Financial Management Core Concepts (Global Edition) 3rd Edition By Raymond Brooks

thE studEnt FrOnt And CEntEr

Designed for the nonfinance major, Financial Management: Core Concepts structures a studentcentric learning environment built around three major competencies:

■ Using tools

■ Making connections

■ Studying for success

8

Early tVm tools. The author identifies the key

concepts of finance as “tools.” Students first

need to learn how to use these tools of finance

before they can apply them to larger problems.

That’s why the author drills down to basics

quickly by developing time value of money (TVM)

concepts and interest rates early in the course.

4.3 • Present Value of an Annuity 89

Appendix 4 provides PVIFAs for a set of payments (n) and interest or discount

rates (r).

The annuity equations for present value and future value are straightforward. Just as in Chapter 3, we have one equation and four variables, and to

solve for any one of the four variables, we must know the other three. Example

4.2 shows a present value problem solved with the three methods introduced

in Chapter 3.

ExAMPLE 4.2 Making retirement golden (present value

of an annuity)

Problem Ben and Donna determine that upon retirement they will need to

withdraw $50,000 annually at the end of each year for the next thirty years. They

know that they can earn 4% each year on their investment. What is the present

value of this annuity? In other words, how much will Ben and Donna need in

their retirement account (at the beginning of their retirement) to generate this

future cash flow?

Solution In this problem, we assume that Ben and Donna need to have the

present value of the thirty-year annuity in their account at the start of their

retirement, even though they will not make the first withdrawal of $50,000 until

the end of the first year of retirement. They will make thirty withdrawals from

this account during retirement. The investment rate is 4%. It is the same as the

discount rate for the future payments of $50,000 that will come at the end of

each year for the next thirty years. The known variables are r = 4%, n = 30, and

PMT = $50,000. Solve for PV.

method 1 Using the equation

First, calculate the PVIFA value for n = 30 and r = 4%:

1 - 31>(1 + 0.04)304

0.04

=

31 - (0.308319)4

0.04

= 17.292033

Then multiply the annuity payment by this factor:

PV = $50,000 × 17.292033 = $864,601.67

method 2 Using the TVM keys

Again, the calculator must be in END mode so that you treat the payments

as an ordinary annuity. Set the calculator for an ordinary annuity (END

mode). Then

Input 30 4.0 ? -50,000 0

Key N I/Y PV PMT FV

CPT 864,601.67

MyFinanceLab Video

M04_BROO6698_CH04_pp081-113.indd 89 11/19/14 6:31 PM

Later Application and Visual Links.

Students soon begin to see just how

powerful these tools are. They learn

to forge links between

basic principles and

new applications.

A tool icon alerts

students when a new

tool is introduced

and when a tool can

be applied in a new

situation.

150 Chapter 6  •  Bonds and Bond Valuation

Method 1: Using the equation Let’s now proceed through the four main steps 

in pricing a bond. You may want to refer to Figure 6.2 as you read through the 

discussion.

Step 1 is to lay out the timing and amount of the future cash flows promised.

The first future cash flow we need to determine is the annual interest payment. 

Here it is the coupon rate of 6.5% times the par value of the bond. We will use 

$1,000 as the par value of this bond:

annual coupon or interest payment = $1,000 × 0.065 = $65.00

The second future cash flow that we need to determine is the payment of 

the par value or principal—in this case, the $1,000 par value of the bond—at 

the maturity date of July 15, 2018. Recall from Chapter 4 that this payment is 

one method of paying back a loan: interest as you go and principal repaid at 

maturity.

We can set out the future cash flow as shown in Figure 6.3. Note that in the 

time line T0 represents the original issue date of July 15, 2008, and T1 is the first 

annual coupon payment date of July 15, 2009. The annual payments continue for 

ten years, with T10 being the last payment on July 15, 2018. This point is a moment 

of recognition in which we can apply previously learned concepts: the coupon 

payments constitute an annuity stream, the same amount at regular intervals. 

The principal or par value of $1,000 also pays out at maturity. Here we recognize 

another key concept: the final amount is a lump-sum payment. So we now have 

the promised set of future cash flows for the Merrill Lynch bond.

Step 2 is to determine the appropriate discount rate for the cash flow. We 

will jump to the answer now and use the yield of 5.30% from the bond data 

in Table 6.1. Later we will develop the concepts behind an appropriate discount rate.

For step 3, we now apply two of the time value of money equations to find

the present value of the lump-sum principal and the annuity stream of coupons.

Because we know that the coupon payments constitute an annuity stream, we use 

the equation for the present value of an annuity from Chapter 4. To value the par 

value, we use the equation for the present value of a lump-sum payment from 

150 Chapter 6  •  Bonds and Bond Valuation

Method 1: Using the equation Let’s now proceed through the four main steps 

in pricing a bond. You may want to refer to Figure 6.2 as you read through the 

discussion.

Step 1 is to lay out the timing and amount of the future cash flows promised.

The first future cash flow we need to determine is the annual interest payment. 

Here it is the coupon rate of 6.5% times the par value of the bond. We will use 

$1,000 as the par value of this bond:

annual coupon or interest payment = $1,000 × 0.065 = $65.00

The second future cash flow that we need to determine is the payment of 

the par value or principal—in this case, the $1,000 par value of the bond—at 

the maturity date of July 15, 2018. Recall from Chapter 4 that this payment is 

one method of paying back a loan: interest as you go and principal repaid at 

maturity.

We can set out the future cash flow as shown in Figure 6.3. Note that in the 

time line T0 represents the original issue date of July 15, 2008, and T1 is the first 

annual coupon payment date of July 15, 2009. The annual payments continue for 

ten years, with T10 being the last payment on July 15, 2018. This point is a moment 

of recognition in which we can apply previously learned concepts: the coupon 

payments constitute an annuity stream, the same amount at regular intervals. 

The principal or par value of $1,000 also pays out at maturity. Here we recognize 

another key concept: the final amount is a lump-sum payment. So we now have 

the promised set of future cash flows for the Merrill Lynch bond.

Step 2 is to determine the appropriate discount rate for the cash flow. We 

will jump to the answer now and use the yield of 5.30% from the bond data