McGraw-Hill/Irwin ©2008 The McGraw-Hill Companies, All Rights Reserved Chapter 12 Compound Interest and Present Value

12-2 Compare simple interest with compound interest Calculate the compound amount and interest manually and by table lookup Explain and compute the effective rate Compound Interest and Present Value #12 Learning Unit Objectives Compound Interest (Future Value) – The Big Picture LU12.1

12-3 Compare present value (PV) with compound interest (FV) Compute present value by table lookup Check the present value answer by compounding Compound Interest and Present Value #12 Learning Unit Objectives Present Value -- The Big Picture LU12.2

12-4 Compounding Compounding: The process of calculating the interest periodically over the life of the loan (or an investment). After each calculation, the interest is added to the loan, and starts to accrue additional interest for the next period based on the adjusted principal (equal to the previous principal plus the interest).

12-5 Compound Interest Compound interest: Interest on the principal of the loan, plus the interest on all the accrued interests (the interests of all previous periods).

12-6 Future Value (Compound Amount) Future value (or Compound amount): The final amount of the loan or the investment at the end of the last period. Refer to the next slide to explore $1 will grow in the value of at 8% in 4 consecutive years. $1

12-7 Figure 12.1 Future Value of $1 at 8% for Four Periods Number of periods Compounding goes from present value to future value Present value After 1 period $1 is worth $1.08 After 2 periods $1 is worth $1.17 After 3 periods $1 is worth $1.26 Future Value After 4 periods $1 is worth $1.36 $1.00 $1.08 $ $1.2597$1.3605

12-8 Growth of $1 in 4 Years at 8% Present: $1 After 1 year:$1.08 (end of year 1) After 2 years:$1.17 (end of year 2) After 3 years: $1.26 (end of year 3) After 4 years: $1.36 (end of year 4) Future value of $1 at 8% in 4 years: $1.36

12-9 Figure 12.1 Future Value of $1 at 8% for Four Periods Manual Calculation

12-10 Present Value vs Future Value Present value: The value of money as of today. Future value (or Compound amount): The final amount of the money, loan or investment at the end of the last period.

12-11 Compounding Interest (Future Value) Compound interest - the interest on the principal plus the interest of prior periods Compounding - involves the calculation of interest periodically over the life of the loan or investment Present value - the value of a loan or investment today Future value (compound amount) - is the final amount of the loan or investment at the end of the last period

12-12 Compounding Terms Compounding PeriodsInterested Calculated Compounding AnnuallyOnce a year Compounding SemiannuallyEvery 6 months Compounding QuarterlyEvery 3 months Compounding MonthlyEvery month Compounding DailyEvery day

12-13 Simple Interest Formula Simple Interest (I) = Principal (P) x Rate (R) x Time (T) Recall Chapter 10 Stated as a Percent Stated as a Percent

12-14 Simple Versus Compound Interest Al Jones deposited $1,000 in a savings account for 5 years at an annual interest rate of 10%. What is Al’s simple interest and maturity value? I = P x R x T I = $1,000 x.10 x 5 I = $500 MV = $1,000 + $500 MV = $1,500 I = P x R x T I = $1,000 x.10 x 5 I = $500 MV = $1,000 + $500 MV = $1,500 Al Jones deposited $1,000 in a savings account for 5 years at an annual compounded rate of 10%. What is Al’s interest and compounded amount? SimpleCompoundedCompounded Interest: $1, $1,000 = $610.51

12-15 Calculating Compound Amount & Interests 1.Manual Method (As in the previous slide) 2. Look-up Method from a Table Use the formula: Principal x Table factor = Compound Amount (Future Value) How to find Table factor: * Define the number of periods of interest * Define the appropriate rate for each period.

12-16 Tools for Calculating Compound Interest Number of periods (N) Number of years multiplied the number of times the interest is compounded per year Rate for each period (R) Annual interest rate divided by the number of times the interest is compounded per year If you compounded $100 for 3 years at 6% annually, semiannually, or quarterly What is N and R? Annually:3 yrsx 1 = 3 Semiannually: 3 yrs x 2 = 6 Quarterly:3 yrs x 4 = 12 Annually:6% / 1 = 6% Semiannually: 6% / 2 = 3% Quarterly:6% / 4 = 1.5% Periods Rate

12-17 Calculating Compound Amount by Table Lookup Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year Step 3. Go down the period column of the table to the number desired; look across the row to find the rate. At the intersection is the table factor Step 4. Multiply the table factor by the amount of the loan.

