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 ω is X(ω), X ∈ X .
Andrea Mazzon Portfolio optimization 6 / 79 Risk aversion
Suppose X = {X1,X2}, with
X1(ω) = 100 · 1A(ω),X2(ω) = 50,
for every ω ∈ Ω, where A ∈ 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 ∈ X , But again: the world is not risk neutral! Idea: P Maximize E [u(X)] over X ∈ 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 ∪ {∞} 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 ∈ S = R, λ > 0. Logarithmic utility function:
u(x) = ln(x), x ∈ S = (0, ∞).
Power utility function: xα u(x) = , x ∈ S = (0, ∞), 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 ∈ X .
Optimization problem of the investor with initial capital v > 0: P Maximize E [u(X)] over X ∈ 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 ) ∈ R . The final values of admissible portfolios are random variables in a subset X of
d+1 {θ¯ · S¯ > 0 | θ¯ ∈ R }.
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 {θ¯ ∈ R | θ¯ · π¯ = v}, i.e., X = {θ¯ · S¯ > 0 | θ¯ · π¯ = v}. 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 ∈ 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) := {X ∈ X | E [X] = v, E [u(X)] < ∞}. We want to find P W0(v) = sup E [u(X)]. X∈X (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 ≤ +∞, with u invertible and such 0 that limx→a u (x) = +∞. 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 ∈ S = R, γ > 0. ∗ 1 dQ Utility maximizing payoff: X = γ H(Q|P ) − ln dP + v, where ( Q ln dQ if Q P H(Q|P ) := E dP . +∞ otherwise
∗ −γv−H(Q|P ) Maximized expected utility: E[u(X )] = 1 − e . Logarithmic utility function: u(x) = ln(x), x ∈ S = (0, ∞). ∗ dP Utility maximizing payoff: X = v dQ . ∗ Maximized expected utility: E[u(X )] = ln(v) + H(P |Q).
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. Suppose there exist: 0 t A risk free asset defined by St = (1 + r) , t = 0,...,T , with a deterministic interest rate r > 0. A risky asset adapted to F defined by
St = S0 · Y1 ····· Yt, t = 0, 1,...,T,
where Yt can take the two values d, u with 0 < d < 1 + r < u, and Yt are i.i.d. and such that Yt+1 is independent of Ft. It then holds
0 0 St = St−1(1 + r), t = 1,...,T
and
St = St−1Yt, t = 1,...,T.
Andrea Mazzon Portfolio optimization 20 / 79 Equivalent martingale measure
In order for the market to be arbitrage-free and complete, there must exist a measure S Q ∼ P such that S0 is a martingale. It can be seen that it must hold 1 + r − d q := Q(Y = u) = . t u − d Such Q exists and is unique as we have supposed 0 < d < 1 + r < u, and
dQ q n(ω) 1 − q T −n(ω) (ω) = , dP p 1 − p where p := P (Yt = u) and n(ω) is the number of times t = 1,...,T when Yt(ω) = u.
Andrea Mazzon Portfolio optimization 21 / 79 Formulation of the problem
An investor with initial capital v > 0 wants to maximize the expected utility at time T of the value a portfolio made of (S0,S) by a self-financing and adapted strategy θ = (θ0, θ1), where θ 0 0 1 Vt = θt St + θt St, t = 1,...,T. Since θ has to be self-financing it must hold
θ 0 0 1 Vt = θt−1St + θt−1St, t = 1,...,T. (1)
The problem is P θ Maximize E [u(VT )] over all adapted strategies θ satisfying (1). Under the pricing measure Q, the process V θ is also a martingale.
Andrea Mazzon Portfolio optimization 22 / 79 Solution 1: Dynamic programming (1)
θ At every time t = 0,...,T , an investor knows the value Vt of the portfolio at t and θ invests a fraction πt of Vt in the risky asset St, i.e.,
1 θt St πt = θ , t = 1,...,T. Vt Then at time t + 1 it holds (stressing now the dependence of V on π)
π π π Vt+1 = Vt [(1 − πt)(1 + r) + πtYt+1] = Vt [1 + r + πt (Yt+1 − 1 − r)] .
From this representation, the core idea of Dynamic programming is to consider a family of utility maximization problems instead of the original problem.
