How to handle various compounding periods – Chapter 4, Section 4.3 Module 1.3 Copyright © 2013 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

Compounding Periods Compounding an investment m times a year for T years provides for future value of wealth: Note: we are dividing an annual rate by the number of compounding periods per year, and multiplying the exponent T (number of years) by m to get the total number of compounding periods.

4-2 Compounding Periods  For example, if you invest $50 for 3 years at 12% APR compounded semi-annually, your investment will grow to:  Note: “APR” = “annual percentage rate”

4-3 Effective Annual Rates of Interest A reasonable question to ask in the above example is “what is the effective annual rate of interest on that investment?” The Effective Annual Rate (EAR) of interest is the annual rate that would give us the same end-of-investment wealth after 3 years:

4-4 Effective Annual Rates of Interest So, investing at 12.36% compounded annually is the same as investing at 12% compounded semi-annually.

4-5 Effective Annual Rates of Interest  What is the EAR of an 18% APR loan that is compounded monthly?  What we have is a loan with a monthly interest rate rate of 1½%.  This is equivalent to a loan with an annual interest rate of 19.56%.

4-6 Continuous compounding  What will happen to FV of $1000 over 1 year if we keep APR at 10%, but keep increasing the compounding period?  At m=1, FV=1000(1.1) 1 = 1,  At m=2, FV=1000(1.05) 2 = 1,  At m=12, FV=1000(1.0083) 12 = 1,  At m=365, FV=1000( ) 365 = 1,  Then, what if m ∞ ?

4-7 Continuous compounding  As m goes to infinity, we need the limit of (1+r/m) m*T  At this limit, FV=Ce rT  FV = 1000e (.12*1) = $1,127.50

4-8 Continuous Compounding  The general formula for the future value of an investment compounded continuously over many periods can be written as: FV = C 0 ×e rT Where C 0 is cash flow at date 0, r is the stated annual interest rate, T is the number of years, and e is a transcendental number approximately equal to e x is a key on your calculator.

4-9 Rate transformations  Rates in loans and other contracts can be stated a variety of ways.  Typical of a credit card is “9.9% APR compounded daily”  Note that any stated rate can be transformed into an equivalent APR or continuously compounded rate.