- February 9, 2021

General Approach Markowitz Model MATH 11158: Optimization Methods in Finance Portfolio Optimization1 Akshay Gupte School of Mathematics, University of Edinburgh Week 3 : 25 January, 2021 1Chapter 6 in the textbook Email: [email protected] 1 / 40 General Approach Markowitz Model General Approach 2 / 40 General Approach Markowitz Model What is Portfolio Optimization ? In arbitrage detection, we created a “good” portfolio with zero risk But is it realistic to select a portfolio (i.e., make an investment) without accounting for any risk in our decisions ? Financial markets are highly nondeterministic Question: What makes a good investment? • High (expected) return • Low risk • Something else? In fact we would simultaneously like high return and low risk. Improving one often results in worse performance on the other2. Select a portfolio that strikes a balance between these two objectives =⇒ Multicriteria (or Pareto) optimization 2There ain’t no such thing as a free lunch 3 / 40 General Approach Markowitz Model What is Portfolio Optimization ? In arbitrage detection, we created a “good” portfolio with zero risk But is it realistic to select a portfolio (i.e., make an investment) without accounting for any risk in our decisions ? Financial markets are highly nondeterministic Question: What makes a good investment? • High (expected) return • Low risk • Something else? In fact we would simultaneously like high return and low risk. Improving one often results in worse performance on the other2. Select a portfolio that strikes a balance between these two objectives =⇒ Multicriteria (or Pareto) optimization 2There ain’t no such thing as a free lunch 3 / 40 General Approach Markowitz Model What is Portfolio Optimization ? In arbitrage detection, we created a “good” portfolio with zero risk But is it realistic to select a portfolio (i.e., make an investment) without accounting for any risk in our decisions ? Financial markets are highly nondeterministic Question: What makes a good investment? • High (expected) return • Low risk • Something else? In fact we would simultaneously like high return and low risk. Improving one often results in worse performance on the other2. Select a portfolio that strikes a balance between these two objectives =⇒ Multicriteria (or Pareto) optimization 2There ain’t no such thing as a free lunch 3 / 40 General Approach Markowitz Model What is Portfolio Optimization ? In arbitrage detection, we created a “good” portfolio with zero risk But is it realistic to select a portfolio (i.e., make an investment) without accounting for any risk in our decisions ? Financial markets are highly nondeterministic Question: What makes a good investment? • High (expected) return • Low risk • Something else? In fact we would simultaneously like high return and low risk. Improving one often results in worse performance on the other2. Select a portfolio that strikes a balance between these two objectives =⇒ Multicriteria (or Pareto) optimization 2There ain’t no such thing as a free lunch 3 / 40 General Approach Markowitz Model Assumptions on Market Conditions • Rational decision-makers : investors want to maximise return while reducing the risks associated with their investment • No arbitrage : cannot make a costless, riskless profit • Risky securities : S1, . . . , Sn for n ≥ 2, whose future returns are uncertain. There is no risk-free asset S0 in the portfolio • Equilibrium : supply equals demand for securities • Liquidity : any # of units of a security can be bought and sold quickly • Access to information : rapid availability of accurate information • Price is efficient : Price of security adjusts immediately to new information, and current price reflects past information and expected further behaviour • No transaction costs and taxes : transaction costs are assumed to be negligible compared to value of trades and are ignored. No taxes (capital-gains etc.) on transactions 4 / 40 General Approach Markowitz Model Notation We have £1 to invest3 in n risky securities S1, . . . ,Sn Goal is to select a portfolio x = (x1, . . . , xn) ∈ Rn, where xi = amount invested in S i Intuitive to think of xi ≥ 0, however, doing so means we do not allow short-selling on S i , and we generally allow short-selling of assets. Hence, x ≥ 0 is not imposed as a constraint always. • two time periods, now (time 0) and future (time 1) • ri : Ω→ R is random variable for return of asset i (e.g., a 5% return means r = 1.05). Probability distribution of ri usually unknown µi = E[ri ], expected return of S i • r := r(ω) = (r1(ω), . . . , rn(ω)) is vector of random returns µ = (µ1, . . . , µn) ∈ Rn is vector of means 3In our analysis, investment of £1 can be scaled to £b for any b > 0 5 / 40 General Approach Markowitz Model Notation We have £1 to invest3 in n risky securities S1, . . . ,Sn Goal is to select a portfolio x = (x1, . . . , xn) ∈ Rn, where xi = amount invested in S i Intuitive to think of xi ≥ 0, however, doing so means we do not allow short-selling on S i , and we generally allow short-selling of assets. Hence, x ≥ 0 is not imposed as a constraint always. • two time periods, now (time 0) and future (time 1) • ri : Ω→ R is random variable for return of asset i (e.g., a 5% return means r = 1.05). Probability distribution of ri usually unknown µi = E[ri ], expected return of S i • r := r(ω) = (r1(ω), . . . , rn(ω)) is vector of random returns µ = (µ1, . . . , µn) ∈ Rn is vector of means 3In our analysis, investment of £1 can be scaled to £b for any b > 0 5 / 40 General Approach Markowitz Model Notation We have £1 to invest3 in n risky securities S1, . . . ,Sn Goal is to select a portfolio x = (x1, . . . , xn) ∈ Rn, where xi = amount invested in S i Intuitive to think of xi ≥ 0, however, doing so means we do not allow short-selling on S i , and we generally allow short-selling of assets. Hence, x ≥ 0 is not imposed as a constraint always. • two time periods, now (time 0) and future (time 1) • ri : Ω→ R is random variable for return of asset i (e.g., a 5% return means r = 1.05). Probability distribution of ri usually unknown µi = E[ri ], expected return of S i • r := r(ω) = (r1(ω), . . . , rn(ω)) is vector of random returns µ = (µ1, . . . , µn) ∈ Rn is vector of means 3In our analysis, investment of £1 can be scaled to £b for any b > 0 5 / 40 General Approach Markowitz Model • Σ : covariance matrix for random vector r Σij = Cov(ri , rj) for all i 6= j Σij = E[(ri − µi )(rj − µj)] Σii = Var[ri ] for all i = 1, . . . , n Σi = E[(ri − µi )2] = E[r2i ]− (E[ri ])2 = E[r2i ]− µ2i Fact Covariance matrix Σ is symmetric and positive semidefinite. x>Σx = x> E[(r − µ)(r − µ)>]︸ ︷︷ ︸ Σ x = E[x>(r − µ)(r − µ)>x ] = E[(r>x − µ>x)2] = E[(r(x)− E[r(x)])2] = Var[r>x ] ≥ 0 6 / 40 General Approach Markowitz Model • Σ : covariance matrix for random vector r Σij = Cov(ri , rj) for all i 6= j Σij = E[(ri − µi )(rj − µj)] Σii = Var[ri ] for all i = 1, . . . , n Σi = E[(ri − µi )2] = E[r2i ]− (E[ri ])2 = E[r2i ]− µ2i Fact Covariance matrix Σ is symmetric and positive semidefinite. x>Σx = x> E[(r − µ)(r − µ)>]︸ ︷︷ ︸ Σ x = E[x>(r − µ)(r − µ)>x ] = E[(r>x − µ>x)2] = E[(r(x)− E[r(x)])2] = Var[r>x ] ≥ 0 6 / 40 General Approach Markowitz Model Admissible Portfolios The set of feasible portfolios is denoted by the set X ⊂ Rn Budget constraint is always included n∑ i=1 xi = 1, or n∑ i=1 xi = b if initial is $b • No short selling : xi ≥ 0 • Short selling allowed : xi ≥ −`i • Diversification : xi ∈ {0} ∪ [`i , ui ], such a semi-continuous variable can be modeled using integer programming Unless stated otherwise, we assume X = {x ∈ Rn : ∑ni=1 xi = 1} 7 / 40 General Approach Markowitz Model Return of a Portfolio Return of a portfolio x is a random variable that is a linear function of x Random return on x = sum of returns on each asset R(x) = n∑ i=1 rixi = r >x Expected return on x E[R(x)] = n∑ i=1 E[rixi ] = n∑ i=1 E[ri ]xi = n∑ i=1 µixi = µ >x 8 / 40 General Approach Markowitz Model Risk of a Portfolio Risk of a portfolio is given by a risk function (risk measure) Risk : x ∈ X → R, where Risk(x) = Risk(R(x)) Commonly used risk measure is variance Risk(x) = Var[R(x)] More generally, we want risk measures to be convex and a few other properties Convexity implies that diversification reduces risk Risk ( 1 2 x + 1 2 x ′ ) ≤ 1 2 Risk(x) + 1 2 Risk(x ′) 9 / 40 General Approach Markowitz Model Optimizing Risk-Return Tradeoff Want to maximize expected return while minimizing risk There are three kinds of problems we could solve max x∈X E[R(x)] s.t. Risk(x) ≤ “risk budget” min x∈X Risk(x) s.t. E[R(x)] ≥ “target return” max x∈X E[R(x)]− δRisk(x) “risk vs return trade-off” 10 / 40 General Approach Markowitz Model Estimating Data How to calculate mean, covariance, etc. of asset return rates ri for future time T + 1? Sampling : Use historical returns r(t) for t ∈ [t1, t2] with 0 ≤ t1 < t2 ≤ T as a sample and take sample mean and variance. Linear Factor Models : Linear regression using some underlying factors that affect return rates (e.g., Fama-French three factor model) 11 / 40 General Approach Markowitz Model Efficient Portfolios 12 / 40 General Approach Markowitz Model Efficiency Definition Portfolio x is efficient if is satisfies one of the following conditions: (1) it has the maximum expected return among all admissible portfolios of the same (or smaller) risk: max x µ>x s.t. Risk(x) ≤ σ2 (EP1) x ∈ X (2) it has the minimum variance among all admissible portfolios with the same (or larger) expected return: min x Risk(x) s.t. µ>x ≥ R (EP2) x ∈ X 13 / 40 General Approach Markowitz Model Visualization via Efficient Frontier We can plot the efficient portfolios x on a “return-vs-risk” diagram. This is called the Efficient Frontier It shows the highest return achievable for a given level of risk, or the lowest risk achievable for a given (target) return. y-axis : Expected return of portfolio x-axis : Some risk measure of return of portfolio 14 / 40 General Approach Markowitz Model Expected Return (µ>x) vs Risk (Risk(x) = variance of x) for 10000 random portofolios ERet(x) Risk(x) 15 / 40 General Approach Markowitz Model →Efficient Frontier Risk(x) ERet(x) 16 / 40 General Approach Markowitz Model 3rd version of Efficient Portfolio: Risk-Adjusted Return max x µ>x − δRisk(x)︸ ︷︷ ︸ risk-adjusted return s.t. x ∈ X (EP3) • Parameter δ > 0 sets the relative importance of return and risk • Choose a small value of δ if you have small sensitivity to risk • Choose a large value of δ if you have large sensitivity to risk • Problem: neither return nor risk are controlled directly 17 / 40 General Approach Markowitz Model Equivalence of the 3 Models Theorem For any convex risk measure, the three characterisations (EP1), (EP2) and (EP3) of Efficient Portfolios are equivalent, i.e., • A portfolio that is efficient under one characterisation is also efficient under the other two, • they lead to the same efficient frontier. Let us see why EP1 ⇐⇒ EP3. Let x∗ be an efficient portfolio under (EP1), that is, there is σ > 0, s.t. x∗ is the (unique) solution to max x∈X µ>x s.t. Risk(x) ≤ σ2 (EP1) is convex, therefore we can dualise the constraint, i.e. ∃λ∗ > 0 such that x∗ is also the solution to max x∈X µ>x − λ∗ ( Risk(x)− σ2 ) = σ2λ∗ + max x∈X µ>x − λ∗Risk(x) that is x∗ is also a solution to (EP3) for δ = λ. 18 / 40 General Approach Markowitz Model Risk vs Return Curve 19 / 40 General Approach Markowitz Model Minimum Risk as a Function of Target Return Consider this family of optimization problems parameterized by R: σ(R) = min x Risk(x) (†) s.t. µ>x ≥ R x ∈ X • σ(R) is the smallest risk of an admissible portfolio whose expected return is at least R • σ(R) is the functional form of the Efficient Frontier • Define X (R) = {x ∈ Rn : µ>x ≥ R, x ∈ X} This set is nonempty if and only if there exists an admissible portfolio (x ∈ X ) yielding return at least R Let Rmax > 0 be the largest R for which X (R) is nonempty. 20 / 40 General Approach Markowitz Model Convexity of σ(R) Theorem The function σ(R) : (−∞,Rmax ]→ R is convex when Risk is a convex risk measure. 21 / 40 General Approach Markowitz Model Proof of Convexity of σ(R) We proceed directly from definition of convexity. Fix arbitrary 0 < α < 1 and R1, R2 ≤ Rmax . We need to show that σ(αR1 + (1− α)R2︸ ︷︷ ︸ R3 ) ≤ ασ(R1) + (1− α)σ(R2). (1) Let f (x) = Risk(x) and let xi for i = 1, 2, 3 be the optimal solution in (†) for R = Ri . Clearly, f (xi ) = σ(Ri ), and hence (1) becomes f (x3) ≤ αf (x1) + (1− α)f (x2). We will show that the convex combination of portfolios x1 and x2: y = αx1 + (1− α)x2 is feasible for R3 (so that y ∈ X (R3)) but Risk(y) ≥ Risk(x3) 22 / 40 General Approach Markowitz Model Since xi is feasible for (†) with right hand side R = Ri , we have µT xi ≥ Ri , i = 1, 2. (2) xi ∈ X , i = 1, 2. (3) By adding an α multiple of the first inequality in (2) to the (1− α) multiple of the second inequality, we get µ>y = µ>(αx1 + (1− α)x2) ≥ αR1 + (1− α)R2 = R3. (4) Moreover, since X is a convex set, from (3) we get y = αx1 + (1− α)x2 ∈ X . (5) (4) and (5) combined say that y ∈ X (R3). Since x3 is the optimal point for (†) with R = R3, we have f (x3) ≤ f (y). Finally, since f is convex, f (x3) ≤ f (y) = f (αx1 + (1− α)x2) ≤ αf (x1) + (1− α)f (x2). 23 / 40 General Approach Markowitz Model Markowitz Model 24 / 40 General Approach Markowitz Model Portfolio Optimization as a Convex QP Portfolio Theory was pioneered by Harry Markowitz in 1950’s through his seminal paper4 which won a Nobel Prize in Economics in 1990 It uses Risk(x) = Var[R(x)] to find efficient portfolios min x x>Σx s.t. µ>x ≥ R (EP2) x ∈ X Similarly for EP1 and EP3 4Markowtiz, “Portfolio selection”, J. Finance, 1952. https://doi.org/10.2307/2975974 25 / 40 General Approach Markowitz Model Variance vs