Compound Interest
2.6 Future Value: Formula Approach
A. Introduction
In the study of compound interest, future value is the total amount that an initial sum, whether invested or borrowed, will accumulate over time at a specific compound interest rate. This includes the original principal and all the interest that accrues, compounded at set intervals. Present value, on the other hand, is the current equivalent of a future sum of money. It’s the initial amount that, when interest is applied over a period, will grow to a predetermined future value.
The future value of compound interest [asciimath]FV[/asciimath] is determined by
[asciimath]FV=PV(1+i)^N[/asciimath] Formula 2.4a
where [asciimath]PV[/asciimath] represents the present value [asciimath]i[/asciimath] is the periodic interest rate (calculated as [asciimath]i=(I//Y)/(C//Y)[/asciimath], Formula 2.1a) and [asciimath]N[/asciimath] is the number of compounding periods in the term (computed as [asciimath]N=C//Y*t[/asciimath], Formula 2.2a).
B. Computing Interest Amount
The future value of an investment or loan includes both the initial principal and the interest earned or charged over the term. To find the total amount of interest accrued, you subtract the present value (initial amount) from the future value. This interest amount (I), often called compound interest (CI) in the context of compound interest calculations, is given by
[asciimath]I=FV-PV[/asciimath] Formula 2.3
where FV is the future value and PV is the present value. This formula was initially introduced in Section 2.2.
Sarah invested $25,000 for 5 years at 6.9% compounded monthly. (a) How much was the accumulated value after 5 years? (b) How much interest was earned during the investment term?
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Given information:
- Present value: [asciimath]PV = $25,000[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 6.9%[/asciimath]
- Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
- Investment Term: [asciimath]t = 5[/asciimath] years
- Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (6.9%)/12 =0.575%[/asciimath]
- Number of compounding periods in the term: [asciimath]N =C//Y*t=12 (5) = 60[/asciimath]
a)
Substituting the values into Formula 2.4a yields
[asciimath]FV=PV(1+i)^N[/asciimath]
[asciimath]FV=25000(1+0.575%)^60[/asciimath]
[asciimath]=25000(1.410595...)[/asciimath]
[asciimath]=35264.885920...[/asciimath]
[asciimath]~~ $35,264.89[/asciimath] (Rounded to the nearest cents)
b) Substituting the PV and FV values into Formula 2.3 gives the amount of interest earned.
[asciimath]I=FV-PV[/asciimath]
[asciimath]I=35,264.89-25,000[/asciimath]
[asciimath]=$10,264.89[/asciimath]
Ontario Servers Inc. borrows $65,400 from a bank at 5.68% compounded quarterly for 4 years and 7 months. a) How much will the accumulated value of the loan be at the end of the term? b) How much interest will be charged on the loan?
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Given information:
- Present value: [asciimath]PV = $65,400[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 5.68%[/asciimath]
- Interest is compounded quarterly so [asciimath]C//Y = 4[/asciimath]
- Loan term: [asciimath]t =[/asciimath] 4 years and 7 months [asciimath]=4 + 7/12=4.58bar(3)[/asciimath] years (The ‘overbar’ indicates that the digit beneath it is repeated continuously)
- Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (5.68%)/4 =1.42%[/asciimath]
- Number of compounding periods in the term: [asciimath]N =C//Y*t=4(4.58bar(3)) = 18.bar(3)[/asciimath]
a)
Substituting the values into Formula 2.4a yields
[asciimath]FV=PV(1+i)^N[/asciimath]
[asciimath]FV=65400(1+1.42%)^(18.bar(3))[/asciimath]
[asciimath]=65400(1.294989...)[/asciimath]
[asciimath]=84,692.287...[/asciimath]
[asciimath]~~ $84,692.29[/asciimath] (Rounded to the nearest cents)
b) Substituting the PV and FV values into Formula 2.3 yields the amount of interest charged.
[asciimath]I=FV-PV[/asciimath]
[asciimath]I=84,692.29-65,400[/asciimath]
[asciimath]=$19,292.29[/asciimath]
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C. Time Given as Specific Dates
As we saw in the simple interest chapter, sometimes the beginning and end dates of the loan/investment term are given. In that case, unless otherwise stated, the time calculation will be most accurate if the number of days between the given dates is computed and then converted to equivalent years. To convert the number of days to the number of years, divide it by 365 as there are 365 days in a year. There are several ways to compute the days between two dates (DBD), some of which are presented in section 1.1.
