Download Principles of Corporate Finance

Risk and Return
To risk or not to risk, that is the question…
Standard deviation and normal distribution

normal distribution is completely defined by
its mean and standard deviation

the probability of abnormally high or low returns
depends on the standard deviation
Std Devs.
from mean
-3
-2
-1
0
1
2
3
Cumulative
probability
0.1%
2.3
15.9
50.0
84.1
97.7
99.9
Markowitz
Portfolio
Theory
Price changes vs. Normal distribution
IBM - Daily % change 1986-2006
Proportion of Days
6
5
4
3
2
1
0
-6 -5 -5 -4 -3 -2 -1 -1
0
1
2
2
3
Daily % Change
4
5
5
6
Calculating mean (or expected) return
Probability
Return
Probability
x return
.20
-10%
-2%
.50
+10
5
.30
+30
9
Total
12%
Mean or
expected
return
Calculating variance and standard deviation
Probability
Return
.20
-10%
.50
.30
Deviation
from mean
return
Probability
x squared
deviation
-22%
96.8
+10
-2
2.0
+30
+18
97.2
Total
196.0
Variance
Standard deviation = square root of variance = 14%
Calculating variance and standard deviation
of Merck returns from past monthly data
Month
1
2
3
4
5
6
Total
Return
Deviation
from mean
return
5.4%
1.7
- 3.6
13.6
- 3.5
3.2
2.6%
- 1.1
- 6.4
10.8
- 6.3
0.4
16.8
Mean:
16.8/6 = 2.8%
Variance:
205.4/6 = 34.237
Std dev:
Sq root of 34.237 = 5.851% per month
Annualised std dev: 5.9 x square root (12) = 20.3%
Squared
deviation
6.76
1.21
40.96
116.64
39.69
0.16
205.42
Mean and standard deviation

mean measures average (or expected return)

standard deviation (or variance) measures the
spread or variability of returns

risk averse investors prefer high mean & low
standard deviation
20
15
expected
return
better
10
5
0
5
10
15
standard deviation
20
Expected portfolio return
Portfolio
proportion (x)
Expected
return (r)
Proportion x
return (xr)
Merck
.40
10%
4%
McDonald
.60
15
9
Total
1.00
13%
Expected
portfolio
return
Calculating covariance and correlation
Prob.
Return on:
A
B
.15
.05
.45
.05
.25
.05
-10%
-10
+10
+10
+30
+30
Mean
Std dev
+12
+14
-10%
+10
+10
+30
+30
-10
Deviation from
mean return:
A
B
-22%
-22
-2
-2
+18
+18
+12
+14
Probability
x product of
deviations
-22%
-2
-2
+18
+18
-22
+ 72.6
+ 2.2
+ 1.8
- 1.8
+ 81.0
- 19.8
Total
136
Covariance
Correlation
coefficient
=
covariance
(sd A) x (sd B)
=
136
14 x 14
=
.6944
Calculating covariance and correlation between
Merck and McDonald from past monthly data
Month
1
2
3
4
5
6
Total
Mean
Return:
Merck
McD
5.4%
1.7
- 3.6
13.6
- 3.5
3.2
10.7%
- 8.4
1.6
10.2
4.4
- 7.8
16.8
2.8
10.7
1.78
Covariance:
106.1/6 = 17.7
Std dev Merck:
5.9%
Std dev McD:
7.7%
Corr. co-effic: Cov/(sdMe . sdMcD )
= 17.7/(5.9 x 7.7) = .39
Deviation
from mean:
Merck
McD
2.6%
- 1.1
- 6.4
10.8
- 6.3
0.4
8.9%
-10.2
- 0.2
8.4
2.6
- 9.6
Product
of
deviations
23.2
11.2
1.2
90.9
-16.5
- 3.8
106.1
Effect of changing correlations:
Portfolio of Merck & McDonald
15
100% McD
Expected return
%
14
13
corr = .4
corr = 1
12
corr = -1
11
10
100% Merck
9
8
0
1
2
3
4
Std dev %
5
6
7
8
Mean & standard deviation: Portfolio
of Merck & McDonald
15
Expected return
%
14
100% McD
13
40% Merck
12
11
10
100% Merck
9
8
4
4.5
5
5.5
6
Std dev %
6.5
7
7.5
8
The set of portfolios
Expected return
A
B
x
x x
x x
x x
x
x
x
x
x x x
x
x x x
x x
x
x
Standard deviation
The set of portfolios between A and B
are efficient portfolios
Adding a riskless asset to the
efficient frontier
tangency
portfolio
riskless rate
Portfolio composition with a riskless asset

