Chapter 3

Mathematics of FinanceSection 2Compound and Continuous Compound Interest

Barnett/Ziegler/Byleen Finite Mathematics 12e

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Compound Interest Unlike simple interest, compound interest on an amount accumulates at a faster rate than simple interest. The basic idea is that after the first interest period, the amount of interest is added to the principal amount and then the interest is computed on this higher principal. The latest computed interest is then added to the increased principal and then interest is calculated again. This process is completed over a certain number of compounding periods. The result is a much faster growth of money than simple interest would yield.

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Example Suppose a principal of $1.00 was invested in an account paying 6% annual interest compounded monthly. How much would be in the account after one year?

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SolutionSolution: Using the simple interest formula A = P (1 + rt) we obtain:amount after one monthafter two monthsafter three months

After 12 months, the amount is 1.00512 = 1.0616778.With simple interest, the amount after one year would be 1.06.The difference becomes more noticeable after several years.

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Graphical Illustration ofCompound Interest Growth of 1.00 compounded monthly at 6% annual interest over a 15 year period (Arrow indicates an increase in value of almost 2.5 times the original amount.)

Chart1

1

1.012042028

1.0242290664

1.0365628615

1.0490451805

1.0616778119

1.0744625658

1.0874012741

1.1004957906

1.1137479917

1.1271597762

1.1407330658

1.1544698053

1.168371963

1.1824415308

1.1966805248

1.2110909852

1.2256749767

1.2404345891

1.2553719371

1.2704891611

1.2857884271

1.3012719273

1.3169418803

1.3328005313

1.3488501525

1.3650930438

1.3815315325

1.3981679738

1.4150047517

1.4320442785

1.4492889958

1.4667413744

1.4844039151

1.5022791485

1.5203696361

1.5386779698

1.5572067729

1.5759587005

1.5949364392

1.6141427085

1.6335802601

1.6532518793

1.6731603847

1.6933086289

1.7136994988

1.7343359161

1.7552208377

1.7763572561

1.7977481999

1.819396734

1.8413059604

1.8634790183

1.8859190848

1.9086293752

1.9316131435

1.954873683

1.9784143266

2.0022384473

2.0263494587

2.0507508156

2.0754460143

2.1004385932

2.1257321335

2.1513302594

2.1772366385

2.2034549831

2.2299890496

2.2568426401

2.2840196024

2.3115238303

2.339359265

2.3675298947

2.3960397559

2.4248929337

2.4540935622

Growth of 1.00 compounded monthly at 6% annual interest over a 15 year period

Sheet1

01

0.21.012042028

0.41.0242290664

0.61.0365628615

0.81.0490451805

11.0616778119

1.21.0744625658

1.41.0874012741

1.61.1004957906

1.81.1137479917

21.1271597762

2.21.1407330658

2.41.1544698053

2.61.168371963

2.81.1824415308

31.1966805248

3.21.2110909852

3.41.2256749767

3.61.2404345891

3.81.2553719371

41.2704891611

4.21.2857884271

4.41.3012719273

4.61.3169418803

4.81.3328005313

51.3488501525

5.21.3650930438

5.41.3815315325

5.61.3981679738

5.81.4150047517

61.4320442785

6.21.4492889958

6.41.4667413744

6.61.4844039151

6.81.5022791485

71.5203696361

7.21.5386779698

7.41.5572067729

7.61.5759587005

7.81.5949364392

81.6141427085

8.21.6335802601

8.41.6532518793

8.61.6731603847

8.81.6933086289

91.7136994988

9.21.7343359161

9.41.7552208377

9.61.7763572561

9.81.7977481999

101.819396734

10.21.8413059604

10.41.8634790183

10.61.8859190848

10.81.9086293752

111.9316131435

11.21.954873683

11.41.9784143266

11.62.0022384473

11.82.0263494587

122.0507508156

12.22.0754460143

12.42.1004385932

12.62.1257321335

12.82.1513302594

132.1772366385

13.22.2034549831

13.42.2299890496

13.62.2568426401

13.82.2840196024

142.3115238303

14.22.339359265

14.42.3675298947

14.62.3960397559

14.82.4248929337

152.4540935622

Sheet1

Growth of 1.00 compounded monthly at 6% annual interest over a 15 year period

Sheet2

Sheet3

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This formula can be generalized to where A is the future amount, P is the principal, r is the interest rate as a decimal, m is the number of compounding periods in one year, and t is the total number of years. To simplify the formula, General Formula In the previous example, the amount to which 1.00 will grow after n months compounded monthly at 6% annual interest is

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Compound Interest General Formula where i = r/mA = amount (future amount) at the end of n periodsP = principal (present value)r = annual nominal ratem = number of compounding periods per yeari = rate per compounding periodt = total number of compounding periods

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ExampleFind the amount to which $1500 will grow if compounded quarterly at 6.75% interest for 10 years.

