Chapter 6 investments efficient diversification

Chapter 6
Efficient Diversification
6.1 Diversification and Portfolio Risk
6-2

Market risk
- The risk that has to do with general economic
conditions.
- The risk that remains even after diversification.
- Systematic risk or non-diversifiable risk.


Firm-specific risk
- Diversifying into many more securities reduce
exposure to firm-specific factors.
- Unique risk, nonsystematic risk, or diversifiable risk.
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6-4
Figure 6.1 Portfolio risk as a function of the
number of stocks in the portfolio
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Figure 6.2 Portfolio risk decreases as
diversification increases.
6.2 Asset Allocation With Two Risky Assets
6-6
- Need to understand how the uncertainties of asset
returns interact when we form a risky portfolio.
- The key determinant of portfolio risk is the extent to
which the returns on the two assets tend to vary
either in tandem or in opposition.
- Portfolio risk depends on the covariance between

the returns of the assets in the portfolio.
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Asset Allocation With Two Risky Assets
= W1 + 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
Two-Security Portfolio: Return
r1
E( )
rp
r2
r1
r2
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portfolio the in securities # n ;rW)rE(
n
1i
i
i
p
==
∑
=
E(rp) = W1r1 + W2r2
W1 =
W2 =
=
=

Two-Security Portfolio Return
E(rp) = 0.6(9.28%) + 0.4(11.97%) = 10.36%
Wi = % of total money
invested in security i
0.6
0.4
9.28%
11.97%
r1
r2
6-9
Combinations of risky assets
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 E[rp]
What statistics measure the
tendency for r1 to be below
expected when r2 is above
expected?
Covariance and Correlation
n = # securities
in the portfolio
6-10
Portfolio Variance and Standard Deviation
Variance of a Two Stock Portfolio:
6-11
Covariance: A measure of the extent to which the returns tend to
vary with each other, that is, to co-vary.

(The covariance between any stock such as Stock 1 and itself is simply
the variance of Stock 1.)
∑∑
= =
=
Q
1I
Q
1J
JIJI
2
p
)]r,Cov(r W[Wσ
portfolio the in stocks of number total The Q
lyrespective J and I stock in invested portfolio total the of PercentageW,W
JI
=
=
J Stock and I Stock of returns the of Covariance)r,Cov(r
JI
=
)r,r(Cov)r,Cov(r & σ )r,(r Cov then J I If
IJJI
2
IJI
===
2
2
2
22121

2
1
2
1
2
),(2
σσσ
WrrCovWWW
p
++=
Covariance calculations
If when r1 > E[r1], r2 > E[r2], and when
r1 < E[r1], r2 < E[r2], then COV will be positive.
If when r1 > E[r1], r2 < E[r2], and when
r1 < E[r1], r2 > E[r2], then COV will be negative.
Which will “average away” more risk?
6-12
∑
=
−×−
=
N
1T
21
n
)r(r)r(r
)r,Cov(r
2
T2,
1

T1,
n
nsobservatio of # n
2 stock for return expected or averager
1 stock for return expected or averager
2
1
=
=
=
Covariance and correlationThe 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.
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Measuring the correlation coefficientStandardized covariance is called
the correlation coefficient or ρ
Correlation coefficient can range from values of -1 to +1.
Values of -1 indicate perfect negative correlation.
Values of +1 indicate perfect positive correlation.

Values of 0 indicate that the returns on assets are unrelated.
6-14
21
21
12
σσ
)r,Cov(r
ρ
×
=
ρ and diversification in a 2 stock portfolio
ρ is always in the range __________ inclusive.
What does ρ12 = +1.0 imply?
What does ρ12 = -1.0 imply?
The two are perfectly positively correlated. Means?
If ρ12 = +1, then σ12 = W1σ1 + W2σ2
The two are perfectly negatively correlated. Means?
If ρ12 = -1, then σ12 = ±(W1σ1 – W2σ2)
It is possible to choose W1 and W2 such that
σ12 = 0.
-1.0 to +1.0
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There are very large diversification benefits from
combining 1 and 2.
ρ and diversification in a 2 stock portfolio
What does -1 < ρ12 < 1 imply?
If -1 < ρ12 < 1, then
σp2 = W12σ12 + W22σ22 + 2W1W2 Cov(r1r2)
And since Cov(r1r2) = ρ12σ1σ2
σp2 = W12σ12 + W22σ22 + 2W1W2 ρ12σ1σ2

There are some diversification benefits from
combining stocks 1 and 2 into a portfolio.
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The effects of correlation &
covariance on diversification
Asset A
Asset B
Portfolio AB
6-17
6-18
The effects of correlation &
covariance on diversification
Asset C
Asset C
Portfolio CD
6-19
Most of the diversifiable risk
eliminated at 25 or so stocks
The power of diversification
Two-Risky Assets Portfolios
rp = W1r1 +W2r2
E(rp) = W1E(r1) + W2E(r2)
σp2 = W12σ12 + W22σ22 + 2W1W2 Cov(r1r2)
= W12σ12 + W22σ22 + 2W1σ1W2σ2 ρ12
Using scenario analysis with probabilities, the covariance can be calculated wit
h the following formula:
6-20
Linear Function
Not Linear Function
1

( , ) ( ) ( ) ( )
S
S B S S B B
i
Cov r r p i r i r r i r
=
  
= − −
  
∑
σp2 =
σp2 =
σp2 =
σp =
σp <
The benefits of diversification
Two-Security Portfolio Risk
W12σ12 + 2W1W2 Cov(r1r2) + W22σ22
0.36(0.15265) +
0.1115019 = variance of the portfolio
33.39%
Let W1 = 60% and W2 = 40% Stock 1 = ABC; Stock 2 = XYZ
40.20%
2(.6)(.4)(0.05933) +
0.16(0.17543)
33.39% < [0.60(0.3907) + 0.40(0.4188)] =
W1σ1 + W2σ2 (Linear combination: ρ12=1)
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∑ ∑
= =

=
Q
1I
Q
1J
JI
2
p
J)]Cov(I, W[Wσ
ρ = +1
ρ = .3
E(r)
13%
8%
12% 20%
St. Dev
TWO-SECURITY PORTFOLIOS WITH
DIFFERENT CORRELATIONS
Stock A
Stock B
WA = 0%
WB = 100%
WA = 100%
WB = 0%
ρ = -1
50%A
50%B
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ρ = 0
Summary: Portfolio Risk/Return Two Security Portfolio

Amount of risk reduction depends critically on correlations or covariances.
Adding securities with correlations _____ will result in risk reduction.
If risk is reduced by more than expected return, what happens to the return per u
nit of risk (the Sharpe ratio)?
< 1
6-23
Minimum Variance Combinations
-1< ρ < +1
1
2
- Cov(r1r2)
W1
=
+
-
2Cov(r1r2)
2
W2
= (1 - W1)
σ
2
σ 2 σ
2
Choosing weights to minimize the portfolio variance
6-24
1
Minimum Variance
Combinations -1< ρ < +1
2
E(r2) = .14 = .20Stk 2

12
= .2
E(r1) = .10 = .15Stk 1
σ
σ
ρ
Cov(r1r2) =
ρ1,2σ1σ2
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