Andrea Mazzon · 2020-02-14 · 2 Portfolio optimization with Markowitz Introduction Efﬁcient portfolios Portfolio selection Criticism to the Markowitz approach 3 Alternative methods

Portfolio optimization

Andrea Mazzon

Ludwig Maximilians Universitat Munchen

Andrea Mazzon Portfolio optimization 1 / 79

These slides are based on Chapter 5 of the publication

“Leitfaden fur das Grundwissen Fach Finanzmathematik und Risikobewertung”

from the Deutsche Aktuareereinigung, available (in German) online.

Further references are:A. J. McNeil, R. Frey, P. Embrechts: Quantitative Risk Management. PrincetonUniversity Press, 2. Edition, 2015.H. Follmer, A. Schied: Stochastic Finance - An Introduction in Discrete Time. 4. Edition,De Gruyter, 2016.


Main contents

1 Utility maximizationIntroductionOne period model: utility maximization with primary productsUtility maximization with derivativesMulti-period case: dynamic Portfolio optimization in a binomial model

2 Portfolio optimization with MarkowitzIntroductionEfficient portfoliosPortfolio selectionCriticism to the Markowitz approach

3 Alternative methods for Portfolio optimization

4 Asset pricingPortfolio theory with secure investmentCapital asset pricing model












Main idea

Goal of an investor is to maximize the expected value of his/her utility at time T ,starting from an initial capital v > 0.

Further capital injections or withdrawals are not allowed.

The investor can choose within a set X of portfolios/positions, identified by realvalued random variables in a measurable space (Ω,F).

The value of the portfolio at time T for a realization ω is X(ω), X ∈ X .


Risk aversion

Suppose X = X1, X2, with

X1(ω) = 100 · 1A(ω), X2(ω) = 50,

for every ω ∈ Ω, where A ∈ F , P (A) = 0.5 for a reference measure P .

In a risk neutral world, any agent would be indifferent between choosing X1 or X2.

However, humans are in general not risk neutral, but risk averse.


Risk attitude and utility function

When you take a choice (in this case, choose a position in X ) you might be willingto maximize your expected utility.

In a risk neutral world with rational agents, everything is simple:

Maximize EP [X] over X ∈ X ,

But again: the world is not risk neutral! Idea:

Maximize EP [u(X)] over X ∈ X ,

for a function u.What about u?

Risk averse agent→ u increasing, but concave;Risk neutral agent→ u increasing, linear;Risk lover agent→ u increasing, convex (never the case in our context);Fool/masochist agent→ u decreasing! (Never the case for us, of course)


Utility function

Definition: Utility functionA function u : S ⊂ R→ R ∪ ∞ is called utility function of a risk averse agent if u isstrictly increasing and strictly concave. Moreover, here we also suppose u to becontinuous.

ExamplesExponential utility function:

u(x) = 1− e−λx, x ∈ S = R, λ > 0.

Logarithmic utility function:

u(x) = ln(x), x ∈ S = (0,∞).

Power utility function:

u(x) =xα

α, x ∈ S = (0,∞), 0 < α < 1.


Formulation of the problem

Definition: Preference orderLet u be an utility function and P a reference probability measure. The preferenceorder of an investor on X is defined via the von-Neumann-Morgenstern representation

X Y ⇐⇒ EP [u(X)] > EP [u(Y )], X, Y ∈ X .

Optimization problem of the investor with initial capital v > 0:

Maximize EP [u(X)] over X ∈ X ,

where X is the set of portfolios values for portfolios built with initial investment v.In order to realise the positions in X , the investor can construct suitable portfoliosof primary financial products or trade in derivative products.

In a complete market, these two cases result in the same quantity of admissablestrategies.In an incomplete market, derivatives offer more flexibility than primary products,providing a richer set of instruments and help to improve the value of the final asset.







The setting

Consider a one-period model with time points 0, T .The market has d+ 1 liquid traded primary products with strictly positive prices:

π = (π0, π) = (π0, π1, . . . , πd) at time 0 (deterministic)S = (S0, S) = (S0, S1, . . . , Sd) at time T (stochastic).

Suppose the product 0 to be risk-free: in particular, π0 = 1, S0 = 1 + r, r > 0.