A sum of $36,200.00 was deposited into an account on June 30, 2020. What will be the future value of this sum of money on October 9, 2025, if the interest rate is 2.22% compounded annually?
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The given information:
- Present value: [asciimath]PV = $36,200.00[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 2.22%[/asciimath]
- Interest is compounded semi-annually so [asciimath]C//Y = 1[/asciimath]
- Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (2.22%)/1 =2.22%[/asciimath]
- Beginning date: June 30, 2020
- End date: October 9, 2025
I. We first need to find the number of days in the investment and find its equivalent years to find time (t). There are 1927 days between the given dates (DBD). We can find time in years by dividing DBD by 365.
[asciimath]t = 1927/365 = 5.279452...[/asciimath] years
[asciimath]N = C//Y*t= 1(1927/365) =[/asciimath][asciimath]5.279452...[/asciimath] keeping at least six decimal places for intermediate values
II. We substitute the values into Formula 2.4a gives
[asciimath]FV=PV(1+i)^N[/asciimath]
[asciimath]FV=36200(1+2.22%)^5.279452[/asciimath]
[asciimath]=36200(1.122290...)[/asciimath]
[asciimath]=40,649.272124...[/asciimath]
[asciimath]~~ $40,649.27[/asciimath] (Rounded to the nearest cents)
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D. Change in the Interest Rate
Sometimes during the term of an investment (or a loan), there is a change in one of the variables such as the interest rate or principal amount. In those cases, we should split the term into separate time segments at the point of change. The following example illustrates how the future value is computed when there is a change in the interest rate.
Pan’s Consulting Inc. invested $84,000 in a mutual fund at 6.5% compounded semi-annually. After 4 years the interest rate was changed to 4.44% compounded quarterly.
(a) How much was the value of the fund 3 years after the rate change?
(b) How much was the total compound interest earned during 7 years?
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Since there is a change in the interest rate sometime in the middle of the term, we will break the term into two periods at the time of rate change, which is year 4. Then we will calculate the future value of the given principal of $84,000 at year 4. Then, since the money remains in the account, the future value of the first period ([asciimath]t_1[/asciimath]) will become the present value of the second period ([asciimath]t_2[/asciimath]). Finally, we use the new interest rate to calculate the future value of the second period.
- Present value: [asciimath]PV_1 = $84,000[/asciimath]
- Nominal interest rate: [asciimath]I//Y_1 = 6.5%[/asciimath]
- Interest is compounded semi-annually so [asciimath]C//Y_1 = 2[/asciimath]
- Time period: [asciimath]t_1=4[/asciimath] years
- Periodic interest rate: [asciimath]i_1= (I//Y_1)/(C//Y_1) = (6.5%)/2 =3.25%[/asciimath]
- Number of compounding periods in the term: [asciimath]N_1 =C//Y_1*t_1=2 (4) = 8[/asciimath]
Substituting the values into Formula 2.4a yields
[asciimath]FV_1=PV_1(1+i_1)^(N_1)[/asciimath]
[asciimath]FV_1=84,000(1+3.25%)^8[/asciimath]
[asciimath]=108,492.512...[/asciimath] (DO NOT round off the intermediate values)
Second period (after the rate change): 3 years
- Present value: [asciimath]PV_2 = FV_1=108,492.512...[/asciimath]
- Nominal interest rate: [asciimath]I//Y_2 = 4.44%[/asciimath]
- Interest is compounded semi-annually so [asciimath]C//Y_2 = 4[/asciimath]
- Time period: [asciimath]t_2=3[/asciimath] years
- Periodic interest rate: [asciimath]i_2= (I//Y_2)/(C//Y_2) = (4.44%)/4 =1.11%[/asciimath]
- Number of compounding periods in the term: [asciimath]N_2 =C//Y_2*t_2=4 (3) = 12[/asciimath]
Substituting the values into Formula 2.4a yields
[asciimath]FV_2=PV_2(1+i_2)^(N_2)[/asciimath]
[asciimath]FV_2=108,492.512... (1+1.11%)^12[/asciimath]
[asciimath]=123,859.434...[/asciimath]
[asciimath]~~ $123,859.43[/asciimath] (Rounded to the nearest cents)
b) The total interest earned during the 7-year term is the difference between the original principal ([asciimath]PV_1[/asciimath]) and the final future value ([asciimath]FV_2[/asciimath]):
[asciimath]I=FV_2-PV_1[/asciimath]
[asciimath]I=123,859.43-84,000[/asciimath]
[asciimath]=$39,859.43[/asciimath]
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E. Change in the Principal
Just like adjustments made for changes in interest rates, if additional contributions or repayments are made during the period, we need to divide the period into smaller segments that correspond with these transactions. At each transaction point, we calculate the future value up to that moment and then adjust the principal by adding the contribution (if it’s an investment) or subtracting the repayment (if it’s a loan). After this adjustment, we compute the future value based on the updated principal for the next segment. The example below illustrates the process for calculating the future value when there are changes to the principal amount during the investment or loan term.
