EC3050 Investment Analysis Module B

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As the number of assets in the portfolio
increases, note how the number of covariance
terms in the expansion increases as the square of
the number of variance terms
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As we add additional assets, we can lower overall
risk. Lowest achievable risk is termed
systematic,
non-diversifiable or market risk
Standard deviation
Lowest risk with n assets
Diversifiable / idiosyncratic risk
Systematic risk
40
1 2 ...
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No. of shares in portfolio
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Percentage of risk on an individual security that
can be eliminated by holding a random portfolio
of stocks

• US 73
• UK 65
• FR 67
• DE 56
• IT 60

BE 80 CH 56 NE 76 International 89
Source Elton et al. Modern Portfolio Theory
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Add assetsespecially with low correlations

• Even without low correlations, you lower variance
  as long as not perfectly correlated
• Low, zero, or (best) negative correlations help
  lower variance best
• An individual assets total variance doesnt much
  affect the risk of a well-diversified portfolio

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Building the efficient frontier combining two
assets in different proportions
Mean return
0, 1
0.5, 0.5
0.75, 0.25
1, 0
Standard deviation
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Risk and return reduced through diversification
Mean return
r - 1
r 0.5
r 1
r - 0.5
r 0
Standard deviation
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Efficient frontier of risky assets
µp
A
x
x
x
x
x
x
x
B
x
x
x
x
x
x
x
x
x
C
sp
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Capital Market Line and market portfolio (M)
Capital Market Line Tangent from risk-free rate
to efficient frontier
µ
B
M
µm
A
µm - rf
rf
a
s
sm
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So far we said nothing about preferences!
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Individual preferences
Mean return
I2 gt I1
µp
I2
I1
B
A
ERp
Z
s
Y
sp
Standard deviation
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Capital Market Line and market portfolio (M)
µ
B
Investor A reaches most preferred M-V combination
by holding some of the risk-free asset and the
rest in the market portfolio M giving position A
IA
M
µm
A
µm - r
r
a
s
sm
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Capital Market Line and market portfolio (M)
IB
µ
B
B is less risk averse than A. Chooses a point
that requires borrowing some money and investing
everything in the market portfolio
M
µm
A
µm - r
r
a
s
sm
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Some lessons from our toy exercise for daily
returns

• Its laborious to compute the efficient set
• Curvature is not that great except for negatively
  correlated assets
• We know that these means and covariances are
  going to be bad estimates of next weeks
  processso how stable do we think asset returns
  are generally. is it just a question of
  longer samples
• or do covariances etc change over time?

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Issues in using covariance matrix for portfolio
decisions

• Expected returns are very volatile past not a
  good guide
• Covariances also volatile, but less so
• If we try to estimate covariances from past data
• (i) we need a lot of them (almost n2/2 for n
  assets)
• (ii) lots of noise in the estimation
• But a simplifying model seems to fit well

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What is ß?

• Could get it from past historic patterns
• (though experience shows these are not stable and
  tend to revert to mean
• adjustments possible (Blume, Vasicek)
• Could project it from asset characteristics (e.g.
  if no market history)
• Dividend payout rate, asset growth, leverage,
  liquidity, size (total assets), earnings
  variability

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Why use single index model?

• (Instead of projecting full matrix of
  covariances)
• Less information requirements
• It fits better!