Risk Analysis & Modelling
Lecture 2: The Mean-Variance Model
The Mean-Variance Model
In this lecture we will introduce one of the most widespread models used to measure and analyse the nature of Market Risk
This model is the Nobel Prize winning Mean-Variance Model developed by Harry Markowitz in 1952
Although over 60 years old this model is still in widespread use and it is an ideal way to introduce the power of quantitative risk modelling
It is the basis of the SCR calculation in Solvency II
Measuring Risk
Last week we made the statement that risk involves uncertainty over some future outcome
Even when we are not certain about the final outcome we often have an idea of the range of outcomes that might occur
We can place reasonable limits on our uncertainty….
Limits on uncertainty
Current Position
Although we are unsure precisely which outcome will occur we can describe a range of outcomes over which our uncertainty arises
We
are
unce
rtai
n ab
out
the
prec
ise
outc
ome
Our thought experiment
Imagine you have £10,000 invested in a portfolio of 2 stocks£6000 is invested in stock A and £4000 in stock BYou are uncertain about how much your portfolio will be worth in one month, however, as an expert investor you feel you can estimate a range of outcomes that might occurYou will do this by writing down a range of possible values for your portfolio and the probability or likelihood of these outcomes
Limits on Uncertainty
£10,000
10%, £11,000 (+10%)
5%, £9,000 (-10%)
45%, £10,500 (+5%)
20%, £9,500 (-5%)
20%, £10,000 (0%)
1 Month Time Horizon
Probability, Portfolio Value
Histogram of Outcomes
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
-10% -5% 0% 5% 10%
Change In Portfolio Value
Pro
ba
bili
ty
Your Friends PortfolioYour friend has a portfolio of £5000
After seeing your risk chart he decides to create one for his own portfolio
Using his knowledge of the assets in his portfolio he produces a histogram showing the possible changes in portfolio value and their respective probabilities
You note that because you are both using the proportional change in your portfolio values you can compare charts (even though your portfolios are of different sizes)
Friends “Riskier” Portfolio
0%
5%
10%
15%
20%
25%
30%
35%
40%
-20% -15% -10% -5% 0% 5% 10% 15% 20%
Change In Portfolio Value
Pro
ba
bili
ty
Greater spread of outcomes means greater risk
Which Portfolio Earns More?
You want to know which portfolio earns more on average or has the greater Expected Return
)1.0*1.0()45.0*05.0()2.0*0()2.0*05.0()05.0*1.0()(1
PrE
%75.10175.0)(1
PrE
)16.0*05.0()04.0*1.0()03.0*15.0()02.0*2.0()(2
PrE
)07.0*2.0()06.0*15.0()09.0*1.0()37.0*05.0()16.0*0( %303.0)(
2PrE
On average we expect our portfolio to earn 1.75% over the 1 month, while we expect our friends portfolio to earn 3%
Which Portfolio Is Riskier?
