Chapter 3 Compound Interest START EXIT.

Chapter 3 Compound Interest START EXIT

Chapter Outline 3.1 Compound Interest: The Basics
3.2 Compounding Frequencies 3.4 Effective Interest Rates Chapter Summary Chapter Exercises

3.1 Compound Interest: The Basics
Let’s consider a $5,000 loan for 5 years at 8% simple interest. We can easily calculate the simple interest: I = PRT I = $5,000 x 8% x 5 I = $2,000 Maturity Value = $7,000 How does the amount of interest grow over the loan’s term? Year Interest Balance Start N/A $5,000 Year 1 $400 $5,400 Year 2 $5,800 Year 3 $6,200 Year 4 $6,600 Year 5 $7,000

Now imagine that this is a deposit that you have made in a bank certificate. At the end of the first year, you would have $5,400 in your account. By leaving it on deposit at the bank for the second year, you are in effect loaning the bank $5,400. Yet, you are being paid interest only on your original $5,000. That’s how simple interest works. No matter how long the loan continues, under simple interest the borrower pays (and lender receives) interest only on the original principal.

This doesn’t seem quite fair. It seems reasonable that you should receive interest on the entire amount of your account balance. If the bank has the use of $5,400 of your money in year 2, then you have every reason to expect that it should pay you interest on the full amount, $5,400. In other words, you want to receive interest on your accumulated interest. That is precisely the point of compound interest. With compound interest, interest is paid on both the original principal and on any interest that accumulates along the way. When interest is paid on interest, we say that it compounds.

Suppose that we look at the account balance year by year, assuming that interest is credited to the account annually. Year 1 I = PRT I = $5,000 x 8% x 1 = $400 Year 2 I = $5,400 x 8% x 1 = $432 Year 3 I = $5,832 x 8% x 1 = $466.56

SIMPLE INTEREST COMPOUND INTEREST Year Start Interest End Difference 1 $5,000 $400 $5,400 $0 2 $5,800 $432 $5,832 $32 3 $6,200 $466.56 $6,298.56 $98.56 4 $6,600 $6,298 $503.88 $6,802.44 $202.44 5 $7,000 $544.20 $7,346.64 $346.64

Looking at things year by year is a good way to get a sense of how compound interest works, but a tedious and impractical way of doing an actual calculation. Note that 8% interest for a single year on $1 amounts to exactly 8 cents. Therefore, in 1 year $1 turns into $1.08. What happens to $5,000? End of year 1 balance $5,000 x $1.08 = $5,400 End of year 2 balance $5,000 x $1.08 x $1.08 = $5,832 End of year 5 balance $5,000 x $1.08 x $1.08 x $1.08 x $1.08 x $1.08 = $7,346.64

This approach is far more efficient than building an entire table, yet the repeated multiplications are still tedious. We can accomplish the same thing more efficiently by using exponents. An exponent is a way of denoting repeated multiplication of the same number. (1.08)(1.08)(1.08)(1.08)(1.08) = (1.08)5

Most calculators have a key for exponents. Different calculator models, though, may mark their exponent keys differently. It may be labeled “^”, or “xy”, or “yx”. Be careful – “ex” is NOT the key you’re looking for!!! Enter 1.08 Press the exponent key Enter 5 Press an equal sign (=) The result should be If we multiply this result by $5,000, we come up with $7,

FORMULA 3.1.1 The Compound Interest Formula FV = PV(1 + i)n where FV represents the FUTURE VALUE (the ending amount) PV represents the PRESENT VALUE (the starting amount) i represents the INTEREST RATE (per time period) n represents the NUMBER OF TIME PERIODS

Example 3.1.1 Problem Use the compound interest formula to find how much $5,000 will grow to in 50 years at 8% annual compound interest. Solution FV = PV(1 + i)n FV = $5,000( )50 FV = $5,000( ) FV = $234,508.06

Order Operation 1st Parentheses 2nd Exponents 3rd Multiplication Division 4th Addition Subtraction Notice that we used the exponent before multiplying, even though reading from left to right it might appear as though the multiplication should have come first.

