Download IFM7 Chapter 28

28 - 1
WEB CHAPTER 28
Basic Financial Tools: A Review
Time Value of Money
Bond Valuation
Risk and Return
Stock Valuation
Copyright © 2002 South-Western
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Time lines show timing of cash flows.
0
1
2
3
CF1
CF2
CF3
i%
CF0
Tick marks at ends of periods, so Time 0
is today; Time 1 is the end of Period 1;
or the beginning of Period 2.
Copyright © 2002 South-Western
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Time line for a $100 lump sum due at
the end of Year 2.
0
i%
1
2 Year
100
Copyright © 2002 South-Western
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Time line for an ordinary annuity of
$100 for 3 years.
0
1
2
3
100
100
100
i%
Copyright © 2002 South-Western
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What’s the FV of an initial $100 after
1, 2, and 3 years if i = 10%?
0
1
2
3
FV = ?
FV = ?
FV = ?
10%
100
Finding FVs (moving to the right
on a time line) is called compounding.
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After 1 year:
FV1 = PV + INT1 = PV + PV (i)
= PV(1 + i)
= $100(1.10)
= $110.00.
After 2 years:
FV2 = PV(1 + i)2
= $100(1.10)2
= $121.00.
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After 3 years:
FV3 = PV(1 + i)3
= $100(1.10)3
= $133.10.
In general,
FVn = PV(1 + i)n.
Copyright © 2002 South-Western
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What’s the FV in 3 years of $100
received in Year 2 at 10%?
0
10%
1
2
3
100
110
Copyright © 2002 South-Western
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What’s the FV of a 3-year ordinary
annuity of $100 at 10%?
0
1
2
100
100
10%
Copyright © 2002 South-Western
3
100
110
121
FV = 331
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Financial Calculator Solution
INPUTS
3
10
0
-100
N
I/YR
PV
PMT
OUTPUT
FV
331.00
Have payments but no lump sum PV,
so enter 0 for present value.
Copyright © 2002 South-Western
28 - 11
What’s the PV of $100 due in 2 years if
i = 10%?
Finding PVs is discounting, and it’s
the reverse of compounding.
0
1
2
10%
PV = ?
Copyright © 2002 South-Western
100
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Solve FVn = PV(1 + i )n for PV:
PV =
FVn
1 


