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lCARE - localising Conditional AutoRegressive Expectiles Wolfgang Karl Hrdle Xiu Xu Andrija Mihoci Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics HumboldtUniversitt zu Berlin

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lCARE - localising Conditional AutoRegressive Expectiles

Wolfgang Karl Härdle Xiu Xu Andrija Mihoci Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. – Center for Applied Statistics and Economics Humboldt–Universität zu Berlin Brandenburg University of Technology lvb.wiwi.hu-berlin.de case.hu-berlin.de irtg1792.hu-berlin.de b-tu.de

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Motivation 1-1

Motivation

⊡ Risk Exposure

◮ Measure tail events ◮ Conditional autoregressive expectile (CARE) model

Expectiles

⊡ Time-varying parameters

◮ Time-varying parameters in CARE

Parameter Dynamics

◮ Interval length reflects the structural changes in economy

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Motivation 1-2

Objectives

⊡ Localising CARE Models

◮ Local parametric approach (LPA) ◮ Balance between modelling bias and parameter variability

⊡ Tail Risk Dynamics

◮ Estimation windows with varying lengths ◮ Time-varying expectile parameters

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Motivation 1-3

Econometrics and Risk Management

Econometrics ⊡ Modelling bias vs. parameter variability ⊡ Interval length and economic variables Risk Management ⊡ Parameter dynamics and structural changes ⊡ Measuring tail risk

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Motivation 1-4

Risk Exposure

An investor observes daily DAX returns from 20050103 to 20141231 and estimates the underlying risk exposure via expectiles (e.g., 1% and 5%) over a one-year time horizon. Modelling strategies (a) Data windows fixed on an ad hoc basis (b) Adaptively selected data intervals: time-varying parameters

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Motivation 1-5

Portfolio Protection

An investor decides about the daily allocation into a stock portfolio (DAX). Goal: a proportion of the initial portfolio value (100) is preserved at the end of a horizon, i.e., the target floor equals 90. Decision at day t: multiple of the difference between the portfolio value and the discounted floor up to t is invested into the stock portfolio (DAX), the rest into a riskless asset. Multiplier m selection: constant or time-varying (lCARE)

Constant m

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Motivation 1-6

Research Questions

How to account for time-varying parameters in tail event risk measures estimation? What are the typical data interval lengths assessing risk more accurately, i.e., striking a balance between bias and variability? How well does the lCARE technique perform in practice?

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Outline

  • 1. Motivation
  • 2. Conditional Autoregressive Expectile (CARE)
  • 3. Local Parametric Approach (LPA)
  • 4. Empirical Results
  • 5. Applications
  • 6. Conclusions

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Conditional Autoregressive Expectile (CARE) 2-1

Conditional Autoregressive Expectile

⊡ Taylor (2008), Kuan et al. (2009), Engle and Manganelli (2004)

CAViaR

⊡ Random variable Y (e.g. returns), identically distributed, yt, t = 1, ..., n ⊡ CARE specification conditional on information set Ft−1 yt = et,τ + εt,τ

ετ ∼ AND

  • 0, σ2

ε,τ , τ

  • et,τ = α0,τ + α1,τyt−1 + α2,τ
  • y+

t−1

2 + α3,τ

  • y−

t−1

2

◮ Expectile et,τ at τ ∈ (0, 1), θτ =

  • α0,τ, α1,τ, α2,τ, α3,τ, σ2

ε,τ

⊤ ◮ Returns: y +

t−1 = max {yt−1, 0}, y − t−1 = min {yt−1, 0}

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Conditional Autoregressive Expectile (CARE) 2-2

Parameter Estimation

⊡ Data calibration with time-varying intervals ⊡ Observed returns Y = {y1, . . . , yn} ⊡ Quasi maximum likelihood estimate (QMLE)

  • θI,τ = arg max

θτ∈Θ ℓI (Y; θτ)

ℓI (·)

◮ I = [t0 − v, t0] - interval of (v + 1) observations at t0 ◮ ℓI (·) - quasi log likelihood

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Conditional Autoregressive Expectile (CARE) 2-3

Estimation Quality

⊡ Mercurio and Spokoiny (2004), Spokoiny (2009) ⊡ Quality of estimating true parameter vector θ∗

τ by QMLE

θI,τ in terms of Kullback-Leibler divergence; Rr (θ∗

τ) - risk bound

Eθ∗

τ

  • ℓI(Y;

