1. Diversification and Portfolio Risk Asset Allocation With Two Risky Assets
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2. Combinations of risky assets
When we put stocks in a portfolio, p <
Why?
When Stock 1 has a return E[r1] it is likely that Stock 2 has a return E[r2] so that rp that contains stocks 1 and 2 remains close to
What statistics measure the tendency for r1 to be above expected when r2 is below expected?
Covariance and Correlation
(Wii)
Averaging principle
<
>
E[rp]
Why is covariance important?
Discuss the averaging principle, that is the key concept
n = # securities in the portfolio
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3. Portfolio Variance and Standard Deviation
Write it out for two stock case,
Then we will work on why Cov is important, what it represents (systematic risk) and how to measure it.
Variance of a Two Stock Portfolio:
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4. Covariance Calculation
Ex ante. Using scenario analysis with probabilities the covariance can be calculated with the following formula:
Ex post. Using a time series of returns, the covariance can be calculated with the following formula:
Note that 10 returns is too short a time series
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5. Covariance and correlation
The problem with covariance
Covariance does not tell us the intensity of the comovement of the stock returns, only the direction.
We can standardize the covariance however and calculate the correlation coefficient which will tell us not only the direction but provides a scale to estimate the degree to which the stocks move together.
I can’t look at a Covariance and tell you whether it is ‘big’ or not, because its ‘bigness’ is a function of the standard deviations of the two stocks.
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6. Measuring the correlation coefficient
Standardized covariance is called the _____________________
For Stock 1 and Stock 2
correlation coefficient or 
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7. Correlation Coefficients: Possible Values
Range of values for correlation coefficients:
-1.0 < r < 1.0
If r = 1.0, the securities would be perfectly positively correlated.
If r = - 1.0, the securities would be perfectly negatively correlated.
The closer r is to -1, the better diversification.

8. r and diversification in a 2 stock portfolio
Typically r is greater than ____________________
r(1,2) = r(2,1) and the same is true for the COV
The covariance between any stock such as Stock 1 and itself is simply the variance of Stock 1,
r(1,1) = +1.0 by definition
We have no measure for how three or more stocks move together.
zero and less than 1.0
Note WB = 1 – WA; can use this to solve for min var. weights when  = -1.
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9. The effects of correlation & covariance on diversification
Asset A
Asset B
Portfolio AB
Assets A and B have positive standard deviations and the correlation between A and B is +1. Thus, the standard deviation of Portfolio AB is a simple weighted average of the standard deviations of A and B and no risk is reduced by combining the two.
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10. The effects of correlation & covariance on diversification
Asset C
Asset D
Portfolio CD
Assets C and D have positive standard deviations and the correlation between C and D is -1. In this case the standard deviation of Portfolio CD is much less than a simple weighted average of the standard deviations of C and D and in this specific case CD has no risk. All of the risk has been averaged or diversified away.
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11. Two-Security Portfolio: Returnrp
= W W2
W1 = Proportion of funds in Security 1
W2 = Proportion of funds in Security 2
= Expected return on Security 1
= Expected return on Security 2 r1 r2 r1 r2
The first point to understand is how to get the expected return of a portfolio.
The portfolio expected return is just the weight of each security times the expected return of the security. The sum of the weights must always add to one.
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12. Two-Security Portfolio: Risk
sp2 = W12s12 + W22s22 + 2W1W2 Cov(r1r2)
s12 = Variance of Security 1
s22 = Variance of Security 2
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2
To get the variance of the portfolio returns it is not nearly as straightforward. We must understand how the returns of the different securities interact. Do they both go up together, or when one goes up does the other go down.
This idea of how the movements overlap is captured in the covariance portion of the equation.
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13. Example 1: Calculating portfolio risk using a time series of returns
next 4 slides

14. Calculating Variance and Covariance
Ex post
2ABC =
ABC =
2XYZ =
XYZ =
/ (10-1) =
39.07%
/ (10-1) =
41.88%
Note that 10 returns is too short a time series
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15. COV(ABC,XYZ) = rABC,XYZ = 0.533973 / (10-1) = 0.059330
Note that 10 returns is too short a time series
+ Cov, but no scale to interpret the degree to which they move together, cuz Cov number is a function of the standard deviation of the two stocks and needs to be measured relative to those standard deviations so need to calculate rho.
COV / (sABCXYZ) =
/ ( x )
0.3626
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16. Two-Security Portfolio Return
E(rp) = W1r1 + W2r2
W1 =
W2 = =
0.6
0.4
9.28%
11.97%
Wi = % of total money invested in security i r1 r2
E(rp) = 0.6(9.28%) + 0.4(11.97%) = 10.36%
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17. Two-Security Portfolio Risk
sp2 =
sp2 =
sp =
sp <
W12s12 + 2W1W2 Cov(r1r2) + W22s22
0.36( ) +
= variance of the portfolio
33.39%
Let W1 = 60% and W2 = 40% Stock 1 = ABC; Stock 2 = XYZ
2(.6)(.4)( ) +
0.16( )
We already know how to get the variances. To get the covariance we must get the deviation from the mean for each asset in each state. For each state of nature we multiply the deviations of the two assets together. The covariance is the weighted average of this product of the deviations.
Stock 1: Weight =.6 Var=222.6
Stock 2: Weight =.4 Var=60
Covariance is Positive: The stock and bond portfolios move in same directions (positively correlated, but still good diversification benefits)
W1s1 + W2s2
33.39% < [0.60(0.3907) (0.4188)] =
40.20%
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18. Example 2: Calculating portfolio risk using scenario analysis with probabilities
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19. Scenario Probability Stock Fund Return Bond Fund Return
Recession % %
Normal % 6%
Boom % %
Step 1: Calculate the expected return for the each fund using our formula from Chap.5 for discrete random variables:
Spreadsheet #1

