Citibank: basic of corporate finance
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UNIT 5: INTRODUCTION TO CAPITAL BUDGETING
INTRODUCTION
Capital budgeting refers to the techniques used by financial analysts to evaluate,
compare, and select investment alternatives that will maximize the shareholder's equity in
the company. In this unit, the formulas and calculations you have learned will be applied
to new situations that require an analyst to make "invest / don't invest" decisions. We will
also introduce some additional factors, including inflation, that the analyst must consider
in making investment decisions.
UNIT OBJECTIVES
When you complete this unit, you will be able to:
• Recognize the difference between nominal and effective interest rates
• Define net present value (NPV) and internal rate of return (IRR)
• Calculate the net present value of a stream of cash flows
• Calculate the internal rate of return of a stream of cash flows
• Recognize other analytical techniques that may be applied to the capital
budgeting process
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NOMINAL AND EFFECTIVE RATES
Interest paid
more than
annually
In previous units, most of the securities we referred to in the
examples paid annual interest (or dividend) payments. However, in the
real world, there are many securities that pay interest more
frequently. U.S. Treasury securities that pay interest semi-annually
and corporate bonds that pay interest quarterly are some examples.
For these types of securities, there is a difference between the quoted
interest rate and the amount of interest that is actually paid.
Stated vs.
actual rate of
return
The nominal interest rate is the stated or contracted rate of return
that a security is expected to pay. On a bond, the coupon rate is the
nominal rate. The effective interest rate is the rate of return actually
earned on the investment.
Calculating Effective Interest Rates
Example
An example will illustrate the difference between nominal and
effective interest rates. Suppose that you invest $1,000 in an account
that pays a 10% nominal annual interest, compounded semi-annually.
At the end of one year, how much will be in your account? Recall the
future value formula:
FV
T
= P (1 + R/M)
T x M
Where:
P = Principal invested
R = Nominal annual interest rate
M = Number of compounding periods in the year
T = Number of years
The future value of your account at the end of the year is:
FV
1
= $1,000 (1 + (0.10/2))
1 x 2
FV
1
= $1,102.50
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The effective interest rate is the rate of return actually earned on an
investment. This is calculated by dividing the total amount of
interest earned over the year by the original investment. In the
example, the effective interest rate is:
$102.50 / $1,000.00 = 0.1025 or 10.25%
Effective rate
higher than
nominal rate
As you can see, the effective annual rate (10.25%) is higher than the
nominal annual rate (10%). This is always the case when the quoted
annual interest is compounded more frequently than at annual
intervals.
There is a more direct formula for comparing nominal and effective
interest rates:
R
*
= [ 1 + (R / M) ]
M
- 1
Where:
R
*
= Effective interest rate
R = Nominal interest rate
M = Number of compounding periods in the year
Example
Using this formula, find the effective annual interest rate of an
investment that pays a quoted annual rate of 11% per annum (p.a.),
compounded monthly. Use the values R = 0.11 and M = 12.
R
*
= [ 1 + (R / M)]
M
- 1
R
*
= [ 1 + (0.11 / 12)]
12
- 1
R
*
= [ 1.009167]
12
- 1
R
*
= [ 1.115719] - 1
R
*
= 0.115719 or 11.57%
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Adjusting for Inflation
Relationship
between
nominal, real,
and inflation
rates
Inflation is one of the most influential factors in the determination
of nominal interest rates. You may recall that when we discussed
inflation in Unit Two, we said that the quoted (nominal) interest rate is
equal to the risk-free real interest rate plus an inflation premium. A
more precise formula should be used to find the effective interest rate
when making adjustments for longer-term securities or in
environments of high inflation. The relationship between nominal,
real, and inflation rates can be calculated with this formula:
R
R
= [ (1 + R
N
) / (1 + h) ] - 1
Where:
R
R
= Real interest or real discount rate
R
N
= Nominal interest or discount rate
h = Expected inflation rate
Effective
interest rate
after adjusting
for inflation
Let's look at an example to see how we apply the formula to find
out the effective interest rate on an investment after adjusting for
inflation.
Suppose that a Central Treasury Bank issues one-year bonds at a
nominal interest rate of 25% p.a. Economists expect that the average
inflation for the coming year will be 14%. What is the real interest
rate that an investor can expect to earn by buying the bonds? We use
the values R
N
= 0.25 and h = 0.14.
