Module 1 3 How to handle various compounding

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Module 1. 3 How to handle various compounding periods – Chapter 4, Section 4.

Module 1. 3 How to handle various compounding periods – Chapter 4, Section 4. 3 Mc. Graw-Hill/Irwin Copyright © 2013 by the Mc. Graw-Hill Companies, Inc. All rights reserved.

4. 3 Compounding Periods Compounding an investment m times a year for T years

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. 4 -1

Compounding Periods q For example, if you invest $50 for 3 years at 12%

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

Effective Annual Rates of Interest A reasonable question to ask in the above example

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 endof-investment wealth after 3 years: 4 -3

Effective Annual Rates of Interest So, investing at 12. 36% compounded annually is the

Effective Annual Rates of Interest So, investing at 12. 36% compounded annually is the same as investing at 12% compounded semi-annually. 4 -4

Effective Annual Rates of Interest o What is the EAR of an 18% APR

Effective Annual Rates of Interest o What is the EAR of an 18% APR loan that is compounded monthly? o What we have is a loan with a monthly interest rate of 1½%. o This is equivalent to a loan with an annual interest rate of 19. 56%. 4 -5

Continuous compounding o o o What will happen to FV of $1000 over 1

Continuous compounding o o o 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 -6

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

Continuous compounding o o o As m goes to infinity, we need the limit of (1+r/m)m*T At this limit, FV=Cer. T FV = 1000 e(. 12*1) = $1, 127. 50 4 -7

Continuous Compounding o The general formula for the future value of an investment compounded

Continuous Compounding o The general formula for the future value of an investment compounded continuously over many periods can be written as: FV = C 0×er. T 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 2. 718. ex is a key on your calculator. 4 -8

Rate transformations o o o Rates in loans and other contracts can be stated

Rate transformations o o o 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. 4 -9