Example 3.1.2 Problem Suppose you invest $14,075 at 7.5% annually compounded interest. How much will this grow to over 20 years? Solution FV = PV(1 + i)n FV = $14,075( )20 FV = $14,075( ) FV = $59,788.50

Example 3.1.3 Problem Suppose you invest $14,075 at 7.5% annually compounded interest. How much total interest will you earn over 20 years? Solution I = FV – PV I = $59, $14,075 = $45,713.50

Example 3.1.4 Problem Suppose that $2,000 is deposited at a compound interest rate of 6% annually. Find (a) the total account value after 12 years and (b) the total interest earned in those 12 years. Solution (a) FV = PV(1 + i)n FV = $2,000( )12 FV = $4,024.39 (b) Total Interest = $4, $2,000 = $2,024.39

Example 3.1.5 Problem How much money should I deposit today into an account earning 7 3/8% annually compounded in order to have $2,000 in the account 5 years from now? Solution FV = PV(1 + i)n $2,000 = PV( )5 $2,000 = PV( ) PV = $1,401.25

FORMULA 3.1.2 The Rule of 72 The time required for a sum of money to double at a compound interest rate of x% is approximately 72/x years. The interest rate should not be converted to a decimal.

Example 3.1.6 Problem Jim deposited $3,200 into a retirement account, which he expects to earn 7% annually compounded interest. If his expectations about the interest rate is correct, how much will his deposit grow to between now and when he retires 40 years from now? Use the Rule of 72 to obtain an approximate answer, then use the compound interest formula to find the exact value. Solution Using the Rule of 72, we know that his money should double approximately every 10 years, since 72/7 ≈ So in 40 years, his account should experience approximately 40/10 = 4 doublings. FV = PV(1 + i)n FV = $3,200( )40 = $47,918.27

FORMULA 3.1.3 The Rule of 72 (Alternate Form) The compound interest rate required for a sum of money to double in x years is approximately 72/x percent.

Example 3.1.7 Problem What compound interest rate is required to double $50,000 in 5 years? Solution 72/5 = 14.4, so the interest rate would need to be approximately 14.4%.

Basics of Compound Interest
Section 3.1 Exercises Problem 1: Basics of Compound Interest Problem 2: Future Value Problem 3: Present Value Problem 4: The Rule of 72

Problem 1 Suppose that you deposit $5,000 in an account that pays 3% annually compounded interest for 2 years. Calculate (by hand) the balance at the end of Year 2. CHECK YOUR ANSWER

Solution 1 Suppose that you deposit $5,000 in an account that pays 3% annually compounded interest for 2 years. Calculate (by hand) your total balance at the end of Year 2. Year 1 I = PRT I = $5,000 x 3% x 1 = $150 Year 2 I = $5,150 x 3% x 1 = $154.50 $5,150 + $ = $5,304.50 BACK TO GAME BOARD

Problem 2 Joe deposited $2,000 into a 5-year certificate of deposit paying 3% interest compounded annually. How much will his CD be worth at maturity? Use the compound interest formula for the future value (FV). CHECK YOUR ANSWER

Solution 2 Joe deposited $2,000 into a 5-year certificate of deposit paying 3% interest compounded annually. How much will his CD be worth at maturity? Use the compound interest formula for the future value (FV). FV = PV(1 + i)n FV = $2,000( )5 FV = $2,000( ) = $2,318.55 BACK TO GAME BOARD

Problem 3 You are planning a trip to a dream destination. How much would you need to deposit today into an account paying 4.5% annually compounded interest in order to have $2,500 in 2 years? CHECK YOUR ANSWER

Solution 3 You are planning a trip to a dream destination. How much would you need to deposit today into an account paying 4.5% annually compounded interest in order to have $2,500 in 2 years? FV = PV(1 + i)n $2,500 = PV( )2 $2,500 = PV( ) PV = $2,289.32 BACK TO GAME BOARD

Problem 4 Kristen has just invested $5,000 at 4% annually compounded interest. How long will it take for this investment to double (grow to $10,000). CHECK YOUR ANSWER

Solution 4 Kristen has just invested $5,000 at 4% annually compounded interest. How long will it take for this investment to double (grow to $10,000). Use the Rule of 72. It will take 72/4% ≈ 18 years for Kristen’s investment to double. We can check our estimation: FV = PV(1 + i)n FV = $5,000( )18 FV = $10,129.08 BACK TO GAME BOARD

