1. Chapter 6
Efficient Diversification

2. Relationship Between Risk and Return – Let’s
revisit…
Exhibit I
Possible Investment Outcomes
Taxi & Bus Companies
State of the Economy
Poor Average Good
Bus Company 1,300,000 1,210,000 700,000
Taxi Company 600,000 1,210,000 1,400,000
Harry Markowitz -- one of the founders of modern finance – contributed greatly
to modern financial theory and practice. In his dissertation he argued that investors
are 1) risk adverse, and 2) evaluate investment opportunities by comparing
expected returns relative to risk which he defined as the standard deviation of the
expected returns. This example is based on his seminal work.

3. Step 1: Calculate the potential return on each
investment...
Rit = (Priceit+1 - Priceit)/Priceit
Where:
Rit = The holding period return for investment “i” for time
period “t”
Priceit = The price of investment “i” at time period “t”
Priceit+1 = The price of investment “i’ at time period “t+1”
State of the Economy
Poor Average Good
Bus Company 30.00% 21.00% -30.00%
Taxi Company -40.00% 21.00% 40.00%

4. Step 2: Calculate the expected return
(E)Rit = ΣXi Rit
Where: Xi = Probability of a given event
The Expected Return for the Taxi and Bus Companies -
(E)RBus = 1/3(-30%) + 1/3(21%) + 1/3(30%)
= 7%
(E)RTaxi = 7%

5. Step 3: Measure risk

6. Step 5: Compare the alternatives
Expected Return
10%
7% --------------B------T
2%
10% 20% 30% 40% Risk (Standard
Deviation)
We have two investment alternatives with the same
expected return – which one is preferable?
Bus
Company
Taxi
Company
Expected Return 7% 7%
Standard Deviation 26.42% 34.13%

7. Conclusion
Based on our analysis the Bus Company
represents a superior investment alternative to
the Taxi company. Since the Bus company
represents a superior return to the Taxi
company, why would anyone hold the Taxi
company?

8. Portfolio Analysis of Investment Decision
Assume you invest 50 percent of your money in the Bus company
and 50 percent in the Taxi company.
Exhibit IV
Portfolio Analysis of Investment Decision
State of the Economy
Poor Average Good
Bus Company 30.00% 21.00% -30.00%
Taxi Company -40.00% 21.00% 40.00%
50 percent in each -5.00% 21.00% 5.00%

9. Portfolio Return...
(E)Rp = Σwj(E)rit
Where: (E)Rp = The expected return on the portfolio
wj = The proportion of the portfolio’s total value
= .5(7%) + .5(7%)
= 7%
or,
(E)Rp = -5%(1/3) + 21%(1/3) + 5%(1/3)
= 7%

10. Portfolio Risk
The standard deviation of the portfolio:
σp = 10.68%
Note - The expected return of the portfolio is simply a weighted-
average of the of the expected returns for each alternative; the
standard deviation of the portfolio is not a simple weighted-
average. Why?
The formula for the portfolio standard deviation is:
σp = (wa2* σa2 + wb2* σb2 + 2*wa*wb* σa* σb*rab).5
Where:
Wa – weight of security A
Wb – weight of security B
σa = standard deviation of security A’s return
σb = standard deviation of security B’s return
Corrab = correlation coefficient between security A and B

11. Risk Reduction
 Holding more than one asset in a portfolio (with less than a
correlation coefficient of positive 1) reduces the range or spread
of possible outcomes; the smaller the range, the lower the total
risk.
State of the Economy
Poor Average Good
Bus Company 30.00% 21.00% -30.00%
Taxi Company -40.00% 21.00% 40.00%
50 percent in each -5.00% 21.00% 5.00%
Correlation coefficient = CovarianceAB /σAσB
Covariance = ΣpAB(A – E(A))*(B – E(B))
= 1/3(30% - 7%)(-40% - 7%) + 1/3(21% - 7%)(21% - 7%) + 1/3 (-30% - 7%)(40% - 7%)
= -.0702
Correlation coefficient = -.0702/((.2642)*(.3413)) = -.78

12. Standard Deviation of a Two-Asset
Portfolio
 σp = (wa2* σa2 + wb2* σb2 + 2*wa*wb* σa* σb*rab).5
 Where:
 Wa – weight of security A (.5)
 Wb – weight of security B (.5)
 σa = standard deviation of security A’s return (26.42%)
 σb = standard deviation of security B’s return (34.13%)
 rab = correlation coefficient between security A and B (-.78)
σp = ((.5)2 (26.42)2 + (.5)2 (34.13)2 + 2(.5)(.5)(26.42)(34.13)(-.78)).5
σp = (174.50 +291.21 - 351.67).5
σp = 10.68%

13. Risk Reduction
Expected
Return
10%
7% ----P--------B------T
2%
10% 20% 30% 40% Risk (Standard
Deviation)
The net effect is that an investor can reduce their overall risk by
holding assets with less than a perfect positive correlation in a
portfolio relative to the expected return of the portfolio.

