Quantitative Methods in Portfolio Management - LMU · PDF fileQuantitative Methods in Portfolio Management C. Wagner ... Quantitative Methods in Portfolio ManagementContentsMathematisches

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

• µ-σ 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


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

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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, . . . , n

and 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


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, if

differentiable 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


Some Distributions

Uniform Distribution

X ∼ U(Eµ,Σ), Eµ,Σ elipsoid

f (x) =Γ(N2 + 1)

πN/2|Σ|1/21Eµ,Σ(x), ΦU

µ,Σ(ω) = eiω′µΨ(ω′Σω)

Normal Distrubution

X ∼ N(µ,Σ)

f (x) = (2π)−N/2|Σ|−1/2e−12(x−µ)′Σ−1(x−µ), Φ(ω) = eiµ

′ω−12ω′Σω

Student-t Distrubution

Z ∼ N(µ,Σ), W ∼ χ2(ν)

⇒X ≡√

ν

WZ ∼ St(ν,µ,Σ)

C. Wagner 10 WS 2010/2011


Cauchy Distrubution

X ∼ Ca(µ,Σ) ≡ St(1,µ,Σ)

Lognormal Distrubution

X ∼ LogN(µ,Σ)⇔X = eY ,Y ∼ N(µ,Σ)

C. Wagner 11 WS 2010/2011


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 ΦY (t) = E[eit′Y ] = ψn(t′t)

Generator Ψ as function of a scalar variable uniquely describes sperical distribution

⇒ Y ∼ Sn(ψ)

equivalent representation:

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

unit ball

Definition: X has an elliptical distribution if

Xd= µ + AY

where Y ∼ Sm(ψ) and A ∈ Rn×m, µ ∈ Rn. X ∼ El(µ,Σ, ψ), Σ = AA′.

C. Wagner 12 WS 2010/2011


Σ 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 Σ: ∼ 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

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symmetric-α-stable (one dimension) iff

ΦX(t) = E[eitX ] = exp{iµt− c|t|α}

µ location, c scaling, α tail thickness

symmetric-α-stable (multivariate) iff

ΦX(t) = E[eit′X ] = eit

′µ exp{−∫R|t′s|αmΣ(s)ds}

function mΣ is a symmetric measure, mΣ(s) = mΣ(−s) for all s ∈ Rn and

mΣ(s) ≡ 0 for all s such that s′Σs 6= 1

X ∼ SS(α, µ,mΣ)

Examples: Normal, Cauchy

Counterexamples: Lognormal, Student-t

C. Wagner 14 WS 2010/2011


Normal distribution as symmetric-alpha-stable

X ∼ N(µ,Σ)

spectral decomposition of Σ: Σ = EΛ1/2Λ1/2E

define n vectors {v(1), . . . , v(n)} = EΛ1/2

define measure as

mΣ =1

4

n∑i=1

(δ(v(i) + δ(−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 τ

• P T+τ N-dimensional random variable

Modeling the market means modeling P T+τ :

1. modeling market invariants

2. determining the dsitribution of market invariants

3. projecting invariants into the future T + τ

4. mapping of invariants to market prices

dimension of randomness � numbers of securities ⇒ dimension reduction

C. Wagner 16 WS 2010/2011


Market Invariants

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

random variables Xt, t ∈ Dt,τ 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,τ equally-spaced stock price observations

equity prices are not market invariants (exponential growth)

Total return

Ht,τ =PtPt−τ

is a market invariant

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

C. Wagner 17 WS 2010/2011


hence

linear return Lt,τ = Ht,τ − 1 and log-return Ct,τ = ln (Ht,τ )

are also market invariants.

equity invariants: compound returns C

compund returns can be easily projected to any horizon, distribution approximately symmetric

Example:

continuous-time finance, Black/Scholes, Merton

Ct,τ = ln

(PtPt−τ

)∼ N(µ, σ2).

⇒ total return Ht,τ ∼ LogN(µ, σ2).