Andrea Mazzon Portfolio optimization 23 / 79 Solution 1: Dynamic programming (2)
Let’s make the idea more concrete. With
π π Vt = Vt−1 [1 + r + πt−1 (Yt − 1 − r)]
in mind, define
t,π k−1 Vk (x) := xΠj=t [1 + r + πj (Yj+1 − 1 − r)] , k = t + 1,...,T, the value at time k of a strategy starting from time t, when the value of the portfolio at time t is x. Define the set of admissible strategies
t,π At(x) = {π = (πu)t≤u
P t,π Wt(x) := sup E u(VT (x)) . π∈At(x)
Andrea Mazzon Portfolio optimization 24 / 79 Solution 1: Dynamic programming (3)
Have in mind: value function of the utility maximization problem at time t ≤ T is
P t,π Wt(x) := sup E u(VT (x)) , π∈At(x) with t,π k−1 Vk (x) := xΠj=t [1 + r + πj (Yj+1 − 1 − r)] , k = t + 1,...,T. For the family of these value functions, the Bellman principle holds:
P Wt(x) = sup E [Wt+1 (x(1 + r + y(Yt+1 − 1 − r)))], y∈R for all t ≤ T , x ∈ (0, ∞). The Bellman principle allows a recursive calculation of the value functions starting from the final condition WT (v) = u(v).
Andrea Mazzon Portfolio optimization 25 / 79 Solution 2: Martingale method (1)
The idea of the Martingale method is to reduce the problem to the one period problem studied above. It mainly consists of three steps:
Determine the set X (v) of values for the wealth VT at time T , reachable by a self-financing strategy starting with budget v > 0; ∗ determine the optimal reachable wealth VT ; ∗ α∗ ∗ α∗ find a self financing strategy α such that VT = VT where in VT we make explicit the dependence of the terminal wealth on the strategy α (remember that the market is complete).
Andrea Mazzon Portfolio optimization 26 / 79 Solution 2: Martingale method (2)
Step 1 π The value process (Vt )t=0,1,...,T must satisfy for every π ∈ A0(v) the following condition: Q −T π E [(1 + r) VT ] = v. Thus, since the market is complete,
Q −T X (v) := {X > 0 : X FT -measurable with E [(1 + r) X] = v, v > 0}.
Step 2 We have to find V ∗ as ∗ P V (v) = sup E [u(X)]. X∈X (v) This is a static problem can be solved with the Lagrange multipliers method as above.
Step 3 Hedging problem: find a strategy that replicates the optimal value V ∗ found above.
Andrea Mazzon Portfolio optimization 27 / 79 Example: Binomial model with logarithmic utility function
We take u(x) = ln(x), x ∈ S = (0, ∞), and suppose r = 0. dQ First-second step: From the expression of the Radon-Nikodym derivative dP , and applying the Lagrange multipliers method, we find
p nT 1 − p T −nT V ∗(v) = v , q 1 − q
where nT is the number of times when Yt = u, t = 1,...,T . The expected utility from terminal wealth is p 1 − p P [u(V ∗(v))] = ln(v) + pT ln + T (1 − p) ln . E q 1 − q
Third step: The ratio of the wealth invested in St (1 + r)(p − q) π∗ = , t ∈ {1,...,T } t q(1 − q)(u − d)
P ∗ is optimal for E [u(V (v))]. In economic terms, this means that with a logarithmic utility function, a constant proportion of the portfolio value is always invested in the risky asset.
Andrea Mazzon Portfolio optimization 28 / 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 29 / 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 30 / 79 Motivation
Investors in capital markets can invest on a large number of individual financial securities available and can generate portfolios from these. In this context, it is first of all necessary to clarify which effects a portfolio formation has on “risk” and “value”. At the risk level, diversification between the individual securities is central, i.e. the risk of the portfolio can be reduced below the risk of the individual securities by “suitable” portfolio formation. At the risk/value level, aspects of risk/return dominance and efficient portfolios come into play. A second issue concerns the construction of an “optimal” portfolio for a given investor. Both questions are addressed by the classical capital market model of Markowitz. Fields of application of the Markowitz model are the derivation of optimal portfolios from single stocks (standard application: optimal stock portfolios) and the optimal composition of asset classes (asset allocation).