n = FVn 
 1+ i
1+ i
2
n
1 

 = $100PVIFi,n 
PV = $100
 1.10 
= $1000.8264 = $82.64.
Copyright © 2002 South-Western
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What’s the PV of this ordinary annuity?
0
1
2
3
100
100
100
10%
90.91
82.64
75.13
248.69 = PV
Copyright © 2002 South-Western
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INPUTS
3
10
N
I/YR
OUTPUT
PV
100
0
PMT
FV
-248.69
Have payments but no lump sum FV,
so enter 0 for future value.
Copyright © 2002 South-Western
28 - 15
How much do you need to save each
month for 30 years in order to retire on
$145,000 a year for 20 years, i = 10%?
months before retirement years after retirement
0
1
2
360
1
2
...
PMT PMT
19
20
-145k
-145k
...
PMT
Copyright © 2002 South-Western
-145k
-145k
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How much must you have in your
account on the day you retire if
i = 10%?
years after retirement
0
1
2
...
20
-145k
-145k
...
-145k
How much do you need
on this date?
Copyright © 2002 South-Western
19
-145k
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You need the present value of a
20- year 145k annuity--or $1,234,467.
INPUTS
20
10
N
I/YR
OUTPUT
Copyright © 2002 South-Western
-145000 0
PV
PMT
1,234,467
FV
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How much do you need to save each
month for 30 years in order to have the
$1,234,467 in your account?
months before retirement
0
1
2
360
...
PMT PMT
You need
$1,234,467
on this date.
...
PMT
Copyright © 2002 South-Western
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You need a payment such that the
future value of a 360-period annuity
earning 10%/12 per period is
$1,234,467.
INPUTS
360
10/12
0
N
I/YR
PV
OUTPUT
1234467
PMT
FV
546.11
It will take an investment of $546.11 per
month to fund your retirement.
Copyright © 2002 South-Western
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Key Features of a Bond
1.
Par value: Face amount; paid
at maturity. Assume $1,000.
2.
Coupon interest rate: Stated
interest rate. Multiply by par
value to get dollars of interest.
Generally fixed.
(More…)
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3.
Maturity: Years until bond
must be repaid. Declines.
4.
Issue date: Date when bond
was issued.
Copyright © 2002 South-Western
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The bond consists of a 10-year, 10%
annuity of $100/year plus a $1,000 lump
sum at t = 10:
PV annuity
= $ 614.46
PV maturity value =
385.54
PV annuity
= $1,000.00
INPUTS
10
N
OUTPUT
Copyright © 2002 South-Western
10
I/YR
PV
-1,000
100
PMT
1000
FV
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What would happen if expected
inflation rose by 3%, causing k = 13%?
INPUTS
10
N
OUTPUT
13
I/YR
PV
-837.21
100
PMT
1000
FV
When kd rises, above the coupon rate,
the bond’s value falls below par, so it
sells at a discount.
Copyright © 2002 South-Western
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What would happen if inflation fell, and
kd declined to 7%?
INPUTS
10
N
OUTPUT
7
I/YR
100
PV PMT
-1,210.71
1000
FV
If coupon rate > kd, price rises above
par, and bond sells at a premium.
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The bond was issued 20 years ago
and now has 10 years to maturity.
What would happen to its value over
time if the required rate of return
remained at 10%, or at 13%,
or at 7%?
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Bond Value ($)
1,372
1,211
kd = 7%.
kd = 10%.
1,000
M
837
kd = 13%.
775
30
25
20
15
10
5
0
Years remaining to Maturity
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At maturity, the value of any bond
must equal its par value.
The value of a premium bond would
decrease to $1,000.
The value of a discount bond would
increase to $1,000.
A par bond stays at $1,000 if kd
remains constant.
Copyright © 2002 South-Western
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Assume the Following
Investment Alternatives
Economy
Recession
Below avg.
Average
Above avg.
Boom
Prob. T-Bill
0.10
0.20
0.40
0.20
0.10
1.00
Copyright © 2002 South-Western
HT
Coll
8.0% -22.0% 28.0%
8.0
-2.0
14.7
8.0
20.0
0.0
8.0
35.0 -10.0
8.0
50.0 -20.0
USR
MP
10.0% -13.0%
-10.0
1.0
7.0
15.0
45.0
29.0
30.0
43.0
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What is unique about
the T-bill return?
The T-bill will return 8% regardless
of the state of the economy.
Is the T-bill riskless? Explain.
Copyright © 2002 South-Western
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Do the returns of HT and Collections
move with or counter to the economy?
HT moves with the economy, so it
is positively correlated with the
economy. This is the typical
situation.
Collections moves counter to the
economy. Such negative
correlation is unusual.
Copyright © 2002 South-Western
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Calculate the expected rate of return
on each alternative.
^
k = expected rate of return.
k =
n
k P.
i
i
i=1
^
kHT = 0.10(-22%) + 0.20(-2%)
+ 0.40(20%) + 0.20(35%)
+ 0.10(50%) = 17.4%.
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^
k
HT
17.40%
Market
15.00
USR
13.80
T-bill
8.00
Collections
1.74
 HT has the highest rate of return.
 Does that make it best?
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What is the standard deviation
of returns for each alternative?
 = Standard deviation.
 =
Variance
n
=
.
(k

