Pricing Risk

• Short Class Exercise
• 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

Pricing Financial
Assets

How much are you willing to pay for the following two financial securities?
• Risky security with a payment in one year from today that depends on the level of rainfall in Houston in the coming year.
• A risk free security that pays a fixed payment one year from today.

Houston Rainfall
Average annual rainfall is 50.63 inches
Meteorologists say that rainfall this coming year will exceed 50.63 inches with 50% chance

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

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?

Expected (Mean) Return
E [ R ]
=
p
R
R
R
Variance and Standard Deviation
V [ R ]
=
E ( R
-
E ( R ))
2 =
p
R
( 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

Realized Return over year t, t+1
R t
+
1
=
Div t
+
1
+
P t
+
1
-
1
=
P t
Div t
+
1
P t
+
P t
+
1
-
P t
P t
With quarterly dividends
1
+
R t
+
1
=
(1
+
R
Q 1
)(1
+
R
Q 2
)(1
+
R
Q 3
)(1
+
R
Q 4
)
Quarterly returns calculation (example for second quarter):
R
Q 2
=
Div
Q 2
+
P
Q 2 -
1
P
Q 1

Calculating Realized Return for GM

Realized Returns for S&P500, GM, and T bills

The Empirical Distribution 1926-2004

Average Annual Return last T years
R
=
1
T
(
R
1
+
R
2
+
...
+
R
T
)
Variance Estimate s
R
=
1
T
-
1
T
t
=
1
( R t
-
R )
2

Empirical Distributions of different Portfolios

Excess Returns
Excess Return: the difference between the return on the investment and the return on 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

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

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

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

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

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.

What is the volatility of the average return of ten type S firms? Or type I firms?

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.

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 b i
=
Cov ( r i
, r mkt
)
Var ( r mkt
)
The return on the S&P 500 index is considered the return on the market “r mkt
”

Values of Beta in the data

Advanced Micro Devices, Inc.

Newmont Mining Corp

Cisco Systems, Inc.

Market Risk Premium: the excess return from holding the market portfolio
Market Risk Premium
=
E ( R mkt
)
r f
Estimating a Traded Security’s Expected Return
(this is the “Capital Asset Pricing Model” or CAPM)
E(R i
)
r f
= b i
´ éë
E ( R mkt
)
r f
ùû
Expected excess return on stock “i”
Systematic risk of stock “i”
Market risk premium

A stock with beta of one:
E ( R i
)
=
E ( R mkt
)
A stock with beta of zero:
E ( R i
)
= r f
A stock with negative beta:
E ( R i
)
< r f
Is this a good investment?

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
= r f
+ b ´ éë
E ( R mkt
)
r f
ùû

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)
R i
r f
= b i
´ éë
R mkt
r f
ùû+ e i

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.

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%.

Applying CAPM to GE Equity Cost of
Capital

• Using the CAPM we know that the beta of GE is 1.28 (according to Google finance) and 1.34
(according to Yahoo finance)
• If the market risk premium is 6% and the risk free rate is 2.7% then the equity cost of capital of GE is:
– According to Google finance:
2.7%+1.34(6%) = 10.74%
– According to Yahoo finance:
2.7%+1.28(6%) = 10.38%

• Next week during class hours 6:00-9:00
• Exam includes 7 questions (3 of them are short questions worth 5 points each)
• 2 A4 sheets are allowed
• Simple/financial calculator
• Material:
– Class notes,
– homework assignments
– Chapter 10: know the difference between systematic and idiosyncratic risk and how to use the CAPM formula.