Andrea Mazzon Portfolio optimization 31 / 79 Main characteristics of the model
This is a static one-period model with n financial assets. Financial assets can be divided at will and transaction costs are neglected. Financial assets and portfolios are assessed using the standard deviation of the return as a measure of risk and the expected return value as a measure of value. The “drivers” for diversification in relation to the standard deviation are the correlations of the individual assets. The reference measure is the statistical measure.
Andrea Mazzon Portfolio optimization 32 / 79 Efficient portfolios
Investors prefer: the portfolio with the higher expected return value for the same portfolio risk; the portfolio with the lower risk for the same expected return value (risk aversion).
Definition
A portfolio with return R1 dominates a portfolio with return R2 if one the two following conditions holds:
1 V ar(R1) < V ar(R2) and E[R1] ≥ E[R2]; 2 E[R1] > E[R2] and V ar(R1) ≤ V ar(R2).
Definition A portfolio is efficient if it is not dominated by any other portfolio.
Only efficient portfolios can be optimal portfolios.
Andrea Mazzon Portfolio optimization 33 / 79 The setting
The return of the asset i is denoted by Ri, i = 1, . . . , n.
The proportional investment in the asset i is xi, i = 1, . . . , n. The return of the portfolio is thus
n X RP = xiRi. i=1 Restrictions on investments: Short sales allowed: only n X xi = 1. (2) i=1 No short sales allowed: (2) plus
xi ≥ 0, i = 1, . . . , n. Denote by: µi = E[Ri], σi = σ(Ri), ρi,j = ρ(Ri,Rj ), i = 1, . . . , n, for the assets; µP = E[Rp], σP = σ(RP ), for portfolios.
Andrea Mazzon Portfolio optimization 34 / 79 Mean and variance of returns
It holds: n n X X E[RP ]= xiE[Ri]= xiµi, i=1 i=1 n n X X V ar(RP )= xixj Cov(Ri,Rj ) i=1 j=1 n X X = xiV ar(Ri) + 2 xixj Cov(Ri,Rj ) i=1 1≤i Andrea Mazzon Portfolio optimization 35 / 79 Admissible sets Denote by: D the set of the admissible weights; M = {(E[RP ], σ(RP )) : RP returns of portfolios constructed with weights in D}, the set of admissible (µ, σ). Andrea Mazzon Portfolio optimization 36 / 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 37 / 79 Determination of the efficient frontier Goal: minimize the variance n X 2 X x → Z1(x)= V ar(RP )= xiσi + 2 xixj ρi,j σiσj i=1 1≤i under the conditions Pn Pn E[RP ] = xiE[Ri] = xiµi = r; Pn i=1 i=1 i=1 xi = 1 x1, . . . , xn ≥ 0 (if short sales not allowed) In vectorial terms: T Z1(x) = x Cx → min! subject to xT µ = r, xT e = 1, x ≥ 0. (4) where e = (1,..., 1). No short sales: “quadratic programming”; computer-aided numerical methods necessary. Andrea Mazzon Portfolio optimization 38 / 79 Case I: short sales allowed Short sales allowed: Solution with standard Lagrange approach. Determine local extreme value of the Lagrange function. Alternative formulation: for a given ` > 0 1 Z (x) = `µT x − xT Cx → max! 2 2 subject to xT e = 1. Andrea Mazzon Portfolio optimization 39 / 79 Solution to the alternative formulation (1) We construct the Lagrange function 1 L(x, λ) = `xT µ − xT Cx − λ(xT e − 1). 2 From T Lx = `µ − Cx − λe = 0,Lλ = x e − 1 = 0 we get 1 x(`) = C−1e + `h = x + `h, c 0 where 1 a x = C−1e, c = eT C−1e, h = C−1 µ − e , a = eT C−1µ, b = µC−1µ. 0 c c It follows that 2 2 2 2 µP (`) := µ(`) = µ0 + α`, σP (`) := σ (`) = σ0 + α` , a 2 1 a2 where µ0 = x , σ0 = c , α = b − c . For ` = 0 we get the minimum variance portfolio. Andrea Mazzon Portfolio optimization 40 / 79 Solution to the alternative formulation (2) For ` ≥ 0, it must hold 1 σ2 = σ2 + (µ − µ )2, P 0 α P 0 or equivalently q 2 2 µP = µ0 ± α(σP − σ0 ). The set of efficient portfolios in terms of investments is given by ∗ n D = {x ∈ R : x = x0 + `h, ` ≥ 0}, The set of efficient portfolios in terms of expectation/variance is the upper branch of ∗ n 2 2 2 M ={(µP , σP ) ∈ R : µP = µ0 + α`, σP = σ0 + α` , ` ≥ 0} 1 = (µ , σ ) ∈ M : σ2 = σ2 + (µ − µ )2 . P P P 0 α P 0 including the portfolio with minimum variance. M ∗ is the geometric boundary of the set