θI,τ) − ℓI(Y; θ∗

τ)

  • r

≤ Rr (θ∗

τ)

Rr

  • θ∗

τ

  • Gaussian Regression

⊡ ’Modest’ risk, r = 0.5 (shorter intervals of homogeneity) ⊡ ’Conservative’ risk, r = 1 (longer intervals of homogeneity) Solomon Kullback and Richard A. Leibler on BBI:

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Local Parametric Approach (LPA) 3-1

Local Parametric Approach (LPA)

⊡ LPA, Spokoiny (1998, 2009)

◮ Time series parameters can be locally approximated ◮ Finding the interval of homogeneity

Details

◮ Balance between modelling bias and parameter variability

⊡ Time series literature

◮ GARCH(1, 1) models - Čížek et al. (2009) ◮ Realized volatility - Chen et al. (2010) ◮ Multiplicative Error Models - Härdle et al. (2015)

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Local Parametric Approach (LPA) 3-2

Interval Selection

⊡ (K + 1) nested intervals with length nk = |Ik| I0 ⊂ I1 ⊂ · · · ⊂ Ik ⊂ · · · ⊂ IK

  • θ0
  • θ1
  • θk
  • θK

Example: Daily index returns Fix t0, Ik = [t0 − nk, t0], nk =

  • n0ck

, c > 1 {nk}11

k=0 = {20 days, 25 days, . . . , 250 days}, c = 1.25

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Local Parametric Approach (LPA) 3-3

Local Change Point Detection

⊡ Fix t0, sequential test (k = 1, . . . , K) H0 : parameter homogeneity within Ik H1 : ∃ change point within Jk = Ik \ Ik−1 Tk,τ = sup

s∈Jk

  • ℓAk,s
  • Y,

θAk,s,τ

  • + ℓBk,s
  • Y,

θBk,s,τ

  • − ℓIk+1
  • Y,

θIk+1,τ

  • with Ak,s = [t0 − nk+1, s] and Bk,s = (s, t0]

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Local Parametric Approach (LPA) 3-4

Critical Values, zk,τ

⊡ Simulate zk - homogeneity of the interval sequence I1, . . . , Ik ⊡ ’Propagation’ condition Eθ∗

τ

  • ℓIk
  • Y;

θIk,τ

  • − ℓIk
  • Y;

θτ

  • r

≤ ρkRr (θ∗

τ)

ρk = ρk K for a given significance level ρ

  • θτ - adaptive estimate

⊡ Check zk,τ for (six) different θ∗

τ

Parameter Scenarios

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Local Parametric Approach (LPA) 3-5

Critical Values, zk,τ

20 40 60 120 10 20

Length in Days Values 20 40 60 120 10 20 Length in Days 20 40 60 120 10 20 Length in Days

Figure 1: Simulated critical values across different parameter constellations

Parameter Scenarios for the modest case r = 0.5, τ = 0.05 and τ = 0.01.

LCARE_Critical_Values LCARE_Critical_Values_Th1_001 LCARE_Critical_Values_Th1_005 LCARE_Critical_Values_Th2_001 LCARE_Critical_Values_Th2_005 LCARE_Critical_Values_Th3_001 LCARE_Critical_Values_Th3_005

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Local Parametric Approach (LPA) 3-6

Critical Values, zk,τ

20 40 60 120 200 400

Length in Days Values

20 40 60 120 200 400

Length in Days

20 40 60 120 200 400

Length in Days

Figure 2: Simulated critical values across different parameter constellations

Parameter Scenarios for the conservative case r = 1, τ = 0.05 and τ = 0.01

. LCARE_Critical_Values LCARE_Critical_Values_Th1_001 LCARE_Critical_Values_Th1_005 LCARE_Critical_Values_Th2_001 LCARE_Critical_Values_Th2_005 LCARE_Critical_Values_Th3_001 LCARE_Critical_Values_Th3_005

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Local Parametric Approach (LPA) 3-7

Adaptive Estimation

LPA zk,τ - Critical Values

⊡ Compare Tk,τ at every step k with zk,τ ⊡ Data window index of the interval of homogeneity - k ⊡ Adaptive estimate

  • θτ =

θIˆ

k,τ,

  • k = max

k≤K {k : Tℓ,τ ≤ zℓ,τ, ℓ ≤ k}

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Empirical Results 4-1

Data

⊡ Series

◮ DAX, FTSE 100 and S&P 500 returns 20050103-20141231 (2608 days) ◮ Research Data Center (RDC) - Datastream

