Mean-Variance in Financial Decisions under Risk and Uncertainty

Prof. Dr. Thomas Prof. Dr. Thomas GriesGries

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Beyond Mean-Variance in Financial decisions under Risk und Uncertainty

Fakultät Wirtschaftswissenschaften

Lehrstuhl für Wachstums- und Konjunkturtheorie

Prof. Dr. Thomas Gries

Sherif Elkoumy


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• Introduction

• Expected Utility Framework

• Mean-Variance Framework

• Alternatives Risk Measures

• Black-Litterman Framework

• Stochastic Dominance Rules


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The Thesis reviews different frameworks concerning financial decisions under risk and uncertainty.

It reviews as well alternative risk measures to the traditional risk measure, Standard deviation.

the thesis documents advantages and disadvantages of those models and frameworks.


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n

i ii 1

EU p u(π )

EU designed by VNM in 1941 and

affected on decisions theory and

portfolio theory.

VNM assume a set of appealing axioms

on preferences.

EU established two major line of

research

Selecting criteria according this formula;


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Criticism:

observing utility is difficult,

variety of patterns in behavior ,

Independence axiom is violated

Risk measure is a qualitative measure


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The cornerstone of modern finance

theory .

The simplicity form in construction and

selection of portfolios.

The interpretation of the mean as the

anticipated return and the variance as

the risk.

Tradeoff between risk and return.

A quantative risk measure


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The Model Assumptions:

Risk Aversion, Two Parameter, One-

Period, Homogenous expectations.

In case of two Assets, A and B


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The Model: In case of three Assets, A ,

B, C


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Limitation of the model

Error maximization

Unstable optimal solutions

Ignorance of higher moments of

distributions

Standard deviation inefficiency


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Semi-Variance

Lower Partial Moments

Value at Risk

Expected Shortfall


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Returns below the mean

violates the subadditivity

Theoretically, it outperforms

Variance

Empirically, M-SV outperforms M-

V (Non-Normal Distribution)


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General type of risk measure

considers negative deviations

from target outcomes

represents different types of

utility functions and their

characteristics


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How bad can things get?

the worst loss over a time horizon

with a given level of target

probability

Time horizon from 1 day to 2

weeks

Probabilities from 1% to 5%

Efficient under symmetric

distribution

violates the subadditivity.


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If things do get bad, how much

can one expect to lose?.

satisfies (Monotonicity,

Subadditivity, Positive

homogeneity, Translational

invariance.

measures the expected amount

beyond the VaR


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determine optimal asset allocation in a portfolio.

overcomes the problems of estimation error maximization in M-V approach.

incorporates an investor’s own views in determining asset allocations.


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Basic Idea and steps:

Find implied returns

Formulate investor views

Determine what the expected returns

are

Find the asset allocation for the

optimal portfolio


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Implied Returns + Investor Views = Expected Returns

Π= Π= δδ Σ Σ wwmktmkt

Π = The implied excess equilibrium return (N*1 vector)

δ = (E(r) – rf)/σ2 , risk aversion coefficient Σ = A covariance matrix of the assets

(N*N matrix) wmkt = Market capitalization weights of

the Assets(N*1)


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Implied Returns + Investor Views = Expected Returns

P = A matrix with investors views; each row a specific view of the market and each entry of the row represents the portfolio weights of each assets (K*N matrix)

ε= the error term (uncertanity on views)

Ω = A diagonal covariance matrix with error terms on each view (K*K matrix)

Q = The view vector described in matrix P (K*1 vector)


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Breaking down the views

Asset A has an absolute return of 5%Asset B will outperform Asset C by 1%

1 1

. .

K K

Q

Q

Q

1 0 0

0 . 0

0 0 K


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The new combined expected returns views

11 11 1E R P P P V

Assuming there are N-assets in the portfolio, this formula computes E(R), the expected new return.

τ = A scalar number indicating the uncertainty of the CAPM distribution (0.025-0.05


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The new combined expected returns views


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AdvantagesInvestor’s can insert their view.

Control over the confidence level of views.

More intuitive interpretation, less extreme

shifts in portfolio weights.

The reverse optimization techniques do

not generate implausible solutions.


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Disadvantages

Black-Litterman model does not give the best possible portfolio, merely the best portfolio given the views stated

As with any model, sensitive to assumptions Model assumes that views are independent of each other

The normal distribution


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An alternative approach to The M-V to

the ordering of uncertain prospects.

Decision rule for dividing alternatives

into two mutually exclusive groups:

efficient and inefficient.

Consistent with the VNM axioms on

preferences.


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The most general efficiency criteria

relies only on the assumption that utility

is nondecreasing in income, or the

decision maker prefers more of at least

one good to less.

FSD: Given two CDFs F and G, an asset

F will dominate G by FSD independent

of concavity if F(x) ≤ G(x) for all return x

with at least one strict inequality.


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Intuitively, this rule states that F will dominate G if its CDF always lies to the left of G’s:

F x

G x

F x


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SSD implies that the investor is risk

averse

utility function is concave, implying that

the second derivative of the utility

function is negative.

SSD Rule A necessary and sufficient

condition for an alternative F to be

preferred to a second alternative G by

all risk averse decision makers is that


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(

0 and 0U U

),

Mathematically ;

F z dz G z dzx x

( ) ( ) G z F z dz

x

( ) ( ) 0

Graphically; Alternative F dominates

alternative G for all risk averse

individuals if the cumulative area under

F exceeds the area under the cumulative

distribution function G for all values x


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(),

Graphically ;


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(),

TSD refers to a preferences for positive

skewness. The sum of the cumulative

probabilities for all returns is never

more with F than G and sometimes less.

3 where 0, 0 and 0U U U U U

( ) ( ) ,z t z t

F x dxdt G x dxdt z t

3( ) ( ) for all F GE x E x U U


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(),

AdvantagesIt takes the entire distribution into

account

It does not imply any assumptions related

to the return distribution.

Disadvantages

No precise quantifying for the risk

No complete diversification

framework


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Thank you for your attention!

Economy & Finance

Mean-Variance in Financial Decisions under Risk and Uncertainty