Risk, reward, and the CAPM model -- review for my M&B students.

1. Money & Banking
Risk and Reward
Professor: Julio Huato @SFC
juliohuato@gmail.com
Fall 2011

2. Questions
• What is risk? What is return? What’s risk aversion?
• How do we measure the risk of a given asset?
• How can we use statistical measures to quantify the risk of a
given asset?
• What is the effect on risk of diversifying a portfolio of assets?
• Are there different types of risk and what are they?
• What’s the beta of an asset?
• What is the capital asset pricing model (CAPM) and what is
the security market line?
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3. Risk and return
Risk is the chance of gain/loss. (A loss is a negative gain.)
The future is fundamentally unknown. Therefore, there is uncer-
tainty about the future consequences of choices we make today.
The past is a guide to the future only if the future looks like the
past in some way. But if the future doesn’t look like the past,
then the past is not necessarily a good guide.
One way to measure risk in financial assets is to look at the variabil-
ity of returns. Think of financial assets as lotteries. For example,
a lottery gives you return X if the state of the world is A and a
return Y if the state of the world is non − A.
If risk is the variability of returns. What is return?
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4. Risk and return
Return is the total gain/loss experienced on investing in a financial
asset over a period of time. Usually, we express it as a percentage
of the value of the investment at the beginning of the period.
More formally,
Ct + Pt − Pt−1
kt = (1)
Pt−1
where kt is the actual, expected, or required return rate (or just
return) over period t, Ct is the cash flow received from the asset
investment in the time period from t − 1 to t, Pt is the price (value)
of the asset at time t, and Pt−1 is the price (value) of the asset at
time t − 1.
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5. Risk aversion
People have “preferences” for risk. Some like it or tolerate it better
than others. If we look at the behavior of crowds (e.g. markets),
then it’s clear that most people are risk averse, since people place
a return premium on risky assets.
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6. Examples
Work on your assignment.
Download returns on some financial asset and find expected return
(mean), distance from mean, square distance from mean,.
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7. Risk of one single asset
Sensitivity analysis vs. statistics (probability distributions).
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8. Probability distributions
The return on an investment is a random variable because, in
advance, we don’t know for sure what it will be. Its value is
contingent upon the particular state of the world realized.
Say we can list all the possible states of the world. Say there
are only three equally-possible states of the world: (a) Good, (b)
Regular, and (c) Bad. And, under each of th.ese states of the
world, we know (or can guess) the return rate. Then we can form
expectations on the return.
Also, we can compute different measures of dispersion or variability
that could give us a measure of risk.
Discrete and continuous distributions. Show examples.
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9. Probability distributions
Expected return: Mean of returns.
n
¯=
k kj pj (2)
j=1
where ¯ is the expected return, kj for j = 1, . . . , n is the list of
k
returns observed under different states of the world, and pj is the
probability that kj occurs – being the probability a number between
0 and 1, where 0 means absolute impossibility of occurrence and
1 means absolute certainty that it will occur.
The measures of dispersion or variability used as proxies for risk
are formulas (3) the variance, (4) the standard deviation, and (5)
the coefficient of variation – of the returns. The higher each of
these is, the greater the variability of returns and the higher the
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10. risk.
n
2
σk = (kj − ¯)2 pj
k (3)
j=1
n
σk = (kj − ¯)2 pj
k (4)
j=1
σk
CVk = (5)
¯
k