Standard Deviation Standard deviation ( √ x>Σx) instead of the variance x>Σx). Note: √ x>Σx is strictly convex! (try proving it!)) max x µ>x s.t. x>Σx ≤ σ2 (EP1) x ∈ X max x µ>x s.t. √ x>Σx ≤ σ (EP1′) x ∈ X (EP1) and (EP1’) are clearly equivalent min x x>Σx s.t. µ>x ≥ R (EP2) x ∈ X min x √ x>Σx s.t. µ>x ≥ R (EP2′) x ∈ X (EP2) and (EP2’) are clearly equivalent (lead to the same x∗ for given R) 26 / 40 General Approach Markowitz Model max x µ>x − δx>Σx (EP3) s.t. x ∈ X max x µ>x − δ′ √ x>Σx (EP3′) s.t. x ∈ X • (EP3) and (EP3’) are equivalent : lead to the same efficient frontier. • Let x∗ = x∗(δ) be the optimal portfolio in (EP3) for a given δ > 0. Then there exists a δ′ > 0 such that x∗ is the solution to (EP3’). • (EP3’) is dual to (EP1’)/(EP2’), as with variance • In some sense, the standard deviation is the correct risk measure to use. But variance leads to nicer (easier) optimization problems. 27 / 40 General Approach Markowitz Model max x µ>x − δx>Σx (EP3) s.t. x ∈ X max x µ>x − δ′ √ x>Σx (EP3′) s.t. x ∈ X • (EP3) and (EP3’) are equivalent : lead to the same efficient frontier. • Let x∗ = x∗(δ) be the optimal portfolio in (EP3) for a given δ > 0. Then there exists a δ′ > 0 such that x∗ is the solution to (EP3’). • (EP3’) is dual to (EP1’)/(EP2’), as with variance • In some sense, the standard deviation is the correct risk measure to use. But variance leads to nicer (easier) optimization problems. 27 / 40 General Approach Markowitz Model max x µ>x − δx>Σx (EP3) s.t. x ∈ X max x µ>x − δ′ √ x>Σx (EP3′) s.t. x ∈ X • (EP3) and (EP3’) are equivalent : lead to the same efficient frontier. • Let x∗ = x∗(δ) be the optimal portfolio in (EP3) for a given δ > 0. Then there exists a δ′ > 0 such that x∗ is the solution to (EP3’). • (EP3’) is dual to (EP1’)/(EP2’), as with variance • In some sense, the standard deviation is the correct risk measure to use. But variance leads to nicer (easier) optimization problems. 27 / 40 General Approach Markowitz Model max x µ>x − δx>Σx (EP3) s.t. x ∈ X max x µ>x − δ′ √ x>Σx (EP3′) s.t. x ∈ X • (EP3) and (EP3’) are equivalent : lead to the same efficient frontier. • Let x∗ = x∗(δ) be the optimal portfolio in (EP3) for a given δ > 0. Then there exists a δ′ > 0 such that x∗ is the solution to (EP3’). • (EP3’) is dual to (EP1’)/(EP2’), as with variance • In some sense, the standard deviation is the correct risk measure to use. But variance leads to nicer (easier) optimization problems. 27 / 40 General Approach Markowitz Model Positive Definiteness Condition Covariance matrix Σ is always psd. What if we require it to be positive definite ? What does it mean for the Markowitz model ? Positive definite means that x>Σx 6= 0 for all x 6= 0. If x is a portfolio then its variance x>Σx = 0 means that that there are some “redundancies” in the model ? returns of some assets depend deterministically on the returns of others. We can keep removing these assets from the model until we get positive definiteness. 28 / 40 General Approach Markowitz Model Positive Definiteness Condition Covariance matrix Σ is always psd. What if we require it to be positive definite ? What does it mean for the Markowitz model ? Positive definite means that x>Σx 6= 0 for all x 6= 0. If x is a portfolio then its variance x>Σx = 0 means that that there are some “redundancies” in the model ? returns of some assets depend deterministically on the returns of others. We can keep removing these assets from the model until we get positive definiteness. 