⊡ Setup

◮ Expectile levels: τ = 0.05 and τ = 0.01 ◮ Modest (r = 0.5) and conservative (r = 1) risk cases ◮ {nk}11

k=0 = {20 days, 25 days, . . . , 250 days}

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Empirical Results 4-2

Adaptive Estimation

2006 2010 2014 60 120 180 DAX Length 2006 2010 2014 60 120 180 FTSE 100 2006 2010 2014 60 120 180 S&P 500 Figure 3: Estimated length nˆ

k of intervals of homogeneity from 20060103-

20141231 for the modest risk case r = 0.5, at expectile level τ = 0.05. The red line denotes one-month smoothed values.

Parameter Flag

LCARE_Adaptive_Estimation_Length_005 LCARE_Adaptive_Estimation_005

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Empirical Results 4-3

Adaptive Estimation

2006 2010 2014 60 120 180 DAX Length 2006 2010 2014 60 120 180 FTSE 100 2006 2010 2014 60 120 180 S&P 500 Figure 4: Estimated length nˆ

k of intervals of homogeneity from 20060103-

20141231 for the conservative risk case r = 1, at expectile level τ = 0.05. The red line denotes one-month smoothed values.

Parameter Flag

LCARE_Adaptive_Estimation_Length_005 LCARE_Adaptive_Estimation_005

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Empirical Results 4-4

Adaptive Estimation

2006 2010 2014 60 120 180 DAX Length 2006 2010 2014 60 120 180 FTSE 100 2006 2010 2014 60 120 180 S&P 500 Figure 5: Estimated length nˆ

k of intervals of homogeneity from 20060103-

20141231 for the modest risk case r = 0.5, at expectile level τ = 0.01. The red line denotes one-month smoothed values.

Parameter Flag

LCARE_Adaptive_Estimation_Length_001 LCARE_Adaptive_Estimation_001

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Empirical Results 4-5

Adaptive Estimation

2006 2010 2014 60 120 180 DAX Length 2006 2010 2014 60 120 180 FTSE 100 2006 2010 2014 60 120 180 S&P 500 Figure 6: Estimated length nˆ

k of intervals of homogeneity from 20060103-

20141231 for the conservative risk case r = 1, at expectile level τ = 0.01. The red line denotes one-month smoothed values.

Parameter Flag

LCARE_Adaptive_Estimation_Length_001 LCARE_Adaptive_Estimation_001

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Applications 5-1

Risk Exposure

2006 2008 2010 2012 2014 −0.1 0.1 Returns Time Figure 7: DAX index returns (∗) and adaptively estimated expectile et,τ (r = 1 and τ = 0.05) from 20060103-20141231

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Applications 5-2

Risk Exposure

Expected Shortfall ESet,τ

2006 2008 2010 2012 2014 −0.1 0.1 Returns Time Figure 8: DAX index returns (∗), adaptively estimated expectile et,τ and expected shortfall ESet,τ (r = 1 and τ = 0.05) from 20060103-20141231

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Applications 5-3

Portfolio Protection

⊡ Portfolio protection strategy

Details

◮ Aim: Guarantee a proportion level of wealth at the investment horizon. ◮ The investor can reduce the downside risk as well as participating in gains of risky assets.

Example Decision at day t: multiple of the difference between the portfolio value and the discounted floor up to t is invested into the stock portfolio (DAX), the rest into a riskless asset

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Applications 5-4

Portfolio Protection

⊡ Crucial ingredient: the multiplier m

◮ m: the proportion value invested into risky assets ◮ The larger m, the more risky exposure

⊡ How to select the multiplier?

◮ Standard constant value

Constant m

◮ Based on tail risk measure, VaR or ES

Details

⊡ Multiplier selection - Hamidi et al. (2014), lCARE mt,τ =

  • ESet,τ
  • −1

◮ Practice: threshold range for mt,τ, [1, 12]

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Applications 5-5

Multiplier Dynamics

2006 2008 2010 2012 2014 4 8 12 Multiplier Time Figure 9: Time-varying multiplier mt,τ for DAX index returns based on lCARE (r = 1 and τ = 0.05) from 20060103-20141231

Multiplier Density

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Applications 5-6

Performance

One-year rolling details CAViaR-based rolling details Target floor

2006 2008 2010 2012 2014 100 150 200 Price Index Time Figure 10: Portfolio value: (a) DAX index (black), (b) m = 5 , (c) one-year rolling , (d) CAViaR one-year rolling (α = 0.065), (e) mt,τ - lCARE (r = 1 and τ = 0.05) from 20060103-20141231.

