Chapter 5

Time Value of Money 2:

Analyzing Annuity Cash Flows

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

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Chapter 5 Learning GoalsLG1: Compound multiple cash flows to the future

LG2: Compute the future value of frequent, level cash flows

LG3: Discount multiple cash flows to the present

LG4: Compute the present value of an annuity

LG5: Find the present value of a perpetuity

LG6: Recognize and adjust values for beginning-of-period annuity payments as opposed to end-of-period annuity payments

LG7: Explain the impact of compounding frequency and the difference between the annual percentage rate and the effective annual rate

LG8: Compute the interest rate of annuity payments

LG9: Compute payments and amortization schedules for car and mortgage loans

LG10: Calculate the number of payments on a loan

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Introduction

• The previous chapter involved moving a single cash flow from one point in time to another

• Many business situations involve multiple cash flows

• Annuity problems deal with regular, evenly-spaced cash flows– Car loans and home mortgage loans– Saving for retirement– Companies paying interest on debt– Companies paying dividends

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• Consider the following cash flows: you make a $100 deposit today, followed by a $125 deposit next year and a $150 deposit at the end of the second year. If interest rates are 7%, what is the future value of your account at the end of the 3rd year?

1 2

-100 -125 -150

0

...

3

5

• Notice that the first deposit will compound for 3 years, the second deposit will compound for 2 years, and the last deposit will compound for 1 year.

• We can calculate the future value of each deposit individually and add them up to get the total

FV3 = $122.50 + $143.11 + $160.50= $426.11

1 2

-100 -125 -150

0

...

3

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• Now, suppose that the cash flows are the same each period

• Level cash flows are common in finance. These problems are known as annuities

FVAN = PMT( ) (1+i) - 1

i

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Annuities and the Financial Calculator

• In the previous chapter, the level payment button PMT was always set to zero

• Now, for annuities, we can use the PMT key to input the annuity payment

• Example: suppose that $100 deposits are made at the end of each year for five years. If interest rates are 8 percent per year, the future value of the annuity is:

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Calculator Solution

INPUT 5 8 0 -100N I/YR PV PMT FV

OUTPUT 586.66

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• Another example: Calculate the future value if a $50 deposit is made every year for 20 years at a 6 percent interest rate

INPUT 20 6 0 -50N I/YR PV PMT FV

OUTPUT 1,839.28

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• What if the amount deposited doubles to $100 per year?– The future value doubles to $3,678.56

• What if the $100 is deposited every year for 40 years rather than 20 years? Does the future value double as well?– No – remember that the time and interest

rate variables are exponentially related to value

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• The future value more than quadruples when the time is doubled in this example

INPUT 40 6 0 -100N I/YR PV PMT FV

OUTPUT 15,476.20

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• What if the interest rate is increased from 6 percent to 10 percent?

• The size of the periodic payments, the number of years invested, and the interest rate significantly impact the future value of an annuity

INPUT 40 10 0 -100N I/YR PV PMT FV

OUTPUT 44,259.26

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Present Value of Multiple Cash Flows

• Consider the example we started with: you make a $100 deposit today, followed by a $125 deposit next year and a $150 deposit at the end of the second year. Interest rates are 7%

• To find the present value of these cash flows we recognize that we can find their individual present values and add them up

1 2

-100 -125 -150

0

...

3

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Present Value of Level Cash Flows

• The present value of an annuity concept has many practical uses:– Most loans are set up with even payments

throughout the life of the loan

• The general formula for the present value of an annuity is:

PVA = PMT( - ) 1 i

1 i(1+i)n

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• Example: What is the present value of an annuity consisting of $100 payments made at the end of the next 5 years if interest rates are 8 percent per year?

PV = 100 ( ) PV = 100 ( ) 1 .08

1 .08(1+.08)5

PV = 100(3.9927) PV = 100(3.9927)

PV = 399.27 PV = 399.27

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• Calculator solution:

INPUT 5 8 -100 0N I/YR PV PMT FV

OUTPUT 399.27

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Perpetuities• Perpetuities represent a special type of

annuity in which the cash flows go on forever

• Real-life applications of perpetuities– Preferred stock– British 2 ½ % Consolidated Stock (a debt

known as consols)

• The present value of a perpetuity is calculated using a simple equation:

PV of perpetuity = PMT

i

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• Example: Find the present value of a perpetuity that pays $100 per year forever if the discount rate is 10 percent.

