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A stochastic gradient descent algorithm to maximize power utility of large credit portfolios under Marshall–Olkin dependence

  • *Corresponding author: Matthias Scherer

    *Corresponding author: Matthias Scherer
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  • A vector of bankruptcy times with Marshall–Olkin multivariate exponential distribution implies a simple, yet reasonable, continuous-time model for dependent credit-risky assets with an appealing trade-off between tractability and realism. Within this framework, the maximization of expected power utility of