Compound Interest and Present Value Chapter 12 Compound Interest and Present Value

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

Learning Unit Objectives #12 Compound Interest and Present Value Learning Unit Objectives LU12.2 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

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

Compounding Terms 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 Future Value After 4 periods $1 is worth $1.36 After 3 periods $1 is worth $1.26 After 2 periods $1 is worth $1.17 After 1 period $1 is worth $1.08 Present value $1.2597 $1.3605 $1.1664 $1.08 $1.00 Number of periods

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

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 Rate 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%

Simple Versus Compound Interest Compounded 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? 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? I = P x R x T I = $80 x .08 x 4 I = $25.60 MV = $80+ $25.60 MV = $105.60 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

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

Nominal and Effective Rates (APY) of Interest Nominal Rate (Stated Rate) - The rate on which the bank calculates interest. Truth in Savings Law Annual Percentage Yield 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% 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 End Effective rate balance of interest period balance (APY) of interest Annual Semiannual Quarterly Daily $1,060.00 $1,060.90 $1,061.40 $1,061.80 6.00 6.09% 6.14% 6.18% $1,000 + 6%

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

Figure 12.4 Present Value of $1 at 8% for Four Periods Present value goes from the future value to the present value Future Value $1.0000 $.9259 Present value $.8573 $.7938 $.7350 Number of periods

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

Table 12.3 - Present Value of $1 at End Period

Comparing Compound Interest (FV) Table 12 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 Future Table Future Present 12.1 Value Value 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 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 R = 8% = 2% 4 Table Factor = .7284 Compounded Amount: $20,000 x .7284 = $14,568 Invest Today