Compound Interest
2.4 Interest Rate: Calculator Approach
A. Computing the Periodic and Nominal Interest Rates
If the nominal interest rate of a loan or investment is not specified, we can determine it by using the present value, the future value, and the number of compounding periods in the term.
In the calculator approach, the nominal interest rate [asciimath]I//Y[/asciimath]can be computed directly. Then the periodic interest rate [asciimath]i[/asciimath] can be determined by [asciimath]i=(I//Y)/(C//Y)[/asciimath] (Formula 2.1a).
Nina invested $4,900 in an account that grew to $33,500 over a period of 9 years. Assume that the interest in the account was compounded monthly. a) What was the nominal interest rate of the account? b) What was the interest rate per month (periodic interest rate)? Give your answer as a percentage rounded to two decimal places.
Show/Hide Solution
Given information:
- For investment, present value is a cash outflow: [asciimath]PV = -$4900[/asciimath]
- For investment, future value is a cash inflow: [asciimath]FV = $33,500[/asciimath]
- Interest was compounded monthly so [asciimath]C//Y = 12[/asciimath]
- Investment Term:[asciimath]t = 9[/asciimath]
- Number of compounding periods in the term: [asciimath]N = C//Y*t[/asciimath] [asciimath]=12(9)=108[/asciimath]
In the calculator approach, we first compute the nominal interest rate, and then we will find the periodic rate. Note that for investments, PV is considered a cash outflow and entered as a negative value, while FV is a cash inflow and should be entered as a positive value.
a)
Therefore, the nominal interest rate is 21.55% compounded monthly.
b) The periodic (monthly) interest rate [asciimath]i[/asciimath] can be determined by Formula 2.1a:
[asciimath]i=(I//Y)/(C//Y)[/asciimath]
[asciimath]i=(21.55022167%)/12[/asciimath]
[asciimath]=1.795851...%[/asciimath]
[asciimath]~~1.80%[/asciimath]
Try an Example
B. Calculating Effective Interest Rate
The effective interest rate refers to the nominal interest rate that is compounded annually. The effective interest rate is used to make it easier to compare the annual interest rates between loans and investments with different compounding periods.
If the compounding frequency is already one, the given nominal rate is the effective interest rate. Otherwise, we can again use either a formula or a financial calculator to obtain the effective interest rate.
The ICONV (I conversion) worksheet on a financial calculator can be used to compute the effective interest rate. The worksheet is the secondary function of key 2, so it is opened by pressing 2ND and then key 2. Figure 2.4.1 shows the keys on the ICONV worksheet on a financial calculator.
Figure 2.4.1 The keys on the ICONV worksheet on a financial calculator
Find the effective rate of interest of an investment that earns 6.4% compounded quarterly. Express your answer as a percent rounded to two decimal places.
Show/Hide Solution
- Interest is compounded quarterly so [asciimath]C//Y = 4[/asciimath]
- Nominal interest rate: [asciimath]NOM = 6.4%[/asciimath]
Therefore, the effective interest rate is 6.56% compounded annually.
Try an Example
C. Calculating Equivalent Interest Rates
Equivalent interest rates are nominal interest rates with different compounding periods that result in the same future value given the same principal value and time. We may want to find an equivalent interest rate of a given rate with different compounding periods for comparison purposes. Similar to the effective interest rate discussed in the previous section, we can use either a formula or a financial calculator to compute the equivalent interest rates.
The ICONV worksheet on a financial calculator can also be used to compute the equivalent interest rates. To find the equivalent interest rate of a given nominal rate and frequency, first, convert the original nominal rate to the effective interest rate (compounded annually), and then use the obtained effective interest rate to compute the equivalent nominal interest rate at the desired compounding frequency.
Convert the interest rate of 7.42% compounded quarterly to an equivalent interest rate compounded monthly. Express your answer as a percent rounded to two decimal places.
Watch Video
Show/Hide Solution
- Original compounded frequency is quarterly so [asciimath]C//Y_1 = 4[/asciimath]
- Original nominal interest rate: [asciimath]NOM_1 = 7.42%[/asciimath]
- Desired compounded frequency is monthly so [asciimath]C//Y_2 = 12[/asciimath]
Thus, the interest rate of 7.37% compounded monthly is equivalent to the effective interest rate of 7.63% (compounded annually) and 7.42% compounded quarterly. That means they all will yield the same future value over the same period given the same principal value.
Try an Example
Section 2.4 Exercises
- At what nominal rate of interest compounded quarterly will money grow from $18,000.00 to $20,881.85 in 4 years? Round the value of I/Y to two decimal places.
Show/Hide Answer
I/Y = 3.73% compounded quarterly
- Suppose $11,500.00 is deposited into an account today, and it is expected to grow to a maturity value of $13,998.44 in 8 years from now. What is the account nominal interest rate compounded monthly? Round the answer to two decimal places.
Show/Hide Answer
I/Y = 2.46%
- Find the effective rate of interest of an investment that earns 6.34% compounded quarterly. Round your answer to two decimal places.
Show/Hide Answer
EFF = 6.49% compounded annually
- Convert the interest rate of 4.59% compounded monthly to an equivalent interest rate compounded semi-annually. Round your answer to two decimal places.
Show/Hide Answer
I/Y = 4.63% compounded semi-annually
