1. Portfolio Optimization- Chapter 7

2. Measuring Portfolio Risks
For individual securities, one thinks in terms of the standard deviation of the returns to that security.
When we combine two or more securities into a portfolio (diversify), the risk of the portfolio is related to that of the individual securities depending upon their correlation coefficient.
For two securities, say gold and Infosys, the risk of a portfolio of the two will depend upon:
a) the proportion of funds invested in each (portfolio weights, W)
b) the individual risks of each (s).
b) the correlation between the two (rho).
As we talked about earlier, the search for low correlations has what has kept
money sloshing around the globe.
Correlations range from -1 (perfect negative) to +1 (perfect correlation). Discuss bonds and stocks, stocks and commodities, inter- and intra-industry connections.

3. Naïve diversification
The power of diversification
Just randomly picking stocks gets rid of 60% of the risk of the typical individual security by naive diversification
Most of the diversifiable risk eliminated at 25 or so stocks
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4. E(r) 13% r = -1 r = 0 r = .3 8% r = +1 St. Dev
A=12%
B=20%
St. Dev
TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS
WA = 0%
WB = 100%
r = 0
r = -1
r = +1
r = .3
50%A
50%B
Complicated graph, but hopefully it will help.
Imagine two securities 1. Expected return is 8% and SD is 12%
2. Expected return is 13% and SD is 20%
Depending on the amount of correlation in the returns when we combine them we will alter the portfolio standard deviation. If there is perfect correlation the combination of the two securities has no diversification effects. However if the assets are perfectly negatively correlated we can combine the two securities to completely eliminate variance in the combined portfolio. Generally assets will be somewhere in between where the combination can eliminate some risk but not completely remove it.
WA = 100%
WB = 0%
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5. Minimum Variance Combinations -1< r < +1
Choosing weights to minimize the portfolio variance 1 2
- Cov(r1r2) W1 = +
- 2Cov(r1r2) W2
= (1 - W1)
s 2
One question of interest is: With a given level of correlation how can we find the optimal weights of the securities so that we can minimize the variance of the portfolio. It turns out that we can solve for those weights using these equations.
Recall that Covariance(r1,r2) = 1,212
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6. Extending to All Securities
Consider all possible combinations of securities, with all possible different weightings and keep track of combinations that provide more return for less risk or the least risk for a given level of return and graph the result.
The set of portfolios that provide the optimal trade-offs are described as the efficient frontier.
The efficient frontier portfolios are dominant or the best diversified possible combinations.
All investors should want a portfolio on the efficient frontier.
Dominant means they provide the best return for the given risk level.
… Until we add the riskless asset
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7. The minimum-variance frontier of risky assets
Efficient Frontier is the best diversified set of
investments with the highest returns
Efficient
frontier
Found by forming portfolios of securities with the lowest covariances at a given E(r) level.
Individual
assets
Global
minimum
variance
portfolio
Minimum
variance
frontier
Individual assets combining them into portfolios, considering different weights.
So looking at many risky assets using the same techniques it is possible to build a minimum variance frontier. We are only concerned with the upper portion of the curve. Any minimum variance point on the bottom of the curve can be dominated by the similar point on the upper portion of the curve.
St. Dev.
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8. The EF and asset allocation
E(r)
EF including international & alternative investments
Efficient
frontier
100% Stocks
80% Stocks
20% Bonds
60% Stocks
40% Bonds
40% Stocks
60% Bonds
Ex-Post
20% Stocks
80% Bonds
Alternative investments: REITs, mortgage backed, gold, other precious metals, other commodities and then the international investments.
100% Stocks
St. Dev.
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9. Efficient frontier for international diversification
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10. ALTERNATIVE CALS s E(rP) CAL (P) P E(rA) A CAL (A) E(r) P&F
Efficient Frontier
P&F
E(rP&F)
CAL (Global
minimum variance) G P
There will only be one optimal Capital Allocation Line. This line will be the line that passes through the risk free rate at 0 SD and is tangent to the efficient frontier. It will dominate all other Capital Allocation Lines. F
Risk Free s A
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11. The Capital Market Line or CML
CAL (P) = CML
E(r)
Efficient Frontier
E(rP&F)
The optimal CAL is called the Capital Market Line or CML
The CML dominates the EF P
E(rP)
E(rP&F)
There will only be one optimal Capital Allocation Line. This line will be the line that passes through the risk free rate at 0 SD and is tangent to the efficient frontier. It will dominate all other Capital Allocation Lines. F
Risk Free s
P&F P
P&F
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12. The Capital Market Line or CML
E(r)
Efficient Frontier
E(rP&F) P
Both investors choose the same well diversified risky portfolio P and the risk free asset F, but they choose different proportions of each.
E(rP)
A=4
E(rP&F)
There will only be one optimal Capital Allocation Line. This line will be the line that passes through the risk free rate at 0 SD and is tangent to the efficient frontier. It will dominate all other Capital Allocation Lines. F
Risk Free s
P&F P
P&F
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13. Risk Premium Format Let: Ri = (ri - rf) Risk premium format
Rm = (rm - rf)
The Model:
Rewritten by substituting in these risk premium variables this formula can be written as given.
Ri = ai + ßi(Rm) + ei
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14. Estimating the Index Model
Scatter Plot
Excess Returns (i)
Security
Characteristic
Line . . . . . . . . . . . . . . . . . . . . . . . . . . .
Excess returns
on market index . . . . . . . .
Each point would represent a sample pair of returns observed for a particular holding period. A regression analysis will find the “best fit” line to fit the data. The expected return for the security when the market has zero excess return is the point where the line crosses the vertical axes. Beta is the slope of the regression line. Higher beta means higher systematic risk. Beta above 1 is riskier than the market. . . . . . . . . . . . . . . .
Ri = a i + ßiRm + ei
Slope of SCL = beta
y-intercept = alpha
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15. Estimating the Index Model
Scatter Plot
Excess Returns (i)
Security
Characteristic
Line
Ri = a i + ßiRm + ei . . . . . . . . . . . . . . . . . . . . . . . . . . .
Excess returns
on market index . . . . . . . .
Each point would represent a sample pair of returns observed for a particular holding period. A regression analysis will find the “best fit” line to fit the data. The expected return for the security when the market has zero excess return is the point where the line crosses the vertical axes. Beta is the slope of the regression line. Higher beta means higher systematic risk. Beta above 1 is riskier than the market. . . . . .
Variation in Ri explained by the line is the stock’s _____________
Variation in Ri unrelated to the market (the line) is ________________ . . . . . . . . . .
systematic risk
unsystematic risk
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16. Advantages of the Single Index Model
Reduces the number of inputs needed to account for diversification benefits
If you want to know the risk of a 25 stock portfolio you would have to calculate 25 variances and (25x24) = 600 covariance terms
With the index model you need only 25 betas
Rather than calculating all pairwise covariances, you can calculate covariances of securities versus the index which is a lot easier. For 100 securities you would have (100 x 99 =) 990 covariances to calculate. Against the index you have only 100 covariances to calculate.
This type of Beta model is extremely popular. We will be talking about the single factor CAPM model in the next chapter.
Easy reference point for understanding stock risk.
βM = 1, so if βi > 1 what do we know?
If βi < 1?
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