1. Efficient Diversification
Chapter 6
Efficient Diversification
Describes the financial instruments traded in primary and secondary markets.
Discusses Market indexes. Discusses options and futures.
McGraw-Hill/Irwin
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. 1

2. 6. 1 Diversification and Portfolio Risk 6
6.1 Diversification and Portfolio Risk 6.2 Asset Allocation With Two Risky Assets

3. 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.

4. 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%

5. 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

6. 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:

7. Expost Covariance Calculations
If when r1 > E[r1], r2 > E[r2], and when r1 < E[r1], r2 < E[r2], then COV will be _______.
If when r1 > E[r1], r2 < E[r2], and when r1 < E[r1], r2 > E[r2], then COV will be _______.
positive
Sample covariance, adjusted for loss of df
negative
Which will “average away” more risk?

8. 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.

9. Measuring the Correlation Coefficient
Standardized covariance is called the _____________________
For Stock 1 and Stock 2
correlation coefficient or 

10. r and Diversification in a 2 Stock Portfolio
 is always in the range __________ inclusive.
What does (1,2) = +1.0 imply?
What does (1,2) = -1.0 imply?
-1.0 to +1.0
The two are perfectly positively correlated. Means?
If (1,2) = +1, then (1,2) = W11 + W22
There are very large diversification benefits from combining 1 and 2.
Are there any diversification benefits from combining 1 and 2?
Note WB = 1 – WA; can use this to solve for min var. weights when  = -1.
The two are perfectly negatively correlated. Means?
If (1,2) = -1, then (1,2) = ±(W11 – W22)
It is possible to choose W1 and W2 such that (1,2) = 0.

11. r and Diversification in a 2 Stock Portfolio
What does -1 < (1,2) < 1 imply?
If -1 < (1,2) < 1 then
There are some diversification benefits from combining stocks 1 and 2 into a portfolio.
sp2 = W12s12 + W22s22 + 2W1W2 Cov(r1r2)
And since Cov(r1r2) = r1,2s1s2
sp2 = W12s12 + W22s22 + 2W1W2 r1,2s1s2
Note WB = 1 – WA; can use this to solve for min var. weights when  = -1.

12. 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.

13. 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.

14. The Effects of Correlation & Covariance on Diversification
Asset C
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.

15. 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

16. 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.

17. 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

18. 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

19. Ex ante Covariance Calculation
Using scenario analysis with probabilities the covariance can be calculated with the following formula:
Note that 10 returns is too short a time series

20. 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% 20

21. 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

22. 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

23. 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.

24. 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

25. 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

26. 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.

27. 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

28. 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%

29. 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

30. 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
Minimum
variance
frontier
Global
minimum
variance
portfolio
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.

31. 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.

32. Efficient frontier for international diversification Text Table 6.1

33. Efficient frontier for international diversification Text Figure 6.11

34. 6. 3 The Optimal Risky Portfolio With A Risk-Free Asset 6
6.3 The Optimal Risky Portfolio With A Risk-Free Asset 6.4 Efficient Diversification With Many Risky Assets

35. Including Riskless Investments
The optimal combination becomes linear
A single combination of risky and riskless assets will dominate
When we add in a risk-free asset it will start looking much more like the two asset case we discussed in the last chapter.

36. 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

37. 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 37

38. Dominant CAL with a Risk-Free Investment (F)
CAL(P) = Capital Market Line or CML dominates other lines because it has the the largest slope
Slope = (E(rp) - rf) / sp
(CML maximizes the slope or the return per unit of risk or it equivalently maximizes the Sharpe ratio)
Regardless of risk preferences some combinations of P & F dominate
So now it will just be back to the question of how much of your money you want in the risky portfolio and how much you want in the risk free asset. Where the slope of the Capital Allocation line is the risk premium divided by the SD of the risky portfolio at the tangent point. The CAL at the tangent point P has the best returns for all levels of risk.

39. 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 39

40. Practical Implications
The analyst or planner should identify what they believe will be the best performing well diversified portfolio, call it P.
P may include funds, stocks, bonds, international and other alternative investments. o
This portfolio will serve as the starting point for all their clients. o
The planner will then change the asset allocation between the risky portfolio and “near cash” investments according to risk tolerance of client.
P may include funds, stocks, bonds, international and other Alternative investments. o
The risky portfolio P may have to be adjusted for individual clients for tax and liquidity concerns if relevant and for the client’s opinions.

41. 6.5 A Single Index Asset Market

42. Individual Securities
We have learned that investors should diversify.
Individual securities will be held in a portfolio.
What do we call the risk that cannot be diversified away, i.e., the risk that remains when the stock is put into a portfolio?
How do we measure a stock’s systematic risk?
Consequently, the relevant risk of an individual security is the risk that remains when the security is placed in a portfolio.
From this concept of covariance, or how one asset varies with respect to another, came the idea for the single factor model.
Here the idea is that for each security i, it’s expected return will somehow be correlated with some macro factor in the economy.
Generally this macro factor is considered the entire market return or a proxy such as the S&P 500.
The excess return on an asset above the risk free rate is the expected excess holding period return + beta which is the sensitivity of this asset to any macroeconomic surprise + e which is the impact of any unanticipated firm-specific events.
Systematic risk

43. Systematic risk
Systematic risk arises from events that effect the entire economy such as a change in interest rates or GDP or a financial crisis such as occurred in 2007and 2008.
If a well diversified portfolio has no unsystematic risk then any risk that remains must be systematic.
That is, the variation in returns of a well diversified portfolio must be due to changes in systematic factors.
From this concept of covariance, or how one asset varies with respect to another, came the idea for the single factor model.
Here the idea is that for each security i, it’s expected return will somehow be correlated with some macro factor in the economy.
Generally this macro factor is considered the entire market return or a proxy such as the S&P 500.
The excess return on an asset above the risk free rate is the expected excess holding period return + beta which is the sensitivity of this asset to any macroeconomic surprise + e which is the impact of any unanticipated firm-specific events.

