HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 9.2 Understanding

HAWKES LEARNING

Students Count. Success Matters.

Copyright © 2015 by Hawkes

Learning/Quant Systems, Inc.

All rights reserved.

Section 9.2

Understanding Interest

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Objectives

o Calculate simple interesto Understand future valueo Calculate compound interesto Calculate annual percentage yield

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Understanding Interest

Albert Einstein is reported to have once said, “the most powerful force in the universe is compound interest.” Whether Einstein actually said this or not, it’s a great financial principle to live by. Without attention, compound interest can rapidly become a force of ill intent and create financial burdens we never planned on. Compound interest can also become the means to our financial dreams through a prudent investment.

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Interest

Interest Interest is the amount charged by a lender for borrowing money.

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Principal

Principal The principal is the sum of money on which interest is charged.

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Interest Rate

Interest Rate The interest rate is the amount charged to the borrower expressed as a percentage of the principal.

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Annual Percentage Rate (APR)

Annual Percentage Rate (APR) Annual percentage rate is the yearly interest rate that is charged for borrowing. APR is normally given as a percentage per year.

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Simple Interest Formula

Simple Interest Formula The amount of interest on a simple interest loan with principal P, annual interest rate r (written as a decimal), and loan term of t (usually in years) is calculated with the following formula.

I = Prt

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Example 1: Calculating Simple Interest

Determine the interest that is accrued on $5500 for five years at a rate of 8.5%. Solution We have that the principal is P = $5500 and the interest rate is 8.5%. Note that we need to change the interest rate to a decimal before substituting it in the formula, so r = 0.85. We also have that t = 5. Using our formula, we can determine the amount of interest owed on the $5500 as follows.

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Example 1: Calculating Simple Interest (cont.)

So, if we borrowed $5500 at the given rate of 8.5% for five years, we would owe $2337.50 in interest along with the original principal of $5500 for a total purchase of $7837.50.

I Prt

5500 0.085 5

$2337.50

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Example 2: Calculating Simple Interest on Purchases

Ian is purchasing a new television with a “deal” from Big Screens R Us. The deal offers 90 days same-as-cash to make the purchase. This means that at the end of 90 days, if Ian has paid off the cost of the television, he owes no interest charge. However, if Ian does not pay off the amount, he owes simple interest for the original purchase amount calculated over the entire 90 days. If the price of the television is $2650 (tax included) with an annual interest rate of 21.99%, how much would Ian owe on the 91st day if he made no payments during the first 90 days?

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Example 2: Calculating Simple Interest on Purchases (cont.)

Solution First, we want to determine the entire amount due after the 90 days has expired assuming that Ian has not paid any money toward the amount he owes. This means the principal of $2650 is still due in addition to the amount of interest that accumulated over the 3 months that he borrowed the money. Since the formula for simple interest requires that time t must be given in years, we need to convert three months into a fraction of a year; that is, of a year. 3 1

12 4

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Remember that when using the formula, we need the interest rate to be in decimal form. Therefore, we have that P = 2650, r = 0.2199, and The amount of interest gained over three months can then be calculated as follows.

14 .t

I Prt

1$2650 0.2199

4

$145.68

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So, the amount of interest due at the end of 90 days would be $145.68. Therefore, the total amount Ian must pay for the television on the 91st day is the sum of the principal and the interest.

$2650 + $145.68 = $2795.68

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Skill Check #1

Skill Check #1Find the total cost of a loan for $4320 that has a simple interest rate of 15% for 18 months.

Answers: $5292

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Compound Interest

Compound Interest Compound interest is interest that is computed based on both the principal and the accrued interest as additional principal at each interval. The future value is the total amount of money A that has been accrued after compounding at an annual percentage rate r based on the initial principal P with n compounding intervals per year for t years.

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Compound Interest (cont.)

Compound Interest (cont.)Future value of a compound interest account is calculated with the following formula.