While your friends portfolio has a greater expected return, we could argue that because the spread of outcomes on your friend’s portfolio is greater that the risk is greater
Although this can be ascertained visually it is useful to have a way of measuring this spread numerically
Statistics provides us with a simple means by which to do this: the Variance
Definition of VarianceThe expected return is defined as follows
2n
1iir
22r ))~(.(rp)))~(-r~E(( )r~Var(
irErE
n
1iir .rp )r~E(
irwhere E is the expectation operator (see appendix) and pr is the probability of that outcome
The Variance is defined as the expected squared difference
The Standard Deviation is simply the square root of the Variance
Var(r) )r~sd( r
Calculating VarianceThe variance on our portfolio is:
)2.0*)0175.00.0(()2.0*)0175.005.0(()05.0*)0175.01.0(()( 2221 PVar
002819.0)1.0*)0175.01.0(()45.0*)0175.005.0(( 22
The Variance of the return on our portfolio is 0.002819
By similar calculation we can see that the Variance on our friends portfolio is 0.00735
The Variance of the return does not have any obvious intuitive meaning (unlike the expected return), but it can be used to compare risk of the portfolios (larger Variance means higher Risk)
Calculating the Mean and Variance From Historical Observations
Normally we estimate the Mean and Variance from historical observations rather than hypothetical outcomes and their estimated probabilitiesTo estimate the average from a set of historical observations of size N we use the following formula (which is calculated using the AVERAGE formula in Excel):
To estimate the variance we can use the following formula** (which is calculated using the VARP formula in Excel):
N
iiXN
X1
.1
2
1
.1
)(
N
iiXN
XVAR
Mean Variance AbstractionThe key insight of the Mean-Variance Model is that an asset or portfolio can be described using two numbers: the Mean and Variance of Return
Although this does not capture the entire behaviour of a portfolio these 2 simple statistics directly relate to and quantify important aspects of a portfolio’s behaviour
They tell us how much profit we expect to make and how risky that profit is
Mean-Variance Abstraction
0%
5%
10%
15%
20%
25%
30%
35%
40%
-20% -15% -10% -5% 0% 5% 10% 15% 20%
Change In Portfolio Value
Pro
ba
bili
ty
Range or Distribution of Outcomes
Abstraction
Mean(Measure of Expected Profit)
Variance(Measure of Risk)
Deriving the Portfolio’s Risk
Returning to our investment portfolio, its Risk (or Variance) and Expected Return is derived from the risk of the two assets it contains
It is often useful to express the Risk and Return of the portfolio in terms of the assets it contains
It is easier to quantify the risks of individual assets rather than the risks of the portfolio
We will now derive some very important results by expressing the statistical behaviour of the portfolio interms of the assets it contains
Relationship between the Assets and the Portfolio
A B
P
The behaviour of the portfolio is derived from the behaviour of the assets it contains – a portfolio containing high return/risk assets will also have a high return/risk
Is it possible to derive the Mean and Variance of the Returns on the Portfolio (P) in terms of the
Mean and Variance of the Returns on the Assets it contains
(A and B)?
Mapping Assets to Portfolio
The known market value today of a portfolio (p) of 2 assets (a and b) is equal to the sum of their values:
ttt bap The unknown or random market value of the portfolio at some future time is equal to:
111
~~~ ttt bap
The ~ above the variables simply denotes that they are random or uncertain variables which can take on a range of values
The relationship between the assets value today and the future date can be expressed as:
tAt ara ).~1(~1
tbt brb ).~1(~
1
Where and denotes the random returns on asset A and asset B respectively
If we want to express the portfolio’s return in terms of the return on assets we have:
tBtAt brarp ).~1().~1(~1
Bt
tA
t
t
t
ttp r
p
br
p
a
p
ppr ~.~.
~~ 1
Ar~
Br~
We can rewrite this as:
BbAap rwrwr ~.~.~ Where wa and wb are the proportion of the portfolios total value invested in asset A and asset B respectively
These proportions are also called the investment weights
The usual restrictions on these investment weights is that they sum to 100% and cannot be negative
For example, a £100 portfolio with £30 invested in A and £70 invested in B has w A is equal to 30% and wb is 70%
Example Calculation
The meaning of this mapping equation is fairly simple.It simply states the obvious fact that the random behaviour of the return on the portfolio is directly related to the random behaviour of the assets in the portfolioFor example, if a portfolio contains 40% in stock A with a return of 5% and 60% in stock B with a return of 10% then the portfolio’s random return will be 8%:
%808.01.0*6.005.0*4.0
Review Question
A portfolio contains two assets A and B
30% of the portfolio is held in asset A and 70% in asset B
The return on Asset A is 10% and the return on Asset B is 20%
What is the return on the portfolio?