44. Individual Securities
How do we measure a stock’s systematic risk?
Systematic Factors 
Returns
well diversified portfolio
Δ interest rates,
Δ GDP,
Δ consumer spending,
etc.
Returns
Stock A
From this concept of covariance, or how one asset varies with respect to another, came the idea for the single factor model.
Here the idea is that for each security i, it’s expected return will somehow be correlated with some macro factor in the economy.
Generally this macro factor is considered the entire market return or a proxy such as the S&P 500.
The excess return on an asset above the risk free rate is the expected excess holding period return + beta which is the sensitivity of this asset to any macroeconomic surprise + e which is the impact of any unanticipated firm-specific events.

45. Single Factor Model Ri = E(Ri) + ßiM + ei
Ri = Actual excess return = ri – rf
E(Ri) = expected excess return
Two sources of Uncertainty M ßi ei
= some systematic factor or proxy; in this case M is unanticipated movement in a well diversified broad market index like the S&P500
= sensitivity of a securities’ particular return to the factor
= unanticipated firm specific events
From this concept of covariance, or how one asset varies with respect to another, came the idea for the single factor model.
Here the idea is that for each security i, it’s expected return will somehow be correlated with some macro factor in the economy.
Generally this macro factor is considered the entire market return or a proxy such as the S&P 500.
The excess return on an asset above the risk free rate is the expected excess holding period return + beta which is the sensitivity of this asset to any macroeconomic surprise + e which is the impact of any unanticipated firm-specific events.

46. Single Index Model Parameter Estimation
Risk Prem
Market Risk Prem
or Index Risk Prem αi
= the stock’s expected excess return if the
market’s excess return is zero, i.e., (rm - rf) = 0
Written more formally this is.
ßi(rm - rf) = the component of excess return due to
movements in the market index
ei = firm specific component of excess return that is not due to market movements

47. 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

48. 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

49. 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 49

50. Components of Risk Market or systematic risk:
ßiM + ei
Market or systematic risk:
Unsystematic or firm specific risk:
Total risk =
risk related to the systematic or macro economic factor in this case the market index
risk not related to the macro factor or market index
The idea is that through the diversification we can eliminate the Unsystematic risk (the firm specific risk) Think Systematic is the risk of the entire “System”
Systematic + Unsystematic
i2 = Systematic risk + Unsystematic Risk

51. Comparing Security Characteristic Lines
Describe   e
for each.

52. Measuring Components of Risk
si2 =
where;
bi2 sm2 + s2(ei)
si2 = total variance
bi2 sm2 = systematic variance
s2(ei) = unsystematic variance
The total risk of security i, is the risk associated with the market + the risk associated with any firm specific shocks.
(its this simple because the market variance and the variance of the residuals are uncorrelated.

53. Examining Percentage of Variance
Total Risk =
Systematic Risk / Total Risk =
Systematic Risk + Unsystematic Risk r2
ßi2 s m2 / si2 = r2
bi2 sm2 / (bi2 sm2 + s2(ei)) = r2
The ratio of the systematic risk to total risk is actually the square of the correlation coefficient between the asset and the market.

54. 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?

55. Sharpe Ratios and Alphas
When ranking portfolios and security performance
we must consider both return & risk
“Well performing” diversified portfolios provide high Sharpe ratios:
Sharpe = (rp – rf) / p
You can also use the Sharpe ratio to evaluate an individual stock if the investor does not diversify

56. Sharpe Ratios and Alphas
“Well performing” individual stocks held in diversified portfolios can be evaluated by the stock’s alpha in relation to the stock’s unsystematic risk.
Skip Treynor-Black Model

57. The Treynor-Black Model
Suppose an investor holds a passive portfolio M but believes that an individual security has a positive alpha.
A positive alpha implies the security is undervalued. Suppose it is Google.
Adding Google moves the overall portfolio away from the diversified optimum but it might be worth it to earn the positive alpha.
What is the optimal portfolio including Google?
What is the resulting improvement in the Sharpe ratio?

58. The Treynor-Black Model
Weight of Google in the optimal portfolio O:
The improvement in the Sharpe ratio (S) over the Sharpe of the passive portfolio M can be found as:
Notice that the improvement in the Sharpe ratio is a function of
This ratio is called the “information ratio”

59. The Treynor-Black Model
For multiple stocks in the active portfolio:
The optimal weight of each security in the active portfolio is found as:
A larger alpha increases the weight of stock i and larger residual variance reduces the weight of stock i.

60. The Treynor-Black Model
If A stands for the “active portfolio,” the active portfolio’s alpha, beta and residual risk can be found from:
&

62. Treynor-Black Allocation
CAL
E(r)
CML P A M Rf s 14

63. 6.6 Risk of Long-Term Investments

64. Are Stock Returns Less Risky in the Long Run?
Consider the variance of a 2-year investment with serially independent returns r1 and r2:
The variance of the 2-year return is double that of the one-year return and σ is higher by a multiple of the square root of 2 )

65. Are Stock Returns Less Risky in the Long Run?
Generalizing to an investment horizon of n years and then annualizing:
For a portfolio:

66. The Fly in the ‘Time Diversification’ Ointment
The annualized standard deviation is only appropriate for short-term portfolios
The variance grows linearly with the number of years
Standard deviation grows in proportion to

67. The Fly in the ‘Time Diversification’ Ointment
To compare investments in two different time periods:
Examine risk of the total rate of return rather than average sub-period returns
Must account for both magnitudes of total returns and probabilities of such returns occurring