1ntr

A Pn

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Table 1: Compounding Intervals

Table 1: Compounding Intervals

Compounding Number per Year

Annually 1

Semi-Annually 2

Quarterly 4

Monthly 12

Weekly 52

Daily 365

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Example 3: Computing Compound Interest

Lilly deposits $12,000 into an account with an annual interest rate of 4.5% compounded monthly. If she leaves the money in the account for 10 years, what will the future value be at the end of this time period? Solution We can use the compound interest formula to compute the future value for Lilly after 10 years. In her case, P = 12,000, r = 0.045, and t = 10. Since interest is compounded monthly, n = 12. So we have the following.

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Example 3: Computing Compound Interest (cont.)

So, after 10 years, Lilly will have accumulated a total of $18,803.91, which includes her initial investment of $12,000.

1ntr

A Pn

12 100.045

12,000 112

12012,000 1.003 $18,803.91

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Skill Check #2

Skill Check #2 Use the information in Example 3 to find the future value of Lilly's account after 20 years.

Answer: $29,465.60

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Example 4: Comparing Compound Interest for Different Compounding Intervals

Thomas uses $4500 to open an IRA (Individual Retirement Account) savings account that earns 3.8% APR. If he leaves the money alone, what is the future value after eight years for the following compounding intervals? a. The interest is compounded yearly. b. The interest is compounded quarterly. c. The interest is compounded weekly.

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Example 4: Comparing Compound Interest for Different Compounding Intervals (cont.)

Solution a. We have that Thomas’ principal is P = 4500. The

interest rate is r = 0.038 and time is t = 8. If the interest is compounded yearly, then n = 1. Therefore, we have

So, if the interest is compounded yearly, Thomas will have approximately $6064.45 at the end of eight years.

1ntr

A Pn

1 80.0384500 1

1

$6064.45.

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b. This time interest will be compounded quarterly. In other words, n = 4. We still have that P = 4500, r = 0.038, and t = 8. Substituting these into the compound interest formula, we have

This means that Thomas will have $6089.97 after eight years if the interest is compounded quarterly.

1ntr

A Pn

4 80.0384500 1

4

$6089.97.

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c. Finally, if the interest is compounded weekly, n = 52. Substituting this into the formula, we have

So, after eight years, Thomas’ investment has a future value of $6098.03 with interest compounded weekly.

1ntr

A Pn

52 80.0384500 1

52

6098.03.

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You might notice that the more compounding intervals that occur in a given year, the larger the total accumulated amount of money. The other factor that has an impact on the accumulated amount is time. The longer an amount is invested, the larger the accumulated amount. Making sound financial decisions requires that we also understand the power of compounding interest over a long period of time.

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Continuous Compound Interest Formula

Continuous Compound Interest Formula The future value A of a continuous compound interest account after t years at an annual interest rate of r and an initial amount, or principal, P is calculated with the following formula.

A = Pert

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Example 5: Calculating Continuous Compound Interest

Find the future value of $8900 invested at a rate of 2.05% that is compounded continuously over 15 years. Solution We know that P = 8900, t = 15, and r = 0.0205. Using the continuous compound interest formula, we have

So, the future value is $12,104.19 when interest is compounded continuously.

rtA Pe 0.0205 158900e 12,104.19.

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Example 6: Finding the Maximum Amount of Interest Possible in One Year

Assume you wish to deposit $2500 into an account bearing 6% interest for 10 years. a. What is the maximum future value possible after

10 years? b. What is the maximum amount of interest possible

after 10 years?

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Example 6: Finding the Maximum Amount of Interest Possible in One Year (cont.)

Solution a. Because we want to know the maximum future

value, we need to use the continuous compound interest formula. We are given that the principal is P = 2500, r = 0.06, and t = 10. Substituting the given values into the formula we have the following.

Therefore, the largest possible future value of $2500 invested at 6% over 10 years is $4555.30.

rtA Pe 0.06 102500e 4555.30

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Example 6: Finding the Maximum Amount of Interest Possible in One Year (cont.)

b. We can find the largest amount of interest the principal can earn over the 10-year period by subtracting the principal amount from the future value as shown below.

