Chapter 12 - Compound interest and present value. After you have mastered the material in this chapter, you will be able to: Compare simple interest with compound interest, calculate the compound amount and interest manually and by table lookup, explain and compute the effective rate (APY), compare present value (PV) with compound interest (FV), compute present value by table lookup, check the present value answer by compounding.
Chapter 12 Compound Interest and Present Value McGrawHill/Irwin ©2011 The McGrawHill Companies, All Rights Reserved #12 Compound Interest and Present Value Learning Unit Objectives Compound Interest (Future Value) – LU12.1 The Big Picture Compare simple interest with compound interest Calculate the compound amount and interest manually and by table lookup Explain and compute the effective rate 122 #12 LU12.2 Compound Interest and Present Value Learning Unit Objectives Present Value The Big Picture Compare present value (PV) with compound interest (FV) Compute present value by table lookup Check the present value answer by compounding 123 Compounding Interest (Future Value) Compounding involves the calculation of interest periodically over the life of the loan or investment Future value (compound amount) is the final amount of the loan or investment at the end of the last period 124 Compound interest the interest on the principal plus the interest of prior periods Present value the value of a loan or investment today Compounding Terms 125 Compounding Periods Interested Calculated Compounding Annually Once a year Compounding Semiannually Every 6 months Compounding Quarterly Every 3 months Compounding Monthly Every month Compounding Daily Every day Figure 12.1 Future Value of $1 at 8% for Four Periods Compounding goes from present value to future value $5.00 $4.50 $4.00 $3.50 $3.00 $2.50 $2.00 $1.50 $1.00 $0.50 $0.00 126 Future Value Present value After 1 period $1 is worth $1.08 $1.00 $1.08 After 2 periods $1 is worth $1.17 $1.166 After 3 periods $1 is worth $1.26 After 4 periods $1 is worth $1.36 $1.259 $1.360 Number of periods Figure 12.1 Future Value of $1 at 8% for Four Periods Manual Calculation Year Year Year Year $ 1.00 $ 1.08 $ 1.17 $ 1.26 0.08 x 10 x 10 x 10 Interest $ 0.08 $ 0.09 $ 0.09 $ 0.10 Beg Bal 1.00 1.08 1.17 1.26 End of year $ 1.08 $ 1.17 $ 1.26 $ 1.36 127 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? Periods Annually: 4 x 1 = 4 Semiannually: 4 x 2 = 8 Quarterly: 4 x 4 = 16 128 Rate Annually: 8% / 1 = 8% Semiannually: 8% / 2 = 4% Quarterly: 8% / 4 = 2% Simple Versus Compound Interest Simple Simple 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 = P x R x T I = $80 x .08 x 4 I = $80 x .08 x 4 I = $25.60 I = $25.60 MV = $80+ $25.60 MV = $80+ $25.60 MV = $105.60 MV = $105.60 129 Compounded 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? Year Year Year Year $ 80.00 $ 86.40 $ 93.31 $ 100.77 x 08 x 08 x 08 x 08 Interest $ 6.40 $ 6.91 $ 7.46 $ 8.06 Beg Bal 80.00 86.40 93.31 100.77 End of year $ 86.40 $ 93.31 $ 100.77 $ 108.83 Interest: $108.83 $80.00 = $28.83 Calculating Compound Amount by Table Lookup Step 4. Multiply the table factor by the amount of the loan. 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 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year 1210 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 = 1.3469 Compounded Amount: $8,000 x 1.3469 = $10,775.20 1211 Nominal and Effective Rates (APY) of Interest Truth in Savings Law Annual ge Percenta Yield 1212 Nominal Rate (Stated Rate) The rate on which the bank calculates interest. Effective Rate = Interest for 1 year (APY) Principal 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% 1.0824 x $8,000 Less $8,659.20 $8,000.00 659.20 APY 659.20 = .0824 $8,000 = 8.24% 1213 Sun, 8% compounded semiannually Periods = 2 (2 x 1) Percent = 8% = 4% 2 Principal = $8,000 Table 12.1 lookup: 2 periods, 4% 1.0816 x $8,000 Less $8,652.80 $8,000.00 652.80 APY 652.80 = .0816 $8,000 = 8.16% Figure 12.3 Nominal and Effective Rates (APY) of Interest Compared Beginning Nominal rate Compounding balance of interest period $1,000 1214 + 6% End Effective rate balance (APY) of interest Annual $1,060.00 6.00 Semiannual $1,060.90 6.09% Quarterly $1,061.40 6.14% Daily $1,061.80 6.18% 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 1.4190 $1,500 x 1.4190 = $2,128.50 1215 Figure 12.4 Present Value of $1 at 8% for Four Periods Present value goes from the future value to the present value $1.20 $1.10 $1.00 $0.90 $0.80 $0.70 $0.60 $0.50 $0.40 $0.30 $0.20 $0.10 $0.00 Present value $.7938 $.8573 $.9259 $.7350 Number of periods 1216 $1.000 Future Value Calculating Present Value by Table Lookup Step 4. Multiply the table factor by the future value. This is the present value 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 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year 1217 Table 12.3 Present Value of $1 at End Period Present value of $1 at end period (Partial) 1218 Period 1% 1.50% 2% 3% 4% 5% 6% 7% 8% 9% 10% 0.9901 0.9852 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091 0.9803 0.9707 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264 0.9706 0.9563 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513 0.9610 0.9422 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830 0.9515 0.9283 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209 0.9420 0.9145 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 0.6302 0.5963 0.5645 0.9327 0.9010 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 0.5835 0.5470 0.5132 0.9235 0.8877 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 0.5403 0.5019 0.4665 0.9143 0.8746 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 0.5002 0.4604 0.4241 10 0.9053 0.8617 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855 11 0.8963 0.8489 0.8043 0.7224 0.6496 0.5847 0.5268 0.4751 0.4289 0.3875 0.3505 12 0.8874 0.8364 0.7885 0.7014 0.6246 0.5568 0.4970 0.4440 0.3971 0.3555 0.3186 13 0.8787 0.8240 0.7730 0.6810 0.6006 0.5303 0.4688 0.4150 0.3677 0.3262 0.2897 14 0.8700 0.8119 0.7579 0.6611 0.5775 0.5051 0.4423 0.3878 0.3405 0.2992 0.2633 15 0.8613 0.7999 0.7430 0.6419 0.5553 0.4810 0.4173 0.3624 0.3152 0.2745 0.2394 Comparing Compound Interest (FV) Table 12.1 with Present Value (PV) Table 12.3 Compound value Table 12.1 Table Present Future 12.1 Value Value Present value Table 12.3 Table Future Present 12.3 Value Value 1.3605 x $80 = $108.84 7350 x $108.84 = $80.00 (N = 4, R = 8) (N = 4, R = 8) We know the present dollar amount and find what the dollar amount is worth in the future 1219 We know the future dollar amount and find what the dollar amount is worth in the present 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 Invest R = 8% = 2% Today 4 Table Factor = .7284 Compounded Amount: $20,000 x .7284 = $14,568 1220 ... Comparing Compound Interest (FV) Table 12. 1 with Present Value (PV) Table 12. 3 Compound value Table 12. 1 Table Present Future 12. 1 Value Value Present value Table 12. 3 Table Future ... 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 1.4190 $1,500 x 1.4190 = $2 ,128 .50 12 15 Figure 12. 4 Present Value of $1 at 8% ... Calculate the compound amount and interest manually and by table lookup Explain and compute the effective rate 12 2 #12 LU12.2 Compound Interest and Present Value Learning Unit Objectives Present Value The Big Picture