Portfolio optimization under dynamic risk constraints: Continuous vs. discrete time trading

Imke Redeker; Ralf Wunderlich

doi:10.1515/strm-2017-0001

Article

Portfolio optimization under dynamic risk constraints: Continuous vs. discrete time trading

Imke Redeker and Ralf Wunderlich

Published/Copyright: August 17, 2017

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From the journal Statistics & Risk Modeling Volume 35 Issue 1-2

Abstract

We consider an investor facing a classical portfolio problem of optimal investment in a log-Brownian stock and a fixed-interest bond, but constrained to choose portfolio and consumption strategies that reduce a dynamic shortfall risk measure. For continuous- and discrete-time financial markets we investigate the loss in expected utility of intermediate consumption and terminal wealth caused by imposing a dynamic risk constraint. We derive the dynamic programming equations for the resulting stochastic optimal control problems and solve them numerically. Our numerical results indicate that the loss of portfolio performance is not too large while the risk is notably reduced. We then investigate time discretization effects and find that the loss of portfolio performance resulting from imposing a risk constraint is typically bigger than the loss resulting from infrequent trading.

Keywords: Consumption-investment problem; stochastic optimal control; dynamic risk measure; Markov decision problem; discrete-time approximation

MSC 2010: 91G10; 93E20; 91G80

A Proof of Lemma 2.3

Under the assumption that the portfolio-proportion strategy (𝝅,c) is kept constant and equal to (𝝅¯,c¯) between time t and t+Δ, the wealth at time t+Δ is given by

Xt+Δ=exp⁡{ln⁡(Xt)+(𝝅¯′⁢(𝝁-𝟏⁢r)+r-c¯-∥𝝅¯′⁢𝝈∥22)⁢Δ+𝝅¯′⁢𝝈⁢(𝑾t+Δ-𝑾t)}.

From the equation above we obtain that Xt+Δ is – conditionally on ℱt – distributed as eZ, where Z is normally distributed with mean m and variance s2. Here,

m:=ln⁡(Xt)+(𝝅¯′⁢(𝝁-𝟏⁢r)+r-c¯-12⁢∥𝝅¯′⁢𝝈∥2)⁢Δ and s2:=∥𝝅¯′⁢𝝈∥2⁢Δ.

Recall, the Expected Loss at time t is defined by

ELt(Lt):=𝔼[(Yt-Xt+Δ)+|ℱt]=𝔼[(Yt-eZ)+|ℱt].

This expectation can be calculated as follows. Let

fZ⁢(z)=12⁢π⁢s⁢exp⁡(-(z-m)22⁢s2)

denote the probability density function of Z; then

ELt⁡(Lt)=∫-∞∞(Yt-ez)+⁢fZ⁢(z)⁢dz=∫-∞ln⁡(Yt)(Yt-ez)⁢fZ⁢(z)⁢dz=Yt⁢I1+I2,

where

I1:=∫-∞ln⁡(Yt)fZ⁢(z)⁢dz and I2:=-∫-∞ln⁡(Yt)ez⁢fZ⁢(z)⁢dz.

For the integral I1 an appropriate change of variables yields

I1=∫-∞d112⁢π⁢e-y22⁢dy=Φ⁢(d1),

where

d1:=ln⁡(Yt)-ms=1∥𝝅¯′⁢𝝈∥⁢Δ⁢[ln⁡(YtXt)-(𝝅¯′⁢(𝝁-𝟏⁢r)+r-c¯-∥𝝅¯′⁢𝝈∥22)⁢Δ]

and Φ⁢(⋅) denotes the cumulative distribution function of the standard normal distribution. The integral I2 can be written as

I2=-∫-∞ln⁡(Yt)12⁢π⁢s⁢exp⁡(z-z2-2⁢z⁢m+m22⁢s2)⁢dz

=-es22+m⁢∫-∞ln⁡(Yt)12⁢π⁢s⁢exp⁡(-(z-(m+s2))22⁢s2)⁢dz.

Using the change of variables technique yields

I2=-es22+m⁢∫-∞d212⁢π⁢e-y22⁢dy=-es22+m⁢Φ⁢(d2),

where

d2:=ln⁡(Yt)-(m+s2)s=1∥𝝅¯′⁢𝝈∥⁢Δ⁢[ln⁡(f~⁢(t,Xt)Xt)-(𝝅¯′⁢(𝝁-𝟏⁢r)+r-c¯+12⁢∥𝝅¯′⁢𝝈∥2)⁢Δ].

