02 Portfolio Theory - KU · PDF fileFinancial Modeling I, js Returns and Risks of Portfolios 18 2. Portfolio Theory Objectives: Understanding of • Return and risk of assets

Financial Modeling I, js Returns and Risks of Portfolios 18

2. Portfolio Theory

Objectives:

Understanding of

• Return and risk of assets

• Return and risk of portfolios

• Efficient and optimal portfolios

• Market portfolio Concepts:

• Perfect and perfect inverse correlated asset returns • Covariance risk • Mean-variance curve • Mean variance frontier • Capital market line

Contents:

2.1 Returns and Risks of Assets

2.2 Returns and Risks of Portfolios of Two Risky Assets

2.3 Portfolios of Many Risky Assets

2.4 Portfolios of Risk-free and Risky Assets

2.5 Efficient and Optimal Portfolios


2.1 Return and Risk of Assets

2.1.1 Rate of Return on Risky Assets

By assumption there are two assets X and Y, with the present values X , Y , the future values X and Y and the random rate of returns

rX =

X − XX

, and rY =Y − Y

Y.

Future values include future prices and all (intra-period) payments, such as dividends or coupons. All statements and results apply to portfolios and their returns as well.

2.1.2 (Expected) Return and Risk of Assets

As a measure of the return of an asset X we use the expected value of the rate of return

µX ≡ E rX( ) .

The variance and the standard deviation of the rate of returns serve as a measure of the risk of the asset X:

σ XX ≡Var rX( ) = E rX − E rX( )⎡⎣ ⎤⎦

2{ }

σ X ≡ SD rX( ) = Var rX( )

These moments can be used for asset Y respectively.

2.1.3 Covariance of the Rate of Returns of Two Risky Assets

The covariance,

�

σ XY , measures how the asset returns rX and rY move with each other:

σ XY ≡ Cov rX , rY( ) = E rX − E rX( )⎡⎣ ⎤⎦ rY − E rY( )⎡⎣ ⎤⎦{ }

Obviously the covariance of a random variable with itself is equal to its variance. Instead of the covariance we use sometimes the correlation coefficient of the asset returns

κ XY ≡ CC rX , rY( ) ≡ σ XY

σ XσY.

The correlation coefficient is normalized to the interval

�

−1,1[ ]. Depending on the covariance of

the returns the correlation coefficient takes the following values:

1. κ XY = +1 , i.e. the returns of two assets are perfectly correlated.

2. κ XY = −1 , i.e. the returns are perfectly inverse correlated.

3. −1 <κ XY < 1 , is the realistic range of the correlation coefficient.


2.2 Returns and Risks of a Portfolio of Two Risky Assets

2.2.1 Specification of the Portfolios

We form a portfolio P consisting a percent of the asset X and 1-a percent of the asset Y. (All statements and results apply to portfolios of portfolios X and Y as well.) All results are based on the properties of linear transformations of random variables. (For details see Appendix B).

Characterization of the assets

We assume that X has a higher return than Y and that X is riskier than Y:

E rX( ) > E rY( ) > 0 (2.1)

Var rX( ) >Var rY( ) > 0 (2.2)

2.2.2 Return and risks of the portfolio

The rate of return of the portfolio P is a weighted average of the returns of the assets X and Y

rP = a rX + 1− a( ) rY .

Return of the portfolio

The (expected) return of the portfolio is defined the expectation of the rate of return of the portfolio or the weighted returns of the assets:

µP = E rP( ) = E a rX + 1− a( ) rY

⎡⎣ ⎤⎦ = aE rX( ) + 1− a( )E rY( ) = aµX + 1− a( )µY (2.3)

Risk of the portfolio

The risk of a portfolio is measured by the variance or the standard deviation of the rate of return of the portfolios:

σ PP = E rP − E rP( )⎡⎣ ⎤⎦

2{ } = E a rX + 1− a( ) rY − E a rX + 1− a( ) rY( )⎡⎣

⎤⎦

2⎧⎨⎩

⎫⎬⎭

= E a2 rX − E rX( )⎡⎣ ⎤⎦

2+ 1− a( )2

rY − E rY( )⎡⎣ ⎤⎦2{ +2a 1− a( ) rX − E rX( )⎡⎣ ⎤⎦ rY − E rY( )⎡⎣ ⎤⎦}

= a2σ XX + 1− a( )2

σYY + 2a 1− a( )σ XY

= a2σ XX + 1− a( )2

σYY + 2a 1− a( )κ XYσ XσY (2.4)

Obviously the risk of the portfolio return is non-linear in the portfolio weights. Moreover, it depends on the second order moments of the asset returns.

By definition, the standard deviation of the portfolio return amounts to


σ P = a2σ XX + 1− a( )2

σYY + 2a 1− a( )κ XYσ XσY . (2.5)

2.2.3 Impact of the composition on the return and the risk of a portfolio

We investigate how an increase of the weight a influences the return and the risk of the portfolio.

Impact of the portfolio composition on the portfolio return

dµP

da= µX − µY > 0 (2.6)

The return increases proportionally to a:

Fig 2.1: Return of a portfolio of two assets

For

�

a < 0 , the return of the portfolio is smaller than

�

µY , for

�

a > 1, greater than

�

µX .

