CO 372 Portfolio Optimization Lecture 1 by ild18893

VIEWS: 173 PAGES: 4

									                                                                                       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).




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