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Course Info & Material Instructor: Romy Shioda, rshioda@uwaterloo.ca, MC 5024 Office Hours: W,Th 3 – 4PM TAs: (more info will be posted on website) – Marina Potaptchik C&O 372: Portfolio Course Website: http://www.student.math.uwaterloo.ca/~co372 Optimization – The syllabus, lecture notes, assignments, solutions, office hour information, etc. available on course website Lecture 1: Text: Course Overview & Logistics • Course Notes available at Pixel Planet for $22.94 Basic Portfolio Optimization GAMS: Optimization Modeling software, available by: LP Formulations • Free download of trial version • Unix student.math machines Winter 2005 • NEOS server online Lecture Outline Course Policy • Course Description Assignments (10%): – Weekly • Course Administration and Logistics – Lowest score dropped • Course Outline – Exams inspired by assignments so do them! • Introduction to Portfolio Optimization – Cheating policy: if you’re caught, you get no credit for all previous • Linear Programming Formulation assignments & get reported to the assoc dean Midterm (30%): Tentatively: Wed Feb 16, Thur Feb 17 Project (15%): – Develop your own optimal portfolio model – Group of 4-5 people Final (45%): – During Finals season CO372 Portfolio Optimization Course Outline C&O 372 presents the mathematical theory and optimization techniques behind portfolio optimization. • Linear Optimization Models • We will mainly focus on the Markowitz mean-variance optimal asset – Formulations allocation model. • Basic Nonlinear Programming – Minimize Risk v.s. Maximize Returns – Optimality Conditions • Every financial institutions, from investment banks, mutual funds, and • Portfolio Optimization insurance companies, utilize this model to this day. – Efficient Frontier • We explore mathematical modeling techniques that may arise in these – Capital Market Line contexts • Since the theory behind quadratic programming is essential to fully • Recent Innovations in Portfolio Optimization grasping the behavior of an optimal portfolio, we will extensively study the – Diversification limits, Value-at-Risk, Price Impact Cost theory and solution methods for Quadratic Programming. • Quadratic Programming • We will also explore modern alternate models in portfolio optimization – Optimality Conditions – Solution Methods Prerequisite: CO350, Calculus 1 What is Portfolio Optimization? Variance as Measure of Risk Suppose you want to invest in the market to make money How would we measure the risk in the portfolio? • Which one should you choose out of the enormous number of • Risk is often measured by how much the actual total return can differ investment assets (stocks, bonds, real assets, etc)? from the average • How much of your wealth should you invest in each of them? – E.g, the average return maybe 10%, but it may be possible that the actual return can be as low as -5% Most investors also wish to construct a portfolio (collection) of assets • The most common measure of dispersion is the variance or standard with the following criteria: deviation of the total return • Maximize the expected return of the portfolio • Minimize the risk associated with the portfolio Recall that given random variables X and Y and constants a and b, the However, in almost every situation encountered in practice, the above variance of Z=aX+bY is: two criteria oppose each other Var ( Z ) = Var (aX + bY ) = a 2Var ( X ) + b 2Var (Y ) + 2ab Cov ( X , Y ) • Higher the return, the higher the risk • Lower the risk, lower the return Thus the variance of the total returns is: • Thus you need to decide your risk averseness Var ( R ) = A 2Var ( R A ) + G 2Var ( RG ) + D 2Var ( RD ) Out of a collection of investment assets, how can you best + 2 AG Cov ( R A , RG ) + 2 AD Cov ( R A , RD ) + 2GD Cov ( RG , RD ) allocate your wealth given your criteria? Olympia Investments, Inc. Markowitz Mean-Variance Model A portfolio manager for Olympia Investments wishes to assemble a portfolio of The risk of the portfolio, measured by the variance of the total returns, is: assets of stock in Advent Communications, General Space Systems (GSS), and Digital Devices Var ( R) = 16 A 2 + 22G 2 + 10 D 2 + 6 AG − 10 AD + 2GD • Let the random variables RA, RG, RD be the annual return for Advent, And the expected total return is: GSS and Digital Devices, respectively. • The statistics for RA, RG, RD are illustrated below: E[ R] = 11A + 14G + 7 D Covariance The famous Markowitz model finds the optimal portfolio allocation by Asset Expected Variance RA RG RD minimizing variance while maintaining expected return at a given Value constant level r RA 11.0% 8 This results in the following quadratic programming problem: RG 14.0% 22 3 min 16 A 2 + 22G 2 + 10 D 2 + 6 AG − 10 AD + 2GD, RD 7.0% 10 -5 1 subject to : 11A + 14G + 7 D = r How should the manager allocate Olympia’s investment dollars to maximize A+G + D =1 returns and minimize risk? A, G , D Optimization Problem Markowitz Mean-Variance Model Let us define the following decision variables: For example, for r=11, the optimal value is: 5.76 and A = fraction of investment dollars invested in Advent G = fraction of investment