Some Background Assumptions
A. Risk Aversion – Given a choice between two assets with equal rates of
return, most investors will select the asset with the lower level of risk
B. Definition of Risk – uncertainty of future outcomes or the probability of an
adverse outcome
Markowitz Portfolio Theory
Definition of an efficient asset or portfolio of assets
Under certain assumptions, a single asset or portfolio of assets is considered
to be efficient if no other asset or portfolio of assets offers higher expected
return with the same (or lower) risk, or lower risk with the same (or higher)
expected return
A. Alternative Measures of Risk
B. Expected Rates of Return
 For an individual asset - sum of the potential returns multiplied with the
corresponding probability of the returns (see Exhibit 7.1)
 For a portfolio of assets - weighted average of the expected rates of
return for the individual investments in the portfolio (see Exhibit 7.2)
C. Variance (or Standard Deviation) of Returns for an Individual Investment –
a measure of the variation of possible rates of return from the expected rate
of return (see Exhibit 7.3)
D. Variance (Standard Deviation) of Returns for a Portfolio
1. Covariance of Returns – a measure of the degree to which two variables
“move together” relative to their individual mean values over time
a. Discussion and Example
b. Coca-Cola and Home Depot (see Exhibits 7.5, 7.6, 7.7, 7.8, and 7.9)
2. Covariance and Correlation – The correlation coefficient is obtained by
standardizing (dividing) the covariance by the product of the individual
standard deviations
3. Correlation coefficient – can vary only in the range –1 to +1. A value of
+1 would indicate perfect positive correlation. This means that returns
for the two assets move together in a completely linear manner. A value
of –1 would indicate perfect correlation. This means that the returns for
two assets have the same percentage movement, but in opposite
directions (see Exhibit 7.10)
E. Standard Deviation of a Portfolio
1. Portfolio Standard Deviation Formula – the standard deviation for a
portfolio of assets is a function of the weighted average of the individual
variances (where the weights are squared), plus the weighted covariances
between all the assets in the portfolio
2. Demonstration of the Portfolio Standard Deviation Calculation
a. Equal risk and return - changing correlations (See Exhibit 7.11)
b. Combining stocks with different returns and risks (See Exhibit 7.12)
c. Constant correlation with changing weights (See Exhibit 7.13)
F. A Three-Asset Portfolio
G. Estimation Issues
H. The Efficient Frontier (See Exhibit 7.15)
 The efficient frontier represents that set of portfolios that has the
maximum rate of return for every given level of risk, or the minimum
risk for every level of return
 The slope of efficient frontier curve decreases steadily as you move
upward. This implies that adding equal increments of risk as you move
up the efficient frontier gives you diminishing increments of expected
I. The Efficient Frontier and Investor Utility (See Exhibit 7.16)
 The optimal portfolio is the portfolio on the efficient frontier that has the
highest utility for a given investor. It lies at the point of tangency
between the efficient frontier and the curve with the highest possible
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