Portfolio optimization Andrea Mazzon Ludwig Maximilians Universitat¨ Munchen¨ Andrea Mazzon Portfolio optimization 1 / 79 These slides are based on Chapter 5 of the publication “Leitfaden fur¨ das Grundwissen Fach Finanzmathematik und Risikobewertung” from the Deutsche Aktuareereinigung, available (in German) online. Further references are: A. J. McNeil, R. Frey, P. Embrechts: Quantitative Risk Management. Princeton University Press, 2. Edition, 2015. H. Follmer,¨ A. Schied: Stochastic Finance - An Introduction in Discrete Time. 4. Edition, De Gruyter, 2016. Andrea Mazzon Portfolio optimization 2 / 79 Main contents 1 Utility maximization Introduction One period model: utility maximization with primary products Utility maximization with derivatives Multi-period case: dynamic Portfolio optimization in a binomial model 2 Portfolio optimization with Markowitz Introduction Efficient portfolios Portfolio selection Criticism to the Markowitz approach 3 Alternative methods for Portfolio optimization 4 Asset pricing Portfolio theory with secure investment Capital asset pricing model Andrea Mazzon Portfolio optimization 3 / 79 1 Utility maximization Introduction One period model: utility maximization with primary products Utility maximization with derivatives Multi-period case: dynamic Portfolio optimization in a binomial model 2 Portfolio optimization with Markowitz Introduction Efficient portfolios Portfolio selection Criticism to the Markowitz approach 3 Alternative methods for Portfolio optimization 4 Asset pricing Portfolio theory with secure investment Capital asset pricing model Andrea Mazzon Portfolio optimization 4 / 79 1 Utility maximization Introduction One period model: utility maximization with primary products Utility maximization with derivatives Multi-period case: dynamic Portfolio optimization in a binomial model 2 Portfolio optimization with Markowitz Introduction Efficient portfolios Portfolio selection Criticism to the Markowitz approach 3 Alternative methods for Portfolio optimization 4 Asset pricing Portfolio theory with secure investment Capital asset pricing model Andrea Mazzon Portfolio optimization 5 / 79 Main idea Goal of an investor is to maximize the expected value of his/her utility at time T , starting from an initial capital v > 0. Further capital injections or withdrawals are not allowed. The investor can choose within a set X of portfolios/positions, identified by real valued random variables in a measurable space (Ω; F). The value of the portfolio at time T for a realization ! is X(!), X 2 X . Andrea Mazzon Portfolio optimization 6 / 79 Risk aversion Suppose X = fX1;X2g, with X1(!) = 100 · 1A(!);X2(!) = 50; for every ! 2 Ω, where A 2 F, P (A) = 0:5 for a reference measure P . In a risk neutral world, any agent would be indifferent between choosing X1 or X2. However, humans are in general not risk neutral, but risk averse. Andrea Mazzon Portfolio optimization 7 / 79 Risk attitude and utility function When you take a choice (in this case, choose a position in X ) you might be willing to maximize your expected utility. In a risk neutral world with rational agents, everything is simple: P Maximize E [X] over X 2 X , But again: the world is not risk neutral! Idea: P Maximize E [u(X)] over X 2 X , for a function u. What about u? Risk averse agent ! u increasing, but concave; Risk neutral agent ! u increasing, linear; Risk lover agent ! u increasing, convex (never the case in our context); Fool/masochist agent ! u decreasing! (Never the case for us, of course) Andrea Mazzon Portfolio optimization 8 / 79 Utility function Definition: Utility function A function u : S ⊂ R ! R [ f1g is called utility function of a risk averse agent if u is strictly increasing and strictly concave. Moreover, here we also suppose u to be continuous. Examples Exponential utility function: −λx u(x) = 1 − e ; x 2 S = R; λ > 0: Logarithmic utility function: u(x) = ln(x); x 2 S = (0; 1): Power utility function: xα u(x) = ; x 2 S = (0; 1); 0 < α < 1: α Andrea Mazzon Portfolio optimization 9 / 79 Formulation of the problem Definition: Preference order Let u be an utility function and P a reference probability measure. The preference order of an investor on X is defined via the von-Neumann-Morgenstern representation P P X Y () E [u(X)] > E [u(Y )]; X; Y 2 X : Optimization problem of the investor with initial capital v > 0: P Maximize E [u(X)] over X 2 X , where X is the set of portfolios values for portfolios built with initial investment v. In order to realise the positions in X , the investor can construct suitable portfolios of primary financial products or trade in derivative products. In a complete market, these two cases result in the same quantity of admissable strategies. In an incomplete market, derivatives offer more flexibility than primary products, providing a richer set of instruments and help to improve the value of the final asset. Andrea