28 / 40 General Approach Markowitz Model Illustrative Example 29 / 40 General Approach Markowitz Model Example Consider investing into an index fund of stocks (S&P 500), bonds (10y US Treasury Bond) and money market CDs (1-day Federal Funds Rate). Step 1. Get Historical Data Ii,t = price of asset i = 1, . . . , n at time t = 0, 1, . . . ,T S (asset 1) B (asset 2) MM (asset 3) 1960 (t = 0) 20.26 262.94 100.00 1961 (t = 1) 25.69 268.73 102.33 1962 (t = 2) 23.43 284.09 105.33 1963 (t = 3) 28.75 289.16 108.89 … … … … 2003 (t = 43) 1622.94 5588.19 1366.73 30 / 40 General Approach Markowitz Model Step 2. Transform into Historical (yearly) Return Rates ri,t = Ii,t Ii,t−1 S (asset 1) B (asset 2) MM (asset 3) 1960 (t = 0) – – – 1961 (t = 1) 1.2681 1.0220 1.0233 1962 (t = 2) 0.9122 1.0572 1.0293 1963 (t = 3) 1.2269 1.0179 1.0338 … … … … 31 / 40 General Approach Markowitz Model Step 3. Estimate Mean Return Rates r¯i = 1 T >∑ t=1 ri,t︸ ︷︷ ︸ arithmetic mean µi = >∏ t=1 ri,t 1/T ︸ ︷︷ ︸ geometric mean S (asset 1) B (asset 2) MM (asset 3) r¯i 1.1206% 1.0785% 1.0632% µi 1.1073% 1.0737% 1.0627% µ = (1.1073, 1.0737, 1.0627)> Note: Stocks have highest expected return 32 / 40 General Approach Markowitz Model Step 4. Estimate Covariance Matrix Σij = 1 T >∑ i=1 (ri,t − r¯i )(rj,t − r¯j), i , j ∈ {1, 2, . . . , n} Σ = 0.02778 0.00387 0.000210.00387 0.01112 −0.00020 0.00021 −0.00020 0.00115 Note: MM has lowest variance (risk) Step 5. Find Efficient Portfolio without Short-Selling min x x>Σx (EP2) s.t. µ>x ≥ R x ∈ X := {x ∈ R3 : x ≥ 0, x1 + x2 + x3 = 1} for R ∈ [6.5%, 10.5%]. 33 / 40 General Approach Markowitz Model 2 4 6 8 10 12 14 16 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 Efficient Frontier Standard deviation (%) 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Composition of efficient portfolios Expected return of efficient portfolios (%) Pe rc en t i nv es te d in d iff er en t a ss et c la ss es Stocks Bonds Money Market 34 / 40 General Approach Markowitz Model Analysis of Efficient Portfolios 35 / 40 General Approach Markowitz Model Allowing Short Selling Take the minimum variance portfolio problem min x f (x) = 1 2 x>Σx s.t. e>x = 1, µ>x = R assuming some positive definite covariance matrix Σ Since we have equality constraints5, Method of Lagrange Multipliers can be used to derive analytical form of optimal portfolio x Lagrange function is L(x , λ) = f (x) + λ1(1− e>x) + λ2(R − µ>x) Lagrange Multiplier Theorem and positive definiteness of Σ (which is the Hessian matrix of L(x , λ) w.r.t. x) says that we need to solve first-order gradient conditions ∂L ∂x = 0 =⇒ Σx − λ1e − λ2µ = 0 ∂L ∂λ = 0 =⇒ e>x = 1, µ>x = R 5If we had µ>x ≥ R, we would need to use Karush-Kuhn-Tucker (KKT) optimality conditions 36 / 40 General Approach Markowitz Model Allowing Short Selling Take the minimum variance portfolio problem min x f (x) = 1 2 x>Σx s.t. e>x = 1, µ>x = R assuming some positive definite covariance matrix Σ Since we have equality constraints5, Method of Lagrange Multipliers can be used to derive analytical form of optimal portfolio x Lagrange function is L(x , λ) = f (x) + λ1(1− e>x) + λ2(R − µ>x) Lagrange Multiplier Theorem and positive definiteness of Σ (which is the Hessian matrix of L(x , λ) w.r.t. x) says that we need to solve first-order gradient conditions ∂L ∂x = 0 =⇒ Σx − λ1e − λ2µ = 0 ∂L ∂λ = 0 =⇒ e>x = 1, µ>x = R 5If we had µ>x ≥ R, we would need