This document summarizes key concepts related to bond valuation and alternative investment rules. It begins by providing examples of how bond prices relate to interest rates and defines terms like yield to maturity. It then discusses the term structure of interest rates and the pure expectations and liquidity preference hypotheses for forecasting future rates. The document concludes by explaining four alternative investment rules - payback period, average accounting return, internal rate of return (IRR), and profitability index - and identifying their advantages and disadvantages, especially in comparison to net present value (NPV) analysis. Special attention is given to potential problems with using the IRR approach.

1. Corporate Finance 3-0
© Professor Ho-Mou Wu
Bond Market and Alternative
Investment Rules
3.1 Valuation of Bonds
3.2 The Term Structure of Interest Rates
3.3 Alternative Investment Rules
(1) The Payback Period Rule
(2) The Average Accounting Return
(3) The Internal Rate of Return
(4) The Profitability Index
3.4 Why Use Net Present Value?
(RWJ Ch.5,6)

2. 3-1
© Professor Ho-Mou Wu Corporate Finance
3.1 Valuation of Bonds
Example 1：
Suppose we observe the following bond prices for default-free zero coupon
bonds (pure discount bond, with face value $1,000)：
How are the bond prices related with interest rates?
1 year zero: Price = 926
2 year zero: Price = 842
3 year zero: Price = 758
4 year zero: Price = 683
1 year bond
2 year bond
3 year bond
4 year bond
y1
y2
y3
y4
i1 i2? i4?
i3?
(maturity date)

3. 3-2
© Professor Ho-Mou Wu Corporate Finance
The Present Value Formulas for Bonds
Pure Discount Bonds
Level Coupon Bonds
Consols
for T-maturity bonds with face value F.
T
T
y
F
y
C
y
C
y
C


































1
1
1
1
PV
P 2
B

 
,
1
PV
PB T
T
y
F



Such a rate y is known as the yield to maturity (YTM). The yield to
maturity is a complicated average of different rates of interest. It can be a
useful summary measure.
y
C

 PV
PB
‥ ‥ ‥ ‥ ‥ ‥ ‥ ‥ ‥ ‥(3.1)
‥ ‥(3.2)
‥ ‥ ‥ ‥ ‥ ‥ ‥ ‥ ‥ ‥ ‥ ‥(3.3)

4. 3-3
© Professor Ho-Mou Wu Corporate Finance
Yield to Maturity
Example 1.(continued): we can convert bond prices into “yield to
maturity” ( )
hence,
yn= yield of bonds with n periods as time to maturity, also called
“spot rates.” Plot yn against time to maturity (n) ” yield curve” to
summarize information about bond prices (diagram 1).
n
y
)
1
(
1000
926
1
y


2
2 )
1
(
1000
842
y


3
3 )
1
(
1000
758
y


4
4 )
1
(
1000
683
y


)
(
%
8 1
1
1 y
i
y 

%
9
2 
y
%
66
.
9
3 
y
%
10
4 
y

5. 3-4
© Professor Ho-Mou Wu Corporate Finance
3.2 The Term Structure of Interest Rates
From bond prices, we can compute yields , plot the “yield
curve”, and compute the implied forward rates, .
n
y
n
f
Yield Curve
Maturity
%
10
8
6
4
2
2
1 3 4
implied
“forward rates”
yield curve or
“spot rates”

6. 3-5
© Professor Ho-Mou Wu Corporate Finance
Forward Rates
is the “break-even” interest rate that equates the returns on a n-
period bond to that of a (n – 1) period bond rolled over into a one-
year bond in year n.
For example, , (geometric mean) or
, so as an approximation (arithmetic mean).
Similarly,
n
f
.
1
1
1
,
1
1
1
,
1
1
1
4
4
4
3
3
3
3
3
2
2
2
2
2
1


































































y
f
y
y
f
y
y
f
y
1
.
1
)
08
.
1
(
)
09
.
1
(
1
2
2 

 f %
10
2 
f
%
10
2 
f
 
2
1
2 2
1 f
y
y 

.
%
11
so
),
(
3
1
3
3
2
1
3 


 f
f
f
y
y
.
%
11
so
),
(
3
1
4
4
3
2
1
4 



 f
f
f
f
y
y

7. 3-6
© Professor Ho-Mou Wu Corporate Finance
Forecast of Future Interest
Can we use forward rates fn to forecast future short-term interest rates
in, also called “short rates”？ Assume that the investment horizon is
one year, and investors are risk neutral.
Example 2: Consider two investment alternatives：
(A) buy 1-year zero-coupon bond (safe, no risk).
(B) buy 2-year zero-coupon bond and sell it at the end of 1st year
(risky, subject to price risk at the end of 1st year.)
842 ? 1000
926 1000
(A)
(B)
i2=?
)
1
(
1000
2
i
PB