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Applications 5-7

Performance

Figure 11: Portfolio return moments comparison. Returns and volatility are annualized. The investment strategy is on a one-year investment basis.

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Applications 5-8

Performance - Summary

⊡ lCARE vs empirical data

◮ Slightly lower return (7.36% vs 8.79%) largely lower volatility (13.60% vs 22.54%) ◮ Guarantee the target floor value

⊡ lCARE vs other strategies

◮ higher return than the candidates with CAViaR-based or expectile one-year rolling ◮ Outperform typical constant multiplier benchmarks

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Conclusions 6-1

Conclusions

⊡ Localising CARE Model

◮ Balance between modelling bias and parameter variability ◮ Parameter dynamics

⊡ Tail Risk Dynamics

◮ Expectile levels τ = 0.05 and τ = 0.01 ◮ Expectile and Expected Shortfall

⊡ Asset Allocation

◮ Portfolio insurance on DAX at level τ = 0.05 ◮ Outperform one-year rolling window and other benchmarks

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lCARE - localising Conditional AutoRegressive Expectiles

Wolfgang Karl Härdle Xiu Xu Andrija Mihoci Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. – Center for Applied Statistics and Economics Humboldt–Universität zu Berlin Brandenburg University of Technology lvb.wiwi.hu-berlin.de case.hu-berlin.de irtg1792.hu-berlin.de b-tu.de

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References 7-1

References

Acerbi, C. and Tasche, D. Expected Shortfall: a natural coherent alternative to Value at Risk Economic notes 31(2): 379–388, 2002 Ameur, H.B., and Prigent J.L. Portfolio insurance: Gap risk under conditional multiples European Journal of Operational Research 236(1): 238–253, 2014 Breckling, J. and Chambers, R. M-quantiles Biometrica 75(4): 761-771, 1988 DOI: 10.1093/biomet/75.4.761

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References 7-2

References

Chen, Y. and Härdle, W. and Pigorsch, U. Localized Realized Volatility Journal of the American Statistical Association 105(492): 1376–1393, 2010 Čížek, P., Härdle, W. and Spokoiny, V. Adaptive Pointwise Estimation in Time-Inhomogeneous Conditional Heteroscedasticity Models Econometrics Journal 12: 248–271, 2009

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References 7-3

References

Estep, T. and Kritzman, M. Time-invariant portfolio protection:insurance without complexity Journal of Portfolio Management 14(4): 38–42, 1988 Föllmer, H. and Leukert, P. Quantile hedging Finance and Stochastics 3: 251–273, 1999

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References 7-4

References

Gerlach, R.H., Chen, C.W.S. and Lin, L.Y. Bayesian GARCH Semi-parametric Expected Shortfall Forecasting in Financial Markets Business Analytics Working Paper No. 01/2012, 2012 Hamidi, B., Jurczenko, E. and Maillet, B. A CAViaR modelling for a simple time-varying proportion portfolio insurance strategy Bankers, Markets & Investors 102: 4–21, 2009 Hamidi, B., Maillet, B. and Prigent, J.L. A dynamic autoregressive expectile for time-invariant portfolio protection strategies Journal of Economic Dynamics & Control 46: 1–29, 2014

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References 7-5

References

Härdle, W. K., Hautsch, N. and Mihoci, A. Local Adaptive Multiplicative Error Models for High-Frequency Forecasts Journal of Applied Econometrics, 2014 Jiang, C.H., Ma, Y.K. and An, Y.B. The effectiveness of the VaR-based portfolio insurance strategy: An empirical analysis International Review of Financial Analysis, 18(4): 185–197, 2009 Kuan, C.M., Yeh, J.H. and Hsu, Y.C. Assessing value at risk with CARE, the Conditional Autoregressive Expectile models Journal of Econometrics 150(2): 261–270, 2009

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References 7-6

References

Mercurio, D. and Spokoiny, V. Statistical inference for time-inhomogeneous volatility models The Annals of Statistics 32(2): 577–602, 2004 Spokoiny, V. Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice The Annals of Statistics 26(4): 1356–1378, 1998

