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  • Quantitative Methods

    in Portfolio Management

    C. Wagner

    WS 2010/2011

    Mathematisches Institut,

    Ludwig-Maximilians-Universitat Munchen

  • Quantitative Methods in Portfolio Management Mathematisches Institut, LMU Munchen

    Markowitz

    - efficient

    Return

    Efficient Frontier

    Alpha

    CAPM

    BARRA

    Sharpe

    Shortfall

    Information Ratio

    C. Wagner 2 WS 2010/2011

  • Quantitative Methods in Portfolio Management Contents Mathematisches Institut, LMU Munchen

    Contents

    1 Introduction 5

    2 Utility Theory 8

    3 Modeling the Market 16

    4 Estimating the Distribution of Market Invariants 40

    5 Evaluating Allocations 41

    6 Optimizing Allocations 42

    7 Estimating the Distribution of Market Invariants with Estimation Risk 43

    8 Evaluating Allocations under Estimation Risk 44

    9 Optimizing Allocations under Estimation Risk 45

    C. Wagner 3 WS 2010/2011

  • Quantitative Methods in Portfolio Management Contents Mathematisches Institut, LMU Munchen

    Literature

    introductory

    Modern Portfolio Theory and Investment Analysis ; Elton, Gruber, Brown, Goetzmann; Wi-ley

    Portfoliomanagement ; Breuer, Guertler, Schumacher; Gabler

    quantitative

    Risk and Asset Allocation; Meucci; Springer

    Quantitative Equity Portfolio Management ; Qian, Hua, Sorensen; CRC

    Robust Portfolio Optimization and Management ; Fabozzi, Kolm, Pachamanova, Focardi;Wiley

    C. Wagner 4 WS 2010/2011

  • Quantitative Methods in Portfolio Management 1 Introduction Mathematisches Institut, LMU Munchen

    1 Introduction

    Investor Choice Under Certainty

    Investor will receive EUR 10000 with certainty in each of two periods

    Only investment vailable is savings account (yield 5%)

    Investor can borrow money at 5%

    How much should the investor save or spend in each period?

    Separate problem into two steps:

    1. Specify options

    2. Specify how to choose between options

    C. Wagner 5 WS 2010/2011

  • Quantitative Methods in Portfolio Management 1 Introduction Mathematisches Institut, LMU Munchen

    Opportunity Set:

    A save nothing, spend all when received, (10000, 10000)

    B save first period and consume all in the second (0, 10000 (1 + 0.05) + 10000)

    C consume all in the first, i.e borrow the maximum in from the second in the first period (10000 +

    10000/(1 + 0.05), 0)

    xi income in period i, yi consumption in period i

    y2 = x2 + (x1 y1) 1.05

    Indifference Curve:

    Iso-Happiness Curve (see graph):

    assumption: each additional euro of consumption forgone in periode 1 requires greater consumption

    in period 2

    ordering due to investor prefers more to less

    C. Wagner 6 WS 2010/2011

  • Quantitative Methods in Portfolio Management 1 Introduction Mathematisches Institut, LMU Munchen

    Solution:

    Opportunity set is tangent to indifference set

    C. Wagner 7 WS 2010/2011

  • Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen

    2 Utility Theory

    Use utility function to formalise investors preferences to arrive at optimal portfolio

    Base Modell

    Investor has initial wealth W0 at t = 0 and an investment universe of n+1 assets (n risky assets,one riskless asset)

    Investment horizon t = 1, short-selling, uncertain returns ri (rv) for risky assets i = 1, . . . , nand deterministic r0 for bank account.

    Describe portfolio through asset weights, i.e. P = (x0, x1, . . . , xn), W0 = W0

    i xi

    Uncertain wealth (i.e. rv) W1 at t = 1, W P1 =

    i xiW0(1 + ri)

    Assign preference through utility function to each possible opportunity set, U(W P1 ), and prob-ability to arrive at expected utility E[U(W P1 )].

    optimization problem: maxx0,...,xn E[U(W P1 )]

    C. Wagner 8 WS 2010/2011

  • Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen

    Properties of Utility Function

    determined only up to positiv linear transformations (ranking)

    investor prefers more to less, W P < WQ then U(W P ) < U(WQ), U strictly increasing, ifdifferentiable then U > 0

    risk appetite: for W0 = E[W1]

    risk averse: E[U(W0)] > E[U(W1)], U concave (Jensen)

    risk neutral: E[U(W0)] = E[U(W1)], U linear

    risk seeking: E[U(W0)] < E[U(W1)] U convex

    C. Wagner 9 WS 2010/2011

  • Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen

    Some Distributions

    Uniform Distribution

    X U(E,), E, elipsoid

    f (x) =(N2 + 1)