Other Choices:

multivariate case

Ct,τ ∼ EL(µ,Σ, g) or Ct,τ ∼ SS(α, µ,mΣ)

C. Wagner 18 WS 2010/2011


Fixed-Income Market

zero-coupon bonds as building blocks, Z(E)t , E maturity

normalization Z(E)E ≡ 1

consider set of bond prices

Z(E)t , t ∈ Dt0,τ

pull-to-par effect ⇒ bond prices are not market invariants

consider set of non-overlapping total returns

H(E)t,τ ≡

Z(E)t

Z(E)t−τ

, t ∈ Dt0,τ

pull-to-par also breaks time homogeneity of total return ⇒ total returns are not market invariants

consider total return of bonds with same time to maturity ν

R(ν)t,τ ≡

Z(t+ν)t

Z(t+ν−τ)t−τ

, t ∈ Dt0,τ

ratio of prices of two different securities

C. Wagner 19 WS 2010/2011


test shows that R(ν)t,τ is acceptable as market invariant, hence also every function g of R.

define yield to maturity ν as (annualized return of bond)

Y(ν)t ≡ −1

νln(Z

(t+ν)t

)consider now changes in yield to maturity

X(ν)t,τ ≡ Y

(ν)t − Y (ν)

t−τ = −1

νln(R

(ν)t,τ

)

fixed-income invariants: changes in yield to maturity X

invariant is specific to a given sector ν of the yield curve

Examples:

X(ν)t,τ ∼ N(µ, σ), X

(ν)t,τ ∼ El(µ,Σ, g), X

(ν)t,τ ∼ SS(α, µ,mΣ)

C. Wagner 20 WS 2010/2011


Derivatives

vanilla european call option

C(K,E)t = CBS

(E − t,K, Ut, Z(E)

t , σ(K,E)t

)boundary condition C

(K,E)E = max (UE −K, 0)

E expiry date, K strike

market variables:

Ut underlying, Z(E)t zero bond, σ

(K,E)t implied percentage volatility (vol surface)

for zero bond Z and underlying U market invariants are know

what about implied vol?

consider at-the-money-forward (ATMF) implied vol, i.e. implied vol at strike equals forward price

σ(Kt,E)t , t ∈ Dt0,τ where Kt ≡

Ut

Z(E)t

implied vol is not a market invariant as expiry convergence breack time-homogeneity

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eliminate expiry date dependency through considering set of implied vols (rolling ATMF vols)

σ(Kt,t+ν)t , t ∈ Dt0,τ

σ(Kt,t+ν)t ≈

√2π

ν

C(Kt,t+ν)t

Utsill has time dependence

consider changes in ATMF implied vols

X(ν)t,τ = σ

(Kt,t+ν)t − σ(Kt−τ ,t−τ+ν)

t−τ , t ∈ Dt0,τ

derivatives invariants: changes in roll. ATMF implied vol

distribution of changes in roll.ATMF implied vol is symmetrical, hence modeling as

X(ν)t,τ ∼ N(µ, σ), X

(ν)t,τ ∼ El(µ,Σ, g), X

(ν)t,τ ∼ SS(α, µ,mΣ)

C. Wagner 22 WS 2010/2011


Projection of the Invariants to the Investment Horizon

invariants Xt,τ relative to estimation interval τ

representation of distribution in form of probability density fXt,τ or characteristic function ΦXt,τ

in general, investment horizon τ is different than estimation interval τ

estimation interval t

time series analysis investment decision T

investment horizon t

FXT+t,t

FXT+t,t

Figure 1: asset swap.

C. Wagner 23 WS 2010/2011


Projection to investment horizon:

determine distribution of XT+τ,τ , i.e. fXT+τ,τor ΦXT+τ,τ

, from estimated distribution

assume τ is an integer multiple of τ

invariants are additive, hence

XT+τ,τ = XT+τ,τ + XT+τ−τ ,τ + · · · + XT+τ ,τ

since invariants are in the form if differences:

equity return: Xt,τ = ln(Pt)− ln(Pt−τ )

yield to maturity: Xt,τ = Yt − Yt−τATFM impl. vol: Xt,τ = σt − σt−τ

XT+nτ ,τ are invariants wrt non-overlapping intervalls, i.e. iid.