At time 0, the investor chooses a strategy

θ = (θ0, θ) = (θ0, θ1, . . . , θd) ∈ Rd+1.

The final values of admissible portfolios are random variables in a subset X of

θ · S > 0 | θ ∈ Rd+1.


Budget conditions

If the investor has initial budget v > 0, the portfolio strategy has to fulfil thecondition

θ · π ≤ v.

The above condition can be replaced with an equality, thinking that in optimum noresources are “wasted”:

θ · π = v.

Thus the optimal problem can be formulated as

Maximize E[u(θ · S)] over θ ∈ Rd+1 | θ · π = v,i.e.,

X = θ · S > 0 | θ · π = v.

The last observation allows to replace the optimization problem on Rd+1 with oneconstraint by an optimization problem on Rd without constraints.


Unconstrained problem

Consider

Y i =Si

1 + r− πi, i = 1, . . . , d.

Since θ · π = v, it holdsθ · S = (1 + r)(θ · Y + v).

The problem can be thus written as

Maximize E[u(θ · Y )] over θ1, . . . , θd ∈ R,

with u(y) = u ((1 + r)(y + v)).







Utility maximization with derivatives

While portfolios in the one-period model are composed of primary products on alinear basis, derivatives represent all payout profiles that can be contractuallyagreed.

The space X is now constituted by random variables representing the payoffs attime T of all the derivatives in the market.

To simplify the notation, it is assumed that all the payoffs have already beendiscounted with respect to a suitable numeraire.

The prices of the payoffs at time t = 0 are calculated using a pricing measure Qequivalent to the reference measure P : if a derivative has payoff X at time t = T ,it is valued as EQ[X] in t = 0.

For utility function u and initial budget v > 0, we define

X (v) := X ∈ X | EQ[X] = v,EP [u(X)] <∞.

We want to findW0(v) = sup

X∈X (v)

EP [u(X)].


Heuristic approach via Lagrange-Ansatz

The Lagrangian functional is given by

L(λ,X) =EP [u(X)]− λ(EQ[X]− v)

=λv + EP[u(X)− λX dQ

dP

].

Suppose that u : S = (a, b)→ R, −∞ ≤ a < b ≤ +∞, with u′ invertible and suchthat limx→a u

′(x) = +∞. Thus the maximization problem is solved by

X∗ := (u′)−1

(λdQ

dP

),

for λ such that EQ[X∗] = v.


Examples

Exponential utility function, u(x) = 1− e−γx, x ∈ S = R, γ > 0.

Utility maximizing payoff: X∗ = 1γ

(H(Q|P )− ln dQ

dP

)+ v, where

H(Q|P ) :=

EQ[ln dQ

dP

]if Q P

+∞ otherwise.

Maximized expected utility: E[u(X∗)] = 1− e−γv−H(Q|P ).

Logarithmic utility function: u(x) = ln(x), x ∈ S = (0,∞).

Utility maximizing payoff: X∗ = v dPdQ

.

Maximized expected utility: E[u(X∗)] = ln(v) +H(P |Q).







The setting

Take a multi-period model with times t = 0, 1, . . . , T , and consider a probabilityspace (Ω,F ,F, P ), where F = (Ft)t=0,...,T is a filtration representing information.Suppose there exist:

A risk free asset defined by S0t = (1 + r)t, t = 0, . . . , T , with a deterministic interest

rate r > 0.A risky asset adapted to F defined by

St = S0 · Y1 · · · · · Yt, t = 0, 1, . . . , T,

where Yt can take the two values d, u with 0 < d < 1 + r < u, and Yt are i.i.d. andsuch that Yt+1 is independent of Ft.

It then holds

S0t = S0

t−1(1 + r), t = 1, . . . , T

and

St = St−1Yt, t = 1, . . . , T.


Equivalent martingale measure

In order for the market to be arbitrage-free and complete, there must exist a measureQ ∼ P such that S

S0 is a martingale. It can be seen that it must hold

q := Q(Yt = u) =1 + r − du− d .