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References 7-7

References

Spokoiny, V. Multiscale Local Change Point Detection with Applications to Value-at-Risk The Annals of Statistics 37(3): 1405–1436, 2009 Taylor, J.W. Estimating Value at Risk and Expected Shortfall Using Expectiles Journal of Financial Econometrics 6(2): 231–252, 2008 Yao, Q. and Tong, H. Asymmetric least squares regression estimation: a nonparametric approach Journal of Nonparametric Statistics 6(2): 273–292, 1996

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Appendix 8-1

Why Expectiles? Quantile VaR

Motivation

Figure 12: Distribution of returns, the 5% quantile remains unchanged under the changing tail structure

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Appendix 8-2

Expectile v.s. Quantile

Motivation

⊡ Tail inference

◮ Quantile: zero-moment of tail structure - probability Central quantile: median ◮ Expectile: first moment of tail structure Central expectile: mean

⊡ Expectiles are sensitive to extreme magnitude, outliers ⊡ Expectiles link to expected shortfall (ES) nicely

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Appendix 8-3

M-Quantiles

Motivation

⊡ Loss function, Breckling and Chambers (1988) zα = arg min

θ E ρα,γ (Y − θ)

where ρα,γ (u) = |α − 1 {u < 0}| |u|γ, γ ≥ 1

◮ Quantile - ALD location estimate qα = arg min

θ E ρα,1 (Y − θ)

◮ Expectile - AND location estimate eα = arg min

θ E ρα,2 (Y − θ)

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Appendix 8-4

Loss Function

Motivation

u 2 4 −6 −3 3 6 u 2 4 −6 −3 3 6 Figure 13: Expectile and quantile loss functions at α = 0.01 (left) and α = 0.50 (right)

LQRcheck

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Appendix 8-5

Expectiles and Quantiles

Motivation

⊡ M-Quantile α 1 − α = eα

−∞ |y − eα|γ−1 dF(y)

∞

eα |y − eα|γ−1 dF(y)

◮ Expectile - Global influence, obtained from γ = 2, α 1 − α = eα

−∞ |y − eα| dF(y)

∞

eα |y − eα| dF(y)

◮ Quantile - Local influence, obtained from γ = 1, α 1 − α = P (Y ≤ qα) P (Y > qα)

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Appendix 8-6

CAViaR - Conditional Autoregressive Value at Risk by Regression Quantiles

CARE CAViaR performance

⊡ Engle and Manganelli (2004) ⊡ Asymmetric slope specification, conditional on information set Ft−1 at time t yt = qt,α + εt,α Quantα(εt,α|Ft−1) = 0 qt,α = β0 + β1qt−1,α + β2y+

t−1 + β3y− t−1

◮ Quantile (VaR) qt,α at α ∈ (0, 1), Quantα(εt,α|Ft−1) is the α-quantile of εt,α conditional on information set Ft−1 ◮ With AND, set α = 0.065 such that eτα = qα when τα = 0.05

Details

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Appendix 8-7

Asymmetric Normal Distribution (AND)

CARE

⊡ AND

  • µ, σ2, τ
  • pdf:

f (w) = 2 σ

  • π

|τ − 1| + π τ −1 exp

  • −ρτ

w − µ σ

  • ◮

Check function: ρτ (u) = |τ − I {u ≤ 0}| u2 ◮ AND

  • µ, σ2, 1/2
  • = N(µ, σ2), Gerlach et al. (2012)

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Appendix 8-8

PDF

−10 −5 5 10 15 20 0.1 0.2 0.3 0.4 PDF w

Figure 14: Density function for selected ANDs: (a) µ = 0, τ = 0.5 (b) µ = −1, τ = 0.25 (c) µ = −2, τ = 0.05 (d) µ = −3, τ = 0.01, with σ2

ετ = 1

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Appendix 8-9

Quasi Log Likelihood Function

Parameter Estimation

⊡ If ετ ∼ AND

  • µ, σ2

ε, τ

  • with pdf fε (·)

then Y ∼ AND

  • eτ + µ, σ2

ε, τ

  • ⊡ Quasi log likelihood function for observed data

Y = {y1, . . . , yn} over a fixed interval I ℓI (Y; θτ) =

  • t∈I

log fε (yt − et,τ)

