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Corporate Finance
Introduction to risk
Prof. André Farber
SOLVAY BUSINESS SCHOOL
UNIVERSITÉ LIBRE DE BRUXELLES
Introduction to risk
• Objectives for this session :
– 1. Review the problem of the opportunity cost of capital
– 2. Analyze return statistics
– 3. Introduce the variance or standard deviation as a measure of risk for a
portfolio
– 4. See how to calculate the discount rate for a project with risk equal to
that of the market
– 5. Give a preview of the implications of diversification
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Setting the discount rate for a risky project
•
Stockholders have a choice:
– either they invest in real investment projects of companies
– or they invest in financial assets (securities) traded on the capital market
• The cost of capital is the opportunity cost of investing in real assets
• It is defined as the forgone expected return on the capital market with
the same risk as the investment in a real asset
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Three key ideas
• 1. Returns are normally distributed random variables
 Markowitz 1952: portfolio theory, diversification
• 2. Efficient market hypothesis
 Movements of stock prices are random
 Kendall 1953
• 3. Capital Asset Pricing Model
 Sharpe 1964 Lintner 1965
 Expected returns are function of systematic risk
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Preview of what follow
•
First, we will analyze past markets returns.
•
We will:
– compare average returns on common stocks and Treasury bills
– define the variance (or standard deviation) as a measure of the risk of a portfolio of
common stocks
– obtain an estimate of the historical risk premium (the excess return earned by
investing in a risky asset as opposed to a risk-free asset)
•
The discount rate to be used for a project with risk equal to that of the
market will then be calculated as the expected return on the market:
Expected
return on the
market
= Current risk- +
free rate
Historical risk
premium
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Implications of diversification
•
The next step will be to understand the implications of diversification.
• We will show that:
– diversification enables an investor to eliminate part of the risk of a stock
held individually (the unsystematic - or idiosyncratic risk).
– only the remaining risk (the systematic risk) has to be compensated by a
higher expected return
– the systematic risk of a security is measured by its beta (), a measure of
the sensitivity of the actual return of a stock or a portfolio to the
unanticipated return in the market portfolio
– the expected return on a security should be positively related to the
security's beta
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Capital Asset Pricing Model (CAPM)
•
Risk – expected return relationship:
R j  RF  ( RM  RF )   j
Expected
return
Risk-free
interest
rate
Market
risk
premium
Risk
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Returns
• The primitive objects that we will manipulate are percentage returns
over a period of time: R  divt  Pt  Pt 1
t
Pt 1
Pt 1
• The rate of return is a return per dollar (or £, DEM,...) invested in the
asset, composed of
– a dividend yield
– a capital gain
• The period could be of any length: one day, one month, one quarter,
one year.
• In what follow, we will consider yearly returns
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Ex post and ex ante returns
•
Ex post returns are calculated using realized prices and dividends
•
Ex ante, returns are random variables
– several values are possible
– each having a given probability of occurence
• The frequency distribution of past returns gives some indications on
the probability distribution of future returns
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Frequency distribution
•
Suppose that we observe the following frequency distribution for past annual
returns over 50 years. Assuming a stable probability distribution, past relative
frequencies are estimates of probabilities of future possible returns .
Realized Return
Absolute
frequency
Relative
frequency
-20%
2
4%
-10%
5
10%
0%
8
16%
+10%
20
40%
+20%
10
20%
+30%
5
10%
50
100%
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Mean/expected return
• Arithmetic Average (mean)
– The average of the holding period returns for the individual years
R1  R2  ...  RN
Mean  R 
N
• Expected return on asset A:
– A weighted average return : each possible return is multiplied or weighted
by the probability of its occurence. Then, these products are summed to
get the expected return.
E ( R)  p1 R1  p 2 R2  ...  p n Rn
with pi  probabilit y of return Ri
p1  p 2  ...  p n  1
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Variance -Standard deviation
•
Measures of variability (dispersion)
•
Variance
•
Ex post: average of the squared deviations from the mean
(R1  R) 2 (R 2  R) 2 ...(R T  R) 2
2
Var   
T 1
•
Ex ante: the variance is calculated by multiplying each squared deviation from
the expected return by the probability of occurrence and summing the
products
Var(R A ) A2  Expected val ue of (RA  RA ) 2
2
Var(RA) A
 p1(RA,1  RA)2  p2(RA,2  RA)2 ... pN(RA,N  RA)2
•
Unit of measurement : squared deviation units. Clumsy..
•
Standard deviation : The square root of the variance
SD (R A ) A  Var(R A )
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Return Statistics - Example
Return
-20%
-10%
0%
10%
20%
30%
Exp.Return
Variance
Standard deviation
Proba
4%
10%
16%
40%
20%
10%
Squared Dev
0.08526
0.03686
0.00846
0.00006
0.01166
0.04326
9.20%
0.01514
12.30%
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Normal distribution
•
Realized returns can take many, many different values (in fact, any
real number > -100%)
•
Specifying the probability distribution by listing:
– all possible values
– with associated probabilities
•
as we did before wouldn't be simple.