Such Q exists and is unique as we have supposed 0 < d < 1 + r < u, and

dQ

dP(ω) =

(q

p

)n(ω)(1− q1− p

)T−n(ω),

where p := P (Yt = u) and n(ω) is the number of times t = 1, . . . , T when Yt(ω) = d.


Formulation of the problem

An investor with initial capital v > 0 wants to maximize the expected utility at timeT of the value a portfolio made of (S0, S) by a self-financing and adapted strategyθ = (θ0, θ1), where

V θt = θ0tS0t + θ1tSt, t = 1, . . . , T.

Since θ has to be self-financing it must hold

V θt = θ0tS0t−1 + θ1tSt−1, t = 1, . . . , T. (1)

The problem is

Maximize EP [u(V θT )] over all adapted strategies θ satisfying (1).

Under the pricing measure Q, the process V θ is also a martingale.


Solution 1: Dynamic programming (1)

At every time t = 0, . . . , T , an investor knows the value V θt of the portfolio at t andinvests a fraction πt of V θt in the risky asset St, i.e.,

πt =θ1tStV θt

, t = 1, . . . , T.

Then at time t+ 1 it holds (stressing now the dependence of V on π)

V πt+1 = V πt [(1− πt)(1 + r) + πtYt+1] = V πt [1 + r + πt (Yt+1 − 1− r)] .

From this representation, the core idea of Dynamic programming is to consider afamily of utility maximization problems instead of the original problem.



Let’s make the idea more concrete. With

V πt = V πt−1 [1 + r + πt−1 (Yt − 1− r)]

in mind, define

V t,πk (x) := xΠk−1j=t [1 + r + πj (Yj+1 − 1− r)] , k = t+ 1, . . . , T ,

the value at time k of a strategy starting from time t, when the value of the portfolioat time t is x.

Define the set of admissible strategies

At(x) = π = (πu)t≤u<T : π adapted and s.-f., s.t. V t,πk (x) > 0, k = t+ 1, . . . , T.

The value function of the utility maximization problem at time t ≤ T is

Wt(x) := supπ∈At(x)

EP[u(V t,πT (x))

].



Have in mind: value function of the utility maximization problem at time t ≤ T is

Wt(x) := supπ∈At(x)

EP[u(V t,πT (x))

],

withV t,πk (x) := xΠk−1

j=t [1 + r + πj (Yj+1 − 1− r)] , k = t+ 1, . . . , T.

For the family of these value functions, the Bellman principle holds:

Wt(x) = supy∈R

EP [Wt+1 (x(1 + r + y(Yt+1 − 1− r)))],

for all t ≤ T , x ∈ (0,∞).

The Bellman principle allows a recursive calculation of the value functions startingfrom the final condition WT (v) = u(v).


Solution 2: Martingale method (1)

The idea of the Martingale method is to reduce the problem to the one period problemstudied above.It mainly consists of three steps:

Determine the set X (v) of values for the wealth VT at time T , reachable by aself-financing strategy starting with budget v > 0;

determine the optimal reachable wealth V ∗T ;

find a self financing strategy α∗ such that V α∗

T = V ∗T where in V α∗

T we makeexplicit the dependence of the terminal wealth on the strategy α (remember thatthe market is complete).


Solution 2: Martingale method (2)

Step 1The value process (V πt )t=0,1,...,T must satisfy for every π ∈ A0(v) the followingcondition:

EQ[(1 + r)−TV πT ] = v.

Thus, since the market is complete,

X (v) := X > 0 : X FT -measurable with EQ[(1 + r)−TX] = v, v > 0.

Step 2We have to find V ∗ as

V ∗(v) = supX∈X (v)

EP [u(X)].

This is a static problem can be solved with the Lagrange multipliers method asabove.

Step 3Hedging problem: find a strategy that replicates the optimal value V ∗ found above.


Example: Binomial model with logarithmic utility function

We take u(x) = ln(x), x ∈ S = (0,∞), and suppose r = 0.

First-second step: From the expression of the Radon-Nikodym derivative dQdP

, andapplying the Lagrange multipliers method, we find

V ∗(v) = v

(p

q

)nT(

1− p1− q

)T−nT

,

where nT is the number of times when Yt = u, t = 1, . . . , T .

