Diversification and Portfolio
Management (Ch. 8)
05/10/06
How investors view risk and return
• Investors like return. They seek to maximize
return.
• But investors dislike risk. They seek to avoid or
minimize risk. Why?
– Because human beings possess the psychological trait
of “risk aversion” which is a dislike for taking risks.
Implications of risk aversion
• The “risk-return tradeoff” - Risk averse investors require
higher rates of return to induce them to invest in higher
risk securities.
• The higher a security’s risk, the higher the return
investors demand. Thus, the less they are willing to pay
for the investment, i.e. as risk increase, P0 decreases.
• Risk averse investors will diversify their investments in
order to reduce risk.
Diversification
• Definition - An investment strategy designed to
reduce risk by spreading the funds invested across
many securities.
• It is holding a broad portfolio of securities so as “not
to have all your eggs in one basket.”
• Since people hold diversified portfolios of securities,
they are not very concerned about the risk and
return of a single security. They are more concerned
about the risk and return of their entire portfolio.
Two components of an asset’s risk
(standard deviation)
• Unique Risk - Also called “diversifiable risk” and “unsystematic
risk.” The part of a security’s risk associated with random outcomes
generated by events specific to the firm. This risk can be eliminated
by proper diversification.
• Market Risk – Also called “systematic risk.” The part of a security’s
risk that cannot be eliminated by diversification because it is
associated with economic or market factors that systematically
affect most firms.
Market risk reflects economy-wide sources of risk that affect
most firms and, hence, the overall stock market.
The expected return on a portfolio
of stocks
• Assume N stocks are held in the portfolio.
• Stock i is held in the proportion, wi
N
 w  1.00
j
j 1
• Then the expected return on the portfolio of stocks is
the weighted average of the individual stock
expected returns:
N
E (rp )   wjE (rj )
j 1
The standard deviation of
returns for a portfolio of stocks
• The standard deviation of returns for the
portfolio of stocks is given by:
 
