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:
 Portfolio  w1 1  w2  2
n
 Portfolio   wi  i
i 1
2
 Portfolio
 w12 12  w22 22  2w1 w2 12
2
 Portfolio
 w12 12  (1  w1 ) 2  22  2w1 (1  w1 ) 12 1 2

n
2
Portfolio
n
  w    wi w j  ij i j
i 1
2
i
2
i
i 1 j i
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
σ12
σ13
σ14
σ15
σ21
σ22
σ23
σ24
σ25
σ31
σ32
σ33
σ34
σ35
σ41
σ42
σ43
σ44
σ45
σ51
σ52
σ53
σ54
σ55
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
1
2 ...
20
40
No. of shares in portfolio
Expected portfolio variance
Actual expected portfolio variance from portfolios
of different sizes, NYSE
50
40
30
20
10
0
1
10
100
No. of stocks in portfolio
1000
Percentage of risk on an individual security that can be
eliminated by holding a random portfolio of stocks
US
UK
FR
DE
IT
73
65
67
56
60
BE
CH
NE
International
80
56
76
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
 P2
is
 2w2 22  2w1 12
w2
which is close to 2w1 12 if w2 is small.
Some simple cases: if 0 < w1 < 1
Suppose 1   2   and  1   2  
Then  Portfolio   and
2
 Portfolio
 (w12  2w1 w2 12  w22 ) 2   2
(Proof w12  2w1 w2 12  w22  w12  2w1 w2  w22  (w1  w2 ) 2  1 )
Some simple cases (2)
If ρij=0, and wi=1/n, then the variance of the portfolio is
1 n 2
2
 Portfolio  2   i
n i 1
and if all the σi are equal, then

2
Portfolio

2
n
Some simple cases (3)
If  2  0 riskfree return,
Then
2
 Portfolio
 w12 12 or  Portfolio  w1 1
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
2
 Portfolio
 w2 12  (1  w) 2  22  2w(1  w) 12 1 2
can be obtained by differentiating this expression
with respect to w and setting the result to zero, to get
 22   1 2 12
w 2
.
2
 1   2   1 2 12
But are we compromising on return?
Building the efficient frontier: combining two
assets in different proportions
Mean
return 25
0, 1
Expected return (%)
20
0.5, 0.5
15
10
0.75, 0.25
1, 0
5
0
0
5
10
15
Standard deviation
20
25
Standard
deviation
Risk and return reduced through diversification
Mean
return
Expected return (%)
25
20
=-1
15
 = +0.5
 = +1
10
 = - 0.5
=0
5
0
0
10
20
Std. dev.
30
Standard
deviation
Efficient frontier of risky assets
μp
A
x
B
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
C
p
Capital Market Line and market
portfolio (M)
Capital Market Line
μ
B
=Tangent from risk-free rate to
efficient frontier
M
μm
A
μm - rf
rf
a
m

So far we said nothing about preferences!
Individual preferences
Mean return
μ
I2 > I1
I2
p
I1
B
A
Y

ERp
Z
Standard deviation
p
Capital Market Line and market
portfolio (M)
μ
B
IA
μm
M
A
μm - r
r
a
m
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)
IB
μ
B
M
μm
A
μm - r
r
a
m
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 returns
Jan 18-24, 2008
1.0
FTSE
0.5
Mean
S&P
0.0
Average
-0.5
Eurofirst
-1.0
Nikkei
-1.5
0
1
2
3
Standard deviation
4
5
Mean and standard deviation of daily returns
Jan 18-24, 2008
FTSE and S&P
1.0
Mean
0.5
0.0
-0.5
-1.0
-1.5
0
1
2
3
Standard deviation
4
5
Mean and standard deviation of daily returns
Jan 18-24, 2008
1.0
Mean
0.5
0.0
-0.5
Eurofirst and S&P
-1.0
-1.5
0
1
2
3
Standard deviation
4
5
Mean and standard deviation of daily returns
Jan 18-24, 2008
1.0
Mean
0.5
0.0
-0.5
-1.0
Nikkei & FTSE
-1.5
0
1
2
3
Standard deviation
4
5
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
Ri ,t  a i   i RM ,t   i ,t
What assumption on  i,t ?
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
 i2   i2 M2   2i
 ij   i  j M2
The covariance between assets comes only through
their relationship to the market portfolio
Proof (next slide)
Proof:
 ij  Ri  Ri Ri  Ri 
= a i   i Rm   i  a i   i Rm a j   j Rm   j  a j   j Rm 
=  i ( Rm  Rm )   i  j ( Rm  Rm )   j 

=   i  j ( Rm  Rm ) 2   i  j ( Rm  Rm )   j  i ( Rm  Rm )   i  j
= i  j M2 + 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
 P2   P2 M2
where
1
 P   i
n i
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!