The expected utility from terminal wealth is

EP [u(V ∗(v))] = ln(v) + pT ln

(p

q

)+ T (1− p) ln

(1− p1− q

).

Third step: The ratio of the wealth invested in St

π∗t =(1 + r)(p− q)q(1− q)(u− d)

, t ∈ 1, . . . , T

is optimal for EP [u(V ∗(v))].

In economic terms, this means that with a logarithmic utility function, a constantproportion of the portfolio value is always invested in the risky asset.












Motivation

Investors in capital markets can invest on a large number of individual financialsecurities available and can generate portfolios from these.In this context, it is first of all necessary to clarify which effects a portfolio formationhas on “risk” and “value”.

At the risk level, diversification between the individual securities is central, i.e. the riskof the portfolio can be reduced below the risk of the individual securities by “suitable”portfolio formation.At the risk/value level, aspects of risk/return dominance and efficient portfolios comeinto play.

A second issue concerns the construction of an “optimal” portfolio for a giveninvestor.

Both questions are addressed by the classical capital market model of Markowitz.

Fields of application of the Markowitz model are the derivation of optimal portfoliosfrom single stocks (standard application: optimal stock portfolios) and the optimalcomposition of asset classes (asset allocation).


Main characteristics of the model

This is a static one-period model with n financial assets.

Financial assets can be divided at will and transaction costs are neglected.

Financial assets and portfolios are assessed using the standard deviation of thereturn as a measure of risk and the expected return value as a measure of value.

The “drivers” for diversification in relation to the standard deviation are thecorrelations of the individual assets.

The reference measure is the statistical measure.


Efficient portfolios

Investors prefer:

the portfolio with the higher expected return value for the same portfolio risk;

the portfolio with the lower risk for the same expected return value (risk aversion).

DefinitionA portfolio with return R1 dominates a portfolio with return R2 if one the two followingconditions holds:

1 V ar(R1) < V ar(R2) and E[R1] ≥ E[R2];2 E[R1] > E[R2] and V ar(R1) ≤ V ar(R2).

DefinitionA portfolio is efficient if it is not dominated by any other portfolio.

Only efficient portfolios can be optimal portfolios.


The setting

The return of the asset i is denoted by Ri, i = 1, . . . , n.

The proportional investment in the asset i is xi, i = 1, . . . , n.

The return of the portfolio is thus

RP =n∑i=1

xiRi.

Restrictions on investments:Short sales allowed: only

n∑i=1

xi = 1. (2)

No short sales allowed: (2) plus

xi ≥ 0, i = 1, . . . , n.

Denote by:µi = E[Ri], σi = σ(Ri), ρi,j = ρ(Ri, Rj), i = 1, . . . , n, for the assets;µP = E[Rp], σP = σ(RP ), for portfolios.


Mean and variance of returns

It holds:

E[RP ] =n∑i=1

xiE[Ri]=

n∑i=1

xiµi,

V ar(RP ) =n∑i=1

n∑j=1

xixjCov(Ri, Rj)

=

n∑i=1

xiV ar(Ri) + 2∑

1≤i<j≤n

xixjCov(Ri, Rj)

=n∑i=1

xiσ2i + 2

∑1≤i<j≤n

xixjρi,jσiσj . (3)

In vectorial notation,

µP = E[RP ]= xTµ,

σ2P = V ar(RP )= xTCx,

wherex is the vector of investment weights;µ is the vector of mean returns;C is the covariance matrix.


Admissible sets

Denote by:

D the set of the admissible weights;

M = (E[RP ], σ(RP )) : RP returns of portfolios constructed with weights in D,the set of admissible (µ, σ).







Determination of the efficient frontier

Goal: minimize the variance

x→ Z1(x) = V ar(RP )=n∑i=1

xiσ2i + 2

∑1≤i<j≤n

xixjρi,jσiσj

under the conditionsE[RP ] =

∑ni=1 xiE[Ri] =

∑ni=1 xiµi = r;∑n

i=1 xi = 1x1, . . . , xn ≥ 0 (if short sales not allowed)

In vectorial terms:

Z1(x) = xTCx→ min!

subject to

xTµ = r, xT e = 1, x ≥ 0. (4)

where e = (1, . . . , 1).