20. Scenario Probability Stock Fund Return Bond Fund Return
Recession % %
Normal % 6%
Boom % %
Step 2: Calculate the risk (i.e., variance and standard deviation) for the each fund using formulas for discrete random variables:
Spreadsheet #2

21. Scenario Probability Stock Fund Return Bond Fund Return
Recession % %
Normal % 6%
Boom % %
Step 3: Calculate the covariance and correlation coefficient of the 2 funds’ returns.
These are formulas for discrete random variables that we haven’t seen before:
Spreadsheet #3

22. Step 4: Calculate the expected return of a PORTFOLIO that invests in the stock and bond funds:
rp = wB rB + wS rS
For example, let’s calculate the return for a portfolio that has 60% of its money invested in the stock fund and 40% of the portfolio invested in the bond fund:
= (0.4)(6.0%) + (0.6)(10.0%)
= (2.4%) (6.0%)
= %

23. Step 5: Calculate the portfolio the risk (i. e
Step 5: Calculate the portfolio the risk (i.e., variance and standard deviation) of a PORTFOLIO that invests in the stock and bond funds:
For a portfolio that has 60% of its money invested in the stock fund and 40% of the portfolio invested in the bond fund:
s 2P = (0.4) 2(0.0775) 2 + (0.6) 2(0.1492) 2 + (2)(0.4)(0.6)(0.0775)(0.1492) (-0.99) =
= , or 0.348%
Standard deviation ( s P ) = √ = , or 5.9%

24. TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS
13% 8%
12%
20%
St. Dev
TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS
WA = 0%
WB = 100%
r = -1
r = 0
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%
Stock A
Stock B
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25. Summary: Portfolio Risk/Return Two Security Portfolio
Amount of risk reduction depends critically on _________________________.
Adding securities with correlations _____ will result in risk reduction.
If risk is reduced by more than expected return, what happens to the return per unit of risk (the Sharpe ratio)?
correlations or covariances
< 1
To sum up the idea: The less correlated the greater the risk reduction possible through diversification.
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26. Three-Security Portfolio n or Q = 3
s2p = W12s12
+ W22s22
+ W32s32
For an n security portfolio there would be _ variances and _____ covariance terms.
The ___________ are the dominant effect on
+ 2W1W2
Cov(r1r2)
Cov(r1r3)
+ 2W1W3
Cov(r2r3)
+ 2W2W3 n
n(n-1)
For an n security portfolio you would have to calculate n variances and n x (n-1) covariances.
For a three security portfolio the expected return and variance are calculated in a very similar manner. We must calculate the covariance between each of the securities.
covariances
s2p
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27. Possible Risky Investments
Using data from example 2, we calculate the return and risk (standard deviation) of portfolios that invests in different weights of stock and bond funds:

28. Possible Risky Investments (continued)
Graph the return and risk (standard deviation) of portfolios that invests in different weights of stock and bond funds:

29. Possible Risky Investments (continued)
Question: Would you ever want to invest in a portfolio that had a higher % of $ invested in the bond fund than that of the “minimum variance portfolio?
Answer: No. You would expect a lower return for risk than you expect in other combinations!

30. 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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31. Minimum Variance Combinations -1< r < +12
E(r2) = .14
= .20
Stk 2 12
= .2
E(r1) = .10
= .15
Stk 1 s r 1
Cov(r1r2) = r1,2s1s2
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32. Minimum Variance: Return and Risk with r = .21
E[rp] =
.6733(.10) (.14) = or 11.31%
sp2 =
W12s12 + W22s22 + 2W1W2 r1,2s1s2
Now we can solve for the expected return and the SD of this minimum variance portfolio.
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33. Minimum Variance Combination with r = -.31
So here is the same example with a correlation coefficient of -.3 As you can see the different correlation coefficient changes the optimal weights of the minimum variance portfolio.
Cov(r1r2) = r1,2s1s2
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34. Minimum Variance Combination with r = -.31
E[rp] =
0.6087(.10) (.14) = = 11.57%
sp2 =
W12s12 + W22s22 + 2W1W2 r1,2s1s2
So here is the same example with a correlation coefficient of -.3 As you can see the different correlation coefficient changes the optimal weights of the minimum variance portfolio.
Notice lower portfolio standard deviation but higher expected return with smaller 
12 = .2
E(rp) = 11.31%
p = 13.08%
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35. Extending Concepts 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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36. 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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37. 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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