    N/2||1/21E,(x),

    U,() = e

    i()

    Normal Distrubution

    X N(,)

    f (x) = (2)N/2||1/2e12(x)

    1(x), () = ei12

    Student-t Distrubution

    Z N(,), W 2()

    X

    WZ St(,,)

    C. Wagner 10 WS 2010/2011

  • Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen

    Cauchy Distrubution

    X Ca(,) St(1,,)

    Lognormal Distrubution

    X LogN(,)X = eY ,Y N(,)

    C. Wagner 11 WS 2010/2011

  • Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen

    Distribution Classes

    Elliptical Distribution

    Definition: rv Y = (Y1, . . . , Yn) has a sperical distribution if, for every orthogonal matrix U

    UYd= Y

    Properties: Y spherical function g (generator) such that Y (t) = E[eitY ] = n(t

    t)

    Generator as function of a scalar variable uniquely describes sperical distribution

    Y Sn()

    equivalent representation:

    Y = RU , where R = Y is norm (i.e. univariate) and U = Y /Y Y Sn() R and U are independent rvs and U is uniformly distributed on the surface of theunit ball

    Definition: X has an elliptical distribution if

    Xd= + AY

    where Y Sm() and A Rnm, Rn. X El(,, ), = AA.

    C. Wagner 12 WS 2010/2011

  • Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen

    positive definite, then isoprobability contours are surfaces of centered ellipsoides

    More Properties:

    affine transformation: BX + b EL()marginal distributions: EL()conditional distribution: EL()convolution with same dispertion matrix : EL()

    Examples: Uniform, Normal, Student-t, Cauchy highly symmetric and analytically tractable, yet

    quite flexible

    Stable Distribution

    Definition: X,X1, X2 iid rv. X is called stable if for all non-negative c1, c2 and appropriate

    numbers a = a(c1, c2), b = b(c1, c2) the following holds:

    c1X1 + c2X2d= a + bX

    i.e. closed under linear combinations

    C. Wagner 13 WS 2010/2011

  • Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen

    symmetric--stable (one dimension) iff

    X(t) = E[eitX ] = exp{it c|t|}

    location, c scaling, tail thickness

    symmetric--stable (multivariate) iff

    X(t) = E[eitX ] = eit

    exp{R|ts|m(s)ds}

    function m is a symmetric measure, m(s) = m(s) for all s Rn and

    m(s) 0 for all s such that ss 6= 1

    X SS(, ,m)

    Examples: Normal, Cauchy

    Counterexamples: Lognormal, Student-t

    C. Wagner 14 WS 2010/2011

  • Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen

    Normal distribution as symmetric-alpha-stable

    X N(,)spectral decomposition of : = E1/21/2E

    define n vectors {v(1), . . . , v(n)} = E1/2

    define measure as

    m =1

    4

    ni=1

    ((v(i) + (v(i)))

    Remark: stability additvity, but reverse is not true in general, e.g. Wishart dist.

    Infinitely Divisible Distributions

    X is infinitely divisible if, for any integer M we can decompose it in law

    Xd= Y 1 + + Y M

    where (Y i)i=1,...,M are iid rvs with possibly different common distributions for different M . Exam-

    ples: Normal, Lognormal, Chi2 Counterexamples: Wishart

    C. Wagner 15 WS 2010/2011

  • Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen

    3 Modeling the Market

    Market for an investor is represented by an N-dimensional price vector of traded securities, P t:

    Investment decision (allocation) at T

    Investment horizon

    P T+ N-dimensional random variable

    Modeling the market means modeling P T+ :

    1. modeling market invariants

    2. determining the dsitribution of market invariants

    3. projecting invariants into the future T +

    4. mapping of invariants to market prices

    dimension of randomness numbers of securities dimension reduction

    C. Wagner 16 WS 2010/2011

  • Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen

    Market Invariants

    Dt0, = {t0, t0 + , t0 + 2 , . . . } set of equally spaced observation dates

    random variables Xt, t Dt, are called invariant if rv are iid and time homogeneous

    simple tests: check histograms of two non-overlapping subsets of observations, scatter-plot of values

    vs. lagged values

    Equities

    Pt, t Dt0, equally-spaced stock price observations

    equity prices are not market invariants (exponential growth)

    Total return

    Ht, =PtPt

    is a market invariant

    g any function, then if Xt is invariant g(Xt) is also an invariant

    C. Wagner 17 WS 2010/2011

  • Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen

    hence

    linear return