dollars invested in GSS Asset Decision Optimal Name Variable Solution D = fraction of investment dollars invested in Digital Advent A 0.3769 These variables must satisfy: A+G + D =1 GSS G 0.3561 A, G, D ≥ 0 Digital D 0.2670 Let the random variable R be the total return of the portfolio, i.e., Note that if the manager invested only in Advent, then he will also get an R = R A A + RG G + RD D expected return of 11%, but with variance of 16. So the total expected returns of the portfolio is: • Thus diversification (investing in a collection of assets) can reduce the variance of the portfolio for a given return. E[ R ] = E[ R A ] A + E[ RG ]G + E[ RD ]D or E[ R] = 11A + 14G + 7 D 2 The Efficient Frontier LP Review Suppose we solve for several difference values of Expected Return: Further Definitions reviews: Efficient Frontier • Feasible Solution: a point that satisfies all of the constraints in the LP Expected Optimal 15 • Feasible Region: set of all feasible points Return Variance Advent GSS Digital 14 • In LPs, the feasible region is always a polyhedron 7 10.0000 0.0000 0.0000 1.0000 13 8 4.7500 0.2500 0.0000 0.7500 • Objective Value: value of the objective function at a particular points 12 9 3.5827 0.3755 0.0872 0.5373 • Optimal Solution: a feasible solution that has the optimal objective Returns 11 10 4.0651 0.3762 0.2136 0.4102 10 value 11 5.7638 0.3769 0.3561 0.2670 9 12 8.6872 0.3777 0.4985 0.1238 • Optimal Value: objective value at the optimal solution 8 13 12.8888 0.3333 0.6667 0.0000 7 14 21.9999 0.0000 1.0000 0.0000 6 0 5 10 15 20 25 Variance LP Review Production Problem (Bertsimas & Freund 2000) New Bedford Steel (NBS) is a steel producer that procures coking coal as raw Linear Programming Problem: material to produce steel • Problem of maximizing or minimizing a linear function over a polyhedron • It had solicited and received bids from 8 coal mining companies: Ashley Bedford Consol Dunby Earlam Florence Gaston Hopt max/ min f (x) s.t. x∈P Price ($/ton) 49.50 50.00 61.00 63.50 66.50 71.00 72.50 80.00 Union/Non- Union Union Non Union Non Union Non Non union Truck/Rail Rail Truck Rail Truck Truck Truck Rail Rail Linear Function: f (x) = c1 x1 + c2 x2 + + cn xn Volatility (%) 15 16 18 20 21 22 23 25 Polyhedron: subset of Rn represented by a finite number of linear inequalities and Capacity 300 600 510 655 575 680 450 490 (1000 equalities tons/year) Linear Inequality: a1 x1 + a2 x2 + + a n xn ≤ b • NBS is planning on purchasing 1,255,000 tons of coal this year a1 x1 + a2 x2 + + a n xn ≥ b • • The coal must have an average volatility of at least 19% As a hedge against labor relations, it must procure at least 50% of its coal from unionized companies Linear Equality: a1 x1 + a2 x2 + + a n xn = b • Capacity for bringing coal by rail is 650,000 tons per year • Capacity for bringing coal by truck is 720,000 tons per year LP Review Components of an LP: • Decision Variables: variables in the LP. – It should completely describes all the decisions to be made • Objective Function: function to be optimized (min/max) – Objective function is linear in LP • Constraints: set of linear inequalities and equalities the variables must satisfy • Sign Restriction: constraints on the signs of the decision variables (i.e., nonnegative, nonpositive, free) max/ min c1 x1 + c2 x2 + + cn xn s.t. a1,1 x1 + a1, 2 x2 + + a1,n xn ≤ b1 a2,1 x1 + a2, 2 x2 + + a 2 , n xn ≥ b2 a3,1 x1 + a3, 2 x2 + + a3,n xn = b3 xi ≥ 0, x j ≤ 0, xk free 3 LP in Fixed Income Solodrex is considering investing in four bonds: • $1,000,000 is available for investment. • The expected annual return, the worst-case annual return on each bond, and the “duration” of each bond are give in the table below (The duration of a bond is a measure of the bond's sensitivity to interest rates). • Solodrex wants to maximize the expected return from its bond investments, subject to three constraints:: 1. The worst case return of the bond portfolio must be at least 8% 2. The average duration of the portfolio must be at most 6 3. Because of diversification requirements, at most 40% of the total amount can be invested in a single bond Expected Worst-case Duration Return Return Bond 1 13% 6% 3 Bond 2 8% 8% 4 Bond 3 12% 10% 7 Bond 4 14% 9% 9 Multi-period Financial Models A broker is trying to maximize his profit in the bond market. • Four bonds are available for purchase and sales: Bid Price Ask Price Year Bond 1 Bond 2 Bond 3 Bond 4 Bond 1 980 990 1 100 80 70 60 Bond 2 970 985 Bond 3 960 972 2 110 90 80 50 Bond 4 940 954 3 1100 1120 1090 1110 • The broker can buy up to 1000 units of each bond at the ask price • The broker can sell up to 1000 units of each bond at the bid price • During the next three years, the person who sells a bond will pay the bond wonder the cash payments above • Assume that cash payments are discounted, with a payment $1 one year from now being equivalent to a payment of 90 cents now. • The Broker’s goal is to maximize his net revenue from selling bonds less his payments for buying bonds • His constraint is that after each year’s payments are received, his current cash position (due only to cash payments from bonds, not sales or purchases) is nonnegative (arbitrage). 4