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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
• 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
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, . . . , 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
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, 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
Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen
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
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 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
Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen
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
C. Wagner 13 WS 2010/2011
Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 2 Utility Theory Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
C. Wagner 21 WS 2010/2011
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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.
C. Wagner 26 WS 2010/2011
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
C. Wagner 28 WS 2010/2011
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
Quantitative Methods in Portfolio Management 3 Modeling the Market Mathematisches Institut, LMU Munchen
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
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