k̂
)
P
 i
i
2
i=1
Copyright © 2002 South-Western
=

2
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=
n
 (k
 k̂) Pi .
2
i
i=1
HT:
=
((-22 - 17.4)2 0.10 + (-2 - 17.4)2 0.20
+ (20 - 17.4)2 0.40 + (35 - 17.4)2 0.20
+ (50 - 17.4)2 0.10)1/2 = 20.0%.
T-bills = 0.0%.
HT = 20.0%.
Copyright © 2002 South-Western
Coll = 13.4%.
USR = 18.8%.
M = 15.3%.
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The coefficient of variation (CV)
is calculated as follows:
^
/k.
CVHT = 20.0%/17.4% = 1.15  1.2.
CVT-bills = 0.0%/8.0% = 0.
CVColl = 13.4%/1.74% = 7.7.
CVUSR = 18.8%/13.8% = 1.36  1.4.
CVM = 15.3%/15.0% = 1.0.
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Prob.
T-bill
US
R
0
8
13.8
Copyright © 2002 South-Western
17.4
HT
Rate of Return (%)
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Standard deviation measures the
stand-alone risk of an investment.
The larger the standard deviation,
the higher the probability that
returns will be far below the
expected return.
Coefficient of variation is an
alternative measure of stand-alone
risk.
Copyright © 2002 South-Western
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Expected Return versus Risk
Security
HT
Market
USR13.8
T-bills
Collections
Expected
Risk, 
return
17.4%
20.0%
15.0
15.3
18.8
1.4
8.0
0.0
1.74
13.4
 Which alternative is best?
Copyright © 2002 South-Western
CV
1.2
1.0
0.0
7.7
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Portfolio Risk and Return
Assume a two-stock portfolio with
$50,000 in HT and $50,000 in
Collections.
Calculate ^
kp and
 p.
Copyright © 2002 South-Western
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^
Portfolio Return, kp
^
kp is a weighted
average:
n
^
^
kp =  wiki
i=1
^
kp = 0.5(17.4%) + 0.5(1.74%) = 9.6%.
^
^
^
kp is between kHT and
kColl
.
Copyright © 2002
South-Western
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Alternative Method
Estimated Return
Economy
Recession
Below avg.
Average
Above avg.
Boom
Prob.
HT
Coll.
Port.
0.10
0.20
0.40
0.20
0.10
-22.0%
-2.0
20.0
35.0
50.0
28.0%
14.7
0.0
-10.0
-20.0
3.0%
6.4
10.0
12.5
15.0
^
kp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40
+ (12.5%)0.20 + (15.0%)0.10 = 9.6%.
(More...)
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p = ((3.0 - 9.6)2 0.10 + (6.4 - 9.6)2 0.20 +
(10.0 - 9.6)2 0.40 + (12.5 - 9.6)2 0.20 +
(15.0 - 9.6)2 0.10)1/2 = 3.3%.
p is much lower than:
either stock (20% and 13.4%).
average of HT and Coll (16.7%).
The portfolio provides average return
but much lower risk. The key here is
negative correlation.
Copyright © 2002 South-Western
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Portfolio standard deviation in general
p = Portfolio standard deviation.
p =
w   w   2w 1w 2  1 2r1,2
2
1
2
1
2
2
2
2
Where w1 and w2 are portfolio weights
and r1,2 is the correlation coefficient
between stock 1 and 2.
Copyright © 2002 South-Western
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Two-Stock Portfolios
Two stocks can be combined to form a
riskless portfolio if r = -1.0.
Risk is not reduced at all if the two
stocks have r = +1.0.
In general, stocks have r  0.65, so risk
is lowered but not eliminated.
Investors typically hold many stocks.
What happens when r = 0?
Copyright © 2002 South-Western
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Portfolio beta
bp = Portfolio beta
bp = w1b1 + w2b2
Where w1 and w2 are portfolio weights,
and b1 and b2 are stock betas. For our
portfolio of 50% HT and 50% Collections,
bp = 0.5(1.30) + 0.5(-0.87) = 0.215  0.22.
Copyright © 2002 South-Western
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What would happen to the riskiness of
an average portfolio as more randomly
picked stocks were added?
p would decrease because the
added stocks would not be
^
perfectly correlated, but kp would
remain relatively constant.
Copyright © 2002 South-Western
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Prob.
Large
2
1
0
15
1 35% ; Large 20%.
Copyright © 2002 South-Western
Return
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p (%)
Company-Specific
(Diversifiable) Risk
35
Stand-Alone Risk, p
20
Market Risk
0
10
20
30
40
2,000+
# Stocks in Portfolio
Copyright © 2002 South-Western
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Stand-alone Market Diversifiable
= risk +
.
risk
risk
Market risk is that part of a security’s
stand-alone risk that cannot be
eliminated by diversification.
Firm-specific, or diversifiable, risk is
that part of a security’s stand-alone risk
