1. Prof. Dr. Thomas
Gries
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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2. Prof. Dr. Thomas
Gries
• Introduction
• Expected Utility Framework
• Mean-Variance Framework
• Alternatives Risk Measures
• Black-Litterman Framework
• Stochastic Dominance Rules
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3. Prof. Dr. Thomas
Gries
 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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4. Prof. Dr. Thomas
Gries
 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;
n
EU = ∑ pi u(πi )
i =1
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5. Prof. Dr. Thomas
Gries
Criticism:
observing utility is difficult,
variety of patterns in behavior ,
Independence axiom is violated
Risk measure is a qualitative measure
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6. Prof. Dr. Thomas
Gries
 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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7. Prof. Dr. Thomas
Gries
 The Model Assumptions:
 Risk Aversion, Two Parameter, One-Period,
Homogenous expectations.
 In case of two Assets, A and B
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8. Prof. Dr. Thomas
Gries
 The Model: In case of three Assets, A , B, C
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9. Prof. Dr. Thomas
Gries
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10. Prof. Dr. Thomas
Gries
Limitation of the model
Error maximization
Unstable optimal solutions
Ignorance of higher moments of distributions
Standard deviation inefficiency
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11. Prof. Dr. Thomas
Gries
 Semi-Variance
 Lower Partial Moments
 Value at Risk
 Expected Shortfall
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12. Prof. Dr. Thomas
Gries
 Returns below the mean
 violates the subadditivity
 Theoretically, it outperforms Variance
 Empirically, M-SV outperforms M-V
(Non-Normal Distribution)
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13. Prof. Dr. Thomas
Gries
 General type of risk measure
 considers negative deviations from
target outcomes
 represents different types of utility
functions and their characteristics
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14. Prof. Dr. Thomas
Gries
 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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15. Prof. Dr. Thomas
Gries
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16. Prof. Dr. Thomas
Gries
 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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17. Prof. Dr. Thomas
Gries
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18. Prof. Dr. Thomas
Gries
 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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19. Prof. Dr. Thomas
Gries
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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20. Prof. Dr. Thomas
Gries
Implied Returns + Investor Views =
Expected Returns
Π= δ Σ wmkt
 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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21. Prof. Dr. Thomas
Gries
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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22. Prof. Dr. Thomas
Gries
Breaking down the views
Asset A has an absolute return of 5%
Asset B will outperform Asset C by 1%
 Q1   ε 1  ω 1 0 0 
Q + ε = .  + . 
    Ω = 0
 . 0 
Q K   ε K 
    0
 0 ωK 

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23. Prof. Dr. Thomas
Gries
The new combined expected returns views
−1
E [R] =  ( τ Σ ) + P ′Ω P   ( τ Σ ) −1 Π + P ′Ω − 1V 
−1 −1
   
 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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24. Prof. Dr. Thomas
Gries
The new combined expected returns views
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25. Prof. Dr. Thomas
Gries
Advantages
Investor’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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26. Prof. Dr. Thomas
Gries
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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27. Prof. Dr. Thomas
Gries
 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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28. Prof. Dr. Thomas
Gries
 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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29. Prof. Dr. Thomas
Gries
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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30. Prof. Dr. Thomas
Gries
 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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31. (
),
Prof. Dr. Thomas
Gries
 Mathematically ;
U ′ ≥ 0 and U ′′ ≤ 0
x x x
∫ F ( z) dz ≤ ∫ G ( z) dz ∫[ G ( z) −F ( z)] dz ≥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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32. (
),
Prof. Dr. Thomas
Gries
 Graphically ;
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33. (
),
Prof. Dr. Thomas
Gries
 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.
U ∈U 3 where U ′ ≥ 0, U ′′ ≤ 0 and U ′′′ ≥ 0
z t z t
∫ ∫ F (x )dxdt ≤ ∫ ∫ G (x )dxdt [ z , t ] ∈ℜ
−∞ −∞ −∞ −∞
E F (x ) ≥ E G (x ) for all U ∈U 3
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34. (
),
Prof. Dr. Thomas
Gries
Advantages
It 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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35. Prof. Dr. Thomas
Gries
Thank you for your attention!
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