# 程序代写案例-MATH 11158

February 9, 2021留学咨询

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)  