Impact of the portfolio composition on the portfolio risk

dσ PP

da= 2aσ XX − 2 1− a( )σYY + 2 1− 2a( )κ XYσ XσY (2.7)

or

dσ P

da= 1

2a2σ XX + 1− a( )2

σYY + 2a 1− a( )κ XYσ XσY⎡⎣⎢

⎤⎦⎥−1 2

⋅ 2aσ XX − 2 1− a( )σYY + 2 1− 2a( )κ XYσ XσY⎡⎣ ⎤⎦

(2.8)

Obviously, the sign of the change of the risk of the portfolio depends on the covariance or the correlation coefficient of the asset returns.

Sometimes it is useful to know the risk minimal portfolio. It can be derived from the first order condition

dσ PP

da= 2aσ XX − 2 1− a( )σYY + 2 1− 2a( )κ XYσ XσY = 0 , (2.9)

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Since the radicand in the first line of (2.8) is not zero, the first order condition applies to the second line of (2.8). From (2.9) follows the risk minimizing portfolio weight a* is

a* =

σYY −κ XYσ XσY

σ XX +σYY − 2κ XYσ XσY

=σY σY −κ XYσ X( )


.

(2.10)

It is obvious that a* depends on the covariance or the correlation coefficient of the asset returns. In order to discuss the shape of the curves (2.7) 0r (2.8) we distinguish three cases:

Case A: κ XY = 1 , i.e. perfectly correlated assets

Case B: κ XY = −1 , i.e. perfectly inverse correlated assets

Case C: −1 <κ XY < 1 , normal case

For didactical reasons we start with both extreme cases.

2.2.3.1 Case A: κ XY = 1

As the portfolio return is independent from the correlation of the assets return equations (2.3) and (2.6) remain unchanged. Only the risk (variance and standard deviation) of the portfolio is affected by the correlation coefficient.


For perfectly correlated assets the portfolio variance in equation (2.4) reduces to

σ PP = a2σ XX + 1− a( )2

σYY + 2a 1− a( )σ XσY (2.11)

�

= aσ X + 1− a( )σY[ ]2.

As long as we a ≥ 0 assume the expression in the squared bracket is positive, we would not be short in X and thus the portfolio would be located North East of Y. In this case the standard deviation would be

σ P = aσ X + 1− a( )σY > 0 . In case of a sufficiently small a < 0 ,

aσ X + 1− a( )σY may become negative, and therefore, the standard deviation of the portfolio is

to be defined more generally as

σ P = aσ X + 1− a( )σY ≥ 0 . (2.12)

From (2.12) we can derive the impact of the composition of the portfolio on its risk as

dσ P

da= ± σ X −σY( ) ≥ ≤( ) 0 ⇔ a ≥ ≤( ) − σY

σ X −σY.

(2.13)

The derivative is a constant with a alternating sign since it is σ X − σY > 0 if the weight of the

asset X is greater than the critical value, a > −σY σ X − σY( ) , and it is

− σ X − σY( ) < 0 if

a < −σY σ X − σY( ) . Obviously the critical value is itself negative.


Slope of the Mean Variance Curve

If we divide equation (2.6) by equation (2.13) we receive the slope of the so-called Mean Variance Curve (MVC). This curve is also referred to as µ −σ curve. The slope of this curve tells

us how the Mean of the portfolio return changes if we change the risk of the portfolio by a small change of a:

dµP

dσ P

=dµP / dadσ P / da

= ±µX − µY

σ X −σY

≥ ≤( ) 0 ⇔ a ≥ ≤( )− σY

σ X −σY.

(2.14)

From equation (2.6) we know that the nominator of (2.14) is positive and constant. Thus, the slope of the MVC must be a constant with an alternating sign. We can deduce from (2.13) and (2.14)

sign

dµP

dσ P

⎛

⎝⎜⎞

⎠⎟= sign

dσ P

da⎛

⎝⎜⎞

⎠⎟. (2.15)

Taking this into account we can distinguish different slopes in three different parts of the MVC:

1. a > −

σY

σ X −σY

⇒ σ P = aσ X + 1− a( )σY > 0 ⇒ dµP

dσ P

=µX − µY

σ X −σY

> 0

2. a = −

σY

σ X −σY

⇔ σ P = aσ X + 1− a( )σY = 0

(2.16)

3. a < −

σY

σ X −σY

< 0 ⇒ σ P = − aσ X + 1− a( )σY( ) > 0 ⇒ dµP

dσ P

= −µX − µY

σ X −σY

< 0

The standard deviation and the MVC of this type are illustrated in the following graphs:

Fig 2.2a: STDV of a portfolio of perfectly corr. assets Fig 2.2b: MVC of a portfolio. of perfectly corr. assets

In the first range a > −σY σ X − σY( ) , the standard deviation and the MVC are increasing

straight lines, and in the third range a < −σY σ X − σY( ) decreasing straight lines.

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The critical value a = −σY σ X − σY( ) creates the risk minimal portfolio, where the derivative of

the standard deviation curve and the MVC are not defined. We can confirm this by the first order condition (2.10) for a risk minimizing portfolio weight a* if we respect κ XY = 1 :

a* =σYY −κ XYσ XσY


=σYY −σ XσY

σ XX +σYY − 2σ XσY

=−σY σ X −σY( )

σ X −σY( )2 = −σY

σ X −σY

< 0 . (2.17)

The risk of this portfolio equals zero since

σ P = −

σY

σ X −σY

σ X + 1+σY

σ X −σY

⎛

⎝⎜⎞

⎠⎟σY = −

σYσ X

σ X −σY

+σ X −σY +σY( )σY

σ X −σY

= 0 . (2.18)

We should keep in mind that the risk minimal portfolio is reached for an a* < 0 , i.e. a short position in X. Such a portfolio generates negative (expected) returns. It is in any case inefficient.

