Practice investment management pim3 ch16e

CHAPTER SIXTEEN

WHY DIVERSIFY?

Practical Investment Management
Robert A. Strong


Outline
 Use More Than One Basket for Your Eggs
 The Axiom
 The Concept of Risk Aversion Revisited

 Preliminary Steps in Forming a Portfolio
 The Reduced Security Universe
 Security Statistics
 Interpreting the Statistics

 The Role of Uncorrelated Securities
 The Variance of a Linear Combination
 Diversification and Utility
 The Concept of Dominance
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Outline
 The Efficient Frontier









Optimum Diversification of Risky Assets
The Minimum Variance Portfolio
The Effect of a Riskfree Rate
The Efficient Frontier with Borrowing
Different Borrowing and Lending Rates
Naive Diversification
The Single Index Model

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Use More Than One Basket for Your Eggs



Don’t put all your eggs in one basket.



Failure to diversify may violate the terms of a
fiduciary trust.



Risk aversion seems to be an instinctive trait
in human beings.

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Preliminary Steps in Forming a Portfolio
 Identify a collection of eligible investments
known as the security universe.
 Compute statistics for the chosen
securities.
e.g. mean of return
variance / standard deviation of return
matrix of correlation coefficients

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Preliminary Steps in Forming a Portfolio

Insert Figure 16-1 here.

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Preliminary Steps in Forming a Portfolio

Insert Figure 16-2 here.

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Preliminary Steps in Forming a Portfolio
 Interpret the statistics.
1. Do the values seem reasonable?
2. Is any unusual price behavior expected

to recur?
3. Are any of the results unsustainable?
4. Low correlations: Fact or fantasy?

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The Role of Uncorrelated Securities


The expected return of a portfolio is a

weighted average of the component
expected returns.

E ( R portfolio ) = ∑ x i E ( Ri )
n

i =1

where xi = the proportion invested in security i

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The Role of Uncorrelated Securities

Insert Table 16-5 here.

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The Role of Uncorrelated Securities


The total risk of a portfolio comes from the
variance of the components and from the
relationships among the components.

two-security
portfolio risk = riskA + riskB + interactive risk
n

σ = x σ + x σ + 2 xa xb ρ abσ aσ b , ∑ x i = 1
2
p

2
a

2
a

where σ
xi
σi
ρ ab

2
p

2
b

2
b

i =1

= portfolio variance
= proportion of portfolio invested in stock i
= standard deviation of stock i
= correlation coefficient between a and b

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The Role of Uncorrelated Securities
Investors get added utility from greater
return. They get disutility from greater risk.



The point of diversification is to achieve a
given level of expected return while bearing
the least possible risk.
expected return



better
performance

risk
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The Role of Uncorrelated Securities


A portfolio dominates all others if no other
equally risky portfolio has a higher
expected return, or if no portfolio with the
same expected return has less risk.

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The Efficient Frontier :
Optimum Diversification of Risky Assets
The efficient frontier contains portfolios that
are not dominated.
expected return



Efficient frontier

impossible
portfolios
dominated
portfolios

risk
(standard deviation of returns)
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The Efficient Frontier :
The Minimum Variance Portfolio
The right extreme of the efficient frontier is a
single security; the left extreme is the
minimum variance portfolio.
expected return



single security
with the highest
expected return
minimum variance
portfolio

risk (standard deviation of returns)
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The Efficient Frontier :

The Minimum Variance Portfolio

Insert Figure 16-6 here.

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The Efficient Frontier :
The Effect of a Riskfree Rate
When a riskfree investment complements the
set of risky securities, the shape of the
efficient frontier changes markedly.
expected return



Rf

Efficient frontier:
Rf to M to C

impossible
portfolios

C
E
D

M
dominated
portfolios

risk (standard deviation of returns)
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The Efficient Frontier :
The Effect of a Riskfree Rate
 In capital market theory, point M is called
the market portfolio.
 The straight portion of the line is tangent to
the risky securities efficient frontier at
point M and is called the capital market
line.
 Since buying a Treasury bill amounts to
lending money to the U.S. Treasury, a
portfolio partially invested in the riskfree
rate is often called a lending portfolio.
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expected return

The Efficient Frontier with Borrowing

 Buying on margin involves financial leverage,
thereby magnifying the risk and expected
return characteristics of the portfolio. Such a
portfolio is called a borrowing portfolio.
Efficient frontier:
the ray from Rf through M

impossible
portfolios
le

ing
d
n

Rf

bo

in
w
rro

g

M
dominated
portfolios

risk (standard deviation of returns)

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The Efficient Frontier :
Different Borrowing and Lending Rates
Most of us cannot borrow and lend at the
same interest rate.
expected return



Efficient frontier : RL to M, the curve to
N, then the ray from N

impossible
portfolios

N

M
RB
RL

dominated
portfolios

risk (standard deviation of returns)
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The Efficient Frontier : Naive Diversification


Naive diversification is the random selection
of portfolio components without conducting
any serious security analysis.

total risk



Nondiversifiable risk
20

40

number of securities
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As portfolio size increases,
total portfolio risk, on
average, declines. After a
certain point, however, the
marginal reduction in risk
from the addition of
another security is

modest.
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The Efficient Frontier : Naive Diversification
 The remaining risk, when no further
diversification occurs, is pure market risk.
 Market risk is also called systematic risk
and is measured by beta.
 A security with average market risk has a
beta equal to 1.0. Riskier securities have a
beta greater than one, and vice versa.

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The Efficient Frontier : The Single Index Model
 A pairwise comparison of the thousands of
stocks in existence would be an unwieldy
task. To get around this problem, the single
index model compares all securities to a
benchmark measure.
 The single index model relates security
returns to their betas, thereby measuring
how each security varies with the overall
market.

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The Efficient Frontier : The Single Index Model


Beta is the statistic relating an individual
security’s returns to those of the market
index.

ρ imσ i cov( Ri , Rm )
βi =
=
σm
σ m2
where Rm =
Ri =
σi =
σm =
ρ im =

the return on the market index
the return on security i
standard deviation of security i returns
standard deviation of market returns
correlation between security i returns and market returns

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The Efficient Frontier : The Single Index Model


The relationship between beta and expected
return is the essence of the capital asset
pricing model (CAPM), which states that a
security’s expected return is a linear function
of its beta.

(

E(Ri ) − R f = β i E Rm − R f
where R f
Ri
Rm
βi
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=
=
=
=

)

riskless interest rate
return on security i

return on the market
beta of security i
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