of all permissible portfolios. The upper branch of M ∗ is called efficient boundary, or curve of the local minimum variance portfolios. Andrea Mazzon Portfolio optimization 41 / 79 Example 1 Andrea Mazzon Portfolio optimization 42 / 79 Example 2: two assets Andrea Mazzon Portfolio optimization 43 / 79 Example (1) Consider three assets whose returns are uncorrelated, with identical variance 0.1. Suppose µ1 = 0.2, µ2 = 0.1, µ3 = 0.3. The Lagrangian is 1 L(x , x , x , λ) = `(0.2x + 0.1x + 0.3x ) − (0.1x2 + 0.1x2 + 0.1x2) 1 2 3 1 2 3 2 1 2 3 − λ(x1 + x2 + x3 − 1). The Lagrangian equations are therefore L = 0.2` − 0.1x − λ = 0 x1 1 Lx2 = 0.1` − 0.1x2 − λ = 0 Lx3 = 0.3` − 0.1x3 − λ = 0 Lλ = 1 − x1 − x2 − x3 = 0. We find the solution x = 1 1 3 1 x2 = 3 − ` x = 1 + ` 3 3 1 1 λ = 5 ` − 30 . Andrea Mazzon Portfolio optimization 44 / 79 Example (2) The weights for the variance minimizing portfolio are 1 1 1 x = , x = , x = . 1 3 2 3 3 3 Hence, we have 1 1 σ2 = (x2 + x2 + x2) = , µ = 0.2x + 0.1x + 0.3x = 0.2. 0 1 2 3 10 30 0 1 2 3 In general it holds 1 7 µ (`) = 0.2 + 0.2`, σ2 (`) = 0.2`2 + = 5(µ (`))2 − 2µ (`) + . P P 30 P P 30 We obtain s 1 µ (`) = 0.2 ± 0.2 σ2 (`) − P P 30 Andrea Mazzon Portfolio optimization 45 / 79 Efficient frontier Andrea Mazzon Portfolio optimization 46 / 79 No short sales allowed The analysis can be continued using the data of the previous example. Non-negativity is not always guaranteed for those weights (as a function of `). In particular, 1 1 x ≥ 0 ⇐⇒ ` ≤ , x ≥ 0 ⇐⇒ ` ≥ − . 2 3 3 3 1 1 The previous solution can be taken only for the case − 3 ≤ ` ≤ 3 . The general solution is illustrated in the following figure. The marginal portfolios are located on three connected root function segments. Andrea Mazzon Portfolio optimization 47 / 79 Efficient frontier with no short sales Andrea Mazzon Portfolio optimization 48 / 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 49 / 79 Main idea Each investor must perform a “trade-off” between “risk” and “return”, i.e., investor selects a characteristic position (µ, σ) on the efficient frontier. In order to determine this position, the investor must directly or indirectly make his/her preferences explicit regarding the valuation of risky investments. Possible approach: concrete specification of an EV-preference function H(E[R], σ(R)), for example 2 H(E[R], σ(R)) = E[R] − aσ (R), a > 0. Once chosen H(E[R], σ(R)), optimum problem for (µ, σ) under the condition (µ, σ) ∈ M, M set of admissible positions. Andrea Mazzon Portfolio optimization 50 / 79 Example: two assets case Let 2 V (R) = H(E[R], σ(R)) = E[R] − aσ (R). Consider two assets and a portfolio made of the two assets with investments x1 := x, x2 = 1 − x, x ∈ R. Let R be the return of the portfolio and σ12 := Cov(R1,R2). We have: 2 2 2 2 V (R) = µ2 + (µ1 − µ2)x − a(x σ1 + (1 − x) σ2 + 2x(1 − x)σ12). dV (x) 2 2 dx = (µ1 − µ2) − 2axσ1 + 2a(1 − x)σ2 − 2aσ12 + 4axσ12. dV (x) We want to choose x such that dx = 0, and we get 2 2 2a(σ12 − σ2 ) − (µ1 − µ2) µ1 − µ2 − 2a(σ12 − σ2 ) x = 2 2 = 2 . 4aσ12 − 2a(σ1 + σ2 ) 2aσ (R1 − R2) (General) problem: how to choose a? Andrea Mazzon Portfolio optimization 51 / 79 Shortfall restriction In practice, it is often desirable to explicitly limit the risk taken for risk control purposes. For investors it is often more intuitive to specify a target return z or a confidence level 1 − , than to specify a benefit function or a risk tolerance parameter. It is also consistent with the “Solvency II logic” to measure risk separately and to use the Value at Risk as a measure of risk. The level of risk must not exceed a certain tolerated level. In the simplest case, this leads to a limitation of the shortfall probability (“shortfall restriction”) in the form P (R < z) ≤ . If the distribution of R is continuous, the shortfall restriction can also be formulated as an equivalent Value-at-Risk restriction: P (R < z) ≤ ⇐⇒ V aR(R) ≤ −z. Andrea Mazzon Portfolio optimization 52 / 79 Shortfall restriction: normal case 2 If R ∼ N(µ, σ ), it holds −1 V aR(R) = −µ − σΦ (). Thus the shortfall restriction is µ ≥ z − σΦ−1(). This means that the set of admissible (µ, σ) is now bounded by the straight line µ = z − σΦ−1(). Only the sector above the straight line (included) is then still permissible in terms of a controlled shortfall probability. The optimal portfolio then corresponds to the upper intersection of the efficient edge with the shortfall line. Andrea Mazzon Portfolio optimization 53 / 79 Shortfall restriction Andrea Mazzon Portfolio optimization 54 / 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 55 / 79 Main criticism Point 1: use of standard deviation as a risk measure The Markowitz basic model is based on standard deviation. The standard deviation is a measure of risk that is particularly important in the context of elliptical distributions (especially normal), which are distributed symmetrically. In practice, however, there are also cases in which there is a significant skewness. The standard deviation does not take into account these asymmetries in risk measurement. Point 2: analytic solutions far from reality Optimized portfolios often have extreme allocations. Without short selling restrictions, very high short selling positions arise. With short selling restrictions the diversification is low. Andrea Mazzon Portfolio optimization 56 / 79 Main criticism Point 3: Lack of robustness The optimization is very sensitive to the input data. The variation of the input data results in completely different portfolios. Because of this reason the input data, especially the estimated expected values, play a key role for the quality of the portfolio optimization. There exist theoretical approaches towards robustification: treatment of the “estimation error problem”, Black/Litterman method, portfolio heuristics such as minimum variance, equal weight or risk parity. Point 4: Covariance matrix estimation n(n+1) For n assets, one must estimate 2 entries of the covariance matrix. This corresponds to approx. 5, 000 covariances for 100 assets and over 30, 000 covariances for 250 individual titles. Due to this high dimension, in practice no individual estimates are made, but rather a reduction to “common factors” (multifactor models). This considerably reduces the diversity of the (co-)variances to be estimated (and the problem of estimation errors). Andrea Mazzon Portfolio optimization 57 / 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 58 / 79 Extensions to the model A first class of model generalizations uses more general risk measures, but retains the expected return as a value measure. On the other hand, some models use more general value measures. Another alternative is to combine risk and value measures into a risk-adjusted performance measure (standard example: Sharpe ratio) and to maximize this performance measure. Here we retain the expected value as a value measure and focus on the Average Value at Risk as a risk measure (remember: differently from Value at Risk, it is coherent!) An alternative use of AVaR in the portfolio optimization context is to combine the Markowitz EV model with an AVaR restriction. Andrea Mazzon Portfolio optimization 59 / 79 Average Value at Risk as risk measure General form: AV aRα(RP (x)) → min! subject to xT µ = r, xT e = 1, x ≥ 0. (5) Problem: The determination of the AVaR is in general dependent on the distribution assumption. In the literature, based on the results of Uryasev-Rockafellar 1 a sample-based variant has come into place, which also has the advantage that it leads to a linear program. 