Chapter 12 Compound Interest and Present Value McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

Compare simple interest with compound interest 2. Calculate the compound amount and interest manually and by table lookup 3. Explain and compute the effective rate Compound Interest and Present Value #12 Learning Unit Objectives Compound Interest (Future Value) – The Big Picture LU12.1

Compare present value (PV) with compound interest (FV) 2. Compute present value by table lookup 3. 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 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

12-5 Compounding Terms Compounding PeriodsInterested Calculated Compounding AnnuallyOnce a year Compounding SemiannuallyEvery 6 months Compounding QuarterlyEvery 3 months Compounding MonthlyEvery month Compounding DailyEvery day

12-6 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

12-7 Figure 12.1 Future Value of $1 at 8% for Four Periods Manual Calculation

12-8 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 4 years at 8% annually, semiannually, or quarterly What is N and R? Annually:4 x 1 = 4 Semiannually: 4 x 2 = 8 Quarterly:4 x 4 = 16 Annually:8% / 1 = 8% Semiannually: 8% / 2 = 4% Quarterly:8% / 4 = 2% Periods Rate

12-9 Simple Versus Compound Interest Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s simple interest and maturity value? I = P x R x T I = $80 x.08 x 4 I = $25.60 MV = $80+ $25.60 MV = $ I = P x R x T I = $80 x.08 x 4 I = $25.60 MV = $80+ $25.60 MV = $ Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s interest and compounded Amount? Simple Compounded Interest: $ $80.00 = $28.83 Simple

12-10 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.

12-11 Table Future Value of $1 at Compound Interest

12-12 Calculating Compound Amount by Table Lookup Pam Donahue deposits $8,000 in her savings account that pays 6% interest compounded quarterly. What will be the balance of her account at the end of 5 years? N = 4 x 5 = 20 R = 6% = 1.5% 1 Table Factor = Compounded Amount: $8,000 x = $10,775.20

12-13 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.

12-14 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% 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%

12-15 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

12-16 Table Compounding Interest Daily

12-17 Compounding Interest Daily Calculate by Table 12.2 what $1,500 compounded daily for 5 years will grow to at 7% N = 5 R = 7% Factor $1,500 x = $2,128.50

12-18 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

12-19 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.

12-20 Table Present Value of $1 at End Period

12-21 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

12-22 Calculating Present Value Amount by Table Lookup Rene Weaver needs $20,000 for college in 4 years. She can earn 8% compounded quarterly at her bank. How much must Rene deposit at the beginning of the year to have $20,000 in 4 years? N = 4 x 4 = 16 R = 8% = 2% 4 Table Factor =.7284 Compounded Amount: $20,000 x.7284 = $14,568 Invest Today

12-23 Problem 12-13: Solution: 8 years x 2 = 16 periods 6% 2 = 3% $40, x = $64,188.

12-24 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

12-25 Problem 12-16: Solution: 3 years x 2 = 6 periods $15,000 x = $19, , $59, % 2 = 4.5% $59, x = $77,531.78

12-26 Problem 12-27: Solution: 8 years x 2 = 16 periods 6% 2 = 3% $6,000 x.6232 = $3,739.20