14. Extending the example to numerous securities...
Expected
Return
Risk (Standard
Deviation)
Each point represents the expected return/standard deviation
relationship for some number of individual investment
opportunities.

15. Extending the example to numerous securities...
Expected
Return
Risk (Standard
Deviation)
This point represents a new
possible risk - return
combination

16. More on Correlation & the Risk-
Return Trade-Off

17. Efficient Frontier
Expected
Return
Risk (Standard
Deviation)
Each point represents the
highest potential return for a
given level of risk

18. Breakdown of Risk
Total Risk = Diversifiable Risk + Non Diversifiable Risk
Diversifiable Risk = Company specific risk
Nondiversifiable Risk = Market risk
Total Risk ()
Number of
Securities
Company
specific or
diversifiable
risk
Market risk or Non-
diversifiable risk
Total Risk

19. Diversification and Risk

20. Why Diversification Works, I.
 Correlation: The tendency of the returns on two assets to move
together. Imperfect correlation is the key reason why
diversification reduces portfolio risk as measured by the portfolio
standard deviation.
 Positively correlated assets tend to move up and down
together.
 Negatively correlated assets tend to move in opposite
directions.
 Imperfect correlation, positive or negative, is why diversification
reduces portfolio risk.

21. Why Diversification Works, II.
 The correlation coefficient is denoted by Corr(RA, RB) or
simply, rA,B.
 The correlation coefficient measures correlation and ranges
from:
From: -1 (perfect negative
correlation)
Through: 0 (uncorrelated)
To: +1 (perfect positive
correlation)

22. Why Diversification Works, III.

23. Why Diversification Works, IV.

24. Why Diversification Works, V.

25. Correlation and Diversification

26. Minimum Variance Combinations -1< r < +1
1 2
- Cov(r1r2)
W1
=
+ - 2Cov(r1r2)
2
W2 = (1 - W1)
 2
 2  2
Choosing weights to minimize the portfolio variance
6-26

27. 1
Minimum Variance Combinations -1< r < +1
2E(r2) = .14 = .20Stk 2
12 = .2
E(r1) = .10 = .15Stk 1 

r
11 22
- Cov(r1r2)- Cov(r1r2)
W1W1
==
++ - 2Cov(r1r2)- 2Cov(r1r2)
22
W2W2 = (1 - W1)= (1 - W1)
 2 2
 2 2  2 2
11 22
- Cov(r1r2)- Cov(r1r2)
W1W1
==
++ - 2Cov(r1r2)- 2Cov(r1r2)
22
W2W2 = (1 - W1)= (1 - W1)
 2 2
 2 2  2 2
WW11
==
(.2)(.2)22
-- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1= (1 -- .6733) = .3267.6733) = .3267
WW11
==
(.2)(.2)22
-- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1= (1 -- .6733) = .3267.6733) = .3267
WW11
==
(.2)(.2)22
-- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1= (1 -- .6733) = .3267.6733) = .3267
WW11
==
(.2)(.2)22
-- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1= (1 -- .6733) = .3267.6733) = .3267Cov(r1r2) = r1,212
6-27

28. E[rp] =
Minimum Variance: Return and Risk with
r = .2
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2
1212 = .2= .2
E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 

rr
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2
1212 = .2= .2
E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 

rr
1/2
2222
p (0.2)(0.15)(0.2)(0.3267)(0.6733)2)(0.2)(0.3267)(0.15)(0.6733σ



 
p
2 =
%.. /
p 081301710 21 
WW11
==
(.2)(.2)22
-- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1= (1 -- .6733) = .3267.6733) = .3267
WW11
==
(.2)(.2)22
-- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1= (1 -- .6733) = .3267.6733) = .3267
1
.6733(.10) + .3267(.14) = .1131 or 11.31%
W1
21
2 + W2
22
2 + 2W1W2 r1,212
6-28