3.2 Compounding Frequencies
Example 3.2.1 Problem Find the future value of $2,500 at 6% interest compounded monthly for 7 years. Solution Since the interest is compounded monthly, the annual rate of 6% needs to be divided by 12 (since each month is 1/12 of a year) to make it a monthly rate, 6% ÷ 12 = 0.5%. In addition, the term of 7 years must be expressed in months, so there are 7 x 12 = 84 compounding periods. FV = PV(1 + i)n FV = $2,500( )84 FV = $3,800.92

Example 3.2.2 Problem Find the future value of $3,250 at 4.75% interest compounded daily for 4 years. Solution Daily interest rate = 4.75% ÷ 365 = Number of compounding periods = 4 x 365 = 1,460 FV = PV(1 + i)n FV = $3,250( )1460 FV = $3,930.01

Example 3.2.3 Problem Find the future value of $85.75 at 8.37% interest compounded monthly for 15 years. Solution Monthly interest rate = 8.37% ÷ 12 = % Compounding periods = 15 x 12 = 180 FV = PV(1 + i)n FV = $85.75( )180 FV = $85.75( ) = $299.64

Example 3.2.4 Problem How much do I need to deposit today into a CD paying 6.06% compounded monthly in order to have $10,000 in the account in 3 years? Solution Monthly interest rate = 6.06% ÷ 12 Compounding periods = 3 x 12 = 36 FV = PV(1 + i)n $10,000 = PV( )36 $10,000 = PV( ) PV = $8,341.50

FORMULA 3.2.2 The Continuous Compound Interest Formula FV = PVert where FV represents the FUTURE VALUE (the ending amount) PV represents the PRESENT VALUE (the starting amount) r represents the ANNUAL INTEREST RATE t represents the NUMBER OF YEARS

Leonhard Euler, came up with the number e by using….

Problem Find the future value of $4,000 at 7.12% interest for 12 years compounded continuously. FV = 4,000e^( * 12) = $9,399.86 FV = 4,000e^((0.0712)(12)) = $9,399.86 FV = 4,000e^ * 12 = $51,542.21, NOT CORRECT!!!

Section 3.2 Exercises Problem 1: Using the Compound Interest Formula

Problem 1 Sonya deposited $5,000 in a certificate of deposit (CD) paying $3% compounded monthly for 5 years. How much will she end up with in her account? CHECK YOUR ANSWER

Solution 1 Sonya deposited $5,000 in a certificate of deposit (CD) paying 3% compounded monthly for 5 years. How much will she end up with in her account? Monthly interest rate = 3% ÷ 12 = 0.25% Compounding periods = 5 x 12 = 60 FV = PV(1 + i)n FV = $5,000( )60 FV = $5,000( ) = $5,808.08 BACK TO GAME BOARD

Problem 2 Ray wants to have $60,000 in an investment account when his daughter starts college 15 years from now. Assuming that the account pays 4.5% compounded daily, how much should Ray deposit now? CHECK YOUR ANSWER

Solution 2 Ray wants to have $60,000 in an investment account when his daughter starts college 15 years from now. Assuming that the account pays 4.5% compounded daily, how much should Ray deposit now? Daily interest rate = 4.5% ÷ 365 Compounding periods = 15 x 365 = 5,475 FV = PV(1 + i)n $60,000 = PV( /365)5475 $60,000 = PV( ) PV = $30,550.66 BACK TO GAME BOARD

Effective Interest Rates
Section 3.4 Effective Interest Rates

3.4 Effective Interest Rates
Example 1 Problem Which of the following banks is offering the best rate for a certificate of deposit? Bank Rate Compounding Bank of America 3.79% Annual PNC 3% 44

Example 1 Cont. Solution Both rates are compounded annually, so it’s clear that Bank of America and Loan is offering a slightly higher rate. Bank Rate Compounding Bank of America 3.79% Annual PNC 3% 45

Example 2 Problem Which of these two banks is offering the best CD rate? Bank Rate Compounding Bank of America 4.35% Daily PNC Annual 46

