Chapter 6
Discounted Cash Flows and Valuation
Learning Objectives
1. Explain why cash flows occurring at different times must be discounted to a
common date before they can be compared, and be able to compute the present
value and future value for multiple cash flows.
2. Describe how to calculate the present value of an ordinary annuity and how an
ordinary annuity differs from an annuity due.
3. Explain what a perpetuity is and how it is used in business, and be able to calculate
the value of a perpetuity.
4. Discuss growing annuities and perpetuities, as well as their application in business,
and be able to calculate their value.
5. Discuss why the effective annual interest rate (EAR) is the appropriate way to
annualize interest rates, and be able to calculate EAR.
I.
Chapter Outline
6.1
Multiple Cash Flows
A.
Future Value of Multiple Cash Flows

In contrast to Chapter 5, we now consider situations in which there are multiple
cash flows. Solving future value problems with multiple cash flows involves a
simple process.
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
First, draw a time line to make sure that each cash flow is placed in the correct
time period.

Second, calculate the future value of each cash flow for its time period.

Third, add up the future values.
B.
Present Value of Multiple Cash Flows

Many situations in business call for computing the present value of a series of
expected future cash flows. This could be to determine the market value of a
security or business or to decide whether a capital investment should be made.

The process is similar to determining the future value of multiple cash flows.

First, prepare a time line to identify the magnitude and timing of the cash flows.

Next, calculate the present value of each cash flow using Equation 5.4 from the
previous chapter.

Finally, add up all the present values.

The sum of the present values of a stream of future cash flows is their current
market price, or value.
6.2
Level Cash Flows: Annuities and Perpetuities

There are many situations in which both businesses and individuals would be faced
with either receiving or paying a constant amount for a length of period.

When a firm faces a stream of constant payments on a bank loan for a period of time,
we call that stream of cash flows an annuity.
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
Individual investors may make constant payments on their home or car loans, or
invest a fixed amount year after year to save for their retirement.

Any financial contract that calls for equally spaced and level cash flows over a
finite number of periods is called an annuity.

If the cash flow payments continue forever, the contract is called a perpetuity.

Constant cash flows that occur at the end of each period are called ordinary
annuities.
A.
Present Value of an Annuity
 We can calculate the present value of an annuity the same way as we calculated the
present value of multiple cash flows. However, if the number of payments were to
be very large, then this process will be tedious.
 Instead we can simplify Equation 5.4 to obtain an annuity factor. This results in
Equation 6.1, which can be used to calculate the present value of an annuity.
PVA n  CF  Present va lue factor for anannuity
 (1  Present va lue factor ) 
 CF  

i

1 

1  (1  i ) n 
 CF  

i





In addition to using this annuity equation to solve for the present value of an
annuity, financial calculators and spreadsheets may be used. Present value and
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annuity tables created with the help of Equation 6.1 have limited use outside of a
classroom setting.

One problem that is widely solved using a financial calculator is finding the
monthly payment on a car loan or home loan.
B.
Preparing a Loan Amortization Schedule

Amortization refers to the way the borrowed amount (principal) is paid down
over the life of the loan.

The monthly loan payment is structured so that each month a portion of the
principal is paid off and at the time the loan matures, the loan is entirely paid off.

With an amortized loan, each loan payment contains some payment of principal
and an interest payment.

A loan amortization schedule is just a table that shows the loan balance at the
beginning and end of each period, the payment made during that period, and how
much of that payment represents interest and how much represents repayment of
principal.

With an amortized loan, a bigger proportion of each month’s payment goes
toward interest in the early periods. As the loan gets paid down, a greater
proportion of each payment is used to pay down the principal.

C.
Amortization schedules are best done on a spreadsheet (see Exhibit 6.5).
Finding the Interest Rate
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
The annuity equation can also be used to the find the interest rate or discount rate
for an annuity.

To determine the rate of return for the annuity, we need to solve the equation for
the unknown value i.

Other than using a trial-and-error approach, it is easier to solve using this with a
financial calculator.
D.
Future Value of an Annuity

Future value annuity calculations usually involve finding what a savings or an
investment activity is worth at some point in the future.

