The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible Probabilities

Setting the Flexible Probabilities

• How to assign different relative weights to historical scenarios?• time conditioning - weights depend on time• state conditioning - weights depend on market conditions• joint time and state conditioning - Minimum RelativeEntropy approach

• We measure the statistical power of the estimators based on FlexibleProbabilities with the Effective Number of Scenarios

More on advanced ways to set the Flexible Probabilities in quasi-Bayesian approachand Flexible Probabilities as random variables

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible Probabilities

Setting the Flexible Probabilities

• How to assign different relative weights to historical scenarios?• time conditioning - weights depend on time• state conditioning - weights depend on market conditions• joint time and state conditioning - Minimum RelativeEntropy approach

• We measure the statistical power of the estimators based on FlexibleProbabilities with the Effective Number of Scenarios

More on advanced ways to set the Flexible Probabilities in quasi-Bayesian approachand Flexible Probabilities as random variables

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Jan-31-2017 - Last update

The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesExponential decay and time conditioning

Time conditioning

Set time crisp probabilities as

pt|Tt∗1←{

1 if t ∈ Tt∗0 otherwise (2a.13)

where Tt∗ is the time window of interest.

Set time exponential decay probabilities as

pt|t∗1← e− ln(2)τHL|t∗−t| (2a.14)

where τHL > 0 is the half-life.

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesExponential decay and time conditioning

Example 2a.1. Time exponential decay probabilities for the S&P500 returns

• Target time: t∗ ≡ t̄• Invariants: S&P 500 returns

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesKernels and state conditioning

State conditioning

Set state crisp probabilities as

pt|R(z∗)1←{

1 if zt ∈ R(z∗)0 otherwise (2a.15)

for a range R(z∗) ≡ [z, z̄] around a target level z∗.

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesKernels and state conditioning

Example 2a.2. Conditioning via crisp Flexible Probabilities

• Conditioning variable: Zt smoothing and scoring the VIX log-return• Invariants: S&P 500 returns

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesKernels and state conditioning

Kernels

Set smooth kernel probabilities as

pt|z∗1← e−(|zt−z∗|/h)γ (2a.16)

where h is the kernel’s bandwidth.

• γ ≡ 1 exponential kernel probabilities• γ ≡ 2 Gaussian kernel probabilities

Set multivariate Gaussian kernel probabilities as

pt|z∗1← e−(zt−z∗)′(h2)−1(zt−z∗) (2a.17)

where h2 is a symmetric and positive definite bandwidth matrix.

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesKernels and state conditioning

Kernels

Set smooth kernel probabilities as

pt|z∗1← e−(|zt−z∗|/h)γ (2a.16)

where h is the kernel’s bandwidth.

• γ ≡ 1 exponential kernel probabilities• γ ≡ 2 Gaussian kernel probabilities

Set multivariate Gaussian kernel probabilities as

pt|z∗1← e−(zt−z∗)′(h2)−1(zt−z∗) (2a.17)

where h2 is a symmetric and positive definite bandwidth matrix.

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesKernels and state conditioning

Example 2a.3. Gaussian kernel probabilities for the S&P 500returns

• Conditioning variable: Zt smoothing and scoring the VIX log-return• Invariants: S&P 500 returns

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesJoint state and time conditioning

Joint time and state conditioning: minimum relative entropy

• Compute state crisp moments, with pcrispt as in (2a.15)

µ|z∗ =∑t̄

t=1ztpcrispt , σ2|z∗ =

∑t̄t=1z

2t p

crispt − (µ|z∗)2 (2a.18)

• Impose

V|z∗ :

{ ∑t̄t=1ptzt = µ|z∗∑t̄t=1ptz

2t ≤ (µ|z∗)2 + (σ|z∗)2 (2a.19)

• Setp|z∗ ≡ argmin

p∈V|z∗E(p‖pexp) (2a.20)

where E(p‖pexp) is the relative entropy (25.160).

More on the Minimum Relative Entropy approach

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Jan-31-2017 - Last update

The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesJoint state and time conditioning

Joint time and state conditioning: minimum relative entropy

• Compute state crisp moments, with pcrispt as in (2a.15)

µ|z∗ =∑t̄

t=1ztpcrispt , σ2|z∗ =

∑t̄t=1z

2t p

crispt − (µ|z∗)2 (2a.18)

• Impose

V|z∗ :

{ ∑t̄t=1ptzt = µ|z∗∑t̄t=1ptz

2t ≤ (µ|z∗)2 + (σ|z∗)2 (2a.19)

• Setp|z∗ ≡ argmin

p∈V|z∗E(p‖pexp) (2a.20)

where E(p‖pexp) is the relative entropy (25.160).

More on the Minimum Relative Entropy approach

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Jan-31-2017 - Last update

The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesJoint state and time conditioning

Joint time and state conditioning: minimum relative entropy

• Compute state crisp moments, with pcrispt as in (2a.15)

µ|z∗ =∑t̄

t=1ztpcrispt , σ2|z∗ =

∑t̄t=1z

2t p

crispt − (µ|z∗)2 (2a.18)

• Impose

V|z∗ :

{ ∑t̄t=1ptzt = µ|z∗∑t̄t=1ptz

2t ≤ (µ|z∗)2 + (σ|z∗)2 (2a.19)

• Setp|z∗ ≡ argmin

p∈V|z∗E(p‖pexp) (2a.20)

where E(p‖pexp) is the relative entropy (25.160).

More on the Minimum Relative Entropy approach

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Jan-31-2017 - Last update

The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesJoint state and time conditioning

Example 2a.4. Flexible Probabilities conditioned via MinimumRelative Entropy

• Conditioning variable: Zt smoothing and scoring the VIX log-return• Invariants: S&P 500 returns

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesStatistical power of Flexible Probabilities

Effective Number of ScenariosHow many scenarios are we effectively using when relying on FlexibleProbabilities?

The Effective Number of Scenarios

ens(p) ≡ e−∑t pt ln pt (2a.21)

The Effective Number of Scenarios satisfies 1 ≤ ens(p) ≤ t̄ and the twoextreme cases follow

Flexible Probabilities ens(p)

pt ≡ 1t=t∗ 1

pt ≡ 1t̄for any t t̄

Table 2a.2 Effective Number of Scenarios in two extreme cases

More on the Effective Number of Scenarios

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Jan-31-2017 - Last update

The “Checklist” > 2a. Estimation: Flexible Probabilities > Setting the Flexible ProbabilitiesStatistical power of Flexible Probabilities

Effective Number of ScenariosHow many scenarios are we effectively using when relying on FlexibleProbabilities?

The Effective Number of Scenarios

ens(p) ≡ e−∑t pt ln pt (2a.21)

The Effective Number of Scenarios satisfies 1 ≤ ens(p) ≤ t̄ and the twoextreme cases follow

Flexible Probabilities ens(p)

pt ≡ 1t=t∗ 1

pt ≡ 1t̄for any t t̄

Table 2a.2 Effective Number of Scenarios in two extreme cases

More on the Effective Number of Scenarios

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Jan-31-2017 - Last update