Example 2 Cont. Solution Both rates are the same, but Bank of America compounds interest daily. Since more frequent compounding means more interest overall, we know that Bank of America will end up paying more interest, so their rate is better. Bank Rate Compounding Bank of America 4.35% Daily PNC Annual 47

Example 3 Problem Leo has a life insurance policy with Trustworthy Mutual Life of Nebraska. The company credits interest to his policy’s cash value and offers Leo the choice of two different options shown below. Which option would give Leo the most interest? Option Rate Compounding Daily Dividends 8.00% Daily (banker’s rule) Annual Advancement 8.33% Annual 48

Example 3 Cont. Solution It appears that Annual Advancement would give the better interest rate since 8.33% is higher than 8.00%. However, the 8.33% is compounded annually but the 8.00% is compounded daily. We need a method to compare these two rates when they have different compounding periods. 49

Definition 1 The annually compounded rate which produces the same results as a given interest rate and compounding is called the equivalent annual rate (EAR) or the effective interest rate. The original interest rate is called the nominal rate. So, for example #3, we need to find the equivalent annual rate (or the effective interest rate) that corresponds to the nominal rate of 8.00% compounded daily using banker’s rule. 50

FORMULA 1 The Effective Rate Formula Effective Rate = (1 + r/c)c - 1 r = the nominal interest rate c = the number of compoundings per year 51

Example 3 Problem Leo has a life insurance policy with Trustworthy Mutual Life of Nebraska. The company credits interest to his policy’s cash value and offers Leo the choice of two different options shown below. Which option would give Leo the most interest? Option Rate Compounding Daily Dividends 8.00% Daily (banker’s rule) Annual Advancement 8.33% Annual 52

Example 3 Cont. Solution Daily Dividends Effective rate = ( /360) = = 8.33% Annual Advancement Effective rate = Nominal rate = 8.33% 53

Example 4 Problem Find the equivalent annual rate for 7.35% compounded quarterly. Solution Effective Rate = ( /4)4 – 1 = = 7.56% 54

Example 5 Problem Which of these interest rates is actually the highest? Bank Rate Compounding Bank of America 5.95% Annual Bank of Florida 5.85% Monthly PNC 5.75% Daily 55

Example 5 Solution We need to convert each to its effective rate: BOA 5.95% BOF Effective rate = ( /12)12 -1 = = 6.01% PNC Effective rate = ( /365) = = 5.92% Bank Rate Compounding Bank of America 5.95% Annual Bank of Florida 5.85% Monthly PNC 5.75% Daily 56

The Truth in Lending Act Under most circumstances, financial institutions must disclose the effective rate for deposit accounts. The nominal rate may or may not be given, but the effective rate usually must be. The point of requiring this is to enable consumers to readily compare different rates without being confused by the effects of different compounding frequencies. 57

Example 6 Problem Twelve Corners Federal Credit Union compounds interest on all of its accounts daily. The credit union is offering an effective rate of 7.33% on its 5-year certificates of deposit. If someone put $20,000 into one of these CDs, how much would the certificate be worth at maturity? Solution FV = PV(1 + i)n FV = $20,000( )5 = $28,486.27 58

Problem 1 (Comparing Interest Rates)
You are shopping around for the best rate on an auto loan. Which of the following is offering the best rate? Bank Rate Compounding Creekside Trust 7.37% Daily Southwest National 7.11% Monthly O’Lakes Mutual 7.08% Annually 59

Solution 1 Creekside Trust
Effective Rate = ( /365) = 7.65% Southwest National Effective Rate = ( /12)12 -1 = 7.35% O’Lakes Mutual 7.08% Therefore, O’Lakes Mutual is offering the best rate. 60

Problem 2 (Finding Effective Rates)
You are shopping around for the best rates on student loans. Which of the following is offering the best deal? Bank Rate Compounding First Bank of Elanville 9.26% Annually Bank of America 8.45% Monthly Elanville Credit Union 9.00% Daily 18

Solution 2 FBE 9.26% BA Effective Rate = (1 + 0.0845/12)12 -1 = 8.79%
ECU Effective Rate = ( /365) = 9.42% Therefore, Bank of America is offering the best rate. 62