No short sales: “quadratic programming”; computer-aided numerical methodsnecessary.


Case I: short sales allowed

Short sales allowed: Solution with standard Lagrange approach. Determine localextreme value of the Lagrange function.

Alternative formulation: for a given ` > 0

Z2(x) = `µTx− 1

2xTCx→ max!

subject to

xT e = 1.


Solution to the alternative formulation (1)

We construct the Lagrange function

L(x, λ) = `xTµ− 1

2xTCx− λ(xT e− 1).

FromLx = `µ−Cx− λe = 0, Lλ = xT e− 1 = 0

we get

x(`) =1

cC−1e + `h = x0 + `h,

where

x0 =1

cC−1e, c = eTC−1e, h = C−1

(µ− a

ce), a = eTC−1µ, b = µC−1µ.

It follows that

µP (`) := µ(`) = µ0 + α`, σ2P (`) := σ2(`) = σ2

0 + α`2,

where µ0 = ax

, σ20 = 1

c, α = b− a2

c.

For ` = 0 we get the minimum variance portfolio.


Solution to the alternative formulation (2)

For ` ≥ 0, it must hold

σ2P = σ2

0 +1

α(µP − µ0)2,

or equivalently

µP = µ0 ±√α(σ2

P − σ20).

The set of efficient portfolios in terms of investments is given by

D∗ = x ∈ Rn : x = x0 + `h, ` ≥ 0,

The set of efficient portfolios in terms of expectation/variance is the upper branchof

M∗ =(µP , σP ) ∈ Rn : µP = µ0 + α`, σ2P = σ2

0 + α`2, ` ≥ 0

=

(µP , σP ) ∈M : σ2

P = σ20 +

1

α(µP − µ0)2

.

including the portfolio with minimum variance.M∗ is the geometric boundary of the set of all permissible portfolios.The upper branch of M∗ is called efficient boundary, or curve of the localminimum variance portfolios.


Example 1


Example 2: two assets


Example (1)

Consider three assets whose returns are uncorrelated, with identical variance 0.1.Suppose µ1 = 0.2, µ2 = 0.1, µ3 = 0.3.

The Lagrangian is

L(x1, x2, x3, λ) = `(0.2x1 + 0.1x2 + 0.3x3)− 1

2(0.1x21 + 0.1x22 + 0.1x23)

− λ(x1 + x2 + x3 − 1).

The Lagrangian equations are thereforeLx1 = 0.2`− 0.1x1 − λ = 0

Lx2 = 0.1`− 0.1x2 − λ = 0

Lx3 = 0.3`− 0.1x3 − λ = 0

Lλ = 1− x1 − x2 − x3 = 0.

We find the solution x1 = 1

3

x2 = 13− `

x3 = 13

+ `

λ = 15`− 1

30.


Example (2)

The weights for the variance minimizing portfolio are

x1 =1

3, x2 =

1

3, x3 =

1

3.

Hence, we have

σ20 = (x21 + x22 + x23)

1

10=

1

30, µ0 = 0.2x1 + 0.1x2 + 0.3x3 = 0.2.

In general it holds

µP (`) = 0.2 + 0.2`, σ2P (`) = 0.2`2 +

1

30= 5(µP (`))2 − 2µP (`) +

7

30.

We obtain

µP (`) = 0.2±

√0.2

(σ2P (`)− 1

30

)


Efficient frontier


No short sales allowed

The analysis can be continued using the data of the previous example.

Non-negativity is not always guaranteed for those weights (as a function of `).

In particular,

x2 ≥ 0 ⇐⇒ ` ≤ 1

3, x3 ≥ 0 ⇐⇒ ` ≥ −1

3.

The previous solution can be taken only for the case − 13≤ ` ≤ 1

3.

The general solution is illustrated in the following figure. The marginal portfoliosare located on three connected root function segments.


Efficient frontier with no short sales







Main idea

Each investor must perform a “trade-off” between “risk” and “return”, i.e., investorselects a characteristic position (µ, σ) on the efficient frontier.

In order to determine this position, the investor must directly or indirectly makehis/her preferences explicit regarding the valuation of risky investments.