Mazzon Portfolio optimization 10 / 79 1 Utility maximization Introduction One period model: utility maximization with primary products Utility maximization with derivatives Multi-period case: dynamic Portfolio optimization in a binomial model 2 Portfolio optimization with Markowitz Introduction Efficient portfolios Portfolio selection Criticism to the Markowitz approach 3 Alternative methods for Portfolio optimization 4 Asset pricing Portfolio theory with secure investment Capital asset pricing model Andrea Mazzon Portfolio optimization 11 / 79 The setting Consider a one-period model with time points 0, T . The market has d + 1 liquid traded primary products with strictly positive prices: π¯ = (π0; π) = (π0; π1; : : : ; πd) at time 0 (deterministic) S¯ = (S0;S) = (S0;S1;:::;Sd) at time T (stochastic). Suppose the product 0 to be risk-free: in particular, π0 = 1, S0 = 1 + r, r > 0. At time 0, the investor chooses a strategy 0 0 1 d d+1 θ¯ = (θ ; θ) = (θ ; θ ; : : : ; θ ) 2 R : The final values of admissible portfolios are random variables in a subset X of d+1 fθ¯ · S¯ > 0 j θ¯ 2 R g: Andrea Mazzon Portfolio optimization 12 / 79 Budget conditions If the investor has initial budget v > 0, the portfolio strategy has to fulfil the condition θ¯ · π¯ ≤ v: The above condition can be replaced with an equality, thinking that in optimum no resources are “wasted”: θ¯ · π¯ = v: Thus the optimal problem can be formulated as d+1 Maximize E[u(θ¯ · S¯)] over fθ¯ 2 R j θ¯ · π¯ = vg, i.e., X = fθ¯ · S¯ > 0 j θ¯ · π¯ = vg: d+1 The last observation allows to replace the optimization problem on R with one d constraint by an optimization problem on R without constraints. Andrea Mazzon Portfolio optimization 13 / 79 Unconstrained problem Consider Si Y i = − πi; i = 1; : : : ; d: 1 + r Since θ¯ · π¯ = v, it holds θ¯ · S¯ = (1 + r)(θ · Y + v): The problem can be thus written as 1 d Maximize E[~u(θ · Y )] over θ ; : : : ; θ 2 R, with u~(y) = u ((1 + r)(y + v)). Andrea Mazzon Portfolio optimization 14 / 79 1 Utility maximization Introduction One period model: utility maximization with primary products Utility maximization with derivatives Multi-period case: dynamic Portfolio optimization in a binomial model 2 Portfolio optimization with Markowitz Introduction Efficient portfolios Portfolio selection Criticism to the Markowitz approach 3 Alternative methods for Portfolio optimization 4 Asset pricing Portfolio theory with secure investment Capital asset pricing model Andrea Mazzon Portfolio optimization 15 / 79 Utility maximization with derivatives While portfolios in the one-period model are composed of primary products on a linear basis, derivatives represent all payout profiles that can be contractually agreed. The space X is now constituted by random variables representing the payoffs at time T of all the derivatives in the market. To simplify the notation, it is assumed that all the payoffs have already been discounted with respect to a suitable numeraire.´ The prices of the payoffs at time t = 0 are calculated using a pricing measure Q equivalent to the reference measure P : if a derivative has payoff X at time t = T , Q it is valued as E [X] in t = 0. For utility function u and initial budget v > 0, we define Q P X (v) := fX 2 X j E [X] = v; E [u(X)] < 1g: We want to find P W0(v) = sup E [u(X)]: X2X (v) Andrea Mazzon Portfolio optimization 16 / 79 Heuristic approach via Lagrange-Ansatz The Lagrangian functional is given by P Q L(λ, X) =E [u(X)] − λ(E [X] − v) dQ =λv + P u(X) − λX : E dP 0 Suppose that u : S = (a; b) ! R, −∞ ≤ a < b ≤ +1, with u invertible and such 0 that limx!a u (x) = +1. Thus the maximization problem is solved by dQ X∗ := (u0)−1 λ ; dP Q ∗ for λ such that E [X ] = v. Andrea Mazzon Portfolio optimization 17 / 79 Examples −γx Exponential utility function, u(x) = 1 − e ; x 2 S = R; γ > 0. ∗ 1 dQ Utility maximizing payoff: X = γ H(QjP ) − ln dP + v, where ( Q ln dQ if Q P H(QjP ) := E dP : +1 otherwise ∗ −γv−H(QjP ) Maximized expected utility: E[u(X )] = 1 − e . Logarithmic utility function: u(x) = ln(x); x 2 S = (0; 1). ∗ dP Utility maximizing payoff: X = v dQ . ∗ Maximized expected utility: E[u(X )] = ln(v) + H(P jQ). Andrea Mazzon Portfolio optimization 18 / 79 1 Utility maximization Introduction One period model: utility maximization with primary products Utility maximization with derivatives Multi-period case: dynamic Portfolio optimization in a binomial model 2 Portfolio optimization with Markowitz Introduction Efficient portfolios Portfolio selection Criticism to the Markowitz approach 3 Alternative methods for Portfolio optimization 4 Asset pricing Portfolio theory with secure investment Capital asset pricing model Andrea Mazzon Portfolio optimization 19 / 79 The setting Take a multi-period model with times t = 0; 1;:::;T , and consider a probability space (Ω; F; F;P ), where F = (Ft)t=0;:::;T is a filtration representing information.