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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
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
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
0%
-3.5
0%
-3.0
0%
-2.5
0%
-2.0
0%
-1.5
0%
-1.0
0%
-0.5
0%0.
00%
0.50
%1.
00%
1.50
%2.
00%
2.50
%3.
00%
3.50
%4.
00%
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
%
-4.0
%
-3.5
%
-3.0
%
-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
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