29. WW11
==
(.2)(.2)22
-- (.2)(.15)((.2)(.15)(--.3).3)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(.2)(.15)(2(.2)(.15)(--.3).3)
WW11 = .6087= .6087
WW22 = (1= (1 -- .6087) = .3913.6087) = .3913
WW11
==
(.2)(.2)22
-- (.2)(.15)((.2)(.15)(--.3).3)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(.2)(.15)(2(.2)(.15)(--.3).3)
WW11 = .6087= .6087
WW22 = (1= (1 -- .6087) = .3913.6087) = .3913
Minimum Variance Combination with r = -.3
11 22
- Cov(r1r2)- Cov(r1r2)
W1W1
==
++ - 2Cov(r1r2)- 2Cov(r1r2)
22
W2W2 = (1 - W1)= (1 - W1)
 2 2
 2 2  2 2
11 22
- Cov(r1r2)- Cov(r1r2)
W1W1
==
++ - 2Cov(r1r2)- 2Cov(r1r2)
22
W2W2 = (1 - W1)= (1 - W1)
 2 2
 2 2  2 2
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2
1212 = .2= .2
E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 

rr
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2
1212 = .2= .2
E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 

rr -.31
Cov(r1r2) = r1,212
WW11
==
(.2)(.2)22
-- ((--.3)(.15)(.2).3)(.15)(.2)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(2(--.3)(.15)(.2).3)(.15)(.2)
WW11
==
(.2)(.2)22
-- ((--.3)(.15)(.2).3)(.15)(.2)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(2(--.3)(.15)(.2).3)(.15)(.2)
6-29

30. WW11
==
(.2)(.2)22
-- (.2)(.15)((.2)(.15)(--.3).3)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(.2)(.15)(2(.2)(.15)(--.3).3)
WW11 = .6087= .6087
WW22 = (1= (1 -- .6087) = .3913.6087) = .3913
WW11
==
(.2)(.2)22
-- (.2)(.15)((.2)(.15)(--.3).3)
(.15)(.15)22
+ (.2)+ (.2)22
-- 2(.2)(.15)(2(.2)(.15)(--.3).3)
WW11 = .6087= .6087
WW22 = (1= (1 -- .6087) = .3913.6087) = .3913
Minimum Variance Combination with r = -.3
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2
1212 = .2= .2
E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 

rr
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2
1212 = .2= .2
E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 

rr -.3
E[rp] =
1/2
2222
p (0.2)(0.15)(-0.3)(0.3913)(0.6087)2)(0.2)(0.3913)(0.15)(0.6087σ



 
p
2 =
%.. /
p 091001020 21 
0.6087(.10) + 0.3913(.14) = .1157 = 11.57%
W1
21
2 + W2
22
2 + 2W1W2 r1,212
1
Notice lower portfolio
standard deviation but
higher expected return
with smaller r
r12 = .2
E(rp) = 11.31%
p = 13.08%
6-30

31. Individual securities
We have learned that investors should diversify.
Individual securities will be held in a portfolio.
We call the risk that cannot be diversified away, i.e., the risk that
remains when the stock is put into a portfolio – the systematic
risk
Major question -- 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.
6-31

32. 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.
6-32

33. Single Index Model Parameter Estimation
Risk Prem Market Risk Prem
or Index Risk Prem
= the stock’s expected excess return if the
market’s excess return is zero, i.e., (rm - rf) = 0
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
αi
    errrr ifmiifi
 
6-33

34. Estimating the Index Model
Excess Returns (i)
Security
Characteristic
Line
. .
.
. .
.
.
.
. ..
. .
.
. .
. .
.
.
.
.
. .
. .
.
..
.
. .
. .
.
. .
.
. .
.
. ..
. .
. .. .
Excess returns
on market index
Ri =  i + ßiRm + ei
Slope of SCL = beta
y-intercept = alpha
Scatter
Plot
6-34

35. Estimating the Index Model
Excess Returns (i)
Security
Characteristic
Line. .
.
. .
.
.
.
. ..
. .
.
. .
. .
.
.
.
.
. .
. .
.
..
.
. .
. .
.
. .
.
. .
.
. ..
. .
. .. .
Excess returns
on market index
Variation in Ri explained by the line is the stock’s
systematic risk
Variation in Ri unrelated to the market (the line) is
unsystematic risk
Scatter
Plot
Ri =  i + ßiRm + ei
6-35

36. Components of Risk
 Market or systematic risk:
 Unsystematic or firm specific risk:
 Total risk = Systematic + Unsystematic
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
ßiM + ei
i
2 = Systematic risk + Unsystematic Risk
6-36

37. Comparing Security Characteristic Lines
Describe
 
 
 e
for each
6-37

38. Measuring Components of Risk
i
2 =
where;
i
2 m
2 + 2(ei)
i
2 = total variance
i
2 m
2 = systematic variance
2(ei) = unsystematic variance
6-38
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.)

39. Total Risk = Systematic Risk +
Unsystematic Risk
Systematic Risk / Total Risk
Examining Percentage of Variance
ßi
2  m
2 / i
2 = r2
i
2 m
2 / (i
2 m
2 + 2(ei)) = r2
6-39
The ratio of the systematic risk to total risk is actually the
square of the correlation coefficient between the asset
and the market.

40. 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
The Sharpe ratio can also be used to evaluate an
individual stock if the investor does not diversify
6-40