that can be eliminated by
diversification.
Copyright © 2002 South-Western
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Conclusions
As more stocks are added, each new
stock has a smaller risk-reducing
impact on the portfolio.
p falls very slowly after about 40
stocks are included. The lower limit
for p is about 20% = M .
By forming well-diversified portfolios,
investors can eliminate about half the
riskiness of owning a single stock.
Copyright © 2002 South-Western
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Can an investor holding one stock earn
a return commensurate with its risk?
No. Rational investors will minimize
risk by holding portfolios.
They bear only market risk, so prices
and returns reflect this lower risk.
The one-stock investor bears higher
(stand-alone) risk, so the return is less
than that required by the risk.
Copyright © 2002 South-Western
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How is market risk measured for
individual securities?
Market risk, which is relevant for stocks
held in well-diversified portfolios, is
defined as the contribution of a security
to the overall riskiness of the portfolio.
It is measured by a stock’s beta
coefficient, which measures the stock’s
volatility relative to the market.
What is the relevant risk for a stock held
in isolation?
Copyright © 2002 South-Western
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How are betas calculated?
Run a regression with returns on the
stock in question plotted on the Yaxis and returns on the market
portfolio plotted on the X-axis.
The slope of the regression line,
which measures relative volatility, is
defined as the stock’s beta
coefficient, or b.
Copyright © 2002 South-Western
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Illustration
of beta
calculation:
Beta
Illustration
_
ki
.
.
20
15
10
5
-5
0
5
-5
.
-10
Copyright © 2002 South-Western
10
15
Regression line:
^
^ .
ki = -2.59 + 1.44 k
M
Year kM
1
15%
2
-5
3
12
20
ki
18%
-10
16
_
kM
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How is beta calculated?
The regression line, and hence beta,
can be found using a calculator with
a regression function or a
spreadsheet program. In this
example, b = 1.44.
Analysts typically use five years’ of
monthly returns to establish the
regression line.
Copyright © 2002 South-Western
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How is beta interpreted?
If b = 1.0, stock has average risk.
If b > 1.0, stock is riskier than average.
If b < 1.0, stock is less risky than
average.
Most stocks have betas in the range of
0.5 to 1.5.
Can a stock have a negative beta?
Copyright © 2002 South-Western
28 - 57
_
ki
HT
b = 1.30
40
b=0
20
T-Bills
-20
0
20
40
b = -0.87
_
kM
Collections
-20
Regression Lines of Three Alternatives
Copyright © 2002 South-Western
28 - 58
Expected Return versus Market Risk
Security
HT
Market
USR
T-bills
Collections
Expected
return
17.4%
15.0
13.8
8.0
1.74
Risk, b
1.30
1.00
0.89
0.00
-0.87
 Which of the alternatives is best?
Copyright © 2002 South-Western
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Use the SML to calculate each
alternative’s required return.
The Security Market Line (SML) is
part of the Capital Asset Pricing
Model (CAPM).
SML: ki = kRF + (kM - kRF)bi .
^
Assume kRF = 8%; kM = kM = 15%.
RPM = kM - kRF = 15% - 8% = 7%.
Copyright © 2002 South-Western
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Required Rates of Return
kHT
= 8.0% + (15.0% - 8.0%)(1.30)
= 8.0% + (7%)(1.30)
= 8.0% + 9.1%
= 17.1%.
kM
kUSR
kT-bill
kColl
=
=
=
=
8.0% + (7%)(1.00)
8.0% + (7%)(0.89)
8.0% + (7%)(0.00)
8.0% + (7%)(-0.87)
Copyright © 2002 South-Western
= 15.0%.
= 14.2%.
= 8.0%.
= 1.9%.
28 - 61
Expected versus Required Returns
^
HT
Market
USR
T-bills
Coll
k
17.4%
15.0
13.8
8.0
1.74
Copyright © 2002 South-Western
k
17.1%
15.0
14.2
8.0
1.9
Undervalued
Fairly valued
Overvalued
Fairly valued
Overvalued
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SML:
ki = 8% + (15% - 8%) bi.
ki (%)
.
HT
kM = 15
kRF = 8
.
.
Market
.
USR
. T-bills
Coll.
-1
0
1
2
Risk, bi
SML and Investment Alternatives
Copyright © 2002 South-Western
28 - 63
What is the required rate of return
on the HT/Collections portfolio?
kp = Weighted average k
= 0.5(17%) + 0.5(2%) = 9.5%.
Or use SML:
bp = 0.22 (Slide 2-45)
kp = kRF + (kM - kRF) bp
= 8.0% + (15.0% - 8.0%)(0.22)
= 8.0% + 7%(0.22) = 9.5%.
Copyright © 2002 South-Western
28 - 64
Stock Value = PV of Dividends
 
