Compound Interest

2.2 Future Value: Calculator 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 is the initial amount that, when interest is applied over a period, will grow to a predetermined future value.

Although solving compound interest problems algebraically improves your understanding of the subjects and the steps involved, if you have to regularly solve such problems, financial calculators offer a faster and more convenient way as they have all the formulas pre-programmed in them.

In this chapter, we will solve problems using the BAII Plus financial calculator. Alternatively, you can use this online calculator as an alternative to the BAII Plus.

TVM Worksheet

To solve compound interest problems using a financial calculator, we use its time-value of money (TVM) worksheet, as shown in Figure 2.2.1.

TVM worksheet on a financial calculator (Texas Instrument BA II Plus) displaying input variables including number of payments in the term (N), nominal interest rate per year (I/Y), present value (PV), periodic payment (PMT), and future value(FV). Pressing the 2ND key followed by I/Y opens the P/Y-C/Y worksheet.

Figure 2.2.1 Keys on the TVM worksheet on a financial calculator

Cash-Flow Sign Convention

When using a financial calculator, you might notice that the signs of FV and PV are different (one negative and another positive). That is because the calculator uses the following cash flow sign convention, which is also summarized in Table 2.2.1.

  • Cash inflow: when money is received, cash is flowing in (to your pocket), thus the amount should be entered as a positive value.
  • Cash outflow: when money is paid out (flowing out of your pocket), the amount is entered as a negative value.

Table 2.2.1 Financial Calculator Cash-Flow Sign Convention

PV

FV
Investments

Outflow ([asciimath]-[/asciimath])

 Inflow ([asciimath]+[/asciimath])
Loans

Inflow ([asciimath]+[/asciimath])

 Outflow ([asciimath]-[/asciimath])

Note that if try to do calculations with both PV and FV entered with the same sign in the TVM Worksheet, the calculator throws an error, “Error 5”.

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.

It’s important to note that when working with algebraic formulas, such as Formula 2.3 for calculating the amount of interest, both the present value (PV) and future value (FV) are treated as positive values. This approach differs from the cash flow sign convention used in financial calculators, where the direction of cash flow (inflow or outflow) is indicated by the sign (positive or negative) of the values entered. For algebraic calculations, simply use positive values for both PV and FV, regardless of the cash flow direction.

 

Example 2.2.1: Compute FV – Time Given in Years

Sarah invested $25,000 for 5 years at 6.8% 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:

  • For investments, present value is a cash outflow: [asciimath]PV = -$25,000[/asciimath]
  • Nominal interest rate: [asciimath]I//Y  = 6.8%[/asciimath]
  • Interest is compounded monthly, so [asciimath]C//Y = 12[/asciimath]
  • For compound problems, [asciimath]P//Y=C//Y[/asciimath], so [asciimath]P//Y=12[/asciimath]
  • Investment term: [asciimath]t = 5[/asciimath] years
  • Number of compounding periods in the term: [asciimath]N =C//Y*t=12 (5) = 60[/asciimath]

a) Note that for investments, the present value (PV) is a cash outflow and is entered in the calculator as a negative value. Also, the nominal interest rate (I/Y) is entered as a percent without the percent symbol.

The TVM Worksheet with input value to compute FV. Before entering new values, clear TVM worksheet by pressing 2ND key followed by FV key. Once the given values are entered to their corresponding keys, press the "Compute" key followed by the FV key.

Therefore, the future value is $35,090, 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=35,090-25,000[/asciimath]

 [asciimath]=$10,090[/asciimath]

 

Example 2.2.2: Compute FV – Time Given in Years and Months

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:

  • In loans, present value is a cash inflow: [asciimath]PV = $65,400[/asciimath]
  • Nominal interest rate: [asciimath]I//Y = 5.68%[/asciimath]
  • Interest is compounded quarterly, so [asciimath]C//Y = 4[/asciimath]
  • For compound problems, [asciimath]P//Y=C//Y[/asciimath], so [asciimath]P//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)
  • Number of compounding periods in the term: [asciimath]N =C//Y*t=4(4.58bar(3)) = 18.bar(3)[/asciimath]

a) Compute [asciimath]FV[/asciimath]

The TVM Worksheet with input value to compute FV. Once the given values are entered to their corresponding keys, press the "Compute" key followed by the FV key.

Therefore, the accumulated value of the loan is $84,692.29, 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 or 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 1.2.

Here we use the DATE worksheet of the financial calculator to compute DBD. The worksheet is the secondary function of key 1, so it is opened by pressing 2ND and then key 1. Figure 2.2.2 shows the keys on the DATE worksheet on a financial calculator.

The keys on the DATE worksheet on a financial calculator including DT1 for the beginning date, DT2 for the end date, and DBD for the number of days between the dates. To access the Date worksheet, press the 2ND key followed by "1" key.

Figure 2.2.2: The keys on the DATE worksheet on a financial calculator

 

How to Calculate Days Between Dates (DBD) using a Calculator

1. Activate the DATE worksheet by pressing the “2ND” function key followed by “1”.

2. Input the initial date (DT1) using the MM.DDYY format. This means you should type the month (MM) followed by a decimal point, then the day (DD) as a two-digit number, and the last two digits of the year (YY). For instance, enter “11.0308” for November 3, 2008. After pressing “ENTER”, the calculator will display the date in the MM-DD-YYYY format.

