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

4-2

4.3 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.

Tm

m

rCFV

10

4-3

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”

93.70$)06.1(50$2

12.150$ 6

32

FV

4-4

Effective Annual Rates of InterestA 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:

93.70$)06.1(50$)2

12.1(50$ 632 FV

93.70$)1(50$ 3 EAR

4-5

Effective Annual Rates of Interest

So, investing at 12.36% compounded annually is the same as investing at 12% compounded semi-annually.

93.70$)1(50$ 3 EARFV

50$

93.70$)1( 3 EAR

1236.150$

93.70$31

EAR

4-6

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%.

1956.1)015.1(12

18.11 12

12

m

m

r

4-7

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 = 1,100.00 At m=2, FV=1000(1.05)2 = 1,102.50 At m=12, FV=1000(1.0083)12 = 1,104.71 At m=365, FV=1000(1.00027)365 = 1,103.55 Then, what if m ∞ ?

4-8

Continuous compounding As m goes to infinity, we need the limit of

(1+r/m)m*T

At this limit, FV=CerT

FV = 1000e(.12*1) = $1,127.50

4-9

Continuous Compounding The general formula for the future value of an

investment compounded continuously over many periods can be written as:

FV = C0×erT

Where

C0 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 2.718. ex is a key on your calculator.

4-10

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.