Regardless of the investor's attitude to
risk, he should split his portfolio between
the tangency portfolio and the riskless asset.
- the tangency portfolio provides the
maximum reward per unit of risk
- the riskless asset adjusts the level of risk
Two basic ideas about risk and return
1. Investors require compensation
for risk
2. They care only about a stock's
contribution to portfolio risk
Capital asset pricing model
Expected
return
Expected
market
return
Risk
free
rate
0
.5
r = rf +
1.0
(rm - rf )
Beta
Capital asset pricing model - example
If Treasury bill rate = 5.6%
Bristol Myers Squibb beta
=
Expected market risk premium
r = rf + beta (rm - rf )
= 5.6 + .81 (8.4) = 12.4%
.81
=
8.4%
Testing the CAPM
Beta vs. Average Risk Premium
Avg Risk Premium
1931-2005
30
20
SML
Investors
10
Market
Portfolio
0
1.0
Portfolio Beta
Testing
the
CAPM
Return vs. Book-to-Market
Dollars
(log scale)100
High-minus low book-to-market
10
0.1
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
2006
1996
1986
1976
1966
1956
1946
1936
1
1926
Small minus big
Validity of capital asset pricing model
EVIDENCE IS MIXED:
1. Long-run average returns are significantly related
to beta.
2. But beta is not a complete explanation. Low beta
stocks have earned higher rates of return than
predicted by the model. So have small company stocks
and stocks with low price to book value ratios.
Capital asset pricing model
is attractive
Because:
1. It is simple and usually gives sensible
answers.
2. It distinguishes between diversifiable
and non-diversifiable risk.
CAPM is controversial
BECAUSE:
1. No one knows for sure how to define and
measure the market portfolio -- and using
the wrong market index could lead to the
wrong answers.
2. The model is hard to prove -- or disprove.
3. The model has competitors.
The Consumption CAPM
• In standard CAPM, investors are concerned with the level and
uncertainty of their wealth. (Consumption is outside the model)
• In the Consumption CAPM, investors are concerned with the level
and uncertainty of their consumption. Stocks that provide low
consumption (those with low consumption betas) should have low
expected returns.
• Is the Consumption CAPM useful?
- It has the advantage that you don’t have to identify the
market portfolio.
- Unfortunately consumption is difficult to measure (especially
total consumption for everyone).
-Consumption CAPM is difficult to apply for practical use.
Arbitrage Pricing Theory (APT)
APT assumes that
r = a + b1 (rfactor 1 ) + b2 (rfactor 2 ) + .... + noise

Suppose that a diversified portfolio has no exposure to any
factor. It is essentially risk-free and should offer a return of rf.
So a = rf.

The expected risk premium on a portfolio that is exposed only
to factor 1 (say) should vary in proportion to its exposure to
that factor. If a portfolio is exposed to several factors then its risk will
vary in proportion to those factors. So
r - rf = b1 (rfactor 1 - rf ) + b2 (rfactor 2 - rf ) + ...
Arbitrage pricing theory (APT)

Preserves distinction between diversifiable
and non-diversifiable risk

CAPM and APT can both hold - e.g. CAPM
implies one factor APT, with r factor1 = r m

But APT is more general - e.g. unlike CAPM,
market portfolio doesn't have to be efficient

But usefulness of APT requires heavy-duty
statistics to
 identify factors
 measure factor returns