Table 1: The Future Value of $1.00 (Compound Amount of $1.00) (1 + i)n = FVF (i,n) = T1 (i,n)
Example 4
You invested a large sum of money in the stock of TLC Corporation. The company paid a $3 dividend per share. The dividend is expected to increase by 14 percent per year for the next 3 years. You wish to project the dividends for years 1 through 3.
Intrayear Compounding
Interest is often compounded more frequently than once a year. Banks, for example, compound interest quarterly, daily, or even continuously. If interest is compounded m times a year, then the general formula for solving the future value becomes
The formula reflects more frequent compounding (n x m) at a smaller interest rate per period (i/m). For example, in the case of semiannual compounding (m = 2), the above formula becomes
Example 5
You deposit $10,000 in an account offering an annual interest rate of 16 percent. You will keep the money on deposit for five years. The interest rate is compounded quarterly. The accumulated amount at the end of the fifth year is calculated as follows:
Where P = $10,000
Therefore,
Example 6
As the example shows, the more frequently interest is compounded, the greater the amount accumulated. This is true for any interest for any period of time. How often interest is compounded can substantially affect the rate of return. For example, an 8% annual interest compounded daily provides an 8.33% yield, or a difference of 0.33%. The 8.33% is the effective yield, frequently called annual percentage rate (APR). The annual interest rate (8%) is the nominal, stated, coupon, or face rate. When the compounding frequency is greater than once a year, the effective interest rate will always exceed the nominal rate.
The formula for calculating the effective interest rate or annual percentage rate (APR), in situations where the compounding frequency (n) is greater than once a year, is as follows.
APR = (1 + )m. To illustrate, if the stated annual rate is 8% compounded quarterly (or 2% per quarter), the effective annual rate is: Effective rate = (1 + .02)4 - 1 = (1.02)4 - 1 = 1.0824 - 1 = .0824 = 8.24%
Exhibit 4 shows how compounding for five different time periods affects the effective yield and the amount earned by an investment for one year.
Exhibit 4: Nominal and Effective Interest Rates with Different Compounding Periods
Note: Federal law requires the disclosure of interest rates on an annual basis in all contracts. That is, instead of stating the rate as “1% per month,” contracts must state the rate as “12% per year” if it is simple interest or “12.68% per year” if it is compounded monthly.
The Power of Compounding
The power of compounding is evident in two typical cases:
1. Start early.
2. Equally significant, the fact that a small difference in the interest rate makes a big difference in the future value amount.
Start Early
The current debate on Social Security reform provides a great context to illustrate the power of compounding. One proposed idea is for the government to give $1,000 to every citizen at birth. This gift would be deposited in an account that would earn interest tax-free until the citizen retires. Assuming the account earns a modest 8% annual return until retirement at age 65, the $1,000 would grow to $137,759. Why start so early? If the government waited until age 18 to deposit the money, it would grow to only $34,474 with annual compounding. That is, reducing the time invested by a third results in more than almost 75% reduction in retirement money (see Exhibit 5). The example illustrates the importance of starting early when the power of compounding is involved.
Exhibit 5: Future Values of When to Start
Impact of Interest Rate Difference
A small difference in the interest rate makes a big difference in the future value amount. To illustrate this point, assume that you had $1,000 in a tax-free retirement account. All the money is in stocks returning 12 percent or all the money in bonds earning 10 percent. Assuming reinvested profits and annually compounding, your investment will grow to $51,875 from bonds after 10 years. But your stocks, returning 2 percent more, would be worth $62,117, which implies a 2% higher interest would result in some 20% increase in the future value after 10 years. Exhibit 6 illustrates this impact.
Exhibit 6: Impact of Interest Rate Difference
Future Value of an Annuity
Two types of annuities are discussed below: the ordinary annuity (annuity in arrears) with the payment at the end of the year, and the annuity due (or annuity in advance) when the payment is made at the beginning of the year. In addition, the discussion examines the future difference in value between these two annuities.
Ordinary Annuity
An ordinary annuity is defined as a series of payments (or receipts) of a fixed amount for a specified number of periods. Each payment is assumed to occur at the end of the period. The future value of an annuity is a compound annuity which involves depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow.
Then we can write
where FVF-OA(i,n) = T2(i,n) represents the future value of an ordinary annuity of $1 for n years compounded at i percent and can be found in Table 2.
Example 7
You wish to determine the sum of money you will have in a savings account at the end of 6 years by depositing $1,000 at the end of each year for the next 6 years. The annual interest rate is 8 percent. The T2(8%,6 years) is given in Table 2 as 7.336. Therefore,