Remember to use two digits for the day (DD) even if it’s a single-digit day. For example, input “03” for the third day of the month.

3. Navigate to the second date (DT2) using the down arrow key, and enter the second date in the same MM.DDYY format, and then press “ENTER”.

4. To calculate the number of days between the two dates, move down to “DBD” with the down arrow key and press the “CPT” (compute) button to get the result.

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Example 2.2.3: Compute FV Given Specific Dates

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 (cash outflow): [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]
  • For compound problems, [asciimath]P//Y=C//Y[/asciimath], so [asciimath]P//Y=1[/asciimath]
  • Beginning date: June 30, 2020
  • End date: October 9, 2025

I. We first need to use the DATE worksheet to find the number of days in the investment and find its equivalent years to find time (t). Enter the dates in the MM.DDYY format and compute DBD.

  • DT1: 06.3020
  • DT2: 10.0925

DATE worksheet with DT1 and DT2 values entered and DBD computed.

DBD is 1927 days and 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 then enter the given values in the TVM worksheet and compute FV.

The TVM Worksheet with input value to compute FV. Once the given values are entered to their corresponding keys, press the "Compute" key followed by the FV key.

Therefore, the future value is $40,649.27.

 

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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.

 

Example 2.2.4: Compute FV with 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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Timeline image representing a 7-year term for a compound interest calculation, divided into two periods at the 4-year mark when the interest rate changes. The first period covers the initial 4 years, where the future value (FV1) of $84,000 is calculated. The second period starts with this amount, now considered as the present value (PV2), for the remaining 3 years. The end of the 7th year is marked with a question mark, highlighting the objective to calculate the final future value based on FV2.

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.

Given information:

First period: 4 years

  • Present value (cash outflow): [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]
  • For compound problems, [asciimath]P//Y=C//Y[/asciimath], so [asciimath]P//Y_1=2[/asciimath]
  • Time period: [asciimath]t_1=4[/asciimath] years
  • Number of compounding periods in the term: [asciimath]N_1 =C//Y_1*t_1=2 (4) = 8[/asciimath]

Second period (after the rate change): 3 years

  • Present value (cash outflow): [asciimath]PV_2 = FV_1=-108,492.513[/asciimath]
  • Nominal interest rate: [asciimath]I//Y_2 = 4.44%[/asciimath]
  • Interest is compounded semi-annually so [asciimath]C//Y_2 = 4[/asciimath]
  • For compound problems, [asciimath]P//Y=C//Y[/asciimath], so [asciimath]P//Y_2=4[/asciimath]
  • Time period: [asciimath]t_2=3[/asciimath] years
  • Number of compounding periods in the term: [asciimath]N_2 =C//Y_2*t_2=4 (3) = 12[/asciimath]
a) Compute [asciimath]FV[/asciimath]

Two TVM Worksheet visuals for calculating Future Values FV1 and FV2 over separate terms. The first worksheet covers a 4-year term, detailing inputs for FV1 calculation. The second, for a 3-year term, uses FV1 as Present Value (PV2) with a negative sign for cash outflow, showing steps to compute FV2. Instructions include entering each given value, pressing corresponding key, and finalizing with the 'Compute' and FV keys for each future value calculation.

Therefore, the future value at the end of the second period is $123,859.43, 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.

 

Example 2.2.5: Compute FV with Change in principal

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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A timeline illustrating a 9-year compound interest calculation, segmented into two distinct phases. The first phase spans the initial 2 years, during which the future value (FV1) of $8,700 is determined. Following this period, a repayment is made, leading to the second phase. This phase begins with the new principal amount, which is FV1 minus the repayment, now labeled as the present value (PV2). It covers the remaining 7 years of the term. The end of the 9th year is marked with a question mark, highlighting the objective to calculate the final future value based on PV2.

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.

Given information:

Term 1: 2 years

  • Present value (cash inflow): [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]
  • For compound problems, [asciimath]P//Y=C//Y[/asciimath], so [asciimath]P//Y=12[/asciimath]
  • 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]

Term 2: 7 years

  • Present value (cash inflow): [asciimath]PV_2 = 9346.629888-$3,045=$6301.629888[/asciimath] (keep at least 6 decimal places for intermediate values)
  • Nominal interest rate: [asciimath]I//Y = 3.59%[/asciimath]
  • Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
  • For compound problems, [asciimath]P//Y=C//Y[/asciimath], so [asciimath]P//Y=12[/asciimath]
  • 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]

Compute [asciimath]FV[/asciimath]

 Two separate TVM Worksheet calculations to determine Future Values FV1 and FV2. The first worksheet is set up for a 2-year term (up to the repayment time), with specific input values keyed in for this duration. It shows the computation process for FV1. The second worksheet demonstrates the subsequent 7-year term. Here, the repayment is deducted from the previously calculated FV1 and the balance is input as Present Value (PV1), marked with a positive sign to indicate a cash inflow. The worksheet then illustrates the process for calculating the final Future Value, FV2.

 

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Section 2.2 Exercises

  1. 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

  2. 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

  3. 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?
    Show/Hide Answer

     

    a) FV = $53,121.86

    b) I = $23,721.86

  4. 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?
    Show/Hide Answer

     

    Final payment = $57,037.81

  5. 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?
    Show/Hide Answer

     

    FV = $16,668.68

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