Pricing Risk
Chapter 10
Outline
• Measuring risk and return
– Expected return and return Variance
– Realized versus expected return
– Empirical distribution of returns
• The risk return tradeoff
– Computing average returns and volatility of returns from
historical data
• Systematic versus Idiosyncratic risk and Diversification
• Pricing risk
• The CAPM in practice
Measuring Risk and Return
A First Look at Risk and Return
Consider the performance of the following portfolios of securities:
Standard and Poor’s (S&P 500): A portfolio, constructed by Standard and
Poor’s, of 500 U.S. leading stocks.
Small Stocks: A portfolio of stocks of U.S. firms whose market values are in
the bottom 10% of all stocks traded on the NYSE.
World Portfolio: A portfolio of international stocks from all of the world’s
major stock markets in North America, Europe, and Asia
Corporate Bonds: A portfolio of long-term, AAA rated U.S. corporate bonds
with maturity of approximately 20 years
Treasury Bills: An investment in three-month U.S. Treasury Bills
Comparing Portfolios of different Risk
A First Look at Risk and Return
An investment of $100 in small stocks in 1925 would be worth over $8 million in 2005!
An average of 15.2% annual return
An investment of $100 in Treasury bills 1925 would be worth about $2000. An average
of 3.78% annual return.
While small stocks realized the highest return, it was followed by S&P 500 (10.2%),
international stocks (9%), corporate bonds (6.2%) and finally Treasury bills.
Comparing to how prices changed over these years (CPI index) all of the investments
grew faster than inflation (average 3%)
How much do investors demand (in terms of higher
expected return) to bear a given level of risk?
Probability Distribution
Measures of Risk and Return
Expected (Mean) Return
E[R] = å pR R
R
Variance and Standard Deviaiton
V[R] = E(R - E(R))2 = å pR (R - E[R])2
R
SD[R] = s R = Var(R)
Expected Return and Variance for BFI
E[R] = 0.25´ 40%+ 0.5´10%+ 0.25´ (-20%) =10%
Var[R] = 0.25´(40%-10%)2 + 0.25´(-20%-10%)2 = 0.045
s R = 0.045 = 21.2%
Realized versus Expected Performance
Distinguish between realized returns and expected
returns
YTD Apple’s stock price went from $532 to $441 and paid
two dividends of $2.65 and $3.05
Calculating Realized Returns
Realized Return over year t, t+1
Divt+1 + Pt+1
Divt+1 Pt+1 - Pt
Rt+1 =
-1 =
+
Pt
Pt
Pt
With quarterly dividends
1+ Rt+1 = (1+ RQ1 )(1+ RQ2 )(1+ RQ3 )(1+ RQ4 )
Quarterly returns calculation (example for second
quarter):
DivQ2 + PQ2
RQ2 =
-1
PQ1
Calculating Realized Return for GM
Realized Returns for S&P500, GM, and T bills
The Empirical Distribution 1926-2004
Risk and Return Tradeoff
The Empirical Distribution
Average Annual Return last T years
1
R = ( R1 + R2 +... + RT )
T
Variance Estimate
1 T
2
sR =
(R
R)
å
t
T -1 t=1
Empirical Distributions of different Portfolios
Excess Returns
Excess Return: the difference between the return for
the investment and the return fro Treasury bills (a risk
free investment)
Historical Tradeoff Between Risk and
Return in Large Portfolios 1926-2004
Investments with higher volatility have rewarded
investors with higher average returns.
Historical Tradeoff Between Risk and
Return 500 individual stocks 1926-2004
Is volatility a reasonable measure of risk for
individual stocks?
Systematic versus Idiosyncratic risk
and Diversification
Common Versus Independent Risk
The risk of an individual security differs from the risk of a portfolio
composed of similar assets.
Insurance Example
To illustrate this difference consider two types of insurance: theft
insurance and earthquake insurance.
Each year there is about a 1% chance that a given home in the San
Francisco area will be robbed and a 1% chance it will be damaged by
an earthquake
Suppose that the insurance company writes 100,000 policies of each
type for homeowners in San Francisco
Common Versus Independent Risk
The expected number of theft claims is 1000 (or 1% out of all policies
issued). This is also the expected number of claims from an earthquake
(1% chance that the earthquake hits San Francisco, in that case all
100,000 policy holders will file a claim).
Earthquake and theft portfolios lead to very different risk
characteristics!
In the case of earthquake insurance, the insurance company needs to
be ready to cover 100000 claims in a single year!
