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Suppose a portfolio is composed of asset weights wi. Writing the mean and standard deviation of the individual assets and the portfolio as: μi, σi, μPortfolio, σPortfolio. We get:
2211 wwPortfolio
n
iiiPortfolio w
1
122122
22
21
21
2 2 wwwwPortfolio
21121122
21
21
21
2 )1(2)1( wwwwPortfolio
n
i ijjiijji
n
iiiPortfolio www
11
222
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
σ11 σ21 σ31 σ41 σ51 σ12 σ22 σ32 σ42 σ52 σ13 σ23 σ33 σ43 σ53 σ14 σ24 σ34 σ44 σ54 σ15 σ25 σ35 σ45 σ55
Standard
deviation
No. of shares in portfolio
Diversifiable / idiosyncratic risk
Systematic risk
20
40 1 2 ...
As we add additional assets, we can lower overall risk.
Lowest achievable risk is termed “systematic”, “non-diversifiable” or “market” risk
Lowest risk with n assets
Actual expected portfolio variance from portfolios of different sizes, NYSE
0
10
20
30
40
50
1 10 100 1000
No. of stocks in portfolio
Exp
ecte
d po
rtfo
lio v
aria
nce
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 80CH 56NE 76International 89
Source: Elton et al. Modern Portfolio Theory
Add assets…especially 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 asset’s total variance doesn’t much affect the risk of a well-diversified portfolio
Change in portfolio variance by adding a small amount of a new asset 2
is 121222
2
2
22 ww
wP
which is close to 1212 w if w2 is small.
Some simple cases: if 0 < w1 < 1 Suppose 21 and 21
Then Portfolio and 222
2122121
2 )2( wwwwPortfolio
(Proof 1)(22 2
212221
21
221221
21 wwwwwwwwww )
Some simple cases (2) If ρij=0, and wi=1/n, then the variance of the portfolio is
n
iiPortfolio n 1
22
2 1
and if all the σi are equal, then
nPortfolio
22
Some simple cases (3) If 02 riskfree return, Then
21
21
2 wPortfolio or 11 wPortfolio
Standard deviation of a portfolio mixing a riskfree and a risky asset is proportionate to the share of the risky asset in the portfolio
The value of w that minimizes portfolio variance
211222
221
22 )1(2)1( wwwwPortfolio
can be obtained by differentiating this expression with respect to w and setting the result to zero, to get
122122
21
122122
w .
But are we compromising on return?
Building the efficient frontier: combining two assets in different
proportions
0
5
10
15
20
25
0 5 10 15 20 25Standard deviation
Exp
ecte
d r
etu
rn (
%)
1, 0
0, 1
0.5, 0.5
0.75, 0.25
Standard
deviation
Mean return
Risk and return reduced through diversification
0
5
10
15
20
25
0 10 20 30
Std. dev.
Exp
ecte
d r
etu
rn (
%)
= - 0.5
= +1
= - 1 = +0.5
= 0
Mean return
Standard
deviation
Efficient frontier of risky assets
μp
p
x
x
x
x
x
x
xx
xx
xx
x
x
xxB
A
C
Capital Market Line and market portfolio (M)
Capital Market Line=Tangent from risk-free rate
to efficient frontier
rf
A
B
M
μm - rf
m
μ
μm
So far we said nothing about preferences!
Individual preferences
μp
p
AB
ZERp
I2 I1
I2 > I1
Y
Mean return
Standard deviation
Capital Market Line and market portfolio (M)
r
A
B
M
μm - r
m
μ
μm
IA
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
Capital Market Line and market portfolio (M)
r
A
B
M
μm - r
m
μ
μm
IB
B is less risk averse than A. Chooses a point that requires borrowing some money and investing everything in the market portfolio
Mean and standard deviation of daily returnsJan 18-24, 2008
Nikkei
Eurofirst
Average
FTSE
S&P
-1.5
-1.0
-0.5
0.0
0.5
1.0
0 1 2 3 4 5
Standard deviation
Mea
n
Mean and standard deviation of daily returnsJan 18-24, 2008
-1.5
-1.0
-0.5
0.0
0.5
1.0
0 1 2 3 4 5
Standard deviation
Mea
n
FTSE and S&P
Mean and standard deviation of daily returnsJan 18-24, 2008
-1.5
-1.0
-0.5
0.0
0.5
1.0
0 1 2 3 4 5
Standard deviation
Mea
n
Eurofirst and S&P
Mean and standard deviation of daily returnsJan 18-24, 2008
-1.5
-1.0
-0.5
0.0
0.5
1.0
0 1 2 3 4 5
Standard deviation
Mea
n
Nikkei & FTSE
Some lessons from our toy exercise for daily returns
• It’s 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 process…so how stable do we think asset returns are generally…. …is it just a question of longer samples
or do covariances etc change over time?
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:
The market model
titMiiti RR ,,,
What assumption on ti , ?
For the “single index” model we assume that the residual is uncorrelated across assets
Risk and covariance in the single index model: …for assets i and j
2222
iMii 2Mjiij
The covariance between assets comes only through their relationship to the market portfolio Proof (next slide)
Proof: iiiiij RRRR
= mjjjmjjmiiimii RRRR
= jmmjimmi RRRR )()(
= jimmijmmjimmji RRRRRR )()()( 2
= 2Mji + 0 + 0 + 0 (using assumptions (i), (ii) and (iii)
Risk and covariance in the single index model: …for an equally-weighted portfolio of n assets
222MPP
where
i
iP n 1
What is β?
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
Why use single index model?
(Instead of projecting full matrix of covariances)
1. Less information requirements
2. It fits better!
