FINANCE 8. Capital Markets and The Pricing of Risk Professor André Farber Solvay Business School Université Libre de Bruxelles Fall 2007
Dec 19, 2015
FINANCE8. Capital Markets and The Pricing of Risk
Professor André Farber
Solvay Business SchoolUniversité Libre de BruxellesFall 2007
MBA 2007 Risk and return |2April 18, 2023
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
MBA 2007 Risk and return |3April 18, 2023
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
MBA 2007 Risk and return |4April 18, 2023
Uncertainty: 1952 – 1973- the Golden Years
• 1952: Harry Markowitz*
– Portfolio selection in a mean –variance framework
• 1953: Kenneth Arrow*
– Complete markets and the law of one price
• 1958: Franco Modigliani* and Merton Miller*
– Value of company independant of financial structure
• 1963: Paul Samuelson* and Eugene Fama
– Efficient market hypothesis
• 1964: Bill Sharpe* and John Lintner
– Capital Asset Price Model
• 1973: Myron Scholes*, Fisher Black and Robert Merton*
– Option pricing model
MBA 2007 Risk and return |5April 18, 2023
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
MBA 2007 Risk and return |6April 18, 2023
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
= +
MBA 2007 Risk and return |7April 18, 2023
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
MBA 2007 Risk and return |8April 18, 2023
Capital Asset Pricing Model
Expected return
Beta
Risk free interest rate
r
rM
1β
)( FMF rrrrBeta (equity)
Nov. 27, 2006
Source: fi nance.yahoo.com (in key statistics)
Ticker Company Beta
WMT Wal-Mart 0.06
BUD Budweiser 0.32
KO Coca-Cola 0.76
MSFT Microsof t 0.79
SPX S&P 500 I ndex 1.00
SBUX Starbucks 1.17
I NTC I ntel 1.66
ADBE Adobe 1.81
AAPL Apple 2.03
F Ford 2.27
MBA 2007 Risk and return |9April 18, 2023
Returns
• The primitive objects that we will manipulate are percentage returns over a period of time:
• 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
1
1
1
t
tt
t
tt P
PP
P
divR
MBA 2007 Risk and return |10April 18, 2023
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
MBA 2007 Risk and return |11April 18, 2023
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 Absolutefrequency
Relativefrequency
-20% 2 4%
-10% 5 10%
0% 8 16%
+10% 20 40%
+20% 10 20%
+30% 5 10%
50 100%
MBA 2007 Risk and return |12April 18, 2023
Mean/expected return
• Arithmetic Average (mean)
– The average of the holding period returns for the individual years
• 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.
N
RRRRMean N
...21
1...
return ofy probabilit with
...)(
21
2211
n
ii
nn
ppp
Rp
RpRpRpRE
MBA 2007 Risk and return |13April 18, 2023
Variance -Standard deviation
• Measures of variability (dispersion)
• Variance
• Ex post: average of the squared deviations from the mean
• Ex ante: the variance is calculated by multiplying each squared deviation from the expected return by the probability of occurrence and summing the products
• Unit of measurement : squared deviation units. Clumsy..
• Standard deviation : The square root of the variance
• Unit :return
VarR R R R R R
TT
2 12
22 2
1( ) ( ) ... ( )
Var R Expected RA A A( ) ) 2 2 val ue of (RA
Var R p R R p R R p R RA A A A A A N A N A( ) ( ) ( ) ... ( ), , , 21 1
22 2
2 2
SD R Var RA A A( ) ( )
MBA 2007 Risk and return |14April 18, 2023
Return Statistics - Example
Return Proba Squared Dev-20% 4% 0.08526-10% 10% 0.03686
0% 16% 0.0084610% 40% 0.0000620% 20% 0.0116630% 10% 0.04326
Exp.Return 9.20%Variance 0.01514Standard deviation 12.30%
MBA 2007 Risk and return |15April 18, 2023
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
MBA 2007 Risk and return |16April 18, 2023
Belgium - Monthly returns 1951 - 1999
Bourse de Bruxelles 1951-1999
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
180.00
-20.