2.2.3.2 Case B: κ XY = −1

Equations (2.3) and (2.6) remain unchanged. Only the portfolio risk is influenced by the correlation coefficient.


For perfectly inverse correlated assets return the variance of the portfolio returns in equation (2.4) simplifies to

σ PP = a2σ XX + 1− a( )2

σYY − 2a 1− a( )σ XσY (2.19)

�

= aσ X − 1− a( )σY[ ]2

.

As the expression in the squared bracket may be negative, the standard deviation is defined as

σ P = aσ X − 1− a( )σY ≥ 0 . (2.20)

From (2.20) we can derive the impact of the portfolio composition on the risk of the portfolio as

dσ P

da= ± σ X +σY( ) ≥ ≤( ) 0 ⇔ a ≥ ≤( ) σY

σ X +σY.

(2.21)

Again the derivative is constant with alternating sign. If a > σY σ X +σY( ) the derivative is

positive, if a < σY σ X +σY( ) the derivative is negative. At the critical value

a = σY σ X +σY( )

the derivative is undefined.


Dividing (2.6) by (2.21) yields the slope of the MVC:


dµP

dσ P

=dµP / dadσ P / da

= ±µX − µY

σ X +σY

≥ ≤( ) 0 ⇔ a ≥ ≤( ) σY

σ X +σY (2.22)

Again the nominator is positive. Thus the sign of the slope depends on the sign of (2.21). From (2.21) and (2.22) follows

sign

dµP

dσ P

⎛

⎝⎜⎞

⎠⎟= sign

dσ P

da⎛

⎝⎜⎞

⎠⎟.

(2.23)

Moreover, the nominator and the denominator in (2.22) are constant. With these insights we can distinguish three different ranges of the MVC:

1. a >

σY

σ X +σY

⇒ σ P = aσ X − 1− a( )σY > 0 ⇒ dµP

dσ P

=µX − µY

σ X +σY

> 0

2. a =

σY

σ X +σY

⇔ σ P = aσ X − 1− a( )σY = 0

(2.24)

3. a <

σY

σ X +σY

⇒ σ P = − aσ X − 1− a( )σY( ) > 0 ⇒ dµP

dσ P

= −µX − µY

σ X +σY

< 0

The standard deviation and the MVC are illustrated by the subsequent graphs:

Fig 2.3a: STDV of a portfolio of perf. inverse corr. assets Fig 2.3b: MVC of a portfolio of perf. inverse corr. assets

In the first range a > σY σ X +σY( ) the standard deviation and MVC are increasing straight

lines, and in the third range a < σY σ X +σY( ) a decreasing straight lines. For the critical value

a = σY σ X +σY( ) we receive the risk minimal portfolio. Again we can confirm this if we insert

κ XY = −1 into the first order condition (2.10) for the risk minimal portfolio a*:

a* =σYY −κ XYσ XσY


=σYY +σ XσY

σ XX +σYY + 2σ XσY

=σY σ X +σY( )σ X +σY( )2 =

σY

σ X +σY

> 0 . (2.25)

The risk of this portfolio equals zero since

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σ P =

σY

σ X +σY

σ X − 1−σY

σ X +σY

⎛

⎝⎜⎞

⎠⎟σY =

σYσ X

σ X +σY

−σ X +σY −σY( )σY

σ X +σY

= 0 . (2.26)

The risk minimal a* lies in the interval 0,1( ) . For a = 1 any (small) increase of a leads to a

proportional increase of the expected return and the risk as well. For a = 0 a small increase of a causes an increase of the expected return and a decrease of the risk. Obviously, and we will discuss this later in detail this portfolio is inefficient.

2.2.3.3 Case C: −1 <κ XY < 1

Equation (2.6) remains unchanged. However, the analysis of the influence of the portfolio composition is more complex and complicated than the two previous cases. It is recommended to start with a local analysis of the standard deviation and changes of the portfolio composition on the standard deviation at a = 1 and a = 0 .

For

�

a = 1 the portfolio consists only of X, and therefore the standard deviation of the portfolio equals the standard deviation of X. In order to determine the impact of a change of the portfolio composition on the standard deviation we have to evaluate (2.8) for a = 1 :

dσ P

da a=1 =1

2σ X

2σ XX − 2κ XYσ XσY⎡⎣ ⎤⎦ = σ X −κ XYσY⎡⎣ ⎤⎦ > 0 (2.27)

The risk of the portfolio increases with an increasing a.

For

�

a = 0 the portfolio consists only of Y, and therefore the standard deviation equals the standard deviation of Y. Moreover, we evaluate (2.8) for

�

a = 0 in order to investigate the local impact of a on the standard deviation.

dσ P

da a=0 =1

2σY

−2σYY + 2κ XYσ XσY⎡⎣ ⎤⎦ = − σY −κ XYσ X⎡⎣ ⎤⎦ (2.28)

The sign of the reaction of the standard deviation on an increase of a depends on the size of the two terms in the squared brackets. It depends on the size of the standard deviations of X and Y and the correlation of the returns of the assets.