This could be saving periodically for a vacation, car, or house, or even retirement.

We can derive the future value annuity equation from the present value annuity
equation (Equation 6.1). This results in Equation 6.2, as follows.
FVA n  CF  Future value factor for an annuity
Future value factor - 1
i
n
 (1  i )  1
 CF  

i


 CF 

As with present value annuity calculations, future value calculations are made
easier when financial calculators or spreadsheets are used, especially when
lengthy investment periods are involved.
E.
Perpetuities
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
A perpetuity is a constant stream of cash flows that goes on for an infinite period.

In the stock markets, preferred stock issues are considered to be perpetuities, with
the issuer paying a constant dividend to holders.

The equation for the present value of a perpetuity can be derived from the present
value of an annuity equation with n tending to infinity.
PVA   CF  Present va lue factor for anannuity
1

1

 (1  i ) 
 CF  
i


CF

i



(1  0)
  CF 
i


One thing that should be emphasized in the relationship between the present value
of an annuity and a perpetuity is that just as a perpetuity equation was derived
from the present value annuity equation, we could also derive the present value of
an annuity from the equation for a perpetuity.
F.
Annuity Due

When you have an annuity with the payment being incurred at the beginning of
each period rather than at the end, the annuity is called an annuity due.

Rent or lease payments are typically made at the beginning of each period rather
than at the end of each period.

The annuity transformation method (Equation 6.4) shows the relationship between
the ordinary annuity and the annuity due.
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
Each period’s cash flow thus earns an extra period of interest compared to an
ordinary annuity. Thus, the present value or future value of an annuity due is
always higher than that of ordinary annuity.
Annuity due = Ordinary annuity value  (1 + i)
6.3
Cash Flows That Grow at a Constant Rate

In addition to constant cash flow streams, one may have to deal with cash flows that
grow at a constant rate over time.

These cash flow streams are called growing annuities or growing perpetuities.
A.
Growing Annuity

Business may need to compute the value of multiyear product or service contracts
with cash flows that increase each year at a constant rate.

These are called growing annuities.

An example of a growing annuity could be the valuation of a growing business
whose cash flows are increasing every year at a constant rate.

This equation to evaluate the present value of a growing annuity (Equation 6.5)
can be used when the growth rate is less than the discount rate.
PVA n =
n
CF1   1+g  
× 1- 
 
(i - g )   1+i  
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B.
Growing Perpetuity

When the cash flow stream features a constant growing annuity forever, it is
called a growing perpetuity.

This can be derived from Equation 6.5 when n tends to infinity and results in
Equation 6.6.
PVA  =
6.4
CF1
(i - g )
The Effective Annual Interest Rate

Interest rates can be quoted in the financial markets in a variety of ways.

The most common quote, especially for a loan, is the annual percentage rate (APR).

The APR is a rate that represents the simple interest accrued on a loan or an
investment in a single period. This is annualized over a year by multiplying it by the
appropriate number of periods in a year.
A.
Calculating the Effective Annual Interest Rate (EAR)

The correct way to compute an annualized rate is to reflect the compounding that
occurs. This involves calculating the effective annual rate (EAR).

The effective annual interest rate (EAR) is defined as the annual growth rate
that takes compounding into account.

Equation 6.7 shows how the EAR is computed.
EAR = (1 + Quoted rate/m)m – 1,
where, m is the number of compounding periods during a year.
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
The EAR conversion formula accounts for the number of compounding periods
and, thus, effectively adjusts the annualized interest rate for the time value of
money.

B.
The EAR is the true cost of borrowing and lending.
Consumer Protection Acts and Interest Rate Disclosure

Congress passed the Truth-in-Lending Act in 1968 to ensure that the true cost of
credit was disclosed to consumers so that they could make sound financial
decisions.

Similarly, another piece of legislation called the Truth-in-Savings Act was
passed to provide consumers with an accurate estimate of the return they would
earn on an investment.

These two pieces of legislation require by law that the APR be disclosed on all
consumer loans and savings plans and that it be prominently displayed on
advertising and contractual documents.

It is important to note that the EAR, not the APR, is the appropriate rate to use in
present and future value calculations.
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