Note that exp⁡{s22+m}=Xt⁢exp⁡((𝝅¯′⁢(𝝁-𝟏⁢r)+r-c¯)⁢Δ). Finally, we obtain

ELt⁡(Lt)=Yt⁢I1+I2=f~⁢(t,Xt)⁢Φ⁢(d1)-Xt⁢exp⁡((𝝅¯′⁢(𝝁-𝟏⁢r)+r-c¯)⁢Δ)⁢Φ⁢(d2).

B Proof of Lemma 3.1

Given an investment-consumption strategy (φ,η), the wealth process X evolves as follows:

Xtn+1=er⁢Δ⁢(Xtn-ηtn-φtn)+φtn⋅exp⁡{(μ-σ22)⁢Δ+σ⁢(Wtn+1-Wtn)}.

Note that given a probability level α∈(0,1) the Value at Risk at time tn is defined by

VaRtnα(Ltn):=inf{l∈ℝ:P(Ltn≥l|𝒢tn)≤α}.

We have

Ltn=Ytn-Xtn+1=f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)-φtn⋅exp⁡{(μ-σ22)⁢Δ+σ⁢(Wtn+1-Wtn)}

and

P(Ltn≥l|𝒢tn)=P(exp{(μ-σ22)Δ+σ(Wtn+1-Wtn)}≤f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)-lφtn|𝒢tn)

=P(Δ-12(Wtn+1-Wtn)≤z|𝒢tn)=Φ(z),

where

z=1σ⁢Δ⁢(ln⁡{f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)-lφtn}-(μ-σ22)⁢Δ).

We have used that the random variable Δ-12⁢(Wtn+1-Wtn) is standard normally distributed and independent of 𝒢tn. Thus, P(Ltn≥l|𝒢tn)=Φ(z)≤α is satisfied for z≤Φ-1⁢(α) yielding

l≥f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)-exp⁡{Φ-1⁢(α)⁢σ⁢Δ+(μ-σ22)⁢Δ}⁢φtn.

Since the Dynamic Value at Risk is the smallest l satisfying the above inequality, we obtain

VaRtnα⁡(Ltn)=ψ~⁢(tn,Xtn,φtn,ηtn),

where

ψ~⁢(t,x,φ¯,η¯)=f~⁢(t,x)-er⁢Δ⁢(x-η¯-φ¯)-exp⁡{Φ-1⁢(α)⁢σ⁢Δ+(μ-σ22)⁢Δ}⁢φ¯.

The Tail Conditional Expectation at time tn is defined by

TCEtnα(Ltn)=𝔼tn[Ltn|Ltn≥VaRtnα(Ltn)]

=𝔼[Ltn𝑰(Ltn≥VaRtnα(Ltn))|𝒢tn]P(Ltn≥VaRtnα(Ltn)|𝒢tn)

=1α𝔼[Ltn𝑰(Ltn≥VaRtnα(Ltn))|𝒢tn],

where 𝑰⁢(A) denotes the indicator function of the set A. Using

Ltn=Ytn-Xtn+1=f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)-φtn⁢exp⁡{(μ-σ22)⁢Δ+σ⁢(Wtn+1-Wtn)}

and

VaRtnα⁡(Ltn)=f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)-φtn⁢exp⁡{(μ-σ22)⁢Δ+Φ-1⁢(α)⁢σ⁢Δ}

yields that the above inequality VaRtnα⁡(Ltn)≤Ltn is equivalent to Δ-12⁢(Wtn+1-Wtn)≤Φ-1⁢(α). Thus,

𝔼[Ltn𝑰(Ltn≥VaRtnα(Ltn))|𝒢tn]

=𝔼[Ltn𝑰((Wtn+1-Wtn)Δ-12≤Φ-1(α))|𝒢tn]

=(Ytn-er⁢Δ(Xtn-ηtn-φtn))𝔼[𝑰((Wtn+1-Wtn)Δ-12≤Φ-1(α))|𝒢tn]

-φtnexp{(μ-σ22)Δ}𝔼[eσ⁢(Wtn+1-Wtn)𝑰((Wtn+1-Wtn)Δ-12≤Φ-1(α))|𝒢tn]

=(f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn))⁢α

-φtnexp{(μ-σ22)Δ}𝔼[eσ⁢(Wtn+1-Wtn)𝑰((Wtn+1-Wtn)Δ-12≤Φ-1(α))|𝒢tn].