Latasha initially borrowed $8,700 from RBC Bank at 3.59% compounded monthly. After 2 years she repaid $3,045. If she pays off the debt 9 years after the $8,700 was initially borrowed, how much should her final payment be to clear the debt completely?
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Since there is a repayment sometime during the term, we break the term into two periods at the time of repayment, which is in year 2. We calculate the future value of the loan principal of $8,700 in year 2. Then, we find the new principal by deducting the repayment from the future value. This new principal will become the present value of the second period ([asciimath]t_2[/asciimath]). Finally, we calculate the future value of the second period.
Term 1 (before the repayment): 2 years
- Present value: [asciimath]PV_1 = $8,700[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 3.59%[/asciimath]
- Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
- Periodic interest rate: [asciimath]i= (I//Y)/(C//Y) = (3.59%)/12 =0.2991bar(6)%[/asciimath] (The ‘overbar’ indicates that the digit beneath it is repeated continuously.)
- Time period: [asciimath]t_1=2[/asciimath] years
- Number of compounding periods in the term 1: [asciimath]N_1 =C//Y*t_1=12 (2) = 24[/asciimath]
Substituting the values into Formula 2.4a gives the future value at the time of repayment:
[asciimath]FV_1=PV_1(1+i)^(N_1)[/asciimath]
[asciimath]FV_1=8700(1+0.2991bar(6)% )^24[/asciimath]
[asciimath]=9346.629888...[/asciimath] (DO NOT round off the intermediate values)
Term 2 (after the repayment): 7 years
- Present value (cash inflow): [asciimath]PV_2 = 9346.629888-3,045=6301.629888...[/asciimath] (DO NOT round off the intermediate values)
- Same interest rate
- Time period: [asciimath]t_2=7[/asciimath] years
- Number of compounding periods in the term 1: [asciimath]N_2 =C//Y*t_2=12 (7) = 84[/asciimath]
Substituting the values into Formula 2.4a gives the future value at the end of the loan term:
[asciimath]FV_2=PV_2(1+i)^(N_1)[/asciimath]
[asciimath]FV_1=6301.629... (1+0.2991bar(6)% )^24[/asciimath]
[asciimath]=8,098.939...[/asciimath]
[asciimath]~~ $8,098.94[/asciimath] (Rounded to the nearest cents)
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Section 2.6 Exercises
- Indiana Trust Co borrows $15,700 from a bank at 6.4% compounded annually for 7 years and 9 months. a) How much will the accumulated value of the loan be at the end of the term? b) How much interest will be charged on the loan?
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a) FV = $25,392.02
b) I = $9,692.02
- Nevaeh deposited $12,600 in a savings account at 6.8% compounded annually for 6 years and 10 months. a) Calculate the accumulated value of this amount at the end of the term. b) Calculate the amount of compound interest earned.
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a) FV = $19,751.83
b) I = $7,151.83
- Hillary invested $29,400 in a mutual fund at 2.7% compounded semi-annually. After 4 years, the interest rate was changed to 6.1% compounded quarterly. a) How much was the value of the fund 8 years after the rate change? b) How much was the total compound interest earned during the whole 12-year term?
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a) FV = $53,121.86
b) I = $23,721.86
- Erika initially borrowed $46,700 from a bank at 4.3% compounded monthly. After 2 years she repaid $8,651. If she pays off the debt 9 years after the loan was initially borrowed, how much should the final payment be to clear the debt completely?
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Final payment = $57,037.81
- A sum of $13,480 was deposited into an account on June 14, 2013. What will be the future value of this sum of money on August 21, 2016, if the interest rate is 6.77% compounded semi-annually?
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FV = $16,668.68