Projection via convolution:

ΦXT+τ,τ= E

[eiωXT+τ,τ

]= E

[eiωXT+τ,τ+XT+τ−τ ,τ+···+XT+τ ,τ

] iid=(

ΦXT+τ,τ

)τ/τrepresentation involving the density fXT+τ,τ

can be obtained via Fourier transformation

ΦX = F [fX ], fx = F−1[ΦX ]

C. Wagner 24 WS 2010/2011


Example:

Xt,τ ≡

(Ct,τ

Xνt,τ

)=

(lnPt − lnPt−τ

Y νt − Y ν

t−τ

)∼ N(µ,Σ)

Characteristic function:

ΦXt,τ (ω) = eiω′µ−1

2ω′Σω

ΦXT+τ,τ=(

ΦXT+τ,τ

)τ/τ=(eiω′µ−1

2ω′Σω)τ/τ

= eiω′ ττµ−

12ω′ ττΣω

=⇒ XT+τ,τ ∼ N(τ

τµ,τ

τΣ)

Note: normal dist is infinitely divisible, hence τ/τ need not to be an integer

Furthermore for moments:

E[XT+τ,τ ] =τ

τE[Xt,τ ], Cov[XT+τ,τ ] =

τ

τCov[Xt,τ ], Std[XT+τ,τ ] =

√τ

τStd[Xt,τ ]

Remarks:

- simplicity of projection formula due to specific formulation of market invariants

- projection formula hides estimation risk, distribution at horizon can only be estimated (estimation

error)

C. Wagner 25 WS 2010/2011


From Invariants to Market Prices

how to recover prices from invariants?

market prices of securities at horizon T + τ are functions of the the investment-horizon invariants

P T+τ = g(XT+τ,τ )

Equities

PT+τ = PTeX

(see choice of invariants, i.e. compound return)

Fixed-Income

Z(E)T+τ = Z

(E−τ)T e−X

(E−T−τ)(E−T−τ)

in general (equities and fixed-income)

P = eY , with Y ≡ γ + diag(ε)X

γn ≡

ln(PT ), if stock

ln(Z(E−τ)T ) if bond.

, εn ≡

1, if stock

−(E − T − τ ) if bond.

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Y affine transformation of X ⇒ distribution of Y , e.g. as characteristic function

ΦY (ω) = eiω′γΦX(diag(ε)ω)

Example:

two-security market (stock, bond), maturity E = T + τ + ν

P =

(PT+τ

Z(E)T+τ

)= eγ+diag(ε)X , γ =

(ln(PT )

ln(Z(T+ν)T )

), ε =

(1

−ν

).

characteristic function, X multi-normal

ΦY (ω) = eiω′[γ+τ

τ diag(ε)µ]−12ττ ω′diag(ε)Σdiag(ε)ω

⇒ Y multi-normal

Y ∼ N(γ +

τ

τdiag(ε)µ,

τ

τdiag(ε)Σdiag(ε)

)⇒ P log-normal

in most cases not possible to get distribution of future prices in closed form

⇒ usually sufficient to work with moments (Taylor)

E[Pn] = eγnΦX (−εn)

C. Wagner 27 WS 2010/2011


Example: simple stock

E[PT+τ ] = PTeττ µC+τ

τσ2

2

Derivatives

Derivative price is a nonlinear function of several investment-horizon invariants (e.g. call option)

P = g(X), e.g. C(K,E)T+τ = CBS

(E − T − τ,K, UT+τ , Z

(E)T+τ , σ

(K,E)T+τ

)with

UT+τ = UTeX1, Z

(E)T+τ = Z

(E−τ)T e−X2(E−T−τ), σ

(K,E)T+τ = σ

(K,E−τ)T + X3

distribution assumptions for X1, X2, X3, but in general no closed form distribution for P , C.

=⇒ Taylor expansion of P :

P = g(m) + (X −m)′∂xg|x=m +1

2(X −m)′∂xxg|x=m(X −m) + . . .

