Compound Interest
2.8 Interest Rate: Formula 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.
Since the periodic interest rate [asciimath]i[/asciimath] appears in the formula, using the formula approach, we find [asciimath]i[/asciimath] first, and then we calculate the nominal interest rate [asciimath]I//Y[/asciimath] as follows.
Rearranging Formula 2.4a for [asciimath]i[/asciimath] gives
[asciimath]FV=PV(1+i)^N[/asciimath]
[asciimath][/asciimath] [asciimath](1+i)^N=(FV)/(PV)[/asciimath]
[asciimath]1+i=((FV)/(PV) )^(1/N)[/asciimath]
[asciimath]i=((FV)/(PV) )^(1/N)-1[/asciimath]Formula 2.5
The nominal interest rate can be then found by rearranging Formula 2.1a.
[asciimath]I//Y=i*C//Y[/asciimath]Formula 2.1b
Nina invested $4,900 in an account that grew to $33,500 over a period of 9 years. Assuming that the interest in the account was compounded monthly, a) What was the monthly interest rate? b) what was the annual interest rate of the account? Give your answer as a percentage rounded to two decimal places.
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Given information:
- Present value: [asciimath]PV = $4900[/asciimath]
- Future value: [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]
Substituting the values into Formula 2.5 gives the periodic (monthly) interest rate:
a)
[asciimath]i=((FV)/(PV) )^(1/N)-1[/asciimath]
[asciimath]i=((33500)/(4900) )^(1/108)-1[/asciimath]
[asciimath]=1.0179585...\ -1[/asciimath]
[asciimath]=0.0179585...[/asciimath]
[asciimath]=1.79585...%[/asciimath]
[asciimath]~~1.80%[/asciimath] per month
b) The nominal interest rate [asciimath]I//Y[/asciimath] is then given by Formula 2.1b
[asciimath]I//Y=i*C//Y[/asciimath]
[asciimath]=1.79585...% (12)[/asciimath] (Note that the unrounded [asciimath]i[/asciimath] is used here for accuracy)
[asciimath]=21.550221...%[/asciimath]
[asciimath]~~21.55%[/asciimath]
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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.
By definition, given the same principal and time period, the effective interest rate [asciimath]EFF[/asciimath] should result in the same future value as a nominal rate with a given compounding frequency. Therefore, the future values can be made equal, and the equation can be solved for [asciimath]EFF[/asciimath], which yields
[asciimath]EFF=(1+i)^(C//Y)-1[/asciimath]Formula 2.6
where [asciimath]EFF[/asciimath] is the effective interest rate and [asciimath]i[/asciimath] and [asciimath]C//Y[/asciimath] are the stated periodic interest rate and the compounding frequency, respectively.
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.
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Given information:
- Interest is compounded quarterly so [asciimath]C//Y = 4[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 6.4%[/asciimath]
- Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (6.4%)/4 =1.6%[/asciimath]
Substituting the values into Formula 2.6, we obtain
[asciimath]EFF=(1+i)^(C//Y)-1[/asciimath]
[asciimath]EFF=(1+1.6%)^(4)-1[/asciimath]
[asciimath]=0.065552...[/asciimath]
[asciimath]=6.5552...%[/asciimath]
[asciimath]~~6.56%[/asciimath] compounded annually
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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.
By definition, given the same principal value and time period, the equivalent interest rates result in the same future value. Therefore, we can equate the future values and rearrange the equation for an equivalent periodic interest rate [asciimath]i_2[/asciimath] :
[asciimath]i_2=(1+i_1)^((C//Y_1)/(C//Y_2) )-1[/asciimath]Formula 2.7
where [asciimath]i_1[/asciimath] and [asciimath]C//Y_1[/asciimath] are the original periodic interest rate and the compounding frequency, respectively, and [asciimath]C//Y_2[/asciimath] is the compounding frequency of the equivalent interest rate. Note that if [asciimath]C//Y_2=1[/asciimath], Formula 2.7 will become Formula 2.6, and thus [asciimath]i_2[/asciimath] will be the effective interest rate.
Once the periodic interest rate [asciimath]i_2[/asciimath] is known, the equivalent nominal interest rate is then determined by Formula 2.1b.
[asciimath]I//Y_2=C//Y_2*i_2[/asciimath]
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.
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Given information:
- Original compounded frequency is quarterly so [asciimath]C//Y_1 = 4[/asciimath]
- Original nominal interest rate: [asciimath]I//Y_1 = 7.42%[/asciimath]
- Original periodic interest rate: [asciimath]i_1 = (I//Y_1)/(C//Y_1) = (7.42%)/4 =1.855%[/asciimath]
- Desired compounded frequency is monthly so [asciimath]C//Y_2 = 12[/asciimath]
Substituting the values into Formula 2.7 gives
[asciimath]i_2=(1+i_1)^((C//Y_1)/(C//Y_2) )-1[/asciimath]
[asciimath]i_2=(1+1.855%)^(4/12 )-1[/asciimath]
[asciimath]=0.0061454...[/asciimath]
[asciimath]=0.61454...%[/asciimath]
So the equivalent nominal interest rate is obtained by
[asciimath]I//Y_2=C//Y_2*i_2[/asciimath]
[asciimath]I//Y_2=12(0.61454...% )[/asciimath]
[asciimath]=7.374586...%[/asciimath]
[asciimath]~~7.37%[/asciimath] compounded monthly
Therefore, the interest rate of 7.37% compounded monthly is equivalent to 7.42% compounded quarterly, meaning that they will both result in the same future value over the same time period given the same principal value.
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Section 2.8 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.
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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.
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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.
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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.
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I/Y = 4.63% compounded semi-annually