Implying the Portfolio’s Statistical Properties
We will now show it is possible to imply the mean and variance of return for the portfolio from the statistical properties of the assets contained in the portfolio
This will turn out to be extremely useful since it allows us to use simple mathematics to model and explore the nature of Risk in a Portfolio
Portfolio Expected Return
The expected return on the portfolio is equal to:
)~.~.()( BbAap rwrwErE
)~.()~.( BbAa rwErwE
)~(.)~(. BbAa rEwrEw
So the expected return on the portfolio is equal to a weighted average of the returns on the assets
Review Question
A portfolio contains two assets A and B50% of the portfolio is held in asset A and 50% in asset BOver the next year the Expected (Average) Return on Asset A is 20% and the average return on Asset B is expected to be 30%What is the average or expected return on the portfolio over the next year?
Review Question Answer
The formula we just derived states:
Applying the formula:
So on average this portfolio will earn 25% over the next year
)~(.)~(.)~( BbAaP rEwrEwrE
%25%20*%50%30*%50)~( PrE
Portfolio VarianceThe portfolios variance can be defined as:
2PPP ))r~E(-r~()r~Var( E
)~,~cov(..2)~var(.)~var(.)r~Var( 22P BABABBAA rrwwrwrw
Where cov is the covariance
The intuition is that the Variance of the Portfolio depends not just on how the assets it contains move in isolation (Variance) but also on how they move together or co-vary (measured by Covariance).
We note the relationship is quadratic with respect to the investment weights!
CovarianceCovariance measures the co-movement of 2 random variables and is defined as:
)))Y~
(Y~
)).(X~
(X~
(()~
,~
( , EEEYXCov yx A large positive covariance suggest that the random variables have a tendency to move together
A large negative covariance suggests that the random variables have a tendency to move in the opposite direction to each other
A covariance close to zero suggests that the random variables are independent in their movement about the mean
Covariance Diagram
Cov(X,Y)
Covariance measures the simultaneous behaviour of two random variables about their centre. Do they have a tendency to move together,
against each other or independently?
E(X) E(Y)X Y
CorrelationCorrelation is a normalized measure of CovarianceCorrelation must be between –1 and +1 due to the Cauchy-Schwarz inequality (see appendix)A strong positive correlation suggests that the two random variables move about their mean value in unisonA strong negative correlation suggests that the two random variable move in opposite directions about their mean valueCorrelation is defined as:
)(*)(
y)Cov(x,),(C , ysdxsdyxor yx
Calculating Covariance from Historical Observations
To estimate the covariance from a set of historical observations for two random variables X and Y of size N we use the following formula**:
In Excel this can be calculated using the COVAR formula
The related correlation statistic can be calculated in Excel using the CORREL formula
i
N
ii YX
NYXCOV ..
1),(
1
Quadratic Relationship
Let us look at the relationship between the risk on a portfolio of 2 uncorrelated assets (zero covariance) both with a variance of 0.1 the risk of a portfolio containing those 2 assetsOur equation tells us that the variance of the return on the portfolio is:
If we graph this relationship out for different combinations of asset A and asset B we see something interesting…
1.0*1.0*)r~Var( 22p BA ww
Risk for Various Portfolios
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 0.2 0.4 0.6 0.8 1
W(B)
Var
ian
ce P
ort
folio
Minimum Risk Containing 50% of A and 50% of B
100% in A and 0% in B 0% in A and 100% in B
Interpreting the CurveWhat this curve is telling us is that the risk (measured in terms of variance) of a combination of asset A and asset B is lower than the individual risks of the two assetsIn this case it is possible for a portfolio of 50% of A and 50% of B to have half the risk of the individual assetsThis result is not entirely intuitive, and at first sight might seem nonsensicalHowever this is an extremely important resultThe interpretation of this curvature in the risk of the portfolio is that it represents diversification
What if we have perfect correlation?