Problem 3 (Using Effective Rates for Comparison)
You would like to purchase a certificate of deposit (CD). Which of the following is offering the most attractive rate? Bank Rate Compounding Smithfield Financial 2.26% Annually Jamestown Trust 2.45% Daily Bank of Jamestown 2.00% Monthly 63

Solution 3 SF 2.26% JT Effective Rate = (1 + 0.0245/365)365 -1 = 2.48%
BJ Effective Rate = ( /12)12 -1 = 2.02% Therefore, Jamestown Trust is offering the best rate for its customers. 64

Problem 4 (Using Effective Rates in Calculations)
Suzanne invested $3,000 at an effective interest rate of 3% for 3 years. How much interest did she earn? 65

Solution 4 Suzanne invested $3,000 at an effective interest rate of 3% for 3 years. How much interest did she earn? FV = $3,000( )3 = $3,278.18 I = FV – PV = $3, $3,000 = $278.18 66

Chapter 3 Summary The Rule of 72 Compounding Frequencies
The Concept of Compound Interest Calculating Future Value with Compound Interest Calculating Compound Interest Calculating Present Value with Compound Interest The Rule of 72 Compounding Frequencies Comparing Compounding Frequencies

Chapter 3 Exercises $100 $200 Section 3.1 Section 3.2 Section 3.3
EXIT

Section $100 Ashley deposited $2,500 into a 3-year credit union certificate of deposit paying 3.6% interest compounded annually. How much will her CD be worth at maturity? CHECK YOUR ANSWER

Section $100 Ashley deposited $2,500 into a 3-year credit union certificate of deposit paying 3.6% interest compounded annually. How much will her CD be worth at maturity? FV = $2,500( )3 = $2,77984 BACK TO GAME BOARD

Section $200 Salam would like to set up a college fund for his newborn son. If the money will be compounded annually at 5% for 18 years, how much money should he invest today in order to have $50,000 at maturity? CHECK YOUR ANSWER

Section $200 Salam would like to set up a college fund for his newborn son. If the money will be compounded annually at 5% for 18 years, how much money should he invest today in order to have $50,000 at maturity? $50,000 = PV( )18 $50,000 = PV( ) PV = $20,776.03 BACK TO GAME BOARD

Section $100 Rachelle wants to invest $10,000 today in order to have extra retirement money later. If her investment will be compounded monthly at 3.7% for 15 years, how much money will she have at maturity? CHECK YOUR ANSWER

Section $100 Rachelle wants to invest $10,000 today in order to have extra retirement money later. If her investment will be compounded monthly at 3.7% for 15 years, how much money will she have at maturity? FV = $10,000( /12)180 = $17,404.54 BACK TO GAME BOARD

Section $200 Fabulous Treats, Inc. would like to renovate the building in 5 years. How much money should the company invest today, compounded daily at 4.1%, in order to have $100,000 at maturity? CHECK YOUR ANSWER

Section $200 Fabulous Treats, Inc. would like to renovate the building in 5 years. How much money should the company invest today, compounded daily at 4.1%, in order to have $100,000 at maturity? $100,000 = PV( /365)1,825 PV = $81,465.67 BACK TO GAME BOARD

Section $100 Find the effective rate equivalent to this nominal rate: % compounded daily CHECK YOUR ANSWER

Section $100 Find the effective rate equivalent to this nominal rate: % compounded quarterly. FV = $100( /4)4 FV = $119.54 Therefore, the effective rate is ≈19.54%. BACK TO GAME BOARD

Section 3.3 -- $200 Which of the following is offering the best rate?
Bank Rate Compounding Perington Trust 3.97% Monthly Shoemaker Credit 4.19% Daily Arlington Mutual 4.06% Quarterly CHECK YOUR ANSWER

Section 3.3 -- $200 PT FV = $100(1 + 0.0397/12)12 = $104.04
The effective rate is 4.04% SC FV = $100( /365)365 = $104.28 The effective rate is 4.28% AM FV = $100( /4)4 = $104.12 The effective rate is 4.12% If you would like to invest money, you should open an account at Shoemaker Credit since they offer the highest effective rate of 4.28%. However, if you need to borrow money, you should go with Perington Trust at 4.04%, the lowest rate. BACK TO GAME BOARD

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