If the correlation coefficient is sufficiently close to 1, the difference in the brackets is negative, and thus the derivative in (2.8) is positive. If on the contrary, the correlation coefficient is sufficiently low the difference in the bracket is positive, and the derivative is negative.

However, it turns out that we can determine these relations more precisely. Therefore, it is convenient to search for the portfolio composition a, which minimizes the standard deviation. The risk minimizing a must satisfy the first order condition (2.9)

dσ P

da= 2aσ XX − 2 1− a( )σYY + 2 1− 2a( )κ XYσ XσY⎡⎣ ⎤⎦ = 0 .

(2.9)


From (2.10) we have the value of the risk minimizing a* as

a* =

σYY −κ XYσ XσY


=σY σY −κ XYσ X( )

σ XX +σYY − 2κ XYσ XσY.

(2.10)

The denominator of the fraction is positive because of

σ XX +σYY − 2κ XYσ XσY >σ XX +σYY − 2σ XσY = σ X −σY( )2

> 0 . (2.29)

The sign of the risk minimizing a* is the same as the sign of the nominator. A comparison of the equations (2.28) and (2.10) shows

sign

dσ P

da a=0

⎛

⎝⎜⎞

⎠⎟= −sign a*( ) . (2.30)

Thus we can conclude from these equations:

1. σY<σ X κ XY ⇒

dσ P

da a=0 > 0 ∧ a* < 0

2. σY= σ X κ XY ⇒

dσ P

da a=0 = 0 ∧ a* = 0 (2.31)

3. σY>σ X κ XY ⇒

dσ P

da a=0 < 0 ∧ 0 < a* < 1

Three possible shapes of the standard deviation are illustrated in the subsequent graphs:

Fig 2.4: Three possible shapes of the standard deviation curves of portfolios of two assets


Dividing equation (2.6) by (2.8) results the slope of the MVC as

dµP

dσ P

=dµP dadσ P da

=µX − µY⎡⎣ ⎤⎦ a2σ XX + 1− a( )2

σYY + 2a 1− a( )κ XYσ XσY⎡⎣⎢

⎤⎦⎥

1 2

aσ XX − 1− a( )σYY + 1− 2a( )κ XYσ XσY⎡⎣ ⎤⎦

.

(2.32)

The nominator in (2.32) is positive since both factors are positive. Thus, the sign of the derivative (slope of the MVC) in (2.32) is equal to the sign of the denominator.

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From the curve sketched in (2.31) we can conclude derive three possible shapes of the MVC:

1. σY <κ XYσ X The MVC has a positive finite slope for all

�

a ≥ 0 .

2. σY =κ XYσ X The MVC is vertical at

�

a = 0 . The slope is positive for

�

a > 0,

and negative for

�

a < 0.

3. σY >κ XYσ X The MVC is increasing in the neighborhood of

�

a = 1 and decreasing

in the neighborhood of

�

a = 0. Three possible shapes of the MVC are presented in the graphs below:

Fig 2.5: Three possible shapes of the MVCs of portfolios of two assets

2.3 Portfolios of Many Risky Assets

We generalize the analysis to multiple assets and the market as a whole. This can be done easily by the utilization of simple vector representation of the mean and the variance.

2.3.1 Return and Risk of the Portfolio

In a matrix presentation the return (2.3) and the risk (2.4) of a portfolio of two risky assets can be written as:

µP = a 1− a⎡⎣ ⎤⎦

µX

µY

⎡

⎣⎢⎢

⎤

⎦⎥⎥

(2.33)

σ PP = a 1− a⎡⎣ ⎤⎦

σ XX σYX

σ XY σYY

⎡

⎣⎢⎢

⎤

⎦⎥⎥

a1− a⎡

⎣⎢

⎤

⎦⎥

(2.34)

Following this idea we can calculate easily the return and risk of a portfolio of

�

n = 1,… ,N risky assets. In order to do this we have to introduce some useful notations. The returns of assets can

be presented by a N-dimensional vector of random variables, r = r1 r2…rN⎡⎣ ⎤⎦

T, the expected

returns by the vector, µ = µ1 µ2…µN⎡⎣ ⎤⎦

T, and a portfolio X by the vector

x = x1 x2…xN⎡⎣ ⎤⎦T

.

Usually portfolios are defined to satisfy xT e = 1 .

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The pairwise covariance of the returns of different assets can be collected in a covariance matrix

�

∑ . The covariance of a random variable with itself equals the variance of the random variable. Moreover, we should keep in mind that the covariance matrix is symmetric.

∑ =

σ11 σ12 σ1N

σ 21 σ 22 σ 2 N

σ N 1 σ N 2 σ NN

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

Return and Risk of a Portfolio

The rate of return of the portfolio can expressed as:

rX ≡ xT r (2.35)

The return of a portfolio X, denoted as µX and its risk, denoted as σ XX can be expressed as:

µX ≡ E rX( ) = xTµ (2.36)

σ XX ≡Var rX( ) = xT ∑ x (2.37)

Covariance of the Returns of Two Portfolios X and Y

The covariance of the returns of the portfolios can be expressed as

σ XY ≡ Cov rX , rY( ) = xT ∑ y = yT ∑ x . (2.38)

2.3.2 Concept of Mean Variance Frontier

The Mean Variance Frontier (MVF) is the envelope of all feasible risk return combinations available by portfolios of elementary assets. The MVF is thus the convex hull of all possible portfolios and all possible MVCs.