11. Portfolio risk
Correlation is the statistical measure of the association between
any two series of numbers, e.g. returns of two assets under differ-
ent states of the world.
The degree of correlation is measured by the Pearson coefficient:
−1 ≤ ρ ≤ 1, where ρ = −1 means perfect negative correlation,
ρ = 0 means no correlation at all, and rho = 1 means perfect
positive correlation.
In Excel, do example in p. 207.
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12. Diversification
However correlated the returns of two assets may be, the expected
return of a portfolio with the two assets will fall in some midpoint
between the returns of the two assets held in isolation.
If the assets are highly positively correlated, the risk is some mid-
point between the risk of each asset in isolation.
If the assets are largely uncorrelated, the risk is some midpoint
between the risk of the most risky asset and less than the risk of
the least risky asset, but greater than zero.
If the assets are highly negatively correlated, the risk is some mid-
point between the risk of most risky asset and zero.
In no case will a portfolio be riskier than the riskiest asset included
in it.
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13. Types of risk
Say, we measure the risk of a portfolio by its standard deviation,
σkp . Start with a portfolio with one single asset and add assets
randomly to the portfolio, one at a time. What happens to risk as
you keep adding assets? It tends to decline towards a lower limit
or baseline risk.
On average, portfolio risk approaches the lower limit when you’ve
added 15-20 randomly selected securities to your portfolio. So,
total risk can be viewed as the sum of two types of risk:
Total security risk = Nondiversifiable risk + Diversifiable risk.
Diversifiable risk is also called ‘unsystematic,’ because it comes
from random factors that can be eliminated by diversifying the
portfolio. Nondiversifiable risk is also known as ‘systematic’ and it
comes from market-wide factors that affect all securities. It may
also be called ‘market’ or ‘macroeconomic’ risk.
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14. CAPM Model
The model links nondiversifiable risk to returns for all assets.
First, we will discuss the beta, an element of the model. Second,
we will introduce the CAPM equation. Third, we will show how
to use it in concrete applications.
The beta is a measure of the nondiversifiable risk. It indicates to
what extent the return on an asset responds to a change in the
market or average return of all assets.
Who knows about ‘all assets,’ but there are broad indices of se-
curities, e.g. S&P 500. How do we estimate the beta of a given
asset? We need data on the returns on that asset and data on the
returns of a well-diversified portfolio representative of the market
as a whole. We use regression analysis.
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15. Regression analysis
Consider the plot of the returns on two assets. The horizontal
axis shows the return on an asset representative of the whole mar-
ket, e.g. the S&P 500. The vertical axis shows the return on a
particular asset, e.g. the return on Google.
Show plot in Excel.
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16. A linear equation
Consider the simple bi-variate linear equation:
y = a + bx (6)
The equation says that the value of the variable y (the dependent
variable) depends on the value of the variable x (the independent
variable).
The literals a and b are called, respectively, the intercept and the
slope. The intercept indicates the value of y when x = 0. The
slope indicates the change in y associated to a unit change in x or
∆Y
b = ∆X .
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17. Regression analysis
Regression is a statistical procedure to fit a line in a scatterplot.
yi = α + βxi + i (7)
Show regression in Excel.
The beta coefficient (slope) indicates the change in the return
on a particular asset (e.g. Google or GE) when the return on a
market portfolio (e.g. S&P 500) changes in one unit. The beta
shows how sensitive the return on the asset is to performance of
the market as a whole.
What do the betas of Google and GE say?
The beta of a portfolio is the weighted average of its individual
assets’ betas:
n
βp = wi βi (8)
i=1
where n = 1. The beta of a portfolio indicates how responsive
i=1
the portfolio’s return is to changes in the market return.
Example in p. 215.
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18. CAPM Model
The capital asset pricing model (CAPM) gives us the return that
we would require in order to compensate for the risk involved in
holding a given asset i. The model helps us determine the return
over and above a risk-free return that would offset the risk inherent
to that asset. And by risk inherent to that asset, we mean the risk
of holding that asset that is not diversifiable, i.e. the risk that it
shares with a well-diversified market portfolio.
So, to determine that extra return or risk premium, we need to
measure the extent to which an asset is exposed to nondiversifiable
risk. But instead of pulling that measure out of our own heads,
we consider the way crowds (e.g. markets) determine that extra
return.
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19. CAPM Model
We now remember that the beta of an asset tells us the change in
the return on the asset that is due to a one-percent change in the
market return. That beta can be used as an index or measure of
nondiversifiable risk, since it reflects to covariation of the return
on that asset and the market return. So, we plug the beta in the
following equation and get the required return, ki :
ki = RF + [βi (km − RF )] (9)
where, again, ki is the required return on asset i, RF is the risk-free
rate of return (usually the return on a U.S. Treasury bill), βi is the
beta coefficient or index of nondiversifiable risk for asset i, and km
is the return on a market portfolio.
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20. CAPM Model
ki = RF + [βi(km − RF )] (10)
Under the CAPM model, the required return on asset i
has two parts: (1) the risk-free rate of return – some-
thing like a baseline rate of return (say, the return or
interest on a 3-month T bill) – and (2) the risk pre-
mium. In turn, (2) has two parts: (a) the beta or index
of nondiversifiable risk and (b) (km − RF ) or market risk
premium. The market risk premium represents the pre-
mium investors require for taking the average amount
of risk associated with holding a well-diversified market
portfolio of assets.
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21. CAPM Model
The graphical representation of the CAPM model is called the se-
curity market line (SML). We plot our measure of nondiversifiable
risk on the horizontal axis and the required return on the vertical
axis. Note that beta is our independent variable.
Suppose the risk-free return (RF ) is 7% and the expected market
return (km ) is 11%. Then, (km − RF ) = 4%. The intercept of
our graph will be RF . The slope will indicate the change in our
required return when we vary beta in one unit. The slope is given
by (km − RF ) = 4%.
Example in textbook, p. 217. If beta goes from 1 to 1.5, i.e.
∆β = 0.5, the required return increases from 11% to 13% or
2% = 0.5 × 4%.
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22. CAPM Model
In brief, the CAPM model provides us with a way to determine a
return adequate to the (nondiversifiable) risk involved in holding
a given asset. And determining an adequate return is essential
to value assets. It translates (nondiversifiable) risk into a risk
premium or additional return that would compensate (according
to the market) for our taking the risk of holding that given asset.
The CAPM is not foolproof. We can only use historical data to
estimate the betas. But past variability may not reflect future
variability. So, we have to be careful and make adjustments if we
have additional information.
The CAPM, if it is to function well, requires that the market that
prices assets and, therefore, determines returns, be competitive
and efficient – in the sense of being made up by many buyers
and sellers, and capable of absorbing available information quickly.
It is also assumed that government or other types of restrictions
don’t exist, that there are no taxes or transaction costs, and that
investors are rational. Real life is more complicated. Still, the
CAPM is a nice, useful benchmark.
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23. The CAPM performs well with assets such as stocks of highly
traded companies. It doesn’t do as well with real corporate assets,
such as buildings, equipment, etc. But that may be due to the
lack of efficient markets.
All in all, the CAPM is a good starting point. And it is widely used
by finance managers in large companies.

24. What have we learned?
• What is risk? What is return? What’s risk aversion?
• How do we measure the risk of a given asset?
• How can we use statistical measures to quantify the risk of a
given asset?
• What is the effect on risk of diversifying a portfolio of assets?
• Are there different types of risk and what are they?
• What’s the beta of an asset?
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25. • What is the capital asset pricing model (CAPM)?
• What is the security market line?