Mean-Variance in Financial Decisions under Risk and Uncertainty
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Mean-Variance in Financial Decisions under Risk and Uncertainty

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  • Welcome everybody to my presentation. My name is sherif and I will present today my master thesis. It is entitles Beyond Mean-Variance in Financial decisions under risk and Uncertainty. First, I show you my Agenda.

Mean-Variance in Financial Decisions under Risk and Uncertainty Mean-Variance in Financial Decisions under Risk and Uncertainty Presentation Transcript

  • Prof. Dr. ThomasGries Beyond Mean-Variance in Financial decisions under Risk und Uncertainty Fakultät Wirtschaftswissenschaften Lehrstuhl für Wachstums- und Konjunkturtheorie Prof. Dr. Thomas Gries Sherif Elkoumy 16.02.13 1
  • Prof. Dr. ThomasGries • Introduction • Expected Utility Framework • Mean-Variance Framework • Alternatives Risk Measures • Black-Litterman Framework • Stochastic Dominance Rules 16.02.13 2
  • Prof. Dr. ThomasGries  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. 16.02.13 3
  • Prof. Dr. ThomasGries  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 16.02.13 4
  • Prof. Dr. ThomasGries Criticism: observing utility is difficult, variety of patterns in behavior , Independence axiom is violated Risk measure is a qualitative measure 16.02.13 5
  • Prof. Dr. ThomasGries  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 16.02.13 6
  • Prof. Dr. ThomasGries  The Model Assumptions:  Risk Aversion, Two Parameter, One-Period, Homogenous expectations.  In case of two Assets, A and B 16.02.13 7
  • Prof. Dr. ThomasGries  The Model: In case of three Assets, A , B, C 16.02.13 8
  • Prof. Dr. ThomasGries 16.02.13 9
  • Prof. Dr. ThomasGries Limitation of the model Error maximization Unstable optimal solutions Ignorance of higher moments of distributions Standard deviation inefficiency 16.02.13 10
  • Prof. Dr. ThomasGries  Semi-Variance  Lower Partial Moments  Value at Risk  Expected Shortfall 16.02.13 11
  • Prof. Dr. ThomasGries  Returns below the mean  violates the subadditivity  Theoretically, it outperforms Variance  Empirically, M-SV outperforms M-V (Non-Normal Distribution) 16.02.13 12
  • Prof. Dr. ThomasGries  General type of risk measure  considers negative deviations from target outcomes  represents different types of utility functions and their characteristics 16.02.13 13
  • Prof. Dr. ThomasGries  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. 16.02.13 14
  • Prof. Dr. ThomasGries 16.02.13 15
  • Prof. Dr. ThomasGries  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 16.02.13 16
  • Prof. Dr. ThomasGries 16.02.13 17
  • Prof. Dr. ThomasGries  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. 16.02.13 18
  • Prof. Dr. ThomasGries Basic Idea and steps: Find implied returns Formulate investor views Determine what the expected returns are Find the asset allocation for the optimal portfolio 16.02.13 19
  • Prof. Dr. ThomasGries 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) 16.02.13 20
  • Prof. Dr. ThomasGries 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) 16.02.13 21
  • Prof. Dr. ThomasGries 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   16.02.13 22
  • Prof. Dr. ThomasGries 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 16.02.13 23
  • Prof. Dr. ThomasGries The new combined expected returns views 16.02.13 24
  • Prof. Dr. ThomasGries 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. 16.02.13 25
  • Prof. Dr. ThomasGries 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 16.02.13 26
  • Prof. Dr. ThomasGries  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. 16.02.13 27
  • Prof. Dr. ThomasGries  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. 16.02.13 28
  • Prof. Dr. ThomasGries 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) 16.02.13 29
  • Prof. Dr. ThomasGries  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 16.02.13 30
  • ( ),Prof. Dr. ThomasGries  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 16.02.13 31
  • ( ),Prof. Dr. ThomasGries  Graphically ; 16.02.13 32
  • ( ),Prof. Dr. ThomasGries  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 16.02.13 33
  • ( ),Prof. Dr. ThomasGries 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 16.02.13 34
  • Prof. Dr. ThomasGries Thank you for your attention! 16.02.13 35