We will present two different ways to construct the MVF. The first one uses some tangential properties of the MVF. This method is easy to apply. The disadvantage is that the method can’t be applied straightforward in case of short sale restrictions. The second one uses quadratic programming. This method is easy to understand, but it needs some calculation effort.

A third method uses an orthogonal decomposition. We will not discuss this method in this lecture since it needs some more mathematical exposition.

2.3.3 Construction via the Tangential Property

The MVF will be constructed in two steps. At first we construct two elements of the MVF. At second we will span the entire MVF from these two points.


2.3.3.1 Construction of Elements of the MVF

We define

�

cX as an arbitrary constant associated with an envelope portfolio X. Moreover we define e as a N-dimensional unit vector. The vector

�

zX is the solution of the equation

µ − cX e = ∑ zX . (2.39)

As we will see the vector

�

zX has already the composition of a MVF portfolio. However, it is not a well-defined portfolio in the sense that the portfolio weights add up to one. Therefore, it has to be normalized.

Theorem 1: Construction of an element of the MVF

A MVF portfolio has to satisfy both of the following conditions:

i) zX = ∑−1 µ − cX e( ) (2.40)

ii)

xi =zXi

zXjj=1

N

∑=

zXi

zXT e

(2.41)

(2.40) we receive by pre-multiplying (2.39) by the inverse covariance matrix. The product eT zX

in (2.41) sums all elements of the vector zX . Therefore x is a normalization of

�

zX to one.

Proof:

• A MVF portfolio has to lie on a tangent line to MVF running through a corresponding point

cX of the ordinate.

• Any tangential portfolio maximizes or minimizes the risk premium per unit of risk of the portfolio. In other words λ has to attain an extreme value (maximum or minimum):

λ =

xT µ − cX e( )σ XX

=xT µ − cX e( )

xT ∑ x (2.42)

The first order condition for such an optimal portfolio is

µ − cX e( )− 2∑ xλ = 0 . (2.43)

In order to understand this we form in a first step the first order condition of λ with respect to an arbitrary portfolio weight xh :

∂λ∂xh

=xT ∑ x ∂ xT µ − cX e( )⎡⎣ ⎤⎦ ∂xh − xT µ − cX e( )⎡⎣ ⎤⎦∂ xT ∑ x( ) ∂xh

xT ∑ x( )2 = 0 . (2.44)

As the denominator is not (and must not be) equal to zero the nominator has to be equal to zero and therefore the first order condition (2.44) can be expressed as


∂ xT µ − cX e( )⎡⎣ ⎤⎦ ∂xh −

xT µ − cX e( )⎡⎣ ⎤⎦xT ∑ x

∂ xT ∑ x( ) ∂xh= 0. (2.45)

If we substitute the fraction in the second expression by λ and express the matrix products in a more detailed way the first order condition can be rewritten as:

∂ x1 µ1 − cX( ) + x2 µ2 − cX( )…+ xh µh − cX( ) +…+ xN µN − cX( )⎡⎣ ⎤⎦ ∂xh

−λ∂ x12σ11 + x2

2σ 22 +…+ xh2σ hh +…+ xN

2σ NN +⎡⎣ +2x1x2σ12 +…+ 2x1xhσ1h +…+ 2x2xhσ 2h +…+ 2xN xhσ Nh +…⎤⎦ ∂xh = 0

(2.46)

The derivation in (2.45) results the following first order condition for an arbitrary optimal portfolio weight xh :

µh − cX( )− 2∑h xλ = 0 (2.47)

In (2.47) µh and

�

∑h represent the lines h of µ and

�

∑ . If we apply this procedure to all assets

of the portfolio we receive a first order conditions for an optimal portfolio as in (2.43).

• The second order condition corresponding to (2.47) is

∂ µh − cX( )− 2∑h xλ⎡⎣ ⎤⎦ ∂ xh = −2σ hhλ (2.48)

If λ > 0 , the extremum is a maximum, if λ < 0 , it is a minimum. λ > 0 , implies

xT µ − cX e( ) > 0 (ascending tangent), and λ < 0 , implies

xT µ − cX e( ) < 0 (descending

tangent).

• Substituting 2λx by

�

zX and normalizing

�

zX to one shows that any MVF portfolio that maximizes or minimizes

�

λ must fulfill the conditions (2.40) and (2.41).

2.3.3.2 Spanning of the MVF

All elements of the MVF can be constructed by the method of the above paragraph. Moreover, the entire MVF can be spanned as a convex combination of two arbitrary elements of the MVF.

Theorem 2: Spanning of the MVF

The MVF can be spanned by a convex combination of two arbitrary elements of the MVF.

Proof:

• If the vectors x and y contain the weights of two MVF portfolios X and Y, it follows from theorem 1 that there are vectors

�

zX and

�

zY and (vectors of) constants

�

cX and

�

cY that satisfy the conditions (2.49a) - (2.50b):

x =

zX

zXT e

(2.49a)


y =

zY

zYT e

(2.49b)

µ − cX e( )− ∑ zX = 0 (2.50a)

µ − cY e( )− ∑ zY = 0 (2.50b)

• For any real number a, there is a constant

�

cP = acX + 1− a( )cY and a portfolio P with the

composition

�

zP = azX + 1− a( )zY such that a is a solution of the equation (2.51):

µ − cPe − ∑ zP = µ − acx + 1− a( )cy

⎡⎣ ⎤⎦e − ∑ azx + 1− a( ) zy⎡⎣ ⎤⎦ = 0 (2.51)

• From this follows theorem 2.