8. 3-7
© Professor Ho-Mou Wu Corporate Finance
Pure Expectation Hypothesis
Expected return of (A)
Expected return of (B)
Assume that investors are risk-neutral. These two expected returns should
be the same (do not worry about different risks involved in (A) and (B))：
and we know
So , hence
The forward rates are market expectations of future short-term interest
rates . This is called the Pure Expectations Hypothesis：
…… .…….. .……. ……..(3.4)
    1
1
1 1
or
1
1
926
1000 i
y
y
E
E 





   
  



































2
2
2
2
2
2
i
1
1
1
1
1
842
1
1000
2
E
y
i
y
E
i
E
  











2
2
2
1
1
1
1
1
i
E
y
y
   
2
1
2
2 1
1
1
f
y
y




  









 2
2
1
1
1
1
i
E
f
  )
(
2
or
1
1 2
2
2 i
E
f
i
E
f 



n
i






 n
n i
E
f

9. 3-8
© Professor Ho-Mou Wu Corporate Finance
Liquidity Preference Hypothesis
Assume investors are Risk averse. Still with one-year investment
horizon (preference for “liquidity”).
Since (B) is riskier, (B) should have higher expected return to attract
investors： .
Hence .
The Liquidity Preference Hypothesis: + risk premium = f2 .
In general, fn - risk premium = .…………………….……..(3.5)
)
1
(
1
1
)
1
( 1
2
2
2 y
i
E
y 











)
(
2
1
1
1
1
2
2
2
i
E
f
i
E
f













 
2
i
E
 
n
i
E

10. 3-9
© Professor Ho-Mou Wu Corporate Finance
3.3 Alternative Investment Rules
• How long does it take the project to “pay back” its
initial investment?
• Payback Period = number of years to recover
initial costs
• Minimum Acceptance Criteria:
– set by management
• Ranking Criteria:
– set by management
(1) The Payback Period Rule

11. 3-10
© Professor Ho-Mou Wu Corporate Finance
(1) The Payback Period Rule (continued)
• Disadvantages:
– Ignores the time value of money
– Ignores cash flows after the payback period
– Biased against long-term projects
– Requires an arbitrary acceptance criteria
– A project accepted based on the payback criteria may
not have a positive NPV
• Advantages:
– Easy to understand
– Biased toward liquidity

12. 3-11
© Professor Ho-Mou Wu Corporate Finance
The Discounted Payback Period Rule
• How long does it take the project to “pay back” its
initial investment taking the time value of money
into account?
• By the time you have discounted the cash flows,
you might as well calculate the NPV.

13. 3-12
© Professor Ho-Mou Wu Corporate Finance
(2) The Average Accounting Return Rule
• Another attractive but fatally flawed approach.
• Ranking Criteria and Minimum Acceptance Criteria
set by management
• Disadvantages:
– Ignores the time value of money
– Uses an arbitrary benchmark cutoff rate
– Based on book values, not cash flows and market values
• Advantages:
– The accounting information is usually available
– Easy to calculate

14. 3-13
© Professor Ho-Mou Wu Corporate Finance
(3) The Internal Rate of Return (IRR) Rule
• IRR: the discount that sets NPV to zero
• Minimum Acceptance Criteria:
– Accept if the IRR exceeds the required return.
• Ranking Criteria:
– Select alternative with the highest IRR
• Reinvestment assumption:
– All future cash flows assumed reinvested at the IRR.
• Disadvantages:
– Does not distinguish between investing and borrowing.
– IRR may not exist or there may be multiple IRR
– Problems with mutually exclusive investments
• Advantages:
– Easy to understand and communicate

15. 3-14
© Professor Ho-Mou Wu Corporate Finance
(3) The Internal Rate of Return: Example
Example 3
Consider the following project:
0 1 2 3
$50 $100 $150
-$200
The internal rate of return for this project is 19.44%

16. 3-15
© Professor Ho-Mou Wu Corporate Finance
The NPV Payoff Profile for This Example
Discount Rate NPV
0% $100.00
4% $71.04
8% $47.32
12% $27.79
16% $11.65
20% ($1.74)
24% ($12.88)
28% ($22.17)
32% ($29.93)
36% ($36.43)
40% ($41.86)
If we graph NPV versus discount rate, we can see the IRR as
the x-axis intercept.
IRR = 19.44%
($60.00)
($40.00)
($20.00)
$0.00
$20.00
$40.00
$60.00
$80.00
$100.00
$120.00
-1% 9% 19% 29% 39%
Discount rate
NPV

17. 3-16
© Professor Ho-Mou Wu Corporate Finance
Problems with the IRR Approach
• Multiple IRRs.
• Are We Borrowing or Lending?
• The Scale Problem.
• The Timing Problem.