p
n
2
(return
E(r
p)) * probabilit y

i
i
i 1
where returni is the return of the portfolio in
state i and n represents the number of states
of the economy.
The standard deviation of
returns for a portfolio of stocks
• We can also calculate the standard deviation
of returns for the portfolio of stocks as:
p 
N
N
 w w   
i 1 j 1
i
j
ij
i
j
where ρij= the correlation coefficient for
stocks i and j
Correlation coefficient
• The “Correlation Coefficient” is a measure of
•
the extent that two variables move or vary
together.
It ranges between –1.0 and +1.0
– Positive correlation: a high value on one variable is
likely to be associated with a high value on the
other.
– Negative correlation: a high value on one variable
is likely to be associated with a low value on the
other.
– No correlation: values of each are independent of
the other
Correlation coefficient
• It is denoted by the Greek letter, “rho”: ρ
– If ρ = +1.0, perfect positive correlation
– If ρ = -1.0, perfect negative correlation
– If ρ = 0, uncorrelated or independent
How diversification reduces risk
• Combining stocks into a portfolio reduces the
variability of possible returns as long as the
returns on the individual stocks are not
perfectly correlated, i.e. as long as their
correlation coefficients are less than +1.0.
Portfolio standard deviation
Portfolio risk falls as you add securities
0
20
30
Number of Securities
40
Portfolio standard deviation
You can’t eliminate “market risk”
Unique
risk
Market risk
0
20
30
Number of Securities
40
This pattern occurs because of the two
components of a stock’s risk
• Total Risk = Market risk + unique risk
• The unique risk is “diversified away” when individual
stocks are combined in a portfolio.
• Only market risk remains.
• The amount of the market risk is determined by the
market risk of the individual stocks in the portfolio.
How should we measure portfolio risk
now?
• Since diversification eliminates unique risk and leaves
market or non-diversifiable risk, the latter is the only
relevant risk for a diversified investor.
• Therefore, the relevant measure of risk for a portfolio is
a measure of the sensitivity of the portfolio’s returns to
changes in the return on the “market portfolio”.
• This is known as the portfolio’s beta (β)
• By definition, the market portfolio has a beta of 1 and
the risk-free asset has a beta of 0.
How to interpret a beta
• If βi > 1, returns to stock i are amplified
relative to the market.
• If βi is between 0 and 1.0, returns to stock i
tend to move in the same direction as the
market but not as far.
• If βi < 1(very rare), returns to stock i tend to
move in the opposite direction as the market.
How to interpret a beta-cont’d
• A stock with β = 1 has average market risk.
– A well-diversified portfolio of such stocks tends to
move by the same percentage as the overall
market moves.
• A stock with β = +.5 has below average market risk.
– A well-diversified portfolio of these stocks tends to
move half as far as the overall market moves.
Measuring individual security betas
• Security betas are estimated by running a
regression between the historical returns on the
security and the historical returns on the market
portfolio over the same period of time.
• Typically, betas are estimated using 5 years of
historical monthly returns.
• The slope of the regression represents the beta
General comments about risk
• Most stocks are positively correlated with the
market (ρi,m  0.65).
• σ  0.35 for an average stock.
• Combining stocks in a portfolio generally
lowers risk.
Calculating portfolio betas
• Assume N stocks are held in the portfolio.
• Stock i is held in the proportion, wi
N
w
i 1
i
 1.00
• Then the portfolio beta is the weighted
average of the individual stock betas:
N
 p   wi  i
i 1
Risk-return trade-off revisited
• For a diversified investor whose only concern
is non-diversifiable risk, measured by beta,
this investor will now want higher return for
a security with a higher beta.
• This linear relationship between a security’s
expected return and beta is formalized by the
Security Market Line (SML).
The Security Market Line
E(r)
SML
E(rm)
rf
1
BETA
where rf is the return on the risk-free security and E(rm) is
the expected return on the market portfolio
The Security Market Line
E(r)
E(rA)
SML
E(rm)
rf
1 βA
BETA
Reward-to-risk ratio
• The reward-to-risk ratio is calculated as the ratio of
the excess return (beyond that of the risk-free return)
that is required or expected for a particular security
given its level of risk:
Reward to risk ratio

E (rA )  rf
A
• This excess return (E(rA) – rf) is referred to as the
asset’s risk premium, which is the return investors
require beyond that of the risk free rate for security A
Capital Asset Pricing Model (CAPM)
• The CAPM assumes that the reward-to-risk ratio of all
securities are equal….giving us the following model
to estimate the expected or required return on a
stock:
E (rA )  rf   A E (rm )  rf 
where
• rf
•
•
is the return on the risk-free security and is often proxied by
the 3-month U.S. Treasury Bill or Treasury Bond rate;
E(rm) is the expected return on the market portfolio and is often
proxied by the return on the S&P 500 index and;
E(rm) - rf represents the market risk premium
Jensen’s alpha (α)
• Using the CAPM, and assuming that securities are priced based on
this model, one can measure whether a particular security
performed better or worse than expected by this model, i.e., did
the security provide a return greater than that required and
expected by investors?
• This performance measure is called Jensen’s alpha and is
calculated as follows:
  rA  ErA 
where kA represents the actual return achieved by security A and
E(kA) represents the expected return based on CAPM.
The Security Market Line and
Jensen’s alpha
E(r)
E(rA)
SML
E(rm)
Jensen’s alpha
rA
rf
1 βA
BETA
CAPM assumptions
• The CAPM relies on historical data to calculate
its inputs, thus it implicitly assumes that the past
is a good measure of the future
• The model assumes no transaction costs,
identical and complete information, rational
investors and that securities that are mispriced
will self-adjust. These are all efficient market
assumptions.
CAPM limitations
• Theoretically, the market portfolio should
consist of all assets.
• Studies have shown that additional
explanatory (risk) factors must be
considered in explaining security returns.