The computation of the MVF by the spanning method is superior to the method we apply in the next subsection 2.3.4 because solving of linear equations is much easier than solving quadratic programs. However, the spanning method demands the absence of short sales restrictions. Moreover, the latter gives some additional insights into portfolio theory.

2.3.4 Solution of a Quadratic Program

This method can also be applied in case of short sales. However, in that case we have to consider the non-negativity constraint x ≥ 0 . In a first step we construct a single portfolio of the MVF, and in a second step we construct the overall risk minimum portfolio, and in a third step we develop the MVF.

2.3.4.1 Construction of a MVF Portfolio

Any portfolio of the MVF can be found as the solution of the following quadratic program:

Min

x Var rX( ) s. t.

E rX( ) = µX for all µX ≥ 0 I

Fig 2.6: The MVF of many risky assets

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The program can be written in the following way:

Min

x xT ∑ x

s.t. 1 = xT e (2.52)

µX = xTµ (2.53)

In order to calculate an element of the MVF we minimize the variance of a portfolio which assures a given expected portfolio return µX .

In order to find a solution to the optimization problem we optimize the Lagrange function:

L

x ,λ ,ξ = xT ∑ x + λ 1− xT e( ) + ξ µX − xTµ( )

The solution has to satisfy the following first order conditions:

Lx = 2∑ x − λe− ξµ = 0 (2.54a)

Lλ = 1− xT e = 0 (2.54b)

Lξ = µX − xTµ = 0 (2.54c)

If we pre-multiply (2.54a) by the inverse of the covariance matrix we receive

2∑−1 ∑ x = λ∑−1 e+ ξ∑−1 µ , (2.54’)

or

x = λ

2∑−1 e+ ξ

2∑−1 µ . (2.54’’)

Inserting (2.54’’) into (2.54b) and (2.54c) results

1 = xT e = λ

2eT ∑−1 e+ ξ

2µT ∑−1 e (2.55a)

µX = xTµ = λ

2eT ∑−1 µ + ξ

2µT ∑−1 µ (2.55b)

Equations (2.55a) and (2.55b) form a simultaneous equation system in the variables λ 2 and

ξ 2 that can expressed by the following matrix equation:

eT ∑−1 e µT ∑−1 eeT ∑−1 µ µT ∑−1 µ⎡

⎣⎢

⎤

⎦⎥λ 2ξ 2⎡

⎣⎢

⎤

⎦⎥ =

1µX

⎡

⎣⎢

⎤

⎦⎥ (2.56)

Applying Cramer’s Rule yields the following solutions:

λ2=

µT ∑−1 µ − µXµT ∑−1 e

eT ∑−1 eµT ∑−1 µ − µT ∑−1 e( )2 (2.57a)


ξ2=

µX eT ∑−1 e − eT ∑−1 µ

eT ∑−1 eµT ∑−1 µ − µT ∑−1 e( )2 (2.57b)

Reinserting (2.57a) and (2.57b) into (2.54’’) yields the optimal portfolio:

x =µT ∑−1 µ − µXµ

T ∑−1 e

eT ∑−1 eµT ∑−1 µ − µT ∑−1 e( )2 ∑−1 e +

µX eT ∑−1 e − eT ∑−1 µ

eT ∑−1 eµT ∑−1 µ − µT ∑−1 e( )2 ∑−1 µ (2.58)

Note that this is the general formula of MVF portfolios depending on the level of the a priori given portfolio return µX . As all MVF portfolios contains the expressions ∑

−1 e and ∑−1 µ they

can be characterized as

x = φ∑−1 e+ϕ∑−1 µ , (2.59)

where φ and ϕ represent the fractions in the right hand side of (2.58). Envelope portfolios differ because of different values of φ and ϕ .

The corresponding minimal risk of an envelope portfolio can be calculated as

σ XX = xT ∑ x = φeT ∑−1 +ϕµT ∑−1⎡⎣ ⎤⎦∑ φ∑−1 e+ϕ∑−1 µ⎡⎣ ⎤⎦ = φeT +ϕµT⎡⎣ ⎤⎦ φ∑−1 e+ϕ∑−1 µ⎡⎣ ⎤⎦ = φ 2eT ∑−1 e+ 2φϕµT ∑−1 e+ϕ 2µT ∑−1 µ

(2.60)

It is recommended to simplify these expressions using some conventions. Therefore, we define

a = µT ∑−1 µ , b = µT ∑−1 e , and c = eT ∑−1 e . Thus, a − bµX represents the nominator of φ , and

cµX − b the nominator of ϕ .

Equation (2.60) can be rewritten as

σ XX =a2 − 2abµX + b2µX

2

ac − b2( )2 c +2acµX − 2bcµX

2 − 2ab+ 2b2µX

ac − b2( )2 b+c2µX

2 − 2bcµX + b2

ac − b2( )2 a

=a2c − b2a − 2abcµX + 2b3µX + ac2µX

2 − b2cµX2

ac − b2( )2 =ac − b2( ) a − 2bµX + cµX

2( )ac − b2( )2 (2.61)

=

a − 2bµX + cµX2

ac − b2 .

2.3.4.2 The Risk Minimal Portfolio

As in the graphical presentation we will indicate the risk minimal portfolio with D . The risk minimal portfolio is helpful for the construction of an explicit form of the MVF.