00
-18.
00
-16.
00
-14.
00
-12.
00
-10.
00
-8.0
0
-6.0
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2.00
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6.00
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0
24.0
0
26.0
0
28.0
0
30.0
0
Rentabilité mensuelle
Fré
qu
en
ce
MBA 2007 Risk and return |17April 18, 2023
S&P 500
S&P 500 Daily returns (June 96 - Nov 04) StDev = 1.23% n=2,122
0
50
100
150
200
250
300
350
400
450
-8.0
0%
-7.5
0%
-7.0
0%
-6.5
0%
-6.0
0%
-5.5
0%
-5.0
0%
-4.5
0%
-4.0
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-3.5
0%
-3.0
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-2.5
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-2.0
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-1.5
0%
-1.0
0%
-0.5
0%0.
00%
0.50
%1.
00%
1.50
%2.
00%
2.50
%3.
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3.50
%4.
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4.50
%5.
00%
5.50
%6.
00%
6.50
%7.
00%
7.50
%8.
00%
MBA 2007 Risk and return |18April 18, 2023
Microsoft
Microsoft Daily 1996-2003 StDev=2.58% (n=1,850)
0
20
40
60
80
100
120
140
160
180
200
-10.
0%
-9.5
%
-9.0
%
-8.5
%
-8.0
%
-7.5
%
-7.0
%
-6.5
%
-6.0
%
-5.5
%
-5.0
%
-4.5
%
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%
-3.5
%
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%
-2.5
%
-2.0
%
-1.5
%
-1.0
%
-0.5
%
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%
4.5%
5.0%
5.5%
6.0%
6.5%
7.0%
7.5%
8.0%
8.5%
9.0%
9.5%
10.0
%
MBA 2007 Risk and return |19April 18, 2023
Normal distribution illustrated
Normal distribution
0.0000
0.0050
0.0100
0.0150
0.0200
0.0250
68.26%
95.44%
Standard deviation from mean
MBA 2007 Risk and return |20April 18, 2023
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
MBA 2007 Risk and return |21April 18, 2023
Total returns US 1926-2002
Arithmetic Mean
Standard Deviation
Risk Premium
Common Stocks 12.2% 20.5% 8.4%
Small Company Stocks 16.9 33.2 13.1
Long-term Corporate Bonds 6.2 8.7 2.4
Long-term government bonds 5.8 9.4 2.0
Intermediate-term government bond (1926-1999)
5.4 5.8 1.6
U.S. Treasury bills 3.8 3.2
Inflation 3.1 4.4
Source: Ross, Westerfield, Jaffee (2005) Table 9.2
MBA 2007 Risk and return |22April 18, 2023
Market Risk Premium: The Very Long Run
1802-1870 1871-1925 1926-1999 1802-2002
Common Stock 6.8 8.5 12.2 9.7
Treasury Bills 5.4 4.1 3.8 4.3
Risk premium 1.4 4.4 8.4 5.4
Source: Ross, Westerfield, Jaffee (2005) Table 9A.1
The equity premium puzzle:
Was the 20th century an anomaly?
MBA 2007 Risk and return |23April 18, 2023
Diversification
Risk Reduction of Equally Weighted Portfolios
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
30.00%
35.00%
# stocks in portfolio
Po
rtfo
lio
sta
nd
ard
de
via
tio
n
Market risk
Unique risk
MBA 2007 Risk and return |24April 18, 2023
Conclusion
• 1. Diversification pays - adding securities to the portfolio decreases risk. This is because securities are not perfectly positively correlated
• 2. There is a limit to the benefit of diversification : the risk of the portfolio can't be less than the average covariance (cov) between the stocks
• The variance of a security's return can be broken down in the following way:
• The proper definition of the risk of an individual security in a portfolio M is the covariance of the security with the portfolio:
Total risk of individual security
Portfolio risk
Unsystematic or diversifiable risk