So, the principal can earn at most $2055.30 in interest over the 10 years.

interest earned future value principal I A P

4555.30 2500.00I 2055.30

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Annual Percentage Yield (APY)

Annual Percentage Yield (APY) The annual percentage yield (APY) is the effective annual interest rate earned in a given year that accounts for the effects of compounding. APY is calculated with the formula

where r is the annual percentage rate and n is the number of compounding intervals per year.

,APY 1 1 100%

nrn

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Example 7: Annual Percentage Yield

Samantha deposits $5000 in an account paying 5% interest per year. a. Find the APY for Samantha’s investment if the

interest is compounded monthly. b. Find the APY if the interest is compounded daily on

Samantha’s investment.Solution a. Notice, that to determine the APY, we only need the

rate of interest and the number of compound intervals per year.

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Example 7: Annual Percentage Yield (cont.)

The amount of the principal plays no role in finding the APY. So, we need r = 0.05 and n = 12 for monthly compounding intervals.

This means with monthly compounding, the APY for Samantha’s investment is 5.116%.

APY 1 1 100%nr

n

120.051 1 100%

12

0.05116 100%

5.116%

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b. Daily compounding means that n = 365. Therefore, our APY calculation is as follows.

With daily compounding, Samantha has an annual percentage yield of 5.127%.

APY 1 1 100%nr

n

3650.05

1 1 100%365

0.05127 100%

5.127%

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Notice that although the advertised APR for Samantha’s investment was 5%, when interest is compounded, none of the annual percentage yields were actually 5%.

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Example 8: APR vs. APY

Suppose that the APD Bank of the South advertises the following rates for their personal loans.

Find the APY, or effective interest rates, for each of the loan categories.

Table 2: APD Bank of the South Personal Loan RatesLoan Amount APR*

< $20,000 10.49%$20,000–$99,999 9.99%

$100,000 7.50%*interest rates are compounded quarterly

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Example 8: APR vs. APY (cont.)

Solution To find the APY for each loan category we need the published APR as well as the number of compounding intervals per year. In this case, each is compounded quarterly, so n = 4. The APY for an APR of 10.49% compounded quarterly is calculated as follows.

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Therefore, the APY is 10.91%.

APY 1 1 100%nr

n

40.1049

1 1 100%4

0.1091 100%

10.91%

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For an APR of 9.99%, we can calculate the APY as follows.

That gives an APY of 10.37%.

APY 1 1 100%nr

n

40.0999

1 1 100%4

0.1037 100%

10.37%

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Lastly, to calculate the APY for an APR of 7.50%, we have the following.

Therefore, the APY is 7.714%.

APY 1 1 100%nr

n

40.075

1 1 100%4

0.07714 100% 7.714%

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The following shows the comparison of the publicized APR versus the APY.

Therefore, when trying to determine which loan is best, look at the APY to get a clearer picture of the actual cost of the loan.

Table 3: APR vs. APYLoan Amount APR APY

< $20,000 10.49% 10.91%$20,000–$99,999 9.99% 10.37%

$100,000 7.50% 7.71%

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Example 9: Calculating Interest on Payday Loans

Assume you wish to borrow $300 for two weeks in the form of a payday loan and the amount of interest you must pay is $25 per $100 borrowed. This means that at the end of two weeks, you owe $375. What is the APR? Solution Recall that APR is defined as the interest rate over a year. Since the loan was for two weeks, we need to convert this to a yearly rate. We can find the APR by using

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Example 9: Calculating Interest on Payday Loans (cont.)

where the short term rate of interest is 25% for two weeks, and the loan is for two weeks. Now, when we fill in the numbers, we obtain the following.

52 weeksAPR 25% 650%

2 weeks

52 weeks

APR short term interest ratelength of loan

,

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Example 9: Calculating Interest on Payday Loans (cont.)

Thus, the APR is 650%. That is a ridiculous rate of interest for one year. Although this rate is never paid because the loans are for a very short amount of time, the APR is the reason these types of loans have become illegal in many states.

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HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 9.2 Understanding