The risk minimal portfolio can be calculated as the solution of the following optimization problem:


Min

x xT ∑ x

s.t. 1 = xT e (2.62)

In order to find a single solution to the optimization problem we optimize the Lagrange function:

Lx ,λ

= xT ∑ x + λ 1− xT e( )

The solution has to satisfy the following first order conditions:

Lx = 2∑ x − λe = 0 (2.63a)

Lλ = 1− xT e = 0 (2.63b)

If we pre-multiply (2.51a) with the inverse covariance matrix we can receive

2∑−1 ∑ x = λ∑−1 e , (2.63a’)

and

x = λ

2∑−1 e . (2.63a’’)

Inserting (2.63a’’) into (2.63b) results

1 = xT e = λ

2eT ∑−1 e , (2.64)

and

λ2= 1

eT ∑−1 e= 1

c. (2.64’)

Reinserting (2.64’) into (2.63a’’) yields

xD = 1

eT ∑−1 e∑−1 e = 1

c∑−1 e (2.65)

From (2.65) we can calculate the return of the risk minimal portfolio as

µD = xTµ = µT x = µT ∑−1 e

eT ∑−1 e= b

c, (2.66)

and its risk as

σ DD = xT ∑ x = ∑−1 eeT ∑−1 e

⎛

⎝⎜⎞

⎠⎟

T

∑ ∑−1 eeT ∑−1 e

⎛

⎝⎜⎞

⎠⎟= eT ∑−1 ∑∑−1 e

eT ∑−1 e( )2 = 1eT ∑−1 e

= 1c

. (2.67)

2.3.4.3 Construction of the MVF

We can construct an explicit form of the MVF. In order to do so we rewrite (2.61) in the following way:


σ XX =

a − 2bµX + cµX2

ac − b2 =c µX

2 − 2 bcµX + b2

c2 −b2

c2 +ac

⎛⎝⎜

⎞⎠⎟

ac − b2 =c µX − b

c⎛⎝⎜

⎞⎠⎟

2

+ ac − b2

cac − b2

= c

ac − b2 µX − bc

⎛⎝⎜

⎞⎠⎟

2

+ 1c

(2.68)

Obviously, the risk of any element of the MVF can be determined as

σ XX =α µX − µD( )2

+σ DD , (2.69)

where the constant α is defined as α = c ac − b2( ) .

We can derive an explicit form of the MVF from (2.68):

µX = b

c± 1

ασ XX − 1

c⎛⎝⎜

⎞⎠⎟= b

c± ac − b2

cσ XX − 1

c⎛⎝⎜

⎞⎠⎟

(2.70)

Obviously the MVF is not an ordinary function, but a correspondence. The sign before the square root causes the ascending and descending branch of the MVF. In a more explicit way we can express (2.69) and (2.70) as

σ XX =eT ∑−1 e

µT ∑−1 µeT ∑−1 e − µT ∑−1 e( )2 µX −µT ∑−1 eeT ∑−1 e

⎛

⎝⎜⎞

⎠⎟

2

+1

eT ∑−1 e (2.69’)

µX = µT ∑−1 eeT ∑−1 e

±µT ∑−1 µeT ∑−1 e− µT ∑−1 e( )2

eT ∑−1 eσ XX − 1

eT ∑−1 e⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

12

(2.70’)

2.3.5 The Market Portfolio

The universe of the capital market contains all assets. In the equilibrium of the capital market the market portfolio contains all supplied and traded assets in the supplied proportions.

We define the vector of the total market values as VM = V1 V2 VN⎡⎣ ⎤⎦

T, and the proportions

in the market portfolios as xM = xM 1 xM 2 xMN⎡⎣ ⎤⎦

T.

These vectors are related to each other as

xMi =

Vi

eTVM

or xM =

VM

eTVM

. (2.71)

If we assume that the vectors r , µ and the covariance-matrix Σ contain the universe of the assets we can calculate the return rM , the expectation

�

µM and the risk

�

σ MM or σ M of the entire

capital market:


rM = xMT r (2.72)

µM = E rM( ) ≡ xM

Tµ (2.73)

σ MM =Var rM( ) ≡ xM

T ∑ xM (2.74)

σ M ≡ Var rM( )

(2.75)

2.4 Portfolios of Risk-free and Risky Assets

The situation changes substantially if we take a risk free asset into account.

2.4.1 Portfolios of a Single Risky and a Risk-free Asset

Contrary to section 2.2 the asset Y is substituted by a risk-free asset with a fixed return rf . For

convenience we assume µX − rf > 0 .

Returns and Risks of the Portfolio

The return and the risk of a representative portfolio which contains risky and risk-free assets in proportions a and 1-a are

µP = aµX + 1− a( )rf , (2.76)

σ PP = a2σ XX , (2.77)

σ P = a σ X . (2.78)

The risk of the portfolio contains only the weighted risk of the asset X as the variance of rf is of

course zero as well as the covariance of rf with the return of X.

