Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Prepared by Dr. Elena Skliarenko Chapter 14 Compound Interest: Present Value; Future Value; Equivalent Payments; Effective, Nominal, and Equivalent Interest Rates Prepared by Dr. Elena Skliarenko Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Present the timing and amount of payments using a time line diagram #14 Compound Interest LU14.1 Learning Unit Objectives Basic Concepts, Terminology, and Time Line Diagrams Present the timing and amount of payments using a time line diagram Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

#14 Learning Unit Objectives Compound Interest LU14.2 Future Value – The Big Picture Compare simple interest with compound interest Calculate compound amount and interest manually Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

#14 Learning Unit Objectives Compound Interest LU14.3 Present Value – The Big Picture Compare present value (PV) with compound interest (FV) Compute present value Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Learning Unit Objectives #14 Compound Interest Learning Unit Objectives LU14.4 Equivalent Payments Compute the equivalent value of a single payment or serial payments on any date Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

#14 Learning Unit Objectives Compound Interest Effective Interest Rate LU14.5 Effective Interest Rate Explain and compute the effective rate Define an effective interest rate when a nominal interest rate is given Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

#14 Learning Unit Objectives Compound Interest LU14.6 Equivalent Interest Rates Compute the equivalent interest rate when nominal interest rate is given Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

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 Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

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 $1.1664 $1.2597 $1.3605 $1.00 $1.08 Number of periods Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

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 Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Simple Versus Compound Interest Compounded Al Jones deposited $1,000 in a savings account for 5 years at an annual simple interest rate of 10%. What is Al’s simple interest and maturity value? 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? I = P x R x T I = $1,000 x .10 x 5 I = $500 MV = $1,000 + $500 MV = $1,500 Compound amount $1,610.51 - $1,000 = $610.51 Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Tools for Calculating Compound Interest Number of periods (N) Number of years times 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? Periods Rate Annually: 3 x 1 = 3 Semiannually: 3 x 2 = 6 Quarterly: 3 x 4 = 12 Annually: 6% / 1 = 6% Semiannually: 6% / 2 = 3% Quarterly: 6% / 4 = 1.5% Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

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 Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Future Value of $1 at Compound Interest Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Calculating Compound Amount by Table Lookup Steve Smith deposited $1,000 in a savings account for 4 years at an semiannual compounded rate of 8%. What is Steve’s interest and compounded amount? N = 4 x 2 = 8 R = 8% = 4% 2 Table Factor = 1.3686 Compounded Amount: $1,000 x 1.3686 = =$1,368.60 I = $1,368.60 - $1,000 = =$368.60 Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Nominal and Effective Rates of Interest Nominal Rate (Stated Rate) - The rate on which the bank calculates interest. Effective Rate = Interest for 1 year Principal Truth in Savings Law Annual Percentage Yield Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Nominal and Effective Rates of Interest Compared Beginning Nominal rate Compounding End Effective rate balance of interest period balance 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% Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Compounding Interest Daily Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Compounding Interest Daily Calculate what $2,000 compounded daily for 7 years will grow to at 6% N = 7 R = 6% Factor 1.5219 $2,000 x 1.5219 = $3,043.80 Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Present Value of $1 at 8% for Four Periods Future Value Present value goes from the future value to the present value $1.0000 $.9259 Present value $.8573 $.7938 $.7350 Number of periods Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

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 Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Present Value of $1 at End Period Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Calculating Present Value Amount by Table Lookup Steve Smith needs $1,368.80 in 4 years. His bank offers 8% interest compounded semiannually. How much money must Steve put in the bank today (present) to reach his goal in 4 years? N = 4 x 2 = 8 R = 8% = 4% 2 Table Factor = .7307 Compounded Amount: $1,368.80 x .7307 = $1,000.18 Invest Today Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Comparing Compound Interest (FV) with Present Value (PV) Compound value (FV) Present value (PV) Table Present Future Table Future Present Value Value Value Value 1.3688 x $1,000 = $1,368.80 .7307 x $1,368.80 =$1,000 (N = 8, R = 4) (N = 8, R = 4) 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 Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Short Cut Formulas Periodic interest rate is an interest rate per one compounding period and is calculated using formula 14.3 14.3   To calculate nominal interest rate when periodic interest rate and compounding frequency are given, multiply these values: j = i · m 14.4 Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Short Cut Formulas   Where FV is the maturity value at the end of term of an investment or a loan, PV is the present (initial) value or principle of an investment or a loan, i – periodic interest n – number of compounding periods during the term of the investment or loan 14.5 Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Short Cut Formulas You can find the formula for PV through formula rearrangement 14.7 A more common form of this formula may be obtained, using property of powers:   14.8 PV – present value FV – future value n- number of compounding periods in the term i-periodic interest rate   Then Where Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Equivalent Payments We already know how to work with equivalent payments at simple interest. In this unit we will learn how to calculate equivalent payments at compound interest.   The approach is a very similar: it is a two-step procedure. Step 1: Bring all the values of payments to the same point in time and calculate their equivalent values, using formulas 14.5 and 14.8 Step 2. Add them up and calculate the unknown replacement value Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Short Cut Formulas There is a short-cut formula for calculation of an effective interest rate: 14.9 where f is the effective interest rate, m is the compounding frequency and i is the periodic interest rate. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

Short Cut Formulas There is a short-cut formula for calculating equivalent interest rates. To use this formula you have to remember the following two things: To calculate nominal interest rate you should first calculate periodic interest rate. This is a two-step approach. Using shortcut formula assign variables with code “1” to the given values and variables with code “2” to the values to be defined.   i2 = (1+i1)m1/m2 – 1 14.10   Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.