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Appendix 8-10

Gaussian Regression

Estimation Quality

Yi = f (Xi) + εi, i = 1, . . . , n, weights W = {wi}n

i=1

L (W , θ) =

n

  • i=1

ℓ {Yi, fθ (Xi)} wi, log-density ℓ (·), θ = arg max

θ∈Θ L (W , θ)

  • 1. Local constant, f (Xi) ≈ θ∗, εi ∼ N
  • 0, σ2

Eθ∗

  • L(W ,

θ) − L(W , θ∗)

  • r

≤ E |ξ|2r , ξ ∼ N (0, 1)

  • 2. Local linear, f (Xi) ≈ θ∗⊤Ψi, εi ∼ N
  • 0, σ2

, basis functions Ψ = {ψ1 (X1) , . . . , ψp (Xp)}, multivariate ξ Eθ∗

  • L(W ,

θ) − L(W , θ∗)

  • r

≤ E |ξ|2r , ξ ∼ N (0, Ip)

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SLIDE 51

Appendix 8-11

Risk Bound

Estimation Quality

τ = 0.05 τ = 0.01 Low Mid High Low Mid High r = 0.5 0.24 0.33 0.25 0.38 0.38 0.15 r = 1.0 2.40 4.62 2.75 5.90 5.81 1.15

Table 1: Simulated Rr (θ∗

τ), with expectile levels τ = 0.05 and τ =

0.01, for six selected parameter constellation groups

Parameter Scenarios

LCARE_Risk_Bound_Results LCARE_Risk_Bound_Th1_001 LCARE_Risk_Bound_Th1_005 LCARE_Risk_Bound_Th2_001 LCARE_Risk_Bound_Th2_005 LCARE_Risk_Bound_Th3_001 LCARE_Risk_Bound_Th3_005

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SLIDE 52

Appendix 8-12

Parameter Scenarios

Risk Bound Critical Values

τ = 0.05 τ = 0.01 Low Mid High Low Mid High

  • α0,τ
  • 0.0003

0.0003 0.0007

  • 0.0003

0.0003 0.0007

  • α1,τ
  • 0.1058
  • 0.0306

0.0524

  • 0.1035
  • 0.0312

0.0547

  • α2,τ
  • 0.5800
  • 0.5288

0.2438

  • 0.5808
  • 0.5266

0.2089

  • α3,τ

0.5050 0.5852 2.1213 0.5134 0.5871 2.2066

  • σ2

ε,τ

0.0001 0.0001 0.0002 0.0001 0.0001 0.0002

Table 2: Quartiles of estimated CARE parameters based on one-year estimation window, i.e., 250 observations, for stock market returns

  • DAX, FTSE 100 - from 20050103-20141231 (2608 trading days).

LCARE_Parameter_Dynamics_Quartiles

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SLIDE 53

Appendix 8-13

Selected Parameter Scenarios

Adaptive Estimation

Low Mid High 1000 2000 3000 DAX Low Mid High 1000 2000 3000 FTSE 100 Low Mid High 1000 2000 3000 S&P 500 Figure 15: Histogram of the selected parameter scenarios (Low, Mid and High) for adaptive estimation with τ = 0.05 and τ = 0.01.

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SLIDE 54

Appendix 8-14

Expected Shortfall

Risk Exposure CAViaR

⊡ Expectile level τα such that eτα = qα (α-quantile), Yao and Tong (1996), Acerbi and Tasche (2002) τα = α · qα − qα

−∞

ydF(y) E[Y ] − 2 qα

−∞

ydF(y) − (1 − 2α) qα where Y ∼ AND. ⊡ Expected Shortfall (ES), Kuan et al. (2009) ESeτα =

  • 1 + τα (1 − 2τα)−1 α−1
  • eτα

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SLIDE 55

Appendix 8-15

Portfolio Protection Strategy

Strategy Multiplier

⊡ Under certain confidence level, we aim to maintain: Estep and Kritzman (1988) Vt ≥ k × max

  • F ∗ e−rf ∗(T−t), sup

p≤t

Vp

  • = F s

t

◮ Vt: portfolio value at time t, t ∈ (0, T] F s

t : protection value (target floor)

◮ k exogenous parameter (0, 1), set k = 0.9 rf risky free rate, initial value F = 100 ◮ Cushion value Ct = Vt − F s

t ≥ 0

⊡ Allocate Gt = m · Ct proportion into stock portfolio (DAX), and the remaining Vt − Gt into riskless asset, multiplier m ≥ 0.