We can show that the quadratic relation reduces to a linear relationship when the correlation between assets become perfect (ie no diversification effect)
From the definition of correlation we see:
1)(*)(
y)Cov(x,),(C
ysdxsdyxor
)(*)(y)Cov(x, ysdxsd
Using our equation for 2 assets:
)~().~(..2)~(.)~(.)r~var( 2222P BABABBAA rsdrsdwwrsdwrsdw
)~,~cov(..2)~var(.)~var(.)r~var( 22P BABABBAA rrwwrwrw
2P ))~(.)~(.()r~var( BBAA rsdwrsdw
)~(.)~(.)r~sd( P BBAA rsdwrsdw
We can see in the absence of diversification (perfect correlation between assets) the relation between the risk of the portfolio and the investment weights becomes linearThis result explains the effect we saw when we artificially set the correlation between assets to 1 - making them perfectly correlated
DiversificationDiversification is extremely important to financial institutionsThe insurance industry is based on the concept of diversification or pooling of riskRisk is diversified between insurance policies – the risk of a portfolio of policies is less than the sum of the risks on the individual policiesOne of the powerful abilities of the Mean-Variance Framework is that it allows us to measure and analyse diversification in a simple quantitative frameworkDiversification arises when the assets (or liabilities) making up the portfolio move independentlyBy moving independently their movements can offset each other
Diversification and Solvency IIThe regulatory authorities place Solvency Capital requirements on Insurance Companies to protect policyholders from the risk of InsolvencyThe level of capital that an insurance company has to hold depends on the potential losses that might occur on its underwriting and investment portfoliosOne of the weaknesses of the traditional Actuarial Methods used in the Insurance Industry is they measure Risks in isolation rather than across the business as a whole (Enterprise Risk)Using the what is essentially the Mean-Variance Model Solvency II allows insurance companies to model the effect of diversification across different lines of business and across different asset classes using large Matrices…
Larger PortfoliosThe equations we developed can be used to derive the statistical properties of a portfolio of just 2 assets
For a modest portfolio of 30 assets the equation describing the relationship the portfolio variance and the assets contained would contain 465 variance and covariance terms!
Financial institutions hold portfolios containing thousands of assets
In order to perform these calculations on portfolios of this scale we need to use matrices
Matrix Multiplication Review Question
Multiply these 2 matrices together:
9 6
(1,2)
(2,2)
* =
•What is the size/dimension of the resulting matrix?•What are its element(s)
2 1
1 2
Matrix Multiplication Review Question
Multiply these 3 matrices together:
9 6
(1,2)
2 1
1 2
(2,2)
* =
•What is the size/dimension of the resulting matrix?•What are its element(s)
9
6*
(2,1)
Hint!
9 6
(1,2)
2 1
1 2
(2,2)
* = 24 21
(1,2)
9 6
(1,2)
2 1
1 2
(2,2)
* =9
6*
(2,1)
24 219
6*
(1,2)
(2,2)
Expected Return Vector The relationship between the expected return on a portfolio and the statistical properties of the assets it contains can be expressed as follows:
E(rA)
E(rB)
E(rP) = WT.E
Where W is the weight vector (matrix with one column) and E is the expected return vector
wA
wB
W = E =
We can see that this is equivalent to the earlier equation by multiplying out the matrices
E(rA)
E(rB)
wA wB * = wA* E(rA) + wB* E(rB)
Covariance Matrix
The relationship between the variance of a portfolio and the statistical properties of the assets it contains can be expressed as
Var(rP) = WT.C.W
Where W is the weight vector and C is the covariance matrix
wA
wB
W = C =Var(rA) Cov(rA,rB)
Cov(rA,rB) Var(rB)
Again, we can see that this is equivalent to the earlier equation by multiplying out the matrices
Var(rA) Cov(rA,rB)
Cov(rA,rB) Var(rB)wA wB *
wA
wB
*
wA*Var(rA) + wB*Cov(rA, rB) wA*Cov(rA, rB)+ wB*Var(rB) =
wA
wB
*
= wA*wA*Var(rA) + wB*wA*Cov(rA, rB) + wA*wB*Cov(rA, rB) + wB*wB*Var(rB)
The Covariance MatrixWe will be using the Covariance Matrix extensively – it is the main tool we will use to describe the nature of uncertainty or RiskYou can think of it as a specially ordered table containing the covariances and variances of random variablesIt contains information not only on how individual assets behave but also how they behave together (via their covariance)It is a semi-positive definite matrix (ie WT.C.W >=0 for every non-zero column vector W - no negative variances! This is equivalent to saying the Eigen Values of the covariance matrix should be greater than or equal to zero)
Var(RA) Cov(RA,RB)
Cov(RB,RA) Var(RB)
Var(RA) Cov(RA,RB) Cov(RA,RC)
Cov(RB,RA) Var(RB) Cov(RB,RC)
Cov(RC,RA) Cov(RC,RB) Var(RC)
Covariance Matrix for 2 Assets
Covariance Matrix for 3 Assets
IdenticalTerms
A
A B
B
A
A
B
B
C
C
Covariance of a random variable with itself is its variance
Correlation MatrixA matrix closely related to the Covariance Matrix is the Correlation Matrix
As you can probably deduce from the name it contains correlations rather than covariances!