1st order: delta-vega, duration

2nd order: gamma, convexity

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Dimension Reduction

P T+τ = g(XT+τ,τ )

market includes a large number of securities, i.e. market invariant-vectorX t,τ has a large dimension

⇒ dimension reduction to a vector F of few common factors

X t,τ ≡ h(F t,τ ) +U t,τ

with K = dim(F )� N = dim(X)

If X represents market invariant F , U must be invariants too.

common factors F should be responsible for most of the randomness, U should only be a residual,

X ≡ h(X) ≈X

measure goodness of approximation with generalized r-squared :

R2(X,X) ≡ 1−E[(X − X)′(X − X)

]tr(Cov(X))

C. Wagner 29 WS 2010/2011


restrict to linear factor model

X ≡ BF +U

K ×K-matrix B is called factor loadings

Ideally, F and U shouldbe independent variables, but too restrictive for pratical purposes, hence

impose only

Corr(F ,U ) = 0

- explicit factor model: common factors are measurable market variable

- hidden factor model: common factors are synthetic variables

Explicite Factors

factor loadings for linear regression solve

Br ≡ arg maxB

R2(X,BF )

from M ≡ E[(X −BF )(X −BF )′] and ∂M/∂Bij = (0)ij follows

⇒ Br = E[XF ′]E[FF ′]−1

C. Wagner 30 WS 2010/2011


which yields

Xr ≡ BrF , U r ≡X − Xr

In general, residuals U do not have zero expectation and are correlated with F (unless E[F ] = 0).

Enhance linear model with constant factor

X ≡ a +BF +U

minimizing M = E[(X − (a +BF ))(X − (a +BF ))′] yields

Xr ≡ E[X ]Cov[X,F ]Cov[F ]−1(F − E[F ])

perturbartions U now have zero expectation and are uncorelated with F .

quality of regression:

- adding factors trivially improves quality but number should be kept at a minimum

- factors should be chosen as diversified as possible (avoid collinearity)

C. Wagner 31 WS 2010/2011


Hidden Factors

factors are not market invariants (not observable)

X ≡ q +BF (X) +U

Principal Component Analysis (PCA)

assume that hidden factors are affine transformations of invariants:

F ≡ d +A′X,

d is K-dim vector, A is K ×N -dim matrix

recovered invariants are affine transfomation of original invariants

X ≡m +BA′X, with m = q +Bd

PCA solution from

(B,A,m) ≡ arg maxB,A,m

R2(X,m +BA′X)

Impose as additional condition E[F ] = 0

C. Wagner 32 WS 2010/2011


For this, consider spectral decomposition of covariance matrix

Cov[X ] = EΛE′

Λ = diag(λ1, . . . , λN) diag matrix of decreasing positive eigenvalues,

eigenvectors E = (e1, . . . , eN),EE′ = 1N

one hidden factor, K = 1:

guess

F = (e1)′X,

i.e. orthogonal projection of X onto the direction of first eigenvector

recovered invariant is

X = m + e1(e1)′X

impose E[X ] = E[X ]

⇒m =(1N − e1(e1)′

)E[X ]

satisfying E[F ] = 0 yields

F = (e1)′X − (e1)′E[X ]

C. Wagner 33 WS 2010/2011


K hidden factors:

consider N ×K matrix EK =(e1, . . . , eK

)solution to PCA problem is

(B,A,m) = (EK,Ek, (1N −EKE′K)E[X ])

represent orthogonal projection of original invariants onto hyperplane spanned by the K longest

principal axes (i.e. contains the maximum information)

hidden factors:

F = E′K(X − E[X ])

PCA-invariants:

X = E[X ] +EKE′K(X − E[X ])

residuals U = X −X have zero expectation and zero correlation with factors F

E[U ] = 0, Corr[U ,F ] = 0

quality of approximation depends on number of hidden factors, with

R2(X,X) =

∑Kn=1 λn∑Nn=1 λn

C. Wagner 34 WS 2010/2011


Kth eigenvalue is variance of the Kth hidden factor

V[Fn] = (en)′EΛE′en = λn

“n-th eigenvalue is contribution to the total recovered randomness”

Explicit vs. Hidden Factors

explicit factors models are interpretable, hidden factor models tend to have a higher “explanatory”

power

PCA: recovered invariants represent projections of the original invariant onto the K-th dimensional

hyperplane of maximum randmoness spanned by the first K principal axes

Explicit Regression: X =( X1,...,KXK+1,...,N

); recovered invariants represent projections of the original

invariants onto the plane spanned by the K reference invariants

C. Wagner 35 WS 2010/2011


Examples

linear stock returns and index return (explicit)