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ExampleFind the amount to which $1500 will grow if compounded quarterly at 6.75% interest for 10 years. Solution: Use

Helpful hint: Be sure to do the arithmetic using the rules for order of operations.

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Same Problem Using Simple Interest Using the simple interest formula, the amount to which $1500 will grow at an interest of 6.75% for 10 years is given by A = P (1 + rt) = 1500(1+0.0675(10)) = 2512.50 which is more than $400 less than the amount earned using the compound interest formula.

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Changing the number of compounding periods per yearTo what amount will $1500 grow if compounded daily at 6.75% interest for 10 years?

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Changing the number of compounding periods per yearTo what amount will $1500 grow if compounded daily at 6.75% interest for 10 years?

Solution: = 2945.87This is about $15.00 more than compounding $1500 quarterly at 6.75% interest. Since there are 365 days in year (leap years excluded), the number of compounding periods is now 365. We divide the annual rate of interest by 365. Notice, too, that the number of compounding periods in 10 years is 10(365)= 3650.

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Effect of Increasing the Number of Compounding PeriodsIf the number of compounding periods per year is increased while the principal, annual rate of interest and total number of years remain the same, the future amount of money will increase slightly.

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Computing the Inflation RateSuppose a house that was worth $68,000 in 1987 is worth $104,000 in 2004. Assuming a constant rate of inflation from 1987 to 2004, what is the inflation rate?

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Computing the Inflation RateSolutionSuppose a house that was worth $68,000 in 1987 is worth $104,000 in 2004. Assuming a constant rate of inflation from 1987 to 2004, what is the inflation rate?

1. Substitute in compound interest formula.2. Divide both sides by 68,0003. Take the 17th root of both sides of equation 4. Subtract 1 from both sides to solve for r. Solution:

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Computing the Inflation Rate(continued )If the inflation rate remains the same for the next 10 years, what will the house be worth in the year 2014?

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Computing the Inflation Rate(continued )If the inflation rate remains the same for the next 10 years, what will the house be worth in the year 2014? Solution: From 1987 to 2014 is a period of 27 years. If the inflation rate stays the same over that period, r = 0.0253. Substituting into the compound interest formula, we have

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Growth Time of an InvestmentHow long will it take for $5,000 to grow to $15,000 if the money is invested at 8.5% compounded quarterly?

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Growth Time of an InvestmentSolutionHow long will it take for $5,000 to grow to $15,000 if the money is invested at 8.5% compounded quarterly?1. Substitute values in the compound interest formula. 2. divide both sides by 5,0003. Take the natural logarithm of both sides.4. Use the exponent property of logarithms 5. Solve for t. Solution:3 = 1.021254tln(3) = ln1.021254tln(3) = 4t*ln1.02125

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Continuous Compound InterestPreviously we indicated that increasing the number of compounding periods while keeping the interest rate, principal, and time constant resulted in a somewhat higher compounded amount. What would happen to the amount if interest were compounded daily, or every minute, or every second?

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Answer to Continuous Compound Interest QuestionAs the number m of compounding periods per year increases without bound, the compounded amount approaches a limiting value. This value is given by the following formula:

Here A is the compounded amount.

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Example of Continuously Compounded InterestWhat amount will an account have after 10 years if $1,500 is invested at an annual rate of 6.75% compounded continuously?

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Example of Continuously Compounded InterestWhat amount will an account have after 10 years if $1,500 is invested at an annual rate of 6.75% compounded continuously?Solution: Use the continuous compound interest formulaA = Pert with P = 1500, r = 0.0675, and t = 10: A = 1500e(0.0675)(10) = $2,946.05That is only 18 cents more than the amount you receive by daily compounding.

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Annual Percentage YieldThe simple interest rate that will produce the same amount as a given compound interest rate in 1 year is called the annual percentage yield (APY). To find the APY, proceed as follows:This is also called the effective rate. Amount at simple interest APY after one year = Amount at compound interest after one year

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Annual Percentage Yield ExampleWhat is the annual percentage yield for money that is invested at 6% compounded monthly?

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Annual Percentage Yield ExampleWhat is the annual percentage yield for money that is invested at 6% compounded monthly?

General formula:

Substitute values: Effective rate is 0.06168 = 6.168

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Computing the Annual Nominal Rate Given the APY What is the annual nominal rate compounded monthly for a CD that has an annual percentage yield of 5.9%?

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Computing the Annual Nominal Rate Given the APY What is the annual nominal rate compounded monthly for a CD that has an annual percentage yield of 5.9%? 1. Use the general formula for APY. 2. Substitute value of APY and 12 for m (number of compounding periods per year).3. Add one to both sides 4. Take the twelfth root of both sides of equation. 5. Isolate r (subtract 1 and then multiply both sides of equation by 12.

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