Conic portfolio theory Sergi Ferrer Fern andezcomputationalfinance.lsi.upc.edu/wp-content/... · 31 de marzo de 2016. Part I Introduction to portfolio theory. Portfolio theory Measures

Optimisation of conic portfoliosConic portfolio theory

Sergi Ferrer Fernandez

Universitat Politecnica de CatalunyaFacultat de Matematiques i Estadıstica

31 de marzo de 2016

Part I

Introduction to portfolio theory

Portfolio theoryMeasures of performance of a portfolio

Market as the centre of the modelAcceptability setsBid and ask prices

Market as the centre of the model

Ai : an asset.

Zi : the cash flow of Ai between two dates (t0,T ).

Xi : the value of the asset at the horizon date T .

A probability space (Ω,F ,P).

The non-arbitrage hypotesis

Eq[Xi ] = (1 + r)X(0)i

The market accepts to...

Buy: Zi = Xi − (1 + r)w , with w ≤ X(0)i .

Sell: Zi = (1 + r)w − Xi , with w ≥ X(0)i .

Conic portfolio theory Portfolio theory 1 / 20



Acceptability sets

The following condition is satisfied:

Eq[Zi ] ≥ 0.

Acceptability set

A = Z | Eq[Z ] ≥ 0

Given a set of probability measures M.

Generalized acceptability set

A = Z | Eq[Z ] ≥ 0 ∀q ∈M.




Bid and ask prices

The market agrees to buy Ai for b or sell it for a if

Xi − b(1 + r) ∈ A, a(1 + r)− Xi ∈ AWhich means that for all q ∈M

Eq[Xi ]− b(1 + r) ≥ 0,

a(1 + r)− Eq[Xi ] ≥ 0.

Bid and Ask formulas

b(X ) =1

1 + rinf

q∈MEq[X ],

a(X ) =1

1 + rsupq∈M

Eq[X ].




Bid price

The function that we want to maximise is:

b(X ) = infq∈M

Eq[X ].

Assuming comonotone additivity and law invariance:

b(X ) =

∫Rx dΨ(FX (x)).



Coherent risk measuresIndexes of acceptabilityTVaR measureWVaR index of acceptabilityStress levelMinMaxVaR y Wang Transform

Coherent risk measures

A coherent risk measure is a function of the form

ρ(X ) = − infq∈M

Eq[X ] = −b(X ).

Remark: To simplify the notation one can think everything in termsof the bid price b(X ).

Definitions

Set of supporting kernels:

M = q ∈ P | Eq[X ] ≥ b(x) ∀q ∈ L∞(Ω).

Set of extreme measures Q∗(X ) defined as:

Eq[X ] = b(X ) ∀q ∈ Q∗(X ).

Conic portfolio theory Measures of performance of a portfolio 5 / 20



Coherent risk measures

Acceptability set

A = X ∈ L∞(Ω) | b(X ) ≥ 0.




Indexes of acceptability

Acceptability index: coherent risk measure satisfyingQuasi-concavity.Monotonicity.Scale invariance.Fatou property.

Coherent risk measure for an increasingly set (Mx)x∈R+ :

bx(X ) = infq∈Mx

Eq[X ].

Index of acceptability

α(X ) = supx | bx(X ) ≥ 0.

Remark: with this properties the acceptability set is a convex cone.




TVaR measure

TVaR coherent risk measure

TVaRλ(X ) = − infq∈Mλ

Eq[X ].

TVaR equivalent definition

TVaRλ(X ) = Eq[X |X ≤ xλ(X )].




WVaR index of acceptability

A generalisation of TVAR.It is the average of TVARλ with different risk levels λ weighted by aprobability measure µ.

WVaR

WVARµ(X ) =

1∫0

TVARλ(X )µ( dλ).

We can find an alternative definition that looks like the bid price:

WVaR alternative definition

WVARµ(X ) = −∫R

y d(Ψµ(FX (y))).




