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
Portfolio theoryMeasures of performance of a portfolio
Market as the centre of the modelAcceptability setsBid and ask prices
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.
Conic portfolio theory Portfolio theory 2 / 20
Portfolio theoryMeasures of performance of a portfolio
Market as the centre of the modelAcceptability setsBid and ask prices
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 ].
Conic portfolio theory Portfolio theory 3 / 20
Portfolio theoryMeasures of performance of a portfolio
Market as the centre of the modelAcceptability setsBid and ask prices
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)).
Conic portfolio theory Portfolio theory 4 / 20
Portfolio theoryMeasures of performance of a portfolio
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
Portfolio theoryMeasures of performance of a portfolio
Coherent risk measuresIndexes of acceptabilityTVaR measureWVaR index of acceptabilityStress levelMinMaxVaR y Wang Transform
Coherent risk measures
Acceptability set
A = X ∈ L∞(Ω) | b(X ) ≥ 0.
Conic portfolio theory Measures of performance of a portfolio 6 / 20
Portfolio theoryMeasures of performance of a portfolio
Coherent risk measuresIndexes of acceptabilityTVaR measureWVaR index of acceptabilityStress levelMinMaxVaR y Wang Transform
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.
Conic portfolio theory Measures of performance of a portfolio 7 / 20
Portfolio theoryMeasures of performance of a portfolio
Coherent risk measuresIndexes of acceptabilityTVaR measureWVaR index of acceptabilityStress levelMinMaxVaR y Wang Transform
TVaR measure
TVaR coherent risk measure
TVaRλ(X ) = − infq∈Mλ
Eq[X ].
TVaR equivalent definition
TVaRλ(X ) = Eq[X |X ≤ xλ(X )].
Conic portfolio theory Measures of performance of a portfolio 8 / 20
Portfolio theoryMeasures of performance of a portfolio
Coherent risk measuresIndexes of acceptabilityTVaR measureWVaR index of acceptabilityStress levelMinMaxVaR y Wang Transform
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))).
Conic portfolio theory Measures of performance of a portfolio 9 / 20
Portfolio theoryMeasures of performance of a portfolio
Coherent risk measuresIndexes of acceptabilityTVaR measureWVaR index of acceptabilityStress levelMinMaxVaR y Wang Transform
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).
Conic portfolio theory Measures of performance of a portfolio 10 / 20
Portfolio theoryMeasures of performance of a portfolio
Coherent risk measuresIndexes of acceptabilityTVaR measureWVaR index of acceptabilityStress levelMinMaxVaR y Wang Transform
MinMaxVaR y Wang Transform
MinMaxVaR
Ψγ(u) = 1−(
1− u1
1+γ
)1+γ.
Wang Transform
ΨγΦ(u) = Φ(Φ−1(u) + γ).
Conic portfolio theory Measures of performance of a portfolio 11 / 20
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
Discretisation of the bid priceCalibration algorithm for the stress level
Bid price discretisation by MadanAnother bid price discretisation
Another bid price discretisation
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.
Bid price discretisation
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
Discretisation of the bid priceCalibration algorithm for the stress level
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
Discretisation of the bid priceCalibration algorithm for the stress level
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
Optimisation problemsOptimisation problemLong-only portfoliosLong-Short portfolios
Long-only portfolios
Long-only Optimisation problem
find: maxai
~a · ~µ+ b(Rεp),
subject to:N∑i=1
ai = 1.
Conic portfolio theory Optimisation problems 19 / 20
Optimisation problemsOptimisation problemLong-only portfoliosLong-Short portfolios
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.
Conic portfolio theory Optimisation problems 20 / 20