Portfolio Analysis and
Theory
Portfolio Analysis
Definitions
• A portfolio is the collection of securities an
investor holds
• A portfolio weight is the proportion of total
wealth allocated to a given security
Risk and Return for a Portfolio
• The expected portfolio return is the
weighted average of the expected returns on
component securities
• The risk of a portfolio is measured by the
standard deviation of return
Diversification
• Portfolio standard deviation depends upon
the correlation of the returns of component
stocks
• If stocks are not perfectly correlated, the
portfolio standard deviation is less than the
sum of the component standard deviations
• Therefore, diversification reduces risk
Systematic and
Unsystematic Risk
• The risk of a portfolio can be divided into
systematic and unsystematic risk
• Systematic risk can be explained by factors
that are common to all securities
• Unsystematic risk can be explained by
factors that are unique to a given security
• Systematic means part of the system
Diversification and Risk
• Unsystematic risk can be reduced through
diversification
• Systematic risk cannot be reduced through
diversification
• Systematic risk is measured by Beta
Beta and the Market Model
• Beta is the slope of the regression of the historical
stock return on the return of a market index, like
the S&P 500, containing a large number of stocks
• Unsystematic risk is the variation in the stock’s
return that cannot be explained by the variation in
the market index
Market Model (continued)
• The systematic risk of a portfolio is the
weighted sum of the systematic risk of each
component
• You cannot reduce systematic risk through
diversification
• You can only obtain low systematic risk by
choosing securities with low systematic risk
for your portfolio
Systematic Risk
• For very large portfolios unsystematic risk can be
almost eliminated
• In this situation the risk each security contributes to
the portfolio is approximately equal to its systematic
risk or Beta
• Therefore, the relevant risk for an individual
security held within a well-diversified portfolio is
its Beta
• Remember, that for a portfolio the relevant risk is
the standard deviation
Example
• Suppose you put all your wealth in a
General Motors stock. Then the relevant
risk is the standard deviation because your
portfolio consists of a single security
• On the other hand, suppose your holdings of
General Motors is a small fraction of a large
diversified portfolio. Then the relevant risk
is the Beta
Portfolio Theory
Risk Aversion
• Risk averse investors require compensation,
in the form of extra return, for assuming
financial risk
• The risk-free asset has zero risk and is
usually assumed to be the one-year U. S.
treasury bill
• A risk averse investor will hold a risky
portfolio only if its expected return is
greater than the risk-free rate
Risk Aversion (continued)
• Investors are only compensated for bearing
systematic risk
• Investors are not compensated for bearing
unsystematic risk because it can be
eliminated by diversification
Example
• Suppose investors A and B choose portfolios by
throwing darts at the Wall Street Journal. Assume
that the average return and average standard
deviation of stock in the paper is 10% and 14%,
respectively. If investor A throws one dart, then his
expected return and risk will be 10% and 14%. If
investor B throws ten darts, her expected return
will still be 10% but her portfolio standard
deviation will be (probably) less because of the
effect of diversification.
• Should A be compensated for assuming more risk?
Risk-Return Relationship
• There should be a positive relationship
between expected return and the Beta
measure of systematic risk.
• There should be no relationship between the
expected return and unsystematic risk.
• Formal economic model is the Capital Asset
Pricing Model (CAPM)
The Capital Asset Pricing Model
Assumptions
• risk aversion
• rational behavior
• investors choose a portfolio on the basis of
expected returns and standard deviation
• investors have same expectations about the future
expected returns, standard deviations and
correlations among stocks
• existence of risk-free security
• perfect markets: no taxes, transaction costs, or
restrictions on short sales
The security market line (SML)
• SML is the relationship between the
expected return on an asset and its Beta
measure of systematic risk
• Under the CAPM, the relationship is linear
Beta
• The Beta of the risk-free asset is zero and of
course its return is the risk-free rate
• The Beta of the market portfolio is one
• Any stock or portfolio with a Beta = 1 has
the same expected return as the market
• Any stock with a Beta = 0 returns the riskfree rate
Beta (continued)
• A stock with a Beta > 1 has more systematic
risk than the market and has an expected
return that is greater than the market
• A stock with a Beta < 1 has less systematic
risk than the market and an expected return
less than the market.
Beta (continued)
• A stock with high systematic risk (Beta > 1) will
on average (not always) go up by a greater
percentage than the market index when the market
goes up.
• And of course will on average go down by greater
percentage when the market goes down.
• This is what systematic risk means.
• So if you thought the market were going down,
you would buy stocks with low Betas
The Price of Risk
• If systematic risk is priced, than high beta stocks
should have an average return, over the ups and
downs in the market, higher than the market index.
• The high Beta stock has greater systematic risk
because when the market goes down the high Beta
stock will go down even more.
• However, in theory, when you average over the
ups and downs the risk averse investor will earn a
higher average return
Required Return
• An investor demands a positive expected
return for two reasons: time value of money
and a risk aversion.
• The return to compensate for the time value
of money is the risk-free rate.
• The extra return to compensate a risk averse
for bearing risk is called the risk premium.
• The required return equals the risk free rate
plus the risk premium.
Security Market Line
• The market risk premium (the premium on
the market index) by definition equals the
expected market return minus the risk-free
rate
• Under the CAPM, the risk premium on any
security equals the Beta multiplied by
market risk premium
Security Market Line (continued)
• The security market line relates the required
return on a security to its Beta systematic
risk. The required return equals the riskfree rate plus the Beta times the market risk
premium
The benefits of diversification and
portfolio analysis are well
established. The conclusions
drawn from portfolio theory are
tentative and debatable
Alternative Explanation
• A relatively small number of stocks have high
performance
• Most of the return on a diverisified portfolio
can be explained by a small number of high
performing stocks in the portfolio
• If you fail to diversify, then you run the risk of
not holding any high performing stocks
• Your portfolio will have below average return
Alternative Explanation
• A relatively small number of stocks will
perform badly
• Some will go under
• If you fail to diversify, then you might end
up with a large fraction of wealth in a failed
stock
• Your return distribution will have fat tails