The mean-risk portfolio optimization model

The mean-risk portfolio optimization model Consider a portfolio of d risky assets and the random vector X = (X 1,X 2,...,X d ) T of their returns. Let E(X) = µ.

The mean-risk portfolio optimization model Consider a portfolio of d risky assets and the random vector X = (X 1,X 2,...,X d ) T of their returns. Let E(X) = µ. Let P be the family of all portfolios consisting of the obove d assets Any (long-short) portfolio in P is uniquelly determined by its weight vector w = (w i ) IR d with i=1 d w i = 1. w i > 0 (w i < 0) represents a long (short) investment. The return of portfolio w is the r.v. Z(w) = d i=1 w ix i. The expected portfolio return is E(Z(w)) = w T µ. Let P m be the family of portfolios in P with E(Z(w)) = m, for some m IR, m > 0. P m := {w = (w i ) IR d, d i=1 w i = 1,w T µ = m}

The mean-risk portfolio optimization model (contd.)

The mean-risk portfolio optimization model (contd.) If ρ equals the portfolio variance we get min w Pm var(z(w))

The mean-risk portfolio optimization model (contd.) If ρ equals the portfolio variance we get min w Pm var(z(w)) If Cov(x) = Σ and the weights are nonnegative (long-only portfolio) we get the Markovitz portfolio optimization model (Markowitz 1952): min w s.t. w T Σw w T µ = m d i=1 w i = 1 If ρ = VaR α, α (0,1) we get the mean-var pf. optimization model min w P m VaR α (Z(w)). Question: What is the relationship between these three portfolio optimization models?

Mean-risk portfolio optimization in the case of elliptically distributed asset returns

Mean-risk portfolio optimization in the case of elliptically distributed asset returns Theorem: (Embrechts et al., 2002) Let M be the set of returns of the portfolii in P := {w = (w i ) IR d, d i=1 w i = 1}. Let the asset returns X = (X 1,X 2,...,X d ) be elliptically distributed, X = (X 1,X 2,...,X d ) E d (µ,σ,ψ) for some µ IR d, Σ IR d d and ψ: IR IR. Then VaR α ist coherent in M, for any α (0.5,1). Theorem: (Embrechts et al., 2002) Let X = (X 1,X 2,...,X d ) = µ+ay be elliptically distributed with µ IR d, A IR d k and a spherically distributed vector Y S k (ψ). Assume that 0 < E(Xk 2 ) < holds k. If the risk measure ρ has the properties (C1) and (C3) and ρ(y 1 ) > 0 for the first component Y 1 of Y, then argmin{ρ(z(w)): w P m } = argmin{var(z(w)): w P m }

Copulas: Definition and basic properties

Copulas: Definition and basic properties Definition: A d-dimensional copula is a distribution function on [0,1] d with uniform marginal distributions on [0, 1].

Copulas: Definition and basic properties Definition: A d-dimensional copula is a distribution function on [0,1] d with uniform marginal distributions on [0, 1]. Equivalently, a copula C is a function C: [0,1] d [0,1], with the following properties: 1. C(u 1,u 2,...,u d ) is mon. increasing in each variable u i, 1 i d. 2. C(1,1,...,1,u k,1,...,1) = u k for any k {1,...,d} and u k [0,1]. 3. The rectangle inequality holds (a 1,a 2,...,a d ) [0,1] d, (b 1,b 2,...,b d ) [0,1] d with a k b k, k {1,2,...,d}: 2... k 1=1 2 ( 1) k1+k2+...+k d C(u 1k1,u 2k2,...,u dkd ) 0, k d =1 where u j1 = a j and u j2 = b j.

Lemma: Let h: IR IR be a monotone increasing function with h(ir) = IR and h : IR IR be the generalized inverse function of h. Then the following statements hold: 1. h is eine monotone increasing left continuous function. 2. h is continuous h is strictly monotone increasing. 3. h is strictly monotone increasing h is continuous. 4. h (h(x)) x 5. h(h (y)) y 6. h is strictly monotone increasing = h (h(x)) = x.