We obtain

𝔼[eσ⁢(Wtn+1-Wtn)𝑰((Wtn+1-Wtn)Δ-12≤Φ-1(α))|𝒢tn]=∫-∞Φ-1⁢(α)eσ⁢Δ⁢z12⁢πe-12⁢z2dz,

where we have used that the random variable (Wtn+1-Wtn)⁢Δ-12 is standard normally distributed and independent of 𝒢tn. We calculate the above integral by making an appropriate change of variables

∫-∞Φ-1⁢(α)eσ⁢Δ⁢z⁢12⁢π⁢e-12⁢z2⁢dz=eσ2⁢Δ2⁢∫-∞Φ-1⁢(α)-σ⁢Δ12⁢π⁢e-12⁢y2⁢dy=eσ2⁢Δ2⁢Φ⁢(Φ-1⁢(α)-σ⁢Δ).

Finally, the Dynamic Tail Conditional Expectation can be written as

TCEtnα⁡(Ltn)=ψ~⁢(tn,Xtn,φtn,ηtn),

where

ψ~⁢(t,x,φ¯,η¯)=f~⁢(t,x)-er⁢Δ⁢(x-η¯-φ¯)-1α⁢eμ⁢Δ⁢Φ⁢(Φ-1⁢(α)-σ⁢Δ)⁢φ¯.

In order to prove the statement for the Dynamic Expected Loss, we use that the wealth at time Xtn+1 is distributed as er⁢Δ⁢(Xtn-ηtn-φtn)+φtn⋅eZ, where Z is normally distributed with mean m and variance s2, where m:=(μ-σ22)⁢Δ and s2:=σ2⁢Δ. From the definition of the Dynamic Expected Loss we find

ELtn(Ltn)=𝔼[(Ytn-Xtn+1)+|𝒢tn]=𝔼[(f~(tn,Xtn)-er⁢Δ(Xtn-ηtn-φtn)-φtn⋅eZ)+|𝒢tn].

This expectation can be calculated as follows. Let

fZ⁢(z)=12⁢π⁢s⁢exp⁡(-(z-m)22⁢s2)

denote the probability density function of Z; then

ELtn⁡(Ltn)=∫-∞∞(f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)-φtn⋅ez)+⁢fZ⁢(z)⁢dz

=∫-∞d~(f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)-φtn⋅ez)⁢fZ⁢(z)⁢dz

=(f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn))⁢I1-φtn⁢I2,

where

I1=∫-∞d~fZ⁢(z)⁢dz,I2=∫-∞d~ez⁢fZ⁢(z)⁢dz

and

d~=ln⁡(f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)φtn).

The integral I1 can be calculated by making an appropriate change of variables and we obtain

I1=∫-∞d112⁢π⁢e-y22⁢dy=Φ⁢(d1),

where

d1:=1s⁢(ln⁡(f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)φtn)-m)

=1σ⁢Δ⁢[ln⁡(f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)φtn)-(μ-σ22)⁢Δ].

The integral I2 can be written as

I2=∫-∞d~12⁢π⁢s⁢exp⁡(z-z2-2⁢z⁢m+m22⁢s2)⁢dz=es22+m⁢∫-∞d~12⁢π⁢s⁢exp⁡(-(z-(m+s2))22⁢s2)⁢dz.

Using the change of variables method yields

I2=es22+m⁢∫-∞d212⁢π⁢e-y22⁢dy=es22+m⁢Φ⁢(d2),

where

d2=1s⁢[ln⁡(f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)φtn)-(m+s2)]

=1σ⁢Δ⁢[ln⁡(f~⁢(tn,Xtn)-er⁢Δ⁢(Xtn-ηtn-φtn)φtn)-(μ+σ22)⁢Δ].

Note that es22+m=eμ⁢Δ, and thus we obtain ELtn⁡(Ltn)=ψ~⁢(tn,Xtn,φtn,ηtn), where

ψ~⁢(t,x,φ¯,η¯)=(f~⁢(t,x)-er⁢Δ⁢(x-η¯-φ¯))⁢Φ⁢(d1)-eμ⁢Δ⁢Φ⁢(d2)⁢φ¯.