Possible approach: concrete specification of an EV-preference functionH(E[R], σ(R)), for example

H(E[R], σ(R)) = E[R]− aσ2(R), a > 0.

Once chosen H(E[R], σ(R)), optimum problem for (µ, σ) under the condition(µ, σ) ∈M , M set of admissible positions.


Example: two assets case

LetV (R) = H(E[R], σ(R)) = E[R]− aσ2(R).

Consider two assets and a portfolio made of the two assets with investments x1 := x,x2 = 1− x, x ∈ R.Let R be the return of the portfolio and σ12 := Cov(R1, R2). We have:

V (R) = µ2 + (µ1 − µ2)x− a(x2σ21 + (1− x)2σ2

2 + 2x(1− x)σ12).dV (x)dx

= (µ1 − µ2)− 2axσ21 + 2a(1− x)σ2

2 − 2aσ12 + 4axσ12.

We want to choose x such that dV (x)dx

= 0, and we get

x =2a(σ12 − σ2

2)− (µ1 − µ2)

4aσ12 − 2a(σ21 + σ2

2)=µ1 − µ2 − 2a(σ12 − σ2

2)

2aσ2(R1 −R2).

(General) problem: how to choose a?


Shortfall restriction

In practice, it is often desirable to explicitly limit the risk taken for risk controlpurposes.

For investors it is often more intuitive to specify a target return z or a confidencelevel 1− ε, than to specify a benefit function or a risk tolerance parameter.

It is also consistent with the “Solvency II logic” to measure risk separately and touse the Value at Risk as a measure of risk.

The level of risk must not exceed a certain tolerated level.

In the simplest case, this leads to a limitation of the shortfall probability (“shortfallrestriction”) in the form

P (R < z) ≤ ε.

If the distribution of R is continuous, the shortfall restriction can also be formulatedas an equivalent Value-at-Risk restriction:

P (R < z) ≤ ε ⇐⇒ V aRε(R) ≤ −z.


Shortfall restriction: normal case

If R ∼ N(µ, σ2), it holds

V aRε(R) = −µ− σΦ−1(ε).

Thus the shortfall restriction is

µ ≥ z − σΦ−1(ε).

This means that the set of admissible (µ, σ) is now bounded by the straight line

µ = z − σΦ−1(ε).

Only the sector above the straight line (included) is then still permissible in termsof a controlled shortfall probability.

The optimal portfolio then corresponds to the upper intersection of the efficientedge with the shortfall line.


Shortfall restriction







Main criticism

Point 1: use of standard deviation as a risk measure

The Markowitz basic model is based on standard deviation.

The standard deviation is a measure of risk that is particularly important in thecontext of elliptical distributions (especially normal), which are distributedsymmetrically.

In practice, however, there are also cases in which there is a significant skewness.

The standard deviation does not take into account these asymmetries in riskmeasurement.

Point 2: analytic solutions far from reality

Optimized portfolios often have extreme allocations.

Without short selling restrictions, very high short selling positions arise.

With short selling restrictions the diversification is low.


Main criticism

Point 3: Lack of robustness

The optimization is very sensitive to the input data.

The variation of the input data results in completely different portfolios.

Because of this reason the input data, especially the estimated expected values,play a key role for the quality of the portfolio optimization.

There exist theoretical approaches towards robustification: treatment of the“estimation error problem”, Black/Litterman method, portfolio heuristics such asminimum variance, equal weight or risk parity.

Point 4: Covariance matrix estimation

For n assets, one must estimate n(n+1)2

entries of the covariance matrix.

This corresponds to approx. 5, 000 covariances for 100 assets and over 30, 000covariances for 250 individual titles.

Due to this high dimension, in practice no individual estimates are made, butrather a reduction to “common factors” (multifactor models).

This considerably reduces the diversity of the (co-)variances to be estimated (andthe problem of estimation errors).







Extensions to the model

A first class of model generalizations uses more general risk measures, but retainsthe expected return as a value measure.

On the other hand, some models use more general value measures.

Another alternative is to combine risk and value measures into a risk-adjustedperformance measure (standard example: Sharpe ratio) and to maximize thisperformance measure.

Here we retain the expected value as a value measure and focus on the AverageValue at Risk as a risk measure (remember: differently from Value at Risk, it iscoherent!)

