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The rate of return on an investment can be
calculated as follows:
For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is:
($1,100 - $1,000) / $1,000 = 10%.
Two types of investment risk
Calculating the standard deviation for each alternative
standard deviation is the square root of variance. It has
same unit as expected return
In recession, $ return from HT$50,000*(-22%)= -$11,000
$ return from Coll $50,000*(28%)= $14,000
Total $ return is $3,000 so % return is 3%
To examine the relationship between portfolio size and portfolio
risk, consider average annual standard deviations for equally-
weighted portfolios that contain different numbers of randomly
selected NYSE securities.
Principle of diversification: Spreading an investment across many assets will eliminate some of the risk.
Stand-alone risk = Market risk + Firm-specific risk
Models return generating process. It offers significant new insights
into the nature of systematic vs. firm-specific risk
ri - rf = i + i(rM-rf) + i single index model
Single-index (factor) model assumes
Var(ri - rf )= Var(i + i(rM-rf) + i)= i2Var(rM-rf) + Var(i)
Total risk = market risk + firm-specific risk
rp= Σwiri = Σwi(rf +i + i(rM-rf) + i)
=rf + Σwii + (Σwii)(rM-rf) + Σwii
Last term becomes zero
rp-rf = Σwii + (Σwii)(rM-rf) = p + p(rM-rf)
So well-diversified portfolio does not have firm-specific component
Var(rp-rf )= p2Var(rM-rf)
Total risk of well-diversified portfolio is proportional to total risk of
If there are N securities in the market, it can be shown that
So covariance of a security is directly related to the total risk of the
market portfolio. The higher a securities covariance with market,
the higher is security’s contribution to total risk of market.
It is only the market risk of stock A that will affect the risk of the well-diversified portfolio.
Has only market risk
But no firm-specific risk
Has both market and firm-specific risks
Model based upon concept that a stock’s required rate of return is
equal to the risk-free rate of return plus a risk premium that
reflects the riskiness of the stock after diversification.
Recall the efficient set
In equilibrium, the prices for all assets must adjust so that aggregate amount of borrowing equals aggregate amount of lending.
All efficient portfolios are combinations of market with risk free security
rp=wMP rM+wrfP rf
E(rP)=wMP E(rM)+(1-wMP) rf = rf + wMP [E(rM)-rf]
P2=wMP2M2 or P=wMPM
Efficient portfolios’ total risk depends on market risk
E(rP)= rf +P [E(rM)-rf]/M
The last relation is called Capital Market Line.
We need a relation between risk and return that holds for all
securities/portfolios. This is call Security Market Line in CAPM.
Returns of asset i and market are excess returns. Practitioners often use total rather than excess returns. This practice is most common when daily data is used where total and excess returns are almost indistinguishable.
Can the beta of a security be negative?
Slope of the regression line is given by the following formula:
Given our payoff matrix we can calculate:
SecurityExp. Ret. Beta
HT 17.4% 1.30
Market 15.0 1.00
USR 13.8 0.89
T-Bills 8.0 0.00
Coll. 1.7 -0.87
Riskier securities have higher returns, so the rank order is OK.
Recall that CAPM is based upon concept that a stock’s required
rate of return is equal to the risk-free rate of return plus a risk
premium that reflects the riskiness of the stock after diversification.
SML equation states that the risk premium is the product of risk and
extra compensation per unit of risk. Risk is measured by beta, and
extra compensation by excess return on market portfolio.
Ceteris paribus as price rises expected return falls
Example: Equally-weighted two-stock portfolio
The required return of a portfolio is the weighted
average of each of the stock’s required returns.
kP = wHT kHT + wColl kColl
kP = 0.5 (17.1%) + 0.5 (1.9%)
kP = 9.5%
Or, using the portfolio’s beta, CAPM can be used to
solve for required return.
kP = kRF + (kM – kRF) βP
kP = 8.0% + (15.0% – 8.0%) (0.215)
kP = 9.5%
What if investors raise inflation expectations by 3%,
what would happen to the SML?
recall that kRF =k*+IP
kRF will increase by 3%, RPM stays constant since kM also increases by the same amount
What if investors’ risk aversion increased, causing the market risk
premium to increase by 3%, what would happen to the SML?
Investors would require higher risk premium per unit of risk
Again you may assume that kA and kM show excess returns.
recall: cov(x,y)=E(xy) – E(x) E(y)
if cov(x,y)=0 then E(xy) = E(x) E(y)
therefore E(xy) = 0 if E(x)=0 or E(y)=0
so since cov(kM,A)=0 and E(A)=0 then E(kMA)=0