Multiple Year Annual Compounding
A dollar invested today would earn interest, and next year it would be worth more than one dollar. During the second year, the original dollar would earn interest, and “the interest earned during the first year would also earn interest”
To illustrate this process, called compounding, assume Marcia Simon deposited
$1,000 in a bank CD that paid 10 percent interest, compounded annually. How much money would she have at the end of two years?
The future value of a fixed sum is calculated by:
FV = PV (1 + r)n 3.2
where
FV = Future value of the investment at the end of n years PV = Value of the investment today, or present value
r = Interest rate during each compounding period n = Number of years, or compounding periods.
We can use Equation 3.2 to determine the value of $1,000 at the end of two years, compounded annually at 10 percent:
FV = $1,000 (1 + 0.10)2
= $1,000 × 1.21
= $1,210.00
If the CD had a maturity of 10 years, the original investment of $1,000 would grow to $2,593.74:
FV = $1,000 (1 + 0.10)10
= $1,000 × 2.59374
= $2,593.74
Fortunately, we can use financial calculators to establish the future values of a fixed sum for various interest rates and maturity periods. Figure 3.1 shows how a fixed sum of money grows over time at various interest rates. As expected, the higher the interest rate or the longer the compounding period, the larger the future value of a fixed sum at the end of the investment period.
5.0
4.0
3.0
2.0
1.0
0 2 4 6 8 10
Periods
Amount ($)
15%
10%
5%
0%
Figure 3.1 Future Value of a Fixed Sum over Time Source: Author’s own work.
At this point, let us introduce the use of calculators. The most popular financial calculators are HP-12C (Platinum), HP-10B II Plus, HP-17B II Plus, and Texas Instruments BA II Plus. In this chapter, for solving various financial problems we use the HP12C Platinum Financial Calculator because it is considered the best. The HP-12C calculator can be used to answer standard questions in personal finance.
These include: How much does one need to save to achieve future financial goals, such as college costs or retirement? How big a mortgage should one be able to afford, given the amount of funds available to finance it? How would interest rate changes affect monthly payments on an adjustable loan? Should one buy or lease a car? Should tax-exempt investment be preferable to taxable investment for an investor whose income falls in a certain marginal tax bracket?
Details regarding features of other calculators (HP-10B II Plus, HP-17B II Plus, and Texas Instruments BA II Plus) can be found at their manufacturer’s website and on the Internet. They can be used for financial calculations demonstrated in this book using the HP-12C calculator.
HP 12 C
n i PV PMT FV CHS 7 8 9
Yx 1/x % T ∆% % EEX 4 5 6
−
R/S X≷Y CLX NE 1 2 3
T E
RCL R 0 Σ+
STO
Figure 3.2 HP-12C Calculator Key Pad Source: Author’s own work.
The keyboard of the HP-12C calculator is reproduced in Figure 3.2. The calculator keyboard (not shown in this figure) has three colors: white, gold, and blue. Two keys, one labeled “f’’ (gold) and the other “g” (blue), match the colors of the key- board. These keys (shown above the key or the face of a key in the appropriate color code) access specific operation modes of the calculator. The power key is labeled “ON.” When the [ON] key is pressed, several zeros and a decimal point (0.00) appear on the window. The calculator can be cleared by pressing [f] fol- lowed by [CLX]. The gold lettering above CLX reads REG. Above this there is a gold bar with the word CLEAR. Knowing that the gold [f] accesses the gold operation mode, it is possible to recognize that [f] [CLX] “clears the register (display).”
To set the number of decimal places, press [f] and the number of decimal places desired. For example, [f] [4] results in 0.0000; pressing [f]2 gives you 0.00.
The value of a $1,000 CD earning 10 percent and maturing in 10 years can be calculated by using the HP-12C calculator, as demonstrated in Table 3.1.
In a present/future value problem, four variables are involved, namely (a) pres- ent value, (b) future value, (c) interest, and (d) number of years (periods). If any three variables are known, the value of the fourth variable can be calculated.
Rule of 72
“The Rule of 72 provides a speedy method for determining the time or the inter- est rate required for an investment (or any index) to double in value.” This rule, which is an estimate, is now explained.
In order to determine the number of years for an investment to double, 72 is divided by the annual interest rate. For example, if the annual interest rate is
8 percent, the time required for an investment to double in value is 72/8, or nine years. At 5 percent interest, the investment would double in 14.4 years (72/5).
Looking at it another way, the interest rate required for an investment to double over a specified number of years can be determined by dividing 72 by the desired number of years. For instance, if Betty Johnson wishes to double her investment in 10 years, over that period she must invest the principal to earn an interest rate of 7.2 percent (72/10), compounded annually.
Multiple Compound Periods Per Year
The preceding section presented the case of annual compounding. Each year consisted of only one compound period. Often, financial institutions use more frequent compounding periods. For example, semiannually compounding means that there are two compound periods per year, monthly compounding means 12 compound periods per year, and daily compounding indicates 365 compound periods per year.
Equation 3.2 can be modified to calculate the compound value of $1,000 after n years if invested at r percent, and the amount is compounded over multiple periods per year (m):
FV = PV (1 + rnom/m)mn 3.3
Table 3.1 Finding the Future Value (Annual Compounding) Using a Calculator Step 1: Identify known variables:
Future value: $1,000
Interest: 10%
Number of periods: 10 years Step 2: Clear calculator [f] [CLX]
Step 3: Enter data
Press [1] [0] [0] [0] [CHS] [PV]
[10] [i]
[10] [n]
[FV]
Answer: $2,593.74 Source: Author’s own work.
Note: [CHS] key is used when money is paid out. It is not used when money is received.
Table 3.2 Finding the Future Value (Quarterly Compounding) Using a Calculator Step 1: Identify known variables:
Present value: $1,000
Interest: 10%/quarterly (4) Number of periods: 2 years × quarterly (4) Step 2: Clear calculator [f] [CLX]
Step 3: Enter data
Press [1] [0] [0] [0] [CHS] [PV]
[10] E N T E R
[4] [÷] [i]
[2] E N T E R
[4] [X] [n]
[FV]
Answer: $1,218.40 Source: Author’s own work.
where
rnom = Declared interest rate, or nominal interest rate m = Number of compounding periods per year
n = Total number of years.
Using Equation 3.3 we can calculate the FV for $1,000 after two years, which is invested at 10 percent, compounded quarterly:
FV = $1,000 (1 + 0.10/4)4 × 2
= $1,218.40
The same answer could be obtained by using the HP-12C calculator, as demon- strated in Table 3.2.
Note that, as expected, the quarterly compounding resulted in a higher value of
$1,218.40, as compared to the annual compounding value of $1,210.00. Clearly, the more frequent the compounding period, the larger the final compound amount. That is because interest is earned on interest more often. Values of
other compounding periods (PV = $1,000, r = 10 percent, n = 2 years) are given next:
Type of Compounding Future Value
Annual $1,210.00
Semiannual 1,215.51
Quarterly 1,218.40
Daily 1,221.37
These relationships are illustrated in Figure 3.3.