An alternative use of AVaR in the portfolio optimization context is to combine theMarkowitz EV model with an AVaR restriction.


Average Value at Risk as risk measure

General form:

AV aRα(RP (x))→ min!

subject to

xTµ = r, xT e = 1, x ≥ 0. (5)

Problem: The determination of the AVaR is in general dependent on thedistribution assumption.

In the literature, based on the results of Uryasev-Rockafellar 1 a sample-basedvariant has come into place, which also has the advantage that it leads to a linearprogram.

1R. T. Rockafellar, S. Uryasev, Optimization of conditional value-at-risk. The Journal of Risk, 2, 2000.Andrea Mazzon Portfolio optimization 60 / 79

Average Value-at-Risk optimization

Remember that

AV aRα(X) =1

αE[(−V aRα(X)−X)+

]+V aRα(X) =

1

αmin`∈R

(E[(`−X)+

]− α`

).

Considering now the loss L = −X, it makes sense to define

AV aRα(L) := minI∈R

I +

1

αE[(L− I)+].

In particular, the minimization problem allows to compute the AVaR without firstcalculating the VaR.


Average Value-at-Risk optimization with linear programming

Consider a sample r1, . . . , rs of a vector R = R1, . . . , Rn of returns of nindividual assets in the portfolio.If we denote the vector of investments with x = (x1, . . . , xn)T , the loss variablesare given by `i = −xT ri, i = 1, . . . , s.We define the quantities zi = (`i − I)+, i = 1, ..., s.In this way, E

[(`−X)+

]can be estimated by 1

s

∑si=1 zi,

In the same way, E[R] is estimated by r = 1s

∑si=1 ri.

Thus, the minimization problem can be expressed in terms of I, zi, i = 1, . . . , s inthe following way:

I +1

s

s∑i=1

zi → min!

subject to

zi ≥ 0, i = 1, . . . , s,

zi + xT ri + I ≥ 0, i = 1, . . . , s,

xT r = r, xT e = 1, x ≥ 0.


Alternative models are not relevant for Elliptical distributions

TheoremLet ρ be a positive homogenous, translation invariant and distribution invariant riskmeasure. Also let X be a random variable with elliptic distribution. Then it holds

ρ(X) = E[−X] + k(ρ)σ(X),

where k(ρ) is a constant depending on a measure ρ. a

aFor a proof, see Theorem 8.28 in A.J. McNeil, R. Frey, P. Embrechts, Quantitative RiskManagement. Princeton University Press, 2. edition, 2015.

For each fixed expected value, each portfolio minimizes the variance as a functionof the vector of investments x, thus also minimizing the risk measure ρ.

This implies that alternative mean/risk approaches only become relevant outsidethe class of elliptical distributions.












Introduction

Extension of the investment spectrum:

The investment spectrum of the Markowitz model is based only on purely riskyinvestments (variance of the return > 0)

We now extend the model to include a risk-free investment (variance of the return= 0) at the safe interest rate r0.

At the interest rate r0, any amount can be both invested and borrowed (perfectcapital market).

It is important to note that the term “risk-free/sure investment” refers purely tovolatility.


Portfolios’ combination and Sharp ratio

Fix a “risky” portfolio P , with return RP , made of assets with variance > 0.

We consider a portfolio which is a combination of P and of units of the safeinvestment.

Call 0 ≤ x <∞ the investment on P . Thus the return of the combined portfolio is

R = xRP + (1− x)r0.

The return of the combined portfolio has expectation µ and variance σ2 with

µ = xµP + (1− x)(1 + r0) = r0 + x(µP − r0),

σ2 = x2σ2P .

It follows x = σσP

and thus

µ = r0 +µP − r0σP

σ.


The set of reachable portfolios under this enlargement is thus

M =

(µ, σ) : µ = r0 +

µP − r0σP

σ, (µP , σP ) ∈M.

What about the set of the efficient portfolios M∗?

The slope of the straight line in M is

Sharp ratio = µP−r0σP

.

Consider now the efficient frontier for totally risky portfolios, i.e.,

M∗ =

(µP , σP ) ∈M : σ2

P = σ20 +

1

α(µP − µ0)2

.