Lnt,τ =P nt

P nt−τ− 1, n = 1, . . . , N, Ft,τ =

Mt

Mt−τ− 1

Lnt,τ = E[Lnt,τ ] + βnτ (Ft,τ − E(Ft,τ )) with βτ =Cov(Lnt,τ , Ft,τ )

V(Ft,τ )

suppose additinal constraint holds:

E[Lnt,τ ] = βτE[Ft,τ ] + (1− βτ )Rt,τ with Rt,τ =

(1

Z(t)t−τ− 1

)⇒ Lnt,τ = Rt,τ + βτ (Ft,τ −Rt,τ ), CAPM

————————

market-size explicit factors:

(i) broad stock index return

(ii) difference in return between small-cap index and large-cap index (“SmB”)

(iii) difference in return between book-to-marlet value index and small-book-to-market value index

(“HmL”)

C. Wagner 36 WS 2010/2011


Case Study: Modeling the Swap Market

Swap:

plain vanilla interest rate swap (IRS), payer forward start (PFS): committment initiated at t0 to

exchange payments between two different legs, starting from a future time at times t1, . . . , tm,

(“eight-year swap two years forward”).

fixed leg pays

NδiK,

N nominal, K fixed rate, δi day-count fraction

floating leg pays

NδiL(ti−1, ti),

L(ti−1, ti) floating rate between ti−1 − ti reset at ti−1 paid in arrears.

discounted payoff of PFS at t < t1 is

m∑i=1

D(t, ti)Nδi(L(ti−1, ti)−K)

C. Wagner 37 WS 2010/2011


present value of PFS at t:

PV (PFS)t = N

m∑i=1

δiZ(ti)t (F (t; ti−1, ti)−K) = NZ

(t1)t −NZ(tm)

t −NKm∑i=1

δiZ(ti)t

F (t; ti−1, ti) := 1δi

(Z

(ti−1)t

Z(ti)t

− 1

)forward rate

⇒ swap market is completely priced by the set of zero coupon bond prices at all maturities

market invariants for FI:

changes in yield-to-maturity

X(ν)t,τ = Y

(ν)t − Y (ν)

t−τ with yield curve ν 7→ Y(ν)t := −1

νln(Z

(t+ν)t

)data set of zero-coupon bond prices Z

(E)t , τ = one week

C. Wagner 38 WS 2010/2011


dimension reduction:

convariance matrix

C(v, p) = Cov[X(v), X(v+p)

]properties

C(v, p + dt) ≈ C(v + dt, p) smooth

C(v, 0) ≈ C(v + τ, 0) diagonal elements are similar

C(v, p) ≈ C(v + τ, p)

C(v, p) ≈ h(p) approximate structure

with h(p) = h(−p)

C. Wagner 39 WS 2010/2011

Quantitative Methods in Portfolio Management 4 Estimating the Distribution of Market Invariants Mathematisches Institut, LMU Munchen

4 Estimating the Distribution of Market Invariants

C. Wagner 40 WS 2010/2011

Quantitative Methods in Portfolio Management 5 Evaluating Allocations Mathematisches Institut, LMU Munchen

5 Evaluating Allocations

C. Wagner 41 WS 2010/2011

Quantitative Methods in Portfolio Management 6 Optimizing Allocations Mathematisches Institut, LMU Munchen

6 Optimizing Allocations

C. Wagner 42 WS 2010/2011

Quantitative Methods in Portfolio Management 7 Estimating the Distribution of Market Invariants with Estimation RiskMathematisches Institut, LMU Munchen

7 Estimating the Distribution of Market Invariants

with Estimation Risk

C. Wagner 43 WS 2010/2011

Quantitative Methods in Portfolio Management 8 Evaluating Allocations under Estimation Risk Mathematisches Institut, LMU Munchen

8 Evaluating Allocations under Estimation Risk

C. Wagner 44 WS 2010/2011

Quantitative Methods in Portfolio Management 9 Optimizing Allocations under Estimation Risk Mathematisches Institut, LMU Munchen

9 Optimizing Allocations under Estimation Risk

C. Wagner 45 WS 2010/2011

Quantitative Methods in Portfolio Management - LMU · PDF fileQuantitative Methods in Portfolio Management C. Wagner ... Quantitative Methods in Portfolio ManagementContentsMathematisches

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