Correlations are normalised covariances whose value ranges between -1 and 1
The definition of correlation is:
yx
yxyx ysdxsd
yxCov
.)().(
),( ,,
Where px,y is the correlation between x and y and x and y are the standard deviations of x and y.
1 pA,B
pB,A 1
1 pA,B pA,C
pB,A 1 pB,C
pC,A pC,B 1
Correlation Matrix for 2 Assets
Correlation Matrix for 3 Assets
A
A B
B
A
A
B
B
C
C
Identical Terms
Correlation of a random variable with
itself is 1
Real World DataWe will now look at how we might use the Mean Variance Framework to model the risks on a portfolio of 2 stocks: BSkyB and BootsThe statistics for the covariance matrix and expected return vector will be derived directly from a historical dataset – rather than guessed subjectively!In order to calculate the average, variance, covariance and correlation of return on the stocks we will have to use four Excel functions AVERAGE, VARP, COVAR and CORREL respectively.
What We Discover
We notice that the relationship between return and the investment in BSkyB is linearWe notice that the relationship between risk or standard deviation is non linearThe minimum risk portfolio contains a mixture of both BSkyB and Boots even though Boots is a riskier stockThis is due to diversificationThe diversification effect would disappear when we have perfect correlation between the 2 assets
Diversification Graphic
BSkyB
Boots
+ =
Because BSkyB and Boots are independent and do not have perfect correlation their movements can offset each other and reduce the portfolio variance
Portfolio
Implications of DiversificationThe effect of diversification is captured in the quadratic curvature we observe when we graphed the expected return of our portfolio against its variance or risk
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0.0018
0.0001 0.00012 0.00014 0.00016 0.00018 0.0002 0.00022
Portfolio Variance
Po
rtfo
lio
Exp
ecte
d R
etu
rn
Portfolios with the same Risk but different Expected Returns
The implication for investment decisions is that some portfolios are “inefficient” in the mean variance senseLooking at the previous graph we see there are two portfolios which have the same risk, but one has a greater returnAll points below the minimum risk portfolio are said to be inefficientAll points above the minimum risk portfolio are said to be efficient and make up the efficient frontier.The calculation of the efficient frontier is artificially simple when we only have 2 assets in the portfolioFor the case of 3 or more assets we need to use a constrained optimisation tool like Solver
What is SolverSolver is a mathematical programming tool built into Excel which can be used to solve constrained optimisation problemsThere are a large set of problems which involve inequality constraints such as: 0 < X < 1Optimisation problems involving such constraints cannot be solved using the traditional methods of differential calculusInternally Solver uses an algorithm based on Linear Programming called the Feasible Direction Search – but we do not need to know about that to use it!
Where do I Find Solver
It is found under Tools->Solver
If it is not available then you might need to tell Excel to load it by going to Tools->Addins and making sure the “Solver Add-in” is checked
Solver is programmed in VBA!
Using SolverTarget Cell SpecifiesThe output we wish tominimize/maximize
By changing cells specifies the inputs solver will alter to find the minimum or maximum
Constraints specify the conditions that mustBe met in a feasible solution Press to Solve!