From the derivation of the equations (2.76) and (2.78) we learn about the impact of the composition on the return and the risk of the portfolio:

dµP

da= µX − rf > 0 (2.79)

dσ P

da= ±σ X ≥ ≤( ) 0 ⇔ a ≥ ≤( ) 0 (2.80)

From the equations (2.79) and (2.80) we can calculate the slope of the risk-return-curve:

dµP

dσ P

=dµP dadσ P da

= ±µX − rf

σ X

≥ ≤( ) 0 ⇔ a ≥ ≤( ) 0 (2.81)

Thus, we can distinguish three parts of the MVC:


1. a > 0 ⇒ σ P = aσ X > 0 ⇒

dµP

dσ P

=µX − rf

σ X

> 0

2. a = 0 ⇒ σ P = 0 (2.82)

3. a < 0 ⇒ σ P = −aσ X > 0 ⇒

dµP

dσ P

= −µX − rf

σ X

< 0

The MVF can be presented in the subsequent graph:

Fig 2.7: MVF of risky and a risk free asset

2.4.2 Portfolio of the Market Portfolio (of Many Risky Assets) and a Risk-Faree Asset

The tangent to the MVF starting from the risk-free interest rate rf is called the Capital Market

Line (CML). Its slope of the CML is defined as

tgα =

E rM( ) − rf

SD rM( ) =µM − rf

σ M

(2.83)

The slope determines the risk premium of the market per unit market risk.

Fig 2.8: Capital Market Line

X

X P

P

0

X

rf

a >1

0 < a < 1

a < 0

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Returns and Risk of the Portfolio A portfolio of the risk-free and the market portfolio will be indexed with C as it represents the entire Capital Market. Of course these assets are efficient. Their return ad risk amounts to:

µC = aµM + 1− a( )rf

(2.84)

�

σCC = a2σ MM (2.85)

σC = a σ M

(2.86)

All results from section 2.4.1 apply mutatis mutandis to the return and the risk of portfolios out of the market portfolio of risky and the risk free asset.

Returns and Risks of a Portfolio along the Capital Market Line

If investors select the portfolio weights for the market portfolio and the risk-free asset in the interval

�

0 ≤ a ≤ 1 they can realize any portfolio along CML from rf to M. For

�

a > 1 and

�

1− a < 0 , any portfolio along CML northeast of M can be realized.

The (expected) risk premium of any portfolio along CML will be:

E rC( )− rf =

µM − rf

σ M

σ rC( ) (2.87)

2.5 Efficient and Optimal Portfolios

Investments

In section 2.5.1 we restrict the investments in risky assets, short and long positions as well. In section 2.5.2 we allow additionally an investment in a risk-free asset. There is again no restriction on short positions.

2.5.1 Portfolios of Risky Assets

In a first step we derive the efficient portfolios, in a second one the optimal portfolios.

2.5.1.1 Efficient Portfolios of Risky Assets

As we already know, feasible portfolios can be found by the solution of the program:

Min

x Var rX( ) s.t.

E rX( ) = µX for all µX ≥ 0 I

Of course, not all feasible portfolios are efficient. Each portfolio located at the lower branch of the MVF is dominated by a portfolio located on the upper branch of the MVF.

Efficient portfolios are solutions of the program:


Max

x E rX( ) s.t.

Var rX( ) ≤ σ XX for all σ XX ≥ σ DD > 0 II

Fig 2.8: Efficient portfolios of risky assets

• All portfolios on the locus ADB are feasible as they are solutions to the program I. • Portfolios on the locus AD are efficient as they are solutions of the program II. • Portfolios on the locus DB are of course inefficient.

2.5.1.2 Optimal Portfolios of Risky Assets

As all investors are risk-averse we receive distinctive first order conditions of individual optima:

MRSµPσ P

i = MRTµPσ P

i ≠ MRSµPσ P

j = MRTµPσ P

j (2.88)

Obviously each investor realizes different MRS and MRT between returns and risks according to his preferences and the opportunities of the capital market represented by the risk-return-curve.

Fig 2.9: Optimal portfolios of risky assets

Deviations in the MRS and MRT between return and risk always give an incentive for further exchange as long as there is any possibility to equilibrate these entities.

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2.5.2 Portfolios of many risky and one risk-free asset

If a risk free asset exists we can form portfolios of the risky assets and the risk free asset. We will see that this option changes the entire situation significantly. In particular the set of efficient portfolios will form a straight line. Therefore, the differences in the marginal rates of substitution between return and risk, and thus the incentive for further exchange will disappear.

2.5.2.1 Efficient Portfolios

• All portfolios of the set Erf F are feasible.

• Portfolios on the straight line rf F are inefficient.

• Only portfolios on the line rf E are efficient. Portfolios along this line offer the maximal

return for each level of risk. We call this line the Capital Market Line.

Fig 2.10: Efficient portfolios of risky assets and a risk free asset

2.5.2.2 Optimal Portfolios

Fig 2.11: Optimal portfolios of risky assets and a risk free asset

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Optimal portfolios are located at the Capital Market Line. The first-order condition of an optimal portfolio is defined as:

MRSµPσ P

i = MRSµPσ P

j = MRTµPσ P= MRTµMσ M

=µM − rf

σ M

(2.89)

From (2.77) we can conclude some important properties of the capital market equilibrium:

• Despite any differences in the degree of risk-aversion risk investors choose always portfolios along the Capital Market Line (CML).

• Obviously, there are no deviations in the marginal rates of substitution of return and risk among the different investors and thus no incentives for further exchange.

• Moreover, the marginal rates of substitution of the investors equal the marginal rate of transformation of the entire capital market. The latter equals to the slope of the CML.

• Obviously, all portfolios along the CML result as portfolios of the market portfolio and the risk free asset.

Two fund separation theorem

• Optimal portfolios consist of the market portfolio and the risk-free asset.

• Preferences on risk determine the composition of the portfolio.

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