R
R
= [ (1 + R
N
) / (1 + h) ] - 1
R
R
= [ (1 + 0.25) / (1 + 0.14) ] - 1
R
R
= [ (1.25) / (1.14) ] - 1
R
R
= [ 1.0965 ] - 1
R
R
= 0.0965 or 9.65%
This means that although the stated interest rate on the bonds is
25%, after adjusting for inflation, an investor actually earns only
9.65% on the investment. The difference between the two rates
represents the loss of purchasing power resulting from inflation. As
we mentioned before, this concept is particularly important in
economies experiencing high levels of inflation.
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Solving for
nominal rate
By adjusting the formula, we can solve for the nominal rate.
R
N
= R
R
+ h + (h x R
R
)
The following example illustrates how to find the appropriate nominal
rate at which to invest in an inflationary environment when the
targeted real rate of return is known.
Example
An investor would like to earn a real interest rate of 7% p.a. on an
investment. If the expected inflation rate is 8% for the coming year,
at what nominal rate should s/he invest so that the real rate of return
is 7%? Use the second formula with the values R
R
= 0.07 and h =
0.08.
R
N
= R
R
+ h + ( h x R
R
)
R
N
= 0.07 + 0.08 +[ (0.08)(0.07) ]
R
N
= 0.1556 or 15.56%
Given the expected inflation rate of 8%, the investor should invest at
a 15.56% nominal rate to earn a 7% real rate on the investment.
In the Practice Exercise that follows, you will have an opportunity to practice what you
have learned about calculating effective interest rates for multiple compounding periods
and annual interest rates adjusted for inflation. Please complete the practice exercises
and check your answers with the Answer Key before continuing with the next section on
"Net Present Value."
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PRACTICE EXERCISE 5.1
Directions: Calculate the answer to each question, then enter the correct answer. Check
your solution with the Answer Key on the next page.
1. An investor buys $1M of 8% Treasury notes which pay interest after six months and
are redeemed after 1 year. What is the effective interest rate earned by the investor,
assuming that he can reinvest the first coupon payment at 8% for the next six
months?
$_____________________
2. An account pays 1.5% interest on the balance at the end of each month. What is the
effective annual interest rate earned on the account?
$_____________________
3. Treasury bonds are issued at a nominal interest rate of 30% for the following year.
Inflation is expected to be 15% for the same period. What is the real interest rate
earned on the bonds?
$_____________________
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ANSWER KEY
1. An investor buys $1M of 8% Treasury notes which pay interest after six months and
are redeemed after 1 year. What is the effective interest rate earned by the investor,
assuming that he can reinvest the first coupon payment at 8% for the next six
months?
8.16%
R
*
= [ 1 + (R/M)]
M
- 1
R
*
= [1 + (0.08/2)]
2
- 1
R
*
= [1.0816] - 1
R
*
= 0.0816 or 8.16%
2. An account pays 1.5% interest on the balance at the end of each month. What is the
effective annual interest rate earned on the account?
19.56%
R
*
= [ 1 + (R/M)]
M
- 1
R
*
= [1 + (0.015)]
12
- 1
R
*
= [1.1956] - 1
R
*
= 0.1956 or 19.56%
3. Treasury bonds are issued at a nominal interest rate of 30% for the following year.
Inflation is expected to be 15% for the same period. What is the real interest rate
earned on the bonds?
13.04%
R
R
= [ (1 + R
N
) / (1 + h) ] - 1
R
R
= [ (1 + 0.30) / (1 + 0.15) ] - 1
R
R
= [ (1.30) / (1.15) ] - 1
R
R
= [ 1.1304] - 1
R
R
= 0.1304 or 13.04%
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NET PRESENT VALUE
The concept of present value and its calculation should be familiar
to you from our discussion of the time value of money in Unit
Three. In this section, we extend the concept to show another way it
is used: to calculate the net present value of an investment.
Present Value of Cash Flows
Let's begin by reviewing the formula for calculating the present
value of a series of cash flows.
PV = CF
1
[ 1 / (1+R) ]
1
+ CF
2
[ 1 / (1+R) ]
2
+ ... + CF
T
[ 1 / (1+R) ]
T
Sum of a series
of discounted
cash flows
We discount the cash flow in the first period (CF
1
) for one period,
discount the cash flow received in the second period (CF
2
) for two
periods, and so forth until we have discounted all of the cash flows.
The present value is the sum of the discounted cash flows. We can
shorten this formula using the summation symbol Σ (pronounced
sigma) to represent the sum of a series.
T
PV = Σ CF
t
[ 1/ (1 + R) ]
t
t=1
These two equations are equal; they both represent the sum of a series
of discounted cash flows. In the shortened version, the T at
the top of the Σ means that we end with the T (last) cash flow, and the t
= 1 at the bottom means that we start the summation with the first cash
flow.
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