Compounding is an arithmetic process whereby an initial value increases or grows at a compound interest rate over time to reach a value in the future. Compound interest involves earning interest on interest in addition to interest on the principal or initial investment.
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The Concept of Compound Interest
To understand compounding, let’s assume that you leave the investment with a bank for more than one year. For example, the first bank accepts your $1,000 deposit now, adds $80 at the end of one year, retains the $1,080 for the second year, and pays you interest at an 8 percent rate.
The bank returns your initial deposit plus accumulated interest at the end of the second year. How much will you receive as a future value?
In word terms, we have the following calculation:
Future value = Present value × [(1 + Interest rate) × (1 + Interest rate)]
For our two-year investment example, we have,
Future value = $1,000 × (1.08) × (1.08) = $1,000 × 1.1664
= $1,166.40
= $1,166 (rounded)
For a one-year investment, the return would be $1,080 ($1,000 × 1.08), which is the same as the return on a simple interest investment, as was previously shown.
However, a two-year investment at an 8 percent compound interest rate will return $1,166.40, compared to $1,160 using an 8 percent simple interest rate.
The compounding concept also can be expressed in equation form as,
FVn = PV(1 + r)n
where FV is the future value, PV is the present value, r is the interest rate, and n is the number of periods in years. For our $1,000 deposit, 8 percent, two-year example, we have,
FV2 = $1,000(1 + 0.08) = $1,000(1.1164) = $1,166.40 = $1,166 (rounded)
If we extend the time period to ten years, the $1,000 deposit would grow to,
FV10 = $1,000(1 + 0.08)10 = $1,000(2.1589) = $2,158.90 = $2,159 (rounded)
How To Find A Future Value?
Now that we have set up an equation-based process for finding a future value when compounding is involved, we will demonstrate other methods that could be used to find a future value.
Texas Instruments (TI) and Hewlett Packard (HP) make two popular types of financial calculators. However, they are programmed differently.1 Reference is made to the use of TI and HP calculators when discussing calculator solutions throughout the remainder of this chapter.
Other available financial calculators are usually programmed like either the TI or HP calculators. What is important is that if you are going to use a financial calculator to solve time value of money problems, you must understand how your particular calculator works.
Most financial calculators are programmed to readily find future values. Typically, financial calculators will have a present value key (PV), a future value key (FV), a number of time periods key (N), an interest rate key (usually designated %i), and a compute key (usually designated as CPT). If you have a financial calculator, you can verify the future value result for the ten-year example.
First, clear any values stored in the calculator’s memory. Next, enter 1000 and press the PV key (some financial calculators require that you enter the present value amount as a minus value because it is an investment or outflow).
Then, enter 8 and press the %i key (most financial calculators are programmed so that you enter whole numbers rather than decimals for the interest rate). Next, enter 10 for the number of time periods (usually years) and press the N key.
Finally, press the CPT key followed by the FV key to calculate the future value of 2,158.93, which rounds to $2,159. Actually, financial calculators are programmed to calculate answers to 12 significant digits.
Computer spreadsheet programs also are available for finding future values.
Inflation or Purchasing Power Implications
The compounding process described in the preceding section does not say anything about the purchasing power of the initial $1 investment at some point in the future. $1 growing at a 10 percent interest rate would be worth $2.59 (rounded) at the end of ten years.
With zero inflation, you could purchase $2.59 of the same quality of goods after ten years relative to what you could purchase now.
However, if the stated, or nominal, the interest rate is 10 percent and the inflation rate is 5 percent, then in terms of increased purchasing power, the “net” or differential compounding rate would be 5 percent (10 percent – 5 percent) and $1 would have an inflation-adjusted value of $1.63 after ten years. This translates into an increased purchasing power of $0.63 ($1.63 – $1.00).
Also note that if the compound inflation rate is equal to the compound interest rate, the purchasing power would not change. For example, if in Figure 9.1 both the inflation and interest rates were 5 percent, the purchasing power of $1 would remain the same over time.
Thus, to make this concept operational, subtract the expected inflation rate from the stated interest rate and compound the remaining (differential) interest rate to determine the change in purchasing power over a stated time period.
For example, if the interest rate is 10 percent and the inflation rate is 3 percent, the savings or investment should be compounded at a differential 7 percent rate. Turning to Table 9.1, we see that $1 invested at a 7 percent interest rate for ten years would grow to $1.967 ($1.97 rounded) in terms of purchasing power. Of course, the actual dollar value would be $2.594 ($2.59 rounded).
Financial contracts(e.g., savings deposits and bank loans) in countries that have experienced high and volatile inflation rates sometimes have been linked to a consumer price or similar inflation index.
Such actions are designed to reduce the exposure to inflation risk for both savers and lenders. Since the interest rate they receive on their savings deposits will vary with the rate of inflation, savers receive purchasing power protection. As inflation rises, so will the rate of interest individuals receive on their savings deposits such that purchasing power will be maintained.
Bank lenders are similarly protected against changing inflation rates since the rates they charge on their loans will also vary with changes in inflation rates.
At least, in theory, banks will be able to maintain a profit spread between the interest rates they pay to savers and the higher interest rates they lend to borrowers because inflation affects both financial contracts.
Of course, if the borrowers are business firms, they need to be able to pass on higher prices for their products and services to consumers to be able to maintain profit margins when interest rates are rising along with increases in inflation.
Inflation in the United States has averaged about 2 percent annually during the first part of the twenty-first century. However, some economists and others have expressed concern that the Fed’s emphasis since late-2008 on near-zero federal funds interest rate targets and its QE1, QE2, and QE3 quantitative easing efforts will result in much higher inflation rates in the future.
The Fed responded in late-2015 with its first of what are likely to be several increases in the federal funds target interest rates to combat possible increases in inflation.