• We will, instead, rely on a theoretical distribution function (the Normal
distribution) that is widely used in many applications.
•
The frequency distribution for a normal distribution is a bellshaped
curve.
• It is a symetric distribution entirely defined by two parameters
• – the expected value (mean)
• – the standard deviation
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Belgium - Monthly returns 1951 - 1999
Bourse de Bruxelles 1951-1999
180.00
160.00
140.00
100.00
80.00
60.00
40.00
20.00
0.00
-2
0.
00
-1
8.
00
-1
6.
00
-1
4.
00
-1
2.
00
-1
0.
00
-8
.0
0
-6
.0
0
-4
.0
0
-2
.0
0
0.
00
2.
00
4.
00
6.
00
8.
00
10
.0
0
12
.0
0
14
.0
0
16
.0
0
18
.0
0
20
.0
0
22
.0
0
24
.0
0
26
.0
0
28
.0
0
30
.0
0
Fréquence
120.00
Rentabilité mensuelle
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Normal distribution illustrated
Normal distribution
0.0250
0.0200
0.0150
68.26%
0.0100
0.0050
95.44%
4.
0
3.
6
3.
2
Standard deviation from mean
2.
8
2.
4
2.
0
1.
6
1.
2
0.
8
0.
4
0.
0
-4
.0
-3
.6
-3
.2
-2
.8
-2
.4
-2
.0
-1
.6
-1
.2
-0
.8
-0
.4
0.0000
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Risk premium on a risky asset
•
The excess return earned by investing in a risky asset as opposed to
a risk-free asset
•
• U.S.Treasury bills, which are a short-term, default-free asset, will be
used a the proxy for a risk-free asset.
• The ex post (after the fact) or realized risk premium is calculated by
substracting the average risk-free return from the average risk return.
•
Risk-free return = return on 1-year Treasury bills
•
Risk premium = Average excess return on a risky asset
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Total returns US 1926-1999
Arithmetic
Mean
Standard
Deviation
Risk Premium
13.3%
20.1%
9.5%
Small Company Stocks
17.6
33.6
13.8
Long-term Corporate Bonds
5.9
8.7
2.1
Long-term government bonds
5.5
9.3
1.7
Intermediate-term
government bond
5.4
5.8
1.6
U.S. Treasury bills
3.8
3.2
Inflation
3.2
4.5
Common Stocks
Source: Ross, Westerfield, Jaffee (2002) Table 9.2
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Market Risk Premium: The Very Long Run
The equity premium puzzle:
1802-1970
1871-1925
1926-1999
1802-1999
Common Stock
6.8
8.5
13.3
9.7
Treasury Bills
5.4
4.1
3.8
4.4
Risk premium
1.4
4.4
9.5
5.3
Source: Ross, Westerfield, Jaffee (2002) Table 9A.1
Was the 20th century an anomaly?
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Notions of Market Efficiency
•
An Efficient market is one in which:
– Arbitrage is disallowed: rules out free lunches
– Purchase or sale of a security at the prevailing market price is never a
positive NPV transaction.
– Prices reveal information
•
Three forms of Market Efficiency
• (a) Weak Form Efficiency
 Prices reflect all information in the past record of stock prices
• (b) Semi-strong Form Efficiency
 Prices reflect all publicly available information
• (c) Strong-form Efficiency
 Price reflect all information
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Efficient markets: intuition
Price
Realization
Expectation
Price change is
unexpected
Time
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Weak Form Efficiency
•
Random-walk model:
– Pt -Pt-1 = Pt-1 * (Expected return) + Random error
– Expected value (Random error) = 0
– Random error of period t unrelated to random component of any past
period
•
Implication:
– Expected value (Pt) = Pt-1 * (1 + Expected return)
– Technical analysis: useless
•
Empirical evidence: serial correlation
– Correlation coefficient between current return and some past return
– Serial correlation = Cor (Rt, Rt-s)
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Random walk - illustration
Bourse de Bruxelles 1980-1999
25.00
20.00
15.00
10.00
Rentabilité mois t+1
5.00
0.00
-30.00
-25.00
-20.00
-15.00
-10.00
0.00
-5.00
5.00
10.00
15.00
20.00
25.00
-5.00
-10.00
-15.00
-20.00
-25.00
-30.00
Rentabilité mois t
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Semi-strong Form Efficiency
•
Prices reflect all publicly available information
•
Empirical evidence: Event studies
•
Test whether the release of information influences returns and when
this influence takes place.
•
Abnormal return AR : ARt = Rt - Rmt
•
Cumulative abnormal return:
•
CARt = ARt0 + ARt0+1 + ARt0+2 +... + ARt0+1
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Strong-form Efficiency
•
How do professional portfolio managers perform?
•
Jensen 1969: Mutual funds do not generate abnormal returns
•
Rfund - Rf =  +  (RM - Rf)
•
Insider trading
•
Insiders do seem to generate abnormal returns
•
(should cover their information acquisition activities)
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