12-18 Table Future Value of $1 at Compound Interest

12-19 Calculating Compound Amount by Table Lookup Steve Smith deposited $80 in a savings account for 4 years at an annual compounded rate of 8%. What is Steve’s interest and compounded amount? N = 4 x 1 = 4 R = 8% = 8% 1 Table Factor = Compounded Amount: $80 x = $ I = $ $80 = $28.84

12-20 Practice John deposits $1,000 in his savings account that pays 6% interest compounded quarterly. What will be the balance of his account at the end of 6 years? Step1: Calculate the numbers of periods: Periods = 4 x 6 years = 24 periods. Step 2: Calculate the appropriate period rate: Rate = 6% / 4 = 1.50% Step 3: Locate the table factor: 24 periods, at 1.5% Look up Table factor = Step 4: Use the formula: Principal x Table factor = $1,000 x = $1,4295

12-21 Problem 12-13: Solution: 7 years x 2 = 14 periods 4% 2 = 2% (Period rate) $25, x = $32, Loan: $25,000 7 years at 4% interest compounded semiannually. Loan amount Lookup table factor Compound amount (Future value) at the end of 7 years

12-22 Problem 12-15: Solution: Mystic 4 years x 2 = 8 periods 10% 2 = 5% $10,000 x = $14, ,000 $ 4,775 Four Rivers 4 years x 4 = 16 periods 8% 4 = 2% $10,000 X = $13, ,000 $ 3,728 Which bank provides higher compound amount? Lookup table factor

12-23 Problem 12-16: Solution: 3 years x 2 = 6 periods $20,000 x = $28,370 Add extra amount for year 5) +30,000 Original deposit: $20,000 $58,370 12% 2 = 6% $58,370 x = $82, Compound amount at end of year 4 Total amount of deposit at the beginning of year 5. Lookup table factor

12-24 Nominal and Effective Rates (APY) of Interest Truth in Savings Law Annual Percentage Yield Effective Rate = Interest for 1 year (APY) Principal Nominal Rate (Stated Rate) - The rate on which the bank calculates interest. Annual Percentage Yield Formula

12-25 Calculating Effective Rate APY Blue, 8% compounded quarterly Periods = 4 (4 x 1) Percent = 8% = 2% 4 Principal = $8,000 Table 12.1 lookup: 4 periods, 2% x $8,000 Less $8, $8, APY =.0824 $8,000 = 8.24% It has a greater APY when the frequency of compounding increases. Sun, 8% compounded semiannually Periods = 2 (2 x 1) Percent = 8% = 4% 2 Principal = $8,000 Table 12.1 lookup: 2 periods, 4% x $8,000 Less $8, $8, APY =.0816 $8,000 = 8.16%

12-26 Figure Nominal and Effective Rates (APY) of Interest Compared Annual Semiannual Quarterly Daily $1, $1, $1, $1, % 6.14% 6.18% $1,000+ 6% Beginning Nominal rate Compounding End Effective rate balance of interest period balance (APY) of interest Daily compounding provides the highest effective rate (APY) of interest.

12-27 Table Compounding Interest Daily

12-28 Compounding Interest Daily Calculate what $2,000 compounded daily for 7 years will grow to at 6% N = 7 R = 6% Factor $2,000 x = $3, Use “Compounding Interest Daily Table

12-29 Figure 12.4 Present Value of $1 at 8% for Four Periods Number of periods Present value goes from the future value to the present value Present value $.7350 $.7938 $.8573 $.9259 $ Future Value

12-30 Calculating Present Value by Table Lookup Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year Step 3. Go down the period column of the table to the number desired; look across the row to find the rate. At the intersection is the table factor. Step 4. Multiply the table factor by the future value. This is the present value.

12-31 Table Present Value of $1 at End Period

12-32 Comparing Compound Interest (FV) Table 12.1 with Present Value (PV) Table 12.3 Compound value Table 12.1 Present value Table 12.3 Table Present FutureTable Future Present 12.1 Value Value12.3 Value Value x $80 = $ x $ = $80.00 (N = 4, R = 8) We know the present dollar amount and find what the dollar amount is worth in the future We know the future dollar amount and find what the dollar amount is worth in the present

12-33 Problem 12-25: Solution: Compounding 5 years x 2 = 10 periods 12% 2 = 6% Yes. Present value 10 periods 6% Future value: $15,000 Calculate present value $15,000 x 0.55 = $8,376 Present value OR Calculate future value: $10,000 x = $17,908

12-34 Calculating Present Value Amount by Table Lookup Steve Smith needs $ in 4 years. His bank offers 8% interest compounded annually. How much money must Steve put in the bank today (present) to reach his goal in 4 years? N = 4 x 1 = 4 R = 8% = 8% 1 Table Factor = Compounded Amount: $ x = $80.00 Invest Today

12-35 Problem 12-27: Solution: 8 years x 2 = 16 periods 6% 2 = 3% Present value of $6,000 is: $6,000 x = $3, Lookup table factor Find present value of a future amount (cost of college tuition):

12-36 Homework