interest, the effective rate of interest is different for each time period and is in fact decreasing. This can be seen by finding the effective rate for the period, as the ratio of the interest earned in the period to the accumulated value at the beginning of the period. Since the interest earned in any period is we have If is thought of as the simple interest rate , this equation simplifies to

which is a decreasing function of
    In contrast, the effective rate of interest for a compound interest pattern is constant and is just the compound interest rate, as can be seen from


which is constant by the definition of compound interest.

Example (Simple and Compound Interest) (a) If 1050 is invested for three years at 4%, find the accumulated value and total interest earned at the end of the period under both simple and compound interest.
    Under simple interest, the 4% simple interest rate determines the constant interest amount by The accumulated value after 3 years is The total interst earned in the three year period is Note that
    Under compound interest, the 4% compound interest rate is the same as the annual effective rate of interest. The accumulated value after 3 years is The total interest earned in the three year period is The extra 5.11 is due to interest compounding, or interest earned on interest.
    (b) If 100 is invested at 3% simple interest, find the accumulated value after 3 years and the effective rates of interest in the third and fourth year.
    The accumulated value after three years is given by the formula where and Therefore,


     The effective rate of interest in the third year is the amount of interest in the third year divided by the accumulated amount at the end of the second year.


     The effective rate of interest in the fourth year is the amount of interest in the fourth year divided by the accumulated amount at the end of the third year.


    (c) If 1115 is invested to earn a constant 44.6 each year, how many years will it take for the account value to double?
    The situation described is simple interest, since the interest amount is constant. We want to solve for given that the accumulated value is 2230.

    (d) If 2130 is invested to earn a constant rate of 4%, what is the accumulated value after 4 years? How many whole years will it take for the account value to become greater than 3000?
    The accumulated value after 4 years is given by the formula where Substituting, we get


    To find the number of years it will take for the account value to grow to over 3000, we must solve the equation


therefore, the number of whole years for the account value to grow to over 3000 is 9.

Definition (Nominal Rate of Interest) The nominal rate of interest (denoted ) is a rate of interest payable times per investment period. That is, the rate earned is for each of a period.

The nominal rate of interest, also called annual percentage rate or APR, is used quite frequently in practice. For example, a credit card may charge 24% interest compounded monthly; that is, an interest charge of 2% is added to the account balance each month. As we will see, the nominal rate of interest is only equal to the effective rate when interest is compounded once per period. Lenders are usually required to disclose the annual effective rate when it differs from the nominal rate.

Example (Nominal Rate of Interest) (a) For an investment of 1,300 at 8% compounded quarterly, find the accumulated value after 3 years.
    The formula can be modified to be used with nominal rates of interst if we generalize to mean the interest rate earned in one compounding period, and to mean the number of compounding periods. As mentioned in Definition 1.11, a nominal rate of interest means that is the interest rate earned in each compounding period. If interest is compounded times per year for years, then the number of compounding periods is Therefore, is generalized to
    To find the accumulated value of 1,300 at 8% compounded quarterly for 3 years, we have with Therefore the equation for the accumulated value is


    (b) Find the equivalent annual effective rate for a nominal rate of 10% compounded monthly.
    Interest rates are equivalent when an amount of principal accumulates to the same value after the same number of investment periods at either rate. If no principal amount is specified, we can take the principal amount to be 1. At the annual effective rate 1 will accumulate to at the end of one year. At a nominal rate of 10% compounded monthly, 1 will accumulate to at the end of one year. Setting these two expressions equal, we have


    First note that the annual effective rate is larger than the nominal rate; this is because the interest added at the end of each quarter earns interest during the next quarter. Intuitively, 10% compounded more than once per year is more favorable to the investor than a 10% annual effective rate because interest is added to the principal more frequently and thus has more time to earn additional interest.
    In addition, the solution to the example can be generalized to show the relationship between any nominal rate and the equivalent annual effective rate. For a nominal rate compounded times per year, the equivalent annual effective rate will be the solution to the equation