Post on 15-Jan-2016
Investment Analysis and Portfolio Management
Lecture 3
Gareth Myles
FT 100 Index
£ and $
Risk
VarianceThe standard measure of risk is the
variance of returnor Its square root: the standard deviation
Sample variance The value obtained from past data
Population variance The value from the true model of the data
Sample Variance
General Motors Stock Price 1962-2008
Sample Variance
Year 93-94 94-95 95-96 96-97 97-98
Return % 36.0 -9.2 17.6 7.2 34.1
Year 98-99 99-00 00-01 01-02 02-03
Return % -1.2 25.3 -16.6 12.7 -40.9
Return on General Motors Stock 1993-2003
Sample Variance
-50
-40
-30
-20
-10
0
10
20
30
40
93-94
94-95
95-96
96-97
97-98
98-99
99-00
00-01
01-02
02-03
Graph of return
Sample Variance
With T observations sample variance is
The standard deviation is
Both these are biased estimators The unbiased estimators are
T
tt rr
T 1
22 1
T
tt rr
T 1
21 0
02
T
ttT rr
T 1
21 1
1
T
ttT rr
T 1
221 1
1
Sample Variance For the returns on the General Motors stock,
the mean return is 6.5 Using this value, the deviations from the
mean and their squares are given by
Year 93-94 94-95 95-96 96-97 97-98
29.5 -15.7 11.1 0.7 27.6
870.25 246.49 123.21 0.49 761.76
Year 98-99 99-00 00-01 01-02 02-03
-7.7 18.8 -23.1 6.2 -47.4
59.29 353.44 533.61 38.44 2246.76
rrt 2rrt
2rrt
rrt
Sample Variance
After summing and averaging, the variance is
The standard deviation is
This information can be used to compare different securities
A security has a mean return and a variance of the return
4.5232
88.224.523
Sample Covariance
The covariance measures the way the returns on two assets vary relative to each other Positive: the returns on the assets tend to rise and fall
together Negative: the returns tend to change in opposite
directions Covariance has important consequences for
portfolios
Asset Return in 2001 Return in 2002
A 10 2
B 2 10
Sample Covariance
Mean return on each stock = 6 Variances of the returns are Portfolio: 1/2 of asset A and 1/2 of asset B
Return in 2001:
Return in 2002:
Variance of return on portfolio is 0
1622 BA
6221
1021 pr
61021
221 pr
Sample Covariance
The covariance of the return is
It is always true that
i.
ii.
T
tBBtAAtAB rrrr
T 1
1
BAAB
2iii
Sample Covariance
Example. The table provides the returns on three assets over three years
Mean returns
Year 1 Year 2 Year 3
A 10 12 11
B 10 14 12
C 12 6 9
9,12,11 CBA rrr
Sample Covariance
Covariance between A and B is
Covariance between A and C is
333.1
12121111121411121210111031
AB
2
991111961112912111031
AC
Variance-Covariance Matrix
Covariance between B and C is
The matrix is symmetric
2
2
2
CBCAC
BAB
A
C
B
A
CBA
4
99121296121491212103
1
CB
Variance-Covariance Matrix
For the example the variance-covariance matrix is
642
66.2333.1
666.0
C
B
A
CBA
Population Return and Variance
Expectations: assign probabilities to outcomes Rolling a dice: any integer between 1 and 6 with
probability 1/6 Outcomes and probabilities are:
{1,1/6}, {2,1/6}, {3,1/6}, {4,1/6}, {5,1/6}, {6,1/6} Expected value: average outcome if experiment
repeated
5.3
661
561
461
361
261
161
][
XE
Population Return and Variance
Formally: M possible outcomes Outcome j is a value xj with probability j
Expected value of the random variable X is
The sample mean is the best estimate of the expected value
M
jjj xXE
1][
Population Return and Variance
After market analysis of Esso an analyst determines possible returns in 2010
The expected return on Esso stock using this data is
E[rEsso] = .2(2) + .3(6) + .3(9) + .2(12)
= 7.3
Return 2 6 9 12
Probability 0.2 0.3 0.3 0.2
Population Return and Variance
The expectation can be applied to functions of X For the dice example applied to X2
And to X3
167.15
3661
2561
1661
961
461
161
][ 2
XE
5.73
21661
12561
6461
2761
861
161
][ 3
XE
Population Return and Variance
The expected value of the square of the deviation from the mean is
This is the population variance
9167.2
5.366
15.35
6
15.34
6
1
5.336
15.32
6
15.31
6
1]][[
222
2222
XEXE
Modelling Returns
States of the world Provide a summary of the information about
future return on an asset A way of modelling the randomness in asset
returns Not intended as a practical description
Modelling Returns
Let there be M states of the world Return on an asset in state j is rj
Probability of state j occurring is j
Expected return on asset i is
M
jjj
MM
r
rrrE
1
11 ...][
Modelling Returns
Example: The temperature next year may be hot, warm or cold
The return on stock in a food production company in each state
If each states occurs with probability 1/3, the expected return on the stock is
State Hot Warm Cold
Return 10 12 18
333.131831
1231
1031
][ rE
Portfolio Expected Return
N assets M states of the world Return on asset i in state j is rij
Probability of state j occurring is j
Xi proportion of the portfolio in asset i Return on the portfolio in state j
N
iijiPj rXr
1
Portfolio Expected Return The expected return on the portfolio
Using returns on individual assets
Collecting terms this is
So
PMMPP rrrE ...11
N
iiMiM
N
iiiP rXrXrE
1111 ...