PV = 100/.10

= $1000

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Ordinary Annuities vs. Annuities Due

• So far we have worked problems where the payment occurs at the end of each period. This is called an ordinary annuity

• Sometimes, however, the annuity payments occur at the beginning of each period. These are called annuities due

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• In calculating the future value of an annuity due, we recognize that the payments all occur one period sooner than for an ordinary annuity, and therefore earn an extra period of interest. We can adjust the FV as follows:

FVAN due = FVAN x (1+i)

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• Likewise, in calculating the present value of an annuity due, we discount each cash flow one less period. We can adjust the PV using the following equation:

PVAN due = PVAN x (1+i)• In our financial calculators, we need to tell

the calculator that the payments occur at the beginning of each period. We do this by putting the calculator in BEGIN mode (represented by BGN)

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• Example: Find the present value of an annuity due that pays 100 per year for 5 years if the interest rate is 8 percent. Before you begin: should the PV be larger or smaller than if the payments occur at the end of each period?– First Step: Place your calculator in BGN

mode

INPUT 5 8 -100 0N I/YR PV PMT FV

OUTPUT 431.21

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Compounding Frequency

• So far we have assumed that interest is compounded once per year

• What happens when interest is compounded more frequently?– Example: What if 12 percent interest is

compounded semiannually? Let’s say that we invest $100. If interest were compounded annually, we would end up with $112. But, semiannual compounding means that our $100 would earn 6 percent halfway through the year and the other 6 percent at the end. We would end up with: FV = $100 x (1+1.06) x (1.06) = $112.36.

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• We end up with more than $112 due to compounding. The $6 interest we earned in the first half earns $0.36 in interest in the second half.

• The quoted, or nominal rate is called the annual percentage rate (APR)

• The rate that incorporates compounding is called the effective annual rate (EAR)

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• The relationship between APR and EAR is as follows:

11

mAPR

m

EAR

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• Example: A bank loan has a quoted rate of 12 percent. Calculate the effective annual rate if the interest is compounded monthly

EAR = 12.68%

11212.0

112

EAR

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• Calculator solution:– Financial calculators have a function that converts

nominal rates to effective rates– These functions have 3 variables: Nominal rate,

Effective rate, and Compounding periods. The user inputs two of them and the calculator solves for the 3rd.

– On the TI BAII Plus calculator the function is ICONV (interest conversion)

– For the example above:• NOM = 12• C/Yr = 12• EFF = 12.68

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• Example: What is the effective rate if the quoted rate is 10 percent compounded daily?

• ICONV– NOM = 10– C/Yr = 365– EFF = 10.5156%

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Annuity Loans• Finding the interest rate

– Often a business will know the cost of something, as well as the associated cash flows.

– For example: A piece of equipment costs $100,000 and provides positive cash flows of $25,000 for 6 years. What rate of return does this opportunity offer?

INPUT 6 -100,000

25,000 0N I/YR PV PMT FV

OUTPUT 12.98

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• Finding Payments on an Amortized Loan– Example: You want a car loan of $10,000. The

loan is for 4 years and interest rates are 9 percent per year. Calculate your monthly payment

– Before we work this problem, we need to discuss how to set our calculator to solve problems that involve payments that are not annual. We typically do this by adjusting the N, I, and PMT to reflect the relevant period (we will assume we leave the calculator set to 1 payment per year, i.e. P/YR=1)

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• For the above problem:

• Solution:

• Since N and I are monthly, we know that the PMT is the monthly payment

Problem Data Calculator input

4 years N = 4 x 12 = 48 months

9% loan I = 9/12 = 0.75% per month

Loan amount = $10,000 PV = 10,000

INPUT 48 0.75 10,000

0N I/YR PV PMT FV

OUTPUT 248.85

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• Amortized Loan Schedules– Amortized loans are characterized by

level payments, with an increasing portion of the payment consisting of principal, and a decreasing proportion of interest

• Example: Your business has received a $150,00 loan that is to be repaid in annual payments over 3 years. The interest rate is 10%. Construct an amortization schedule for the loan.

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• The first step is to calculate the payment.

INPUT 3 10 150,000

0N I/YR PV PMT FV

OUTPUT 60,317.22

Beg Bal Payment Interest Principal End Bal

1 150,000 60,317.22 15,000 45,317.22 104,682.78

2 104,682.78 60,317.22 10,468.28 49,848.94 54,833.84

3 54,833.84 60,317.22 5,483.38 54,833.84 0.00

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• Computing the Time Period– How long will it take to pay off a loan?– Example: How long will it take to pay off a

$5,000 loan with a 19 percent APR which compounds monthly? The payment is $150 per month

• I = 19/12 = 1.58333

INPUT 1.58333 5,000

-150 0N I/YR PV PMT FV

OUTPUT 47.8