View
1.612
Download
0
Category
Tags:
Preview:
Citation preview
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
12.04.23 1
Beyond Mean-Variance in Financial decisions under Risk und Uncertainty
Fakultät Wirtschaftswissenschaften
Lehrstuhl für Wachstums- und Konjunkturtheorie
Prof. Dr. Thomas Gries
Sherif Elkoumy
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
212.04.23
• Introduction
• Expected Utility Framework
• Mean-Variance Framework
• Alternatives Risk Measures
• Black-Litterman Framework
• Stochastic Dominance Rules
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
312.04.23
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.
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
412.04.23
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;
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
512.04.23
Criticism:
observing utility is difficult,
variety of patterns in behavior ,
Independence axiom is violated
Risk measure is a qualitative measure
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
612.04.23
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
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
712.04.23
The Model Assumptions:
Risk Aversion, Two Parameter, One-
Period, Homogenous expectations.
In case of two Assets, A and B
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
812.04.23
The Model: In case of three Assets, A ,
B, C
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
912.04.23
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
1012.04.23
Limitation of the model
Error maximization
Unstable optimal solutions
Ignorance of higher moments of
distributions
Standard deviation inefficiency
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
1112.04.23
Semi-Variance
Lower Partial Moments
Value at Risk
Expected Shortfall
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
1212.04.23
Returns below the mean
violates the subadditivity
Theoretically, it outperforms
Variance
Empirically, M-SV outperforms M-
V (Non-Normal Distribution)
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
1312.04.23
General type of risk measure
considers negative deviations
from target outcomes
represents different types of
utility functions and their
characteristics
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
1412.04.23
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.
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
1512.04.23
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
1612.04.23
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
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
1712.04.23
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
1812.04.23
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.
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
1912.04.23
Basic Idea and steps:
Find implied returns
Formulate investor views
Determine what the expected returns
are
Find the asset allocation for the
optimal portfolio
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
2012.04.23
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)
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
2112.04.23
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)
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
2212.04.23
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
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
2312.04.23
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
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
2412.04.23
The new combined expected returns views
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
2512.04.23
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.
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
2612.04.23
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
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
2712.04.23
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.
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
2812.04.23
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.
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
2912.04.23
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
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
3012.04.23
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
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
3112.04.23
(
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
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
3212.04.23
(),
Graphically ;
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
3312.04.23
(),
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
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
3412.04.23
(),
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
Prof. Dr. Thomas Prof. Dr. Thomas GriesGries
3512.04.23
Thank you for your attention!
Recommended