Stress level

We define a one parameter family of concave functions Ψγ .

γ: stress level.

WVaR acceptability index

AIW(X ) = supγ | bγ(X ) =

∫Ry d(Ψγ(FX (y))) ≥ 0.

Conclusion: Stress level is equivalent to portfolio risk (or acceptabi-lity).




MinMaxVaR y Wang Transform

MinMaxVaR

Ψγ(u) = 1−(

1− u1

1+γ

)1+γ.

Wang Transform

ΨγΦ(u) = Φ(Φ−1(u) + γ).


Part II

Calibration and computations

Discretisation of the bid priceCalibration algorithm for the stress level

Bid price discretisation by MadanAnother bid price discretisation

Bid price discretisation by Madan

The portfolio return over an investment time horitzon is:

Rp =N∑i=0

ai (exi − 1).

Goal: find the optimal ai to maximise the return, or more precisely thebid of it.

Bid price discretisation

b(Rp) =M∑

m=1

N∑i=1

ai (exi,m − 1)

(Ψγ(mM

)−Ψγ

(m − 1

M

)).

Conic portfolio theory Discretisation of the bid price 13 / 20


Bid price discretisation by MadanAnother bid price discretisation

Another bid price discretisation


b(Rp) = infq∈M

Eq[Rp] ≤ infq∈M

N∑i=1

ai

(Eqi [X

(f )i ]

X(0)i

− 1

)Problem: this inequality appears because we are taking the assetsseparated so a correlation should be considered.Problem: this presents a difficulty on the computation and we areassuing a particular form on the distribution.


b(Rp) = infθ1,...,θN

N∑i=1

ai

(Eqθi [X

(f )i ]

X(0)i

− 1

).

Conic portfolio theory Discretisation of the bid price 14 / 20


Calibration algorithm for the stress level

(i) For each of the N stocks estimate the bid price (b′) and the ask price(a′) from market data.

– bid price (b′): the minimum price of the previous 63 days.– ask price (a′): the maximum price of the previous 63 days.

(ii) Relativize each of the previous estimated quantities to the average price(x) of the previous 63 days:

b =b′

x, a =

a′

x.

Conic portfolio theory Calibration algorithm for the stress level 15 / 20


Calibration algorithm for the stress level

1.- For each calibration date t: take the average of the bid b and ask aalong the stocks.

bt =N∑i=1

aib(i)t , bt =

N∑i=1

b(i)t .

2.- We estimate the stress level with least squares minimisation:

γt = arg minγ

(bt − bt(γ)

)2+ (at − at(γ))2, (1)

where

bt(γ) = bt(X , γ) =M∑

m=1

xm

(Ψγ(mM

)−Ψγ

(m − 1

M

)).

Conic portfolio theory Calibration algorithm for the stress level 16 / 20

Part III

Optimisation problems

Optimisation problemsOptimisation problemLong-only portfoliosLong-Short portfolios

Optimisation problem

Let Rp =N∑i=1

aiRi be a portfolio.

General Optimisation problem

find: maxai

b(Rp),

subject to:N∑i=1

ai = 1.

Centred returns

Ri = µi + Rεi ⇒ b(Ri ) = µi + b(Rεi ).

b(Rp) = ~a · ~µ+ b(Rεp).

Conic portfolio theory Optimisation problems 18 / 20


Long-only portfolios

Long-only Optimisation problem

find: maxai

~a · ~µ+ b(Rεp),

subject to:N∑i=1

ai = 1.



Long-short portfolios

Using centred returns as before:

b(Rp) = ~a · ~µ− c(~a).

Long-short optimisation problem

find: minai

c(~a),

subject to: ~a · 1 = 1,

~a · ~µ = µp.

Idea: efficient frontier.


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Conic portfolio theory Sergi Ferrer Fern andezcomputationalfinance.lsi.upc.edu/wp-content/... · 31 de marzo de 2016. Part I Introduction to portfolio theory. Portfolio theory Measures