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SLIDE 56

Appendix 8-16

Example: CPPI - Constant proportion portfolio insurance Consider an insurance strategy under CPPI with constant floor F = 100, constant m = 5, and riskless asset rate rf = 0 (cash). The initial risky asset value is 100, and at each step goes up(down) 15. initial risky asset value F 100 proportion k 0.9 riskless rate rf constant multiplier m 5 steps 4

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SLIDE 57

Appendix 8-17

Risky asset value 160 145 (0.10) 130 (0.12) 130 115 (0.13) 115 (0.13) 100 (0.15) 100 (0.15) 100 85 (0.18) 85 (0.18) (-0.15) 70 (0.21) 70 (-0.18) 55 (0.27) (-0.21 ) 40 (-0.27 )

Table 3: Risky portfolio value and the value in low bracket denotes the asset return.

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SLIDE 58

Appendix 8-18

Portfolio value and the cushion 159.18 135.59 (69.18) 118.91 (45.60) 103.61 107.50 (28.91) 98.24 (13.61) 100 (17.5) 94.71 (8.24) 91.15 (10) 92.50 (4.71) 90.61 (1.15) (2.5) 90.29 (0.61) 89.979 (0.29) 89.979 (0 ) (-0.02 ) 89.979 (0 )

Table 4: Portfolio value and the value in low bracket denotes the cushion.

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SLIDE 59

Appendix 8-19

Multiplier

Multiplier

⊡ Portfolio value Vt Vt+1 = Vt + Gtrt+1 + (Vt − Gt) rft+1 with rt stock index return and rft riskless rate ⊡ Cushion value Ct = Vt − F s

t ≥ 0

Ct+1 = Ct{1 + m · rt+1 + (1 − m) rft+1} ⊡ ∀t ≤ T, since the value Ct ≥ 0 m · rt+1 + (1 − m) rft+1 ≥ −1

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SLIDE 60

Appendix 8-20

Multiplier

Multiplier Gap risk

⊡ If rft is relatively small, and when rt+1 < 0, yield the upper bound on the multiplier: Proposition The guarantee is satisfied at any time of the management period with a probability equal to 1 ≺ ≻ ∀t ≤ T − 1, m ≤

  • −r−

t+1

−1 where r−

t+1 = min {rt+1, 0}.

◮ Multiplier mt, the leverage value on risky assets, is negatively related to the maximum extreme loss of risky assets. ◮ For example, if rt+1 = −10%, m ≤ 10; If rt+1 = −20%, m ≤ 5.

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SLIDE 61

Appendix 8-21

Gap Risk

Multiplier

⊡ In practice, due to the discrete-time rebalancing, the nonnegative cushion value can not be guaranteed perfectly.

Details

⊡ Gap risk: the risk of violating the floor protection, i.e., the tiny level of probability that the cushion values are non-positive. ⊡ How to define the gap risk:

◮ control of the probability of a potential loss - VaR based multiplier ◮ control of the potential loss size - ES based multiplier

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SLIDE 62

Appendix 8-22

Gap Risk - control of the probability of a potential loss - VaR based multiplier

Multiplier

⊡ Given a confidence level 1 − α, the insurance condition, i.e., portfolio value is above floor, is guaranteed, Föllmer and Leukert (1999), P (Ct ≥ 0, ∀t ≤ T) ≥ 1 − α ⊡ Equivalently, (set time-varying multiplier) P

  • mt ≤
  • −r−

t+1

−1 , ∀t ≤ T − 1

  • ≥ 1 − α

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SLIDE 63

Appendix 8-23

Multiplier

Portfolio Protection Gap risk

⊡ Gap risk: control of the probability of a potential loss

Details

Multiplier mt with quantile - Ameur and Prigent (2014) mt,qα = |VaRα(rt+1)|−1 ⊡ Gap risk: control of the potential loss size Multiple mt with expected shortfall - Hamidi et al. (2014) mt,τ =

  • ESet,τ
  • −1

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SLIDE 64

Appendix 8-24

Multiplier Density

Multiplier Dynamics

2 4 6 8 10 12 0.1 0.2 0.3 Density Multiplier Figure 16: Kernel density estimate of the multiplier mt,τ for DAX index returns based on lCARE (r = 1 and τ = 0.05) from 20060103-20141231

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SLIDE 65

Appendix 8-25

CARE-based one-year rolling

Performance

2006 2008 2010 2012 2014 −0.1 0.1 Returns 2006 2008 2010 2012 2014 4 8 12 Multiplier Time