1R. T. Rockafellar, S. Uryasev, Optimization of conditional value-at-risk. The Journal of Risk, 2, 2000. Andrea Mazzon Portfolio optimization 60 / 79 Average Value-at-Risk optimization Remember that 1 + 1 + AV aRα(X) = E (−V aRα(X) − X) +V aRα(X) = min E (` − X) − α` . α α `∈R Considering now the loss L = −X, it makes sense to define 1 + AV aRα(L) := min I + E[(L − I) ]. I∈R α In particular, the minimization problem allows to compute the AVaR without first calculating the VaR. Andrea Mazzon Portfolio optimization 61 / 79 Average Value-at-Risk optimization with linear programming Consider a sample {r1,..., rs} of a vector R = {R1,...,Rn} of returns of n individual assets in the portfolio. T If we denote the vector of investments with x = (x1, . . . , xn) , the loss variables T are given by `i = −x ri, i = 1, . . . , s. + We define the quantities zi = (`i − I) , i = 1, ..., s. + 1 Ps In this way, E (` − X) can be estimated by s i=1 zi, 1 Ps In the same way, E[R] is estimated by ¯r = s i=1 ri. Thus, the minimization problem can be expressed in terms of I, zi,i = 1, . . . , s in the following way: s 1 X I + z → min! αs i i=1 subject to zi ≥ 0, i = 1, . . . , s, T zi + x ri + I ≥ 0, i = 1, . . . , s, xT ¯r = r, xT e = 1, x ≥ 0. Andrea Mazzon Portfolio optimization 62 / 79 Alternative models are not relevant for Elliptical distributions Theorem Let ρ be a positive homogenous, translation invariant and distribution invariant risk measure. Also let X be a random variable with elliptic distribution. Then it holds ρ(X) = E[−X] + k(ρ)σ(X), where k(ρ) is a constant depending on a measure ρ. a aFor a proof, see Theorem 8.28 in A.J. McNeil, R. Frey, P. Embrechts, Quantitative Risk Management. Princeton University Press, 2. edition, 2015. For each fixed expected value, each portfolio minimizes the variance as a function of the vector of investments x, thus also minimizing the risk measure ρ. This implies that alternative mean/risk approaches only become relevant outside the class of elliptical distributions. Andrea Mazzon Portfolio optimization 63 / 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 64 / 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 65 / 79 Introduction Extension of the investment spectrum: The investment spectrum of the Markowitz model is based only on purely risky investments (variance of the return > 0) We now extend the model to include a risk-free investment (variance of the return = 0) at the safe interest rate r0. At the interest rate r0, any amount can be both invested and borrowed (perfect capital market). It is important to note that the term “risk-free/sure investment” refers purely to volatility. Andrea Mazzon Portfolio optimization 66 / 79 Portfolios’ combination and Sharp ratio Fix a “risky” portfolio P , with return RP , made of assets with variance > 0. We consider a portfolio which is a combination of P and of units of the safe investment. Call 0 ≤ x < ∞ the investment on P . Thus the return of the combined portfolio is R = xRP + (1 − x)r0. The return of the combined portfolio has expectation µ and variance σ2 with µ = xµP + (1 − x)r0 = r0 + x(µP − r0), 2 2 2 σ = x σP . It follows x = σ and thus σP µP − r0 µ = r0 + σ. σP Andrea Mazzon Portfolio optimization 67 / 79 The set of reachable portfolios under this enlargement is thus µP − r0 M˜ = (µ, σ): µ = r0 + σ, (µP , σP ) ∈ M . σP What about the set of the efficient portfolios M˜ ∗? The slope of the straight line in M˜ is Sharp ratio = µP −r0 . σP Consider now the efficient frontier for totally risky portfolios, i.e., 1 M ∗ = (µ , σ ) ∈ M : σ2 = σ2 + (µ − µ )2 . P P P 0 α P 0 ∗ It can be seen that for given r0 > 0, there exists one element (=portfolio) T in M with maximum Sharp ratio. It holds ∗ µT − r0 M˜ = (µ, σ): µ = r0 + σ . σT T is called tangential portfolio: the straight line M˜ ∗ is tangent to M ∗ at T . Andrea Mazzon Portfolio optimization 68 / 79 Tangential portfolio Andrea Mazzon Portfolio optimization 69 / 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 70 / 79 The setting The starting point for the one-period model is that of the portfolio theory with safe investment (case r0 < µ0). In addition, the following assumptions are made: There are n assets and m EV investors in the market with budgets Vi > 0, Pm i = 1, . . . , m, V = i=1 Vi. The view of investors on the capital market with regard to the expected values, variances and covariances of assets are homogeneous, i.e. all