Do wewish to Min or Max?
What we will ask solverWe will ask solver to find the lowest risk portfolio for our target level of return given the mean variance model This portfolio will by subject to the constraints that the sum of the weights or investment percentages will equal 1 and that all the investment weights will be positive (greater than or equal to zero)Solver will travel through the set of all possible portfolios that give a return equal to our target level using the feasible direction search to find the portfolio with the lowest riskIt will travel towards the lowest risk portfolio by changing the investment weights and examining the effect they have on risk
The Solver Model
WT.VA.W
WT.E[RA]
Variance Matrix: V Return Vector: R
Fixed Inputs
Weight Vector: W
Variable InputsPortfolioVarianceVar[RP]
PortfolioExp Return
E[RP]
OutputsModel
Solver AlgorithmAlter the weightsgiven the constraints
Check to see if risk can be reducedfurther
Calculating the Efficient Frontier using Solver
When calculating the efficient frontier using Excel’s Solver we must first calculate the range of expected returns over which it spans
The lowest point on the efficient frontier is the expected return related to the minimum standard deviation portfolio
The highest point on the efficient frontier is the expected return related to the maximum return portfolio
The efficient frontier lies between these 2 values
All these are, of course, restricted by any constraints we impose on the frontier
Bounding the Efficient FrontierE
xpec
ted
Ret
urn
Risk (Standard Deviation)
Return on minimum risk portfolio
Maximum Return
Efficient Frontier
Solver Step 1: Calculate Minimum Variance Portfolio
Target Cell is portfolio variance We want to minimize theTarget Cell
We tell Solver it can change the portfolioweights
This is subject to the constraints that each of the weights must be greater than or equal to 0 ($B$12:$B$13 >= 0) and the sum of the weights must equal 1 ($B$15 = 1)
Step 2: Calculate Maximum Return Portfolio
Target Cell is PortfolioReturn
We want to maximize theTarget Cell
We tell Solver it Can change Portfolioweights
This is subject to the constraints that each of the weights must be greater than or equal to 0 ($B$14:$B$15 >= 0) and the sum of the weights must equal 1 ($B$17 = 1)
Calculating Points on the Frontier
To graph out the efficient frontier we need to calculate point between these minimum and maximum returnsIn general, it is best to take evenly spaced returns between these two extremities and calculate the minimum risk portfolio that achieves the target return on the frontier:
Min Var[RP] subject to E[RP] = X where X isour return for which we wish to generate
the corresponding efficient frontier point.
Diagram of Calculation Method
Exp
ecte
d R
etur
n
Risk (Standard Deviation)
Evenly spaced returns used to calculate thepoints on the frontier
Points on theFrontierCalculated
Minimum Variances
Step 3: Calculate Points on the frontier for a set level of return
We want to minimize theTarget Cell
Target Cell is Portfolio variance
We tell Solver it can change Portfolioweights
This is subject to the constraints that each of the weights must be greater than or equal to 0 ($B$14:$B$15 >= 0) and the sum of the weights must equal 1 ($B$17 = 1)and that the return must be equal to some level ($F$14 = $G$42)
Appendix 1: Transformation Between Correlation and Covariance Matrix
The covariance matrix and correlation matrix are obviously linked – they both measure how things move togetherLet D be a square matrix with the standard deviations along the diagonal and zeros everywhere else (a diagonal matrix), let P be the correlation matrix and C be the respective covariance matrix. Then the following relationship is true:
D.P.D = C
2 by 2 Correlation to Covariance Matrix Example
1 pA,B
pA,B 1
A 0
0 B
A 0
0 B
* *
=A* A pA,B*A*B
pA,B*B*A B* B
Var(A) Cov(A,B)
Cov(A,B) Var(B)=
A2 A,B
A,B B2
=
Transformation between Covariance and Correlation Matrix
From our initial relationship we can state:
P= D-1.C .D-1
Where D-1 is the inverse of the standard deviation matrixD is a matrix with the standard deviations along the diagonal and zeros everywhere else (diagonal matrix)Because D is a diagonal matrix its inverse is simply the reciprocal of the elements along the diagonal.