P
0
D1

D2

D3
1  k  1  k  1  k 
1
s
2
s
3
s
. . .
D
1  k 

s
What is a constant growth stock?
One whose dividends are expected to
grow forever at a constant rate, g.
Copyright © 2002 South-Western
.
28 - 65
For a constant growth stock,
D1  D 0 1  g .
1
D 2  D 0 1  g .
2
Dt
t


 Dt 1 g .
If g is constant, then:
D0 1  g
D1
.
P̂0 

ks  g
ks  g
Copyright © 2002 South-Western
28 - 66
$
D t  D 0 1  g
t
Dt
PVD t 
t
1

k


0.25
P0   PVDt
0
Copyright © 2002 South-Western
If g > k, P0 = negative
Years (t)
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What happens if g > ks?
 
P
0
D1
requires k s  g.
ks  g
If ks< g, get negative stock price,
which is nonsense.
We can’t use model unless (1) g  ks
and (2) g is expected to be constant
forever. Because g must be a longterm growth rate, it cannot be  ks.
Copyright © 2002 South-Western
28 - 68
Assume beta = 1.2, kRF = 7%, and kM =
12%. What is the required rate of
return on the firm’s stock?
Use the SML to calculate ks:
ks = kRF + (kM - kRF)bFirm
= 7% + (12% - 7%) (1.2)
= 13%.
Copyright © 2002 South-Western
28 - 69
D0 was $2.00 and g is a constant 6%.
Find the expected dividends for the
next 3 years, and their PVs. ks = 13%.
0
g = 6%
1
D0 = 2.00 2.12
13
1.8761 %
1.7599
1.6508
Copyright © 2002 South-Western
2
2.2472
3
2.3820
4
28 - 70
What’s the stock’s market value?
D0 = 2.00, ks = 13%, g = 6%.
Constant growth model:
D0(1 + g)
D1
P̂0 =
=
ks - g
ks - g
$2.12
$2.12
=
=
= $30.29.
0.13 - 0.06
0.07
Copyright © 2002 South-Western
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Rearrange model to rate of return form:
 
P
0
D1
D1

to k s 
 g.
ks  g
P0
^
Then, ks = $2.12/$30.29 + 0.06
= 0.07 + 0.06 = 13%.
Copyright © 2002 South-Western
28 - 72
If we have supernormal growth of
30% for 3 yrs, then a long-run constant
^ ? k is still 13%.
g = 6%, what is P
0
s
Can no longer use constant growth
model.
However, growth becomes constant
after 3 years.
Copyright © 2002 South-Western
28 - 73
Nonconstant growth followed by constant
growth:
0 k =13%
s
1
g = 30%
D0 = 2.00
2
g = 30%
2.60
3
g = 30%
3.38
4
g = 6%
4.394
4.6576
2.3009
2.6470
3.0453
$4.6576
ˆ 
P
 $66.5377
3
0.13  0.06
46.1140
54.1072
^
= P0
Copyright © 2002 South-Western