It can be seen that for given r0 > 0, there exists one element (=portfolio) T in M∗

with maximum Sharp ratio.

It holds

M∗ =

(µ, σ) : µ = r0 +

µT − r0σT

σ

.

T is called tangential portfolio: the straight line M∗ is tangent to M∗ at T .


Tangential portfolio







The setting

The starting point for the one-period model is that of the portfolio theory with safeinvestment (case r0 < µ0). In addition, the following assumptions are made:

There are n assets and m EV investors in the market with budgets Vi > 0,i = 1, . . . ,m, V =

∑mi=1 Vi.

The view of investors on the capital market with regard to the expected values,variances and covariances of assets are homogeneous, i.e. all investors estimater0 , E[Ri], V ar(Ri) and Cov(Ri, Rj) identically.

Market equilibrium is assumed in t = 0: the prices in t = 0 adjust in such a waythat the market is cleared: supply = demand.


Market portfolio

DefinitionThe Market portfolio is a portfolio consisting of a weighted sum of every asset inthe market, with weights given by the vector

xM := (P1/P, . . . , Pn/P ),

where Pi is the market value of the asset i, i = 1, . . . , n, and P =∑ni=1 Pi.

The return of the Market portfolio is denoted by RM .

RemarkUnder the assumption of market efficiency, the Market portfolio is equal to theTangential portfolio: RT = RM .


Characterization of the efficient frontier: Capital market line

From RM = RT , we have the set of optimal portfolios is characterized by returnssatisfying

E[R] = r0 +E[RM ]− r0σ(RM )

σ(R).

Also called capital market line: expresses returns in terms of standard deviation,for optimal portfolios.

In capital market equilibrium, a linear relationship applies to the optimal portfolios:for a higher expected return, a proportionally higher risk must be accepted.


Market portfolio


Characterization of any portfolio: Security market line

For every portfolio or single asset, it holds:

E[R] = r0 + βR(E[RM ]− r0),

whereβR :=

Cov(R,RM )

V ar(RM )

is the Beta factor.

Exploiting the fact that for an efficient portfolio P it holds σP = xσM , where x is theamount invested on the market / tangential portfolio, it can be seen that theSecurity market line implies the Capital market line.

The Security market line differs with respect to the Capital market line in the “riskmeasure” considered: here expected returns in terms of Beta factor.


Market portfolio


Beta factor and systematic risk

Systematic risk: market risks that cannot be diversified away. Is related to marketfluctuations. Interest rates, recessions, and wars are examples of systematic risks.

Unsystematic Risk: relates to individual stocks. It represents the component of astock’s return that is not correlated with general market fluctuations.

Note that the Beta factor measures how much the price of a particular portfoliojumps up and down compared with how much the entire stock market jumps upand down.

In this way, only systematic risk is assessed by the market and included in prices.

β > 0: movement in the same direction as the market as a whole (both whenmarket prices fall and when they rise).

β < 0: opposite tendency to the market (practically non-existent)


Risk premium and equilibrium price

The quantityE[R]− r0 = βR(E[RM ]− r0),

is called risk premium of the underlying portfolio.

It is the “excess return” demanded by investors in the capital market equilibrium:return in excess with respect to the return on a secure investment, which isrequired for enter into a risky investment in an individual share or a share portfolio.

Let V denote the random end-of-period value of a portfolio of financial instrumentsunder consideration, and R its return.

Thus the market-clearing equilibrium price of this portfolio at the beginning of theperiod is

P =E[V ]

1 + r0 + βR(E[RM ]− r0).

The price equation of the CAPM allows the foundation of a risk-adjusted discountfactor.


Criticism and further developments

Since the CAPM is based on the Markowitz approach, the points of criticismdiscussed above apply to the CAPM.

The central criticism of the CAPM, however, is that the beta factor is “the” centralprice-determining factor.

The criticism of the empirical validity of the CAPM led to further development ofthe CAPM both on a theoretical level and on an empirical level.


Andrea Mazzon · 2020-02-14 · 2 Portfolio optimization with Markowitz Introduction Efﬁcient portfolios Portfolio selection Criticism to the Markowitz approach 3 Alternative methods

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