to use Karush-Kuhn-Tucker (KKT) optimality conditions 36 / 40 General Approach Markowitz Model Multiplying first gradient condition by Σ−1, we get x∗ = λ1Σ−1e + λ2Σ−1µ Substituting this x into the two linear constraints and solving for λ yields λ1 = C − RB AC − B2 , λ2 = RA− B AC − B2 where A = e>Σ−1e, B = µ>Σ−1e and C = µ>Σ−1µ. Hence, x∗ := x∗R = C − RB AC − B2 Σ −1e + RA− B AC − B2 Σ −1µ Markowitz Efficient Frontier is produced by the portfolios{ x∗R : R ≥ B A } 37 / 40 General Approach Markowitz Model Global and Diversified Portfolios • Global Minimum Variance Portfolio is obtained by setting the second Lagrange multiplier to zero λ2 = 0 =⇒ R = B A =⇒ xG = Σ −1e A • Diversified Portfolio is obtained by setting the first Lagrange multiplier to zero λ1 = 0 =⇒ R = C B =⇒ xD = Σ −1µ B 38 / 40 General Approach Markowitz Model Mutual Fund Theorem Theorem Any minimum variance portfolio x∗ can be written as a convex combination of two distinct minimum variance portfolios x ′ and x ′′ where x ′ 6= x ′′, x∗ = αx ′ + (1− α)x ′′, some α ∈ [0, 1]. In particular, we can take x ′ = xG and x ′′ = xD . 39 / 40 General Approach Markowitz Model No Short Selling X = {x ≥ 0 : ∑ni=1 xi = 1} is a simplex in Rn. Do not need Σ to be pd. Exercise Show that the set of optimal portfolios is a compact convex set. Theorem For every minimum variance portfolio x , there exist n + 1 “mutual funds” w1, . . . ,wn+1 such that x is a convex combination of {w1, . . . ,wn+1} x = n+1∑ i=1 αiw i , some α ≥ 0, n+1∑ i=1 αi = 1 Here, compactness of X implies set of optimal solutions X ∗ is bounded, and then the theorem is a consequence of Krein-Milman theorem and Caratheodory theorem for convex sets 40 / 40 General Approach Markowitz Model No Short Selling X = {x ≥ 0 : ∑ni=1 xi = 1} is a simplex in Rn. Do not need Σ to be pd. Exercise Show that the set of optimal portfolios is a compact convex set. Theorem For every minimum variance portfolio x , there exist n + 1 “mutual funds” w1, . . . ,wn+1 such that x is a convex combination of {w1, . . . ,wn+1} x = n+1∑ i=1 αiw i , some α ≥ 0, n+1∑ i=1 αi = 1 Here, compactness of X implies set of optimal solutions X ∗ is bounded, and then the theorem is a consequence of Krein-Milman theorem and Caratheodory theorem for convex sets 40 / 40 General Approach Markowitz Model No Short Selling X = {x ≥ 0 : ∑ni=1 xi = 1} is a simplex in Rn. Do not need Σ to be pd. Exercise Show that the set of optimal portfolios is a compact convex set. Theorem For every minimum variance portfolio x , there exist n + 1 “mutual funds” w1, . . . ,wn+1 such that x is a convex combination of {w1, . . . ,wn+1} x = n+1∑ i=1 αiw i , some α ≥ 0, n+1∑ i=1 αi = 1 Here, compactness of X implies set of optimal solutions X ∗ is bounded, and then the theorem is a consequence of Krein-Milman theorem and Caratheodory theorem for convex sets 40 / 40 General Approach Markowitz Model No Short Selling X = {x ≥ 0 : ∑ni=1 xi = 1} is a simplex in Rn. Do not need Σ to be pd. Exercise Show that the set of optimal portfolios is a compact convex set. Theorem For every minimum variance portfolio x , there exist n + 1 “mutual funds” w1, . . . ,wn+1 such that x is a convex combination of {w1, . . . ,wn+1} x = n+1∑ i=1 αiw i , some α ≥ 0, n+1∑ i=1 αi = 1 Here, compactness of X implies set of optimal solutions X ∗ is bounded, and then the theorem is a consequence of Krein-Milman theorem and Caratheodory theorem for convex sets 40 / 40 欢迎咨询51作业君