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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

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