investors estimate r0 , E[Ri], V ar(Ri) and Cov(Ri,Rj ) identically. Market equilibrium is assumed in t = 0: the prices in t = 0 adjust in such a way that the market is cleared: supply = demand. Andrea Mazzon Portfolio optimization 71 / 79 Market portfolio Definition The Market portfolio is a portfolio consisting of a weighted sum of every asset in the market, with weights given by the vector xM := (P1/P, . . . , Pn/P ), Pn where Pi is the market value of the asset i, i = 1, . . . , n, and P = i=1 Pi. The return of the Market portfolio is denoted by RM . Remark Under the assumption of market efficiency, the Market portfolio is equal to the Tangential portfolio: RT = RM . Andrea Mazzon Portfolio optimization 72 / 79 Characterization of the efficient frontier: Capital market line From RM = RT , we have the set of optimal portfolios is characterized by returns satisfying E[RM ] − r0 E[R] = r0 + σ(R). σ(RM ) Also called capital market line: expresses returns in terms of standard deviation, for optimal portfolios. In capital market equilibrium, a linear relationship applies to the optimal portfolios: for a higher expected return, a proportionally higher risk must be accepted. Andrea Mazzon Portfolio optimization 73 / 79 Market portfolio Andrea Mazzon Portfolio optimization 74 / 79 Characterization of any portfolio: Security market line For every portfolio or single asset, it holds: E[R] = r0 + βR(E[RM ] − r0), where Cov(R,RM ) βR := V ar(RM ) is the Beta factor. Exploiting the fact that for an efficient portfolio P it holds σP = xσM , where x is the amount invested on the market / tangential portfolio, it can be seen that the Security market line implies the Capital market line. The Security market line differs with respect to the Capital market line in the “risk measure” considered: here expected returns in terms of Beta factor. Andrea Mazzon Portfolio optimization 75 / 79 Market portfolio Andrea Mazzon Portfolio optimization 76 / 79 Beta factor and systematic risk Systematic risk: market risks that cannot be diversified away. Is related to market fluctuations. Interest rates, recessions, and wars are examples of systematic risks. Unsystematic Risk: relates to individual stocks. It represents the component of a stock’s return that is not correlated with general market fluctuations. Note that the Beta factor measures how much the price of a particular portfolio jumps up and down compared with how much the entire stock market jumps up and down. In this way, only systematic risk is assessed by the market and included in prices. β > 0: movement in the same direction as the market as a whole (both when market prices fall and when they rise). β < 0: opposite tendency to the market (practically non-existent) Andrea Mazzon Portfolio optimization 77 / 79 Risk premium and equilibrium price The quantity E[R] − r0 = βR(E[RM ] − r0), is called risk premium of the underlying portfolio. It is the “excess return” demanded by investors in the capital market equilibrium: return in excess with respect to the return on a secure investment, which is required for enter into a risky investment in an individual share or a share portfolio. Let V denote the random end-of-period value of a portfolio of financial instruments under consideration, and R its return. Thus the market-clearing equilibrium price of this portfolio at the beginning of the period is [V ] P = E . 1 + r0 + βR(E[RM ] − r0) The price equation of the CAPM allows the foundation of a risk-adjusted discount factor. Andrea Mazzon Portfolio optimization 78 / 79 Criticism and further developments Since the CAPM is based on the Markowitz approach, the points of criticism discussed above apply to the CAPM. The central criticism of the CAPM, however, is that the beta factor is “the” central price-determining factor. The criticism of the empirical validity of the CAPM led to further development of the CAPM both on a theoretical level and on an empirical level. Andrea Mazzon Portfolio optimization 79 / 79