With theft insurance, the insurance company can hold funds sufficient
to cover around 1200 claims since the number of claims will almost
always be between 875 and 1125
Common Versus Independent Risk
The earthquake affects all houses simultaneously, so the risk is
perfectly correlated across homes. We call risk that is
perfectly correlated common risk
Thefts in houses are more or less not related to each other, so
the risk of theft is uncorrelated and independent across
homes. We call this type of risk independent risk.
When risks are independent, the overall number of claims is
quite predictable.
This averaging out of independent risks in a large portfolio is
called diversification
Diversification in Stock Portfolios
Firm specific risk (or “idiosyncratic”, “unique”,
“diversifiable”)
• News about the individual company
Market wide risk (or “systematic”,
“undiversifiable”)
• News that affects all stocks, such as the news
about the economy
Diversification in Stock Portfolios
Diversification in Stock Portfolios
Common Versus Independent Risk
Example
Consider three firm types
Type S firms are affected only by the strength of the
economy, a systematic risk which has 50% chance of
being either strong or weak. If the economy is strong,
type S stocks will earn a return of 40%; if weak, their
return will be -20%.
Type I firms are affected only by idiosyncratic risks. There
returns are equally likely to be 35% or -25% based on
factors specific to each firm’s local market.
Common Versus Independent Risk
What is the volatility of the average return of ten type S
firms? Or type I firms?
Diversification in Action
Pricing Risk
Pricing Risk
The risk premium for diversifiable risk is zero, so
investors are not compensated for holding firmspecific risk
– Why can’t diversifiable risk carry a positive risk
premium in efficient markets?
The risk premium of a security is determined by its
systematic risk and does not depend on its
diversifiable risk
– Stock’s volatility, which is a measure of total risk (that
is, systematic plus diversifiable) is not especially useful
in determining the risk premium that investors will
earn.
Measuring Systematic Risk
A security’s systematic risk is measured by the extent to which its
return is sensitive to economic conditions
We assume that the changing state of the economy must be reflected
in the return on the market portfolio - the market wide portfolio
contains only systematic risk (all firm specific risk has been diversified)
In practice, we do not know return data for many bonds and small
stocks. It is common practice to use the S&P 500 index as the market
portfolio
We measure systematic risk of stock i by its beta
Cov(ri , rmkt )
bi =
Var(rmkt )
The return on the S&P 500
index is considered the
return on the market “rmkt”
Values of Beta in the data
Advanced Micro Devices, Inc.
Advanced Micro Devices, Inc.
Cisco Systems, Inc.
Estimating the Risk Premium
Market Risk Premium: the excess return from holding the
market portfolio
Market Risk Premium = E(Rmkt ) - rf
Estimating a Traded Security’s Expected Return
(this is the “Capital Asset Pricing Model” or CAPM)
E(Ri ) - rf = bi ´ éëE(Rmkt ) - rf ùû
Expected excess
return on stock “i”
Systematic risk
of stock “i”
Market risk
premium
Special Cases
A stock with beta of one:
E(Ri ) = E(Rmkt )
A stock with beta of zero:
E(Ri ) = rf
A stock with negative beta:
E(Ri ) < rf
Is this a good
investment?
Project Cost of Capital
A project’s cost of capital is given by the rate of return required
by investors or their opportunity cost of capital.
From the CAPM the cost of capital “r” is:
r = rf + b ´ éëE(Rmkt ) - rf ùû
Project Cost of Capital
The CAPM in Practice
Measuring Beta
Beta is the expected percent change in the excess return of the
security for a 1% change in the excess return of the market
portfolio (S&P 500 Index)
Ri - rf = bi ´ éëRmkt - rf ùû + ei
Measuring Beta
Forecasting Beta
• Time Horizon: what data do we use? A short horizon leads to weak statistical
power. A long horizon includes outdated data.
• Market Proxy: The theoretical market portfolio includes all risky investments.
Often the S&P 500 index is used, other proxies include the NYSE Composite Index
(a value weighted index of all NYSE stocks), or when considering an international
investment one can use a country or international market index.
• Beta Extrapolation: Often adjustments are made to the beta estimate to reduce
estimation error.
Measuring Beta
Forecasting Beta
• The Risk-Free Interest Rate: used in the CAPM equation on the left-hand side (to
calculate the excess return on the asset) is the current YTM on U.S. Treasury with
maturity similar to our project’s horizon
•
The Market Risk Premium: is estimated from the historical excess return on the
market. The difference between the average return on the market and the average
return on the risk-free asset. Market Risk Premium is around 4%-5%.