Figure 17: Estimated expectile and expected shortfall by CARE based one- year fixed rolling window (upper panel), and the corresponding multiplier (lower panel) for DAX index returns from 20060103 to 20141231

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SLIDE 66

Appendix 8-26

CAViaR-based one-year rolling

Performance CAViaR

2006 2008 2010 2012 2014 −0.1 0.1 Returns 2006 2008 2010 2012 2014 4 8 12 Multiple Time Figure 18: Estimated VaR (α = 0.065) and expected shortfall by CAViaR

  • based one-year rolling (upper panel), and the corresponding multiplier

(lower panel) for DAX from 20060103 to 20141231

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SLIDE 67

Appendix 8-27

Portfolio value and target floor

Performance

2006 2008 2010 2012 2014 100 150 200 Price Index Time

Figure 19: Portfolio value: (a) DAX index (black), (b) mt,τ - lCARE (r = 1 and τ = 0.05), (c) the corresponding target floor F s

t , from 20060103-

20141231.

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SLIDE 68

Appendix 8-28

Portfolio Protection

Motivation Portfolio Protection

08/07 01/08 06/08 11/08 03/09 96 98 100 Value Time 08/07 01/08 06/08 11/08 03/09 40 60 80 100 120 Dax Index 03/09 09/09 03/10 09/10 03/11 100 150 200 Time Value

Figure 20: Portfolio value: (a) DAX index, (b) m = 3, (c) m = 6, (d) m = 9, (e) m = 12 on DAX index in a bull market from 20090309-20110510 (left panel, 567 observations) and in a bear market from 20070716-20090306 (right panel, 431 observations).

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SLIDE 69

Appendix 8-29

Parameter Dynamics

Motivation

2005 2010 2014 −4 −2 2 4

  • α1

Time 1 year 2005 2010 2014 −4 −2 2 4

  • α1

Time 1 month

Figure 21: Estimated α1,0.05 for DAX and FTSE100 using 20 (1 month)

  • r 250 (1 year) observations

more parameters

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SLIDE 70

Appendix 8-30

Parameter Dynamics

Motivation

2005 2010 2014 −4 −2 2 4

  • α1

Time 1 year 2005 2010 2014 −4 −2 2 4

  • α1

Time 1 month

Figure 22: Estimated α1,0.01 for DAX and FTSE100 using 20 (1 month)

  • r 250 (1 year) observations

more parameters

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SLIDE 71

Appendix 8-31

Parameter Distributions

Motivation

Figure 23: Kernel density estimates of α1,0.05 for DAX and FTSE100 using 20, 60, 125 or 250 observations

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SLIDE 72

Appendix 8-32

Parameter Distributions

Motivation

Figure 24: Kernel density estimates of α1,0.01 for DAX and FTSE100 using 20, 60, 125 or 250 observations

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Appendix 8-33

Parameter Dynamics

Parameter Dynamics

2005 2010 2014 −200 −100 100 200

  • α2

Time 1 year 2005 2010 2014 −400 −200 200 400

  • α2

Time 1 month

Figure 25: Estimated α2,0.05 for DAX and FTSE100 using 20 (1 month)

  • r 250 (1 year) observations

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Appendix 8-34

Parameter Dynamics

Parameter Dynamics

2005 2010 2014 −200 −100 100 200

  • α2

Time 1 year 2005 2010 2014 −400 −200 200 400

  • α2

Time 1 month

Figure 26: Estimated α2,0.01 for DAX and FTSE100 using 20 (1 month)

  • r 250 (1 year) observations

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Appendix 8-35

Parameter Dynamics

Parameter Dynamics

2005 2010 2014 −200 −100 100 200

  • α3

Time 1 year 2005 2010 2014 −400 −200 200 400

  • α3

Time 1 month

Figure 27: Estimated α3,0.05 for DAX and FTSE100 using 20 (1 month)

  • r 250 (1 year) observations

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SLIDE 76

Appendix 8-36

Parameter Dynamics

Parameter Dynamics

2005 2010 2014 −200 −100 100 200

  • α3

Time 1 year 2005 2010 2014 −400 −200 200 400

  • α3

Time 1 month

Figure 28: Estimated α3,0.01 for DAX and FTSE100 using 20 (1 month)

  • r 250 (1 year) observations

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