2 by 2 Covariance to Correlation Matrix Example
1/ A 0
0 1/ B
1/ A 0
0 1/ B
* *
=A
2/(A* A) A,B/(A* B)
A,B/(A* B) B2/(B* B)
=
A2
B2
1 pA,B
pA,B 1
Appendix 2: Expectation Operators
The expectation operator measures the average value of some function of a random variableWe will prove the 2 properties of the expectation operator:
)~()~()~~( yExEyxE
)~(.)~.( xEaxaE
Firstly we note:
)..()~.(
Xx
ix
i
ixapxaE
Where X is the set of all possible values of xi and px is the probability of a specific outcome
Since a is a constant
)(...)~.( xEaxpaxaEXx
ix
i
i
For the second proof
).()~~(
Xx Yy
iiyx
i i
iiyxpyxE
Where pxny measures the probability of observing xi and yi together
Since the probabilities are disjoint or mutually exclusive
i
i
ii xYy
yx pp
i
i
ii yXx
yx pp
Xx Yy
iyxXx Yy
iyx
i i
ii
i i
iiypxpyxE ..)~~(
Yy
iyXx
ix
i
i
i
iypxpyxE ..)~~(
)~()~()~~( yExEyxE
This expectation can be re-expressed as
Which is equivalent to
This proof is for discrete random variables, we could produce a similar proof for continuous random variables using integration rather than summation
Appendix 3: Portfolio Variance
The portfolios variance can be defined as:
2vvv ))r~E(-r~()r~Var( E
2BBBBAAAA )))r~(.w r~.w())r~.E(w-r~.E((w E
2BBBB
2AAAA ))r~.E(w-r~.(w))r~.E(w-r~.(w EE
))r~.E(w-r~.(w)).r~.E(w-r~.(w.2 BBBBAAAAE
2BBB
2B
2AA
2 ))r~.E(w-r~(.w))r~E(-r~(. EwA
))r~E(-r~()).r~E(-r~((.w.w.2 BBAABA E
)~,~cov(..2)~var(.)~var(.)r~Var( 22v BABABBAA rrwwrwrw
Where cov is the covariance
The intuition is that the variance of the portfolio depends not just on how the assets it contains move together but also on how they move together or co-vary (measured by covariance).
We note the relationship is quadratic
Appendix 4: Correlation & the Cauchy-Schwarz Inequality
Cauchy-Schwarz inequality is an important result from linear algebra which states:
1*
1
2
1
2
2
1
N
ii
N
ii
N
iii
yx
yx
N
ii
N
ii
N
iii
yEyxEx
yEyxEx
1
22
1
1
))((*))((
))()).(((
12
We can write the correlation coefficient as
From Cauchy-Schwarz we can see
Implying that
11 We can also prove that in the case where p is -1 or 1 (perfect correlation) the random variables x and y are linearly dependent
Appendix 5: Eigen Values of Semi-Positive Definite Matrices
For a semi positive definite matrix C (N by N):
If C is semi positive definite it is possible to decompose it into
Where V is a vector whose columns are the Eigen Vectors of C and D is a diagonal matrix of its Eigen Values
0.. WCW T
TVDVC ..
Making this substitution we see:
X is an arbitrary matrix resulting from this calculation:
We can say:
Since D is a diagonal matrix of Eigen Values (Dii is the ith Eigen Value and Xi is the ith element of the vector X):
Since Xi2 must be greater than or equal to zero for real values of X
then we can see that this sum cannot be negative if the Eigen Values (Dii) are also greater than or equal to zero.This formula can also be used as an extremely efficient way of calculating a Quadratic Form
0)..(.).(...... WVDWVWVDVWWCW TTTTTT
WVX T .
XDXWCW TT ....
N
iiii
T DXWCW1
2...
Top Related