References

[1] D. Akume, Risk-constrained dynamic portfolio management, Ph.D. thesis, University of Yaounde I, 2007. Search in Google Scholar

[2] D. Akume, B. Luderer and R. Wunderlich, Dynamic shortfall constraints for optimal portfolios, Surv. Math. Appl. 5 (2010), 135–149. 10.4314/afst.v4i2.60769Search in Google Scholar

[3] P. Artzner, F. Delbaen, J. Eber and D. Heath, Coherent measures of risk, Math. Finance 9 (1999), no. 3, 203–228. 10.1111/1467-9965.00068Search in Google Scholar

[4] S. Basak and A. Shapiro, Value-at-risk-based risk management: Optimal policies and asset prices, Rev. Financ. Stud. 14 (2001), 371–405. 10.1093/rfs/14.2.371Search in Google Scholar

[5] N. Bäuerle and U. Rieder, Markov Decision Processes with Applications to Finance, Springer, Berlin, 2011. 10.1007/978-3-642-18324-9Search in Google Scholar

[6] N. Bäuerle, S. Urban and L. A. M. Veraart, The relaxed investor with partial information, SIAM J. Financial Math. 3 (2012), no. 1, 304–327. 10.1137/100813646Search in Google Scholar

[7] D. Cuoco, H. He and S. Issaenko, Optimal dynamic trading strategies with risk limits, Oper. Res. 56 (2008), no. 2, 358–368. 10.1287/opre.1070.0433Search in Google Scholar

[8] D. J. Duffy, Finite Difference Methods in Financial Engineering, Wiley Finance Ser., John Wiley & Sons, Chichester, 2006. 10.1002/9781118673447Search in Google Scholar

[9] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd ed., Stoch. Model. Appl. Probab. 25, Springer, New York, 2006. Search in Google Scholar

[10] A. Gabih, J. Sass and R. Wunderlich, Utility maximization under bounded expected loss, Stoch. Models 25 (2009), 375–409. 10.1080/15326340903088495Search in Google Scholar

[11] P. Grandits, F. Hubalek, W. Schachermayer and M. Žigo, Optimal expected exponential utility of dividend payments in a Brownian risk model, Scand. Actuar. J. (2007), no. 2, 73–107. 10.1080/03461230601165201Search in Google Scholar

[12] K. Hinderer, Foundations of Non-Stationary Dynamic Programming with Discrete Time Parameter, Lect. Notes Oper. Res. Math. Systems 33, Springer, Berlin, 1970. 10.1007/978-3-642-46229-0Search in Google Scholar

[13] L. Jichun and C. Yi-Tung, Computational Partial Differential Equations Using MATLAB, CRC Press, Boca Raton, 2008. Search in Google Scholar

[14] P. Jorion, Value at Risk, McGraw–Hill, New York, 2001. Search in Google Scholar

[15] R. Korn, Optimal Portfolios, World Scientific, Singapore, 1997. 10.1142/3548Search in Google Scholar

[16] M. Leippold, F. Trojani and P. Vanini, Equilibrium impact of value-at-risk regulation, J. Econom. Dynam. Control 30 (2006), no. 8, 1277–1313. 10.1016/j.jedc.2005.04.008Search in Google Scholar

[17] S. Moreno-Bromberg, T. A. Pirvu and A. Réveillac, CRRA utility maximization under dynamic risk constraints, Commun. Stoch. Anal. 7 (2013), no. 2, 203–225. 10.31390/cosa.7.2.03Search in Google Scholar

[18] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed., Springer, New York, 2006. Search in Google Scholar

[19] H. Pham, Continuous-Time Stochastic Control and Optimization with Financial Applications, Stoch. Model. Appl. Probab. 61, Springer, Berlin, 2009. 10.1007/978-3-540-89500-8Search in Google Scholar

[20] T. A. Pirvu, Portfolio optimization under the value-at-risk constraint, Quant. Finance 7 (2007), no. 2, 125–136. 10.1080/14697680701213868Search in Google Scholar

[21] W. Putschögl and J. Sass, Optimal consumption and investment under partial information, Decis. Econ. Finance 31 (2008), no. 2, 137–170. 10.1007/s10203-008-0082-3Search in Google Scholar

[22] L. C. G. Rogers, The relaxed investor and parameter uncertainty, Finance Stoch. 5 (2001), no. 2, 131–154. 10.1007/PL00013532Search in Google Scholar

[23] A. A. Shardin and R. Wunderlich, Partially observable stochastic optimal control problems for an energy storage, Stochastics 89 (2017), no. 1, 280–310. 10.1080/17442508.2016.1166506Search in Google Scholar

[24] K. F. C. Yiu, Optimal portfolios under a value-at-risk constraint, J. Econom. Dynam. Control 28 (2004), no. 7, 1317–1334. 10.1016/S0165-1889(03)00116-7Search in Google Scholar

Received: 2017-1-14

Revised: 2017-7-23

Accepted: 2017-7-31

Published Online: 2017-8-17

Published in Print: 2018-1-1

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Articles in the same Issue

https://doi.org/10.1515/strm-2017-0001

Keywords for this article

Consumption-investment problem; stochastic optimal control; dynamic risk measure; Markov decision problem; discrete-time approximation