Compound Interest and Present Value Chapter Twelve Compound Interest and Present Value McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved. 12-

Learning unit objectives LU 12-1 Compound Interest (Future Value) – 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 (APY). LU 12-2 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. 12-

Compounding Interest (Future Value) 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) – 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 12-

Compounding Terms Compounding Periods Interest Calculated Compounding Annually Compounding Semiannually Compounding Quarterly Compounding Monthly Compounding Daily Once a year Every 6 months Every 3 months Every month Every day 12-

Future Value of $1 at 8% for Four Periods (Figure 12.1) Compounding goes from present value to future value Future Value After 4 periods, $1 is worth $1.36 After 1 period, $1 is worth $1.08 After 2 periods, $1 is worth $1.17 After 3 periods, $1 is worth $1.26 Present value $1.00 $1.08 $1.1664 $1.2597 $1.3605 Number of periods 12-

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

Tools for Calculating Compound Interest Number of periods (N) Number of years multiplied by 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: Semiannually: Quarterly: 4 x 1 = 4 Annually: Semiannually: Quarterly: 8% / 1 = 8% 4 x 2 = 8 8% / 2 = 4% 4 x 4 = 16 8% / 4 = 2% 12-

Simple Versus Compound Interest Compounded 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 12-

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 for the compound amount of $1. Step 4. Multiply the table factor by the amount of the loan. This gives the compound amount. 12-

Future Value of $1 at Compound Interest (Table 12.1) 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? Periods (N) = 4 x 5 = 20 Rate (R) = 6%/4 = 1.5% Table Factor = 1.3469 Compounded Amount: $8,000 x 1.3469 = $10,775.20 12-

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

Calculating Effective Rate APY 12-

Nominal and Effective Rates (APY) of Interest Compared (Figure 12.3) 12-

Compounding Interest Daily (Table 12.2) 12-

Compounding Interest Daily Use Table 12.2 to calculate 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-

Present Value of $1 at 8% for Four Periods (Figure 12.4) Present value goes from the future value to the present value Future Value $1.0000 Present value $.9259 $.8573 $.7938 $.7350 Number of periods 12-

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 of the two columns is the table factor for the compound value of $1. Step 4. Multiply the table factor by the future value. This is the present value. 12-

Present Value of $1 at End Period (Table 12.3) 12-

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 Table Present Future 12.1 Value Value 1.3605 x $80 = $108.84 (N = 4, R = 8%) Present value Table 12.3 Table Future Present 12.3 Value Value .7350 x $108.84 = $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-

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? Invest Today Periods (N) = 4 x 4 = 16 Rate (R) = 8%/4 = 2% Table Factor = .7284 Compounded Amount: $20,000 x .7284 = $14,568 12-