N
iiMMiiP rrXrE
111 ...
N
iiiP rXr
1
Portfolio Expected Return
Example: Portfolio of asset A (20%), asset B (80%)
Returns in the 5 possible states and probabilities are:
State 1 2 3 4 5
Probability 0.1 0.2 0.4 0.1 0.2
Return on A 2 6 9 1 2
Return on B 5 1 0 4 3
Portfolio Expected Return
For the two assets the expected returns are
For the portfolio the expected return is
7.132.041.004.012.051.0
5.522.011.094.062.021.0
B
A
r
r
46.27.18.05.52.0 Pr
Population Variance and Covariance
Population variance
The sample variance is an estimate of this Population covariance
The sample covariance is an estimate of this
22iii rErE
jjiiij rErrErE
Population Variance and Covariance
M states of the world, return in state j is rij
Probability of state j is j
Population variance is
Population standard deviation is
M
jiijji rr
1
22
M
jiijji rr
1
2
Population Variance and Covariance
Example: The table details returns in five possible states and the probabilities
The population variance is
State 1 2 3 4 5
Return 5 2 -1 6 3
Probability 0.1 0.2 0.4 0.1 0.2
9.7
332.361.314.322.351. 222222
Portfolio Variance
Two assets A and B Proportions XA and XB
Return on the portfolio rP
Mean return Portfolio variance
Pr
22PPP rErE
Population covariance between A and B is
For M states with probabilities j
M
jBBjAAjjAB rrrr
1
BBAAAB rErrErE
Portfolio Variance
Portfolio Variance
The portfolio return is
So
Collecting terms
BBAAP rXrXr BBAAP rXrXr
22BBAABBAAP rXrXrXrXE
22BBBAAAP rrXrrXE
Squaring
Separate the expectations
Hence
BBAABABBBAAAP rrrrXXrrXrrXE 222222
BBAABA
BBBAAAP
rrrrEXX
rrEXrrEX
2
22222
ABBABBAAP XXXX 222222
Portfolio Variance
Portfolio Variance
Example: Portfolio consisting of 1/3 asset A 2/3 asset B
The variances/covariance are
The portfolio variance is333.3,333.8,333.2 22 ABBA
148.9
333.332
31
2333.832
333.231 22
2
P
Correlation Coefficient
The correlation coefficient is defined by
Value satisfies
perfect positive correlation
BA
ABAB
11 AB
1AB
rA
rB
Correlation Coefficient
perfect negative correlation
Variance of the return of a portfolio
BAABBABBAAP XXXX 222222
rB
rA
1AB
Correlation Coefficient
Example: Portfolio consisting of 1/4 asset A 3/4 asset B
The variances/correlation are
The portfolio variance is
5.0,9,16 22 ABBA
8125.3
)3)(4)(5.0(4
3
4
129
4
316
4
122
2
P
General Formula
N assets, proportions Xi
Portfolio variance is
But so
N
i
N
ikkikkiiiP XXX
1 ,1
222
N
i
N
kikkiP XX
1 1
2
2iii
Effect of Diversification
Diversification: a means of reducing risk Consider holding N assets Proportions Xi = 1/N Variance of portfolio
N
i
N
ikkikiP NN1 ,1
22
22 11
Effect of Diversification
N terms in the first summation, N[ N-1] in the second
Gives
Define
Then
N
i
N
ikk
ikN
iiP NNN
NNN 1 11
22
11111
N
iia N1
22 1
N
i
N
ikk
ikab NN1 1 11
abaP N
N
N
11 22
Effect of Diversification
Let N tend to infinity (extreme diversification) Then
Hence
In a well-diversified portfolio only the covariance between assets counts for portfolio variance
abP 2
01 2
aN ababN
N
1