18. 3-17
© Professor Ho-Mou Wu Corporate Finance
Multiple IRRs
Example 4: There are two IRRs for this project:
0 1 2 3
$200 $800
-$200 - $800
($150.00)
($100.00)
($50.00)
$0.00
$50.00
$100.00
-50% 0% 50% 100% 150% 200%
Discount rate
NPV
100% = IRR2
0% = IRR1
Which one should
we use?

19. 3-18
© Professor Ho-Mou Wu Corporate Finance
The Scale Problem
Would you rather make 100% or 50% on your
investments?
What if the 100% return is on a $1 investment while
the 50% return is on a $1,000 investment?

20. 3-19
© Professor Ho-Mou Wu Corporate Finance
The Timing Problem
0 1 2 3
$10,000 $1,000 $1,000
-$10,000
Project A
0 1 2 3
$1,000 $1,000 $12,000
-$10,000
Project B
The preferred project in this case depends on the discount rate,
not the IRR.
Example 5:

21. 3-20
© Professor Ho-Mou Wu Corporate Finance
The Timing Problem
($4,000.00)
($3,000.00)
($2,000.00)
($1,000.00)
$0.00
$1,000.00
$2,000.00
$3,000.00
$4,000.00
$5,000.00
0% 10% 20% 30% 40%
Discount rate
NPV
Project A
Project B
10.55% = crossover rate
12.94% = IRRB 16.04% = IRRA
Example 5:

22. 3-21
© Professor Ho-Mou Wu Corporate Finance
Calculating the Crossover Rate
Compute the IRR for either project “A-B” or “B-A”
Year Project A Project B Project A-B Project B-A
0 ($10,000) ($10,000) $0 $0
1 $10,000 $1,000 $9,000 ($9,000)
2 $1,000 $1,000 $0 $0
3 $1,000 $12,000 ($11,000) $11,000
($3,000.00)
($2,000.00)
($1,000.00)
$0.00
$1,000.00
$2,000.00
$3,000.00
0% 5% 10% 15% 20%
Discount rate
NPV
A-B
B-A
10.55% = IRR
Example 5:

23. 3-22
© Professor Ho-Mou Wu Corporate Finance
Mutually Exclusive vs. Independent Project
• Mutually Exclusive Projects: only ONE of several
potential projects can be chosen, e.g. acquiring an
accounting system.
– RANK all alternatives and select the best one.
• Independent Projects: accepting or rejecting one project
does not affect the decision of the other projects.
– Must exceed a MINIMUM acceptance criteria.

24. 3-23
© Professor Ho-Mou Wu Corporate Finance
(4) The Profitability Index (PI) Rule
• Minimum Acceptance Criteria:
– Accept if PI > 1
• Ranking Criteria:
– Select alternative with highest PI
• Disadvantages:
– Problems with mutually exclusive investments
• Advantages:
– May be useful when available investment funds are
limited
– Easy to understand and communicate
– Correct decision when evaluating independent projects

25. 3-24
© Professor Ho-Mou Wu Corporate Finance
3.4 Why Use Net Present Value?
• Accepting positive NPV projects benefits
shareholders.
 NPV uses cash flows
 NPV uses all the cash flows of the project
 NPV discounts the cash flows properly

26. 3-25
© Professor Ho-Mou Wu Corporate Finance
The Net Present Value (NPV) Rule
• Net Present Value (NPV) = Total PV of future CF’s +
Initial Investment
• Estimating NPV:
– 1. Estimate future cash flows: how much? and when?
– 2. Estimate discount rate
– 3. Estimate initial costs
• Minimum Acceptance Criteria: Accept if NPV > 0
• Ranking Criteria: Choose the highest NPV

27. 3-26
© Professor Ho-Mou Wu Corporate Finance
Good Attributes of the NPV Rule
• 1. Uses cash flows
• 2. Uses ALL cash flows of the project
• 3. Discounts ALL cash flows properly
• Reinvestment assumption: the NPV rule assumes
that all cash flows can be reinvested at the discount
rate.

28. 3-27
© Professor Ho-Mou Wu Corporate Finance
The Practice of Capital Budgeting
• Varies by industry:
– Some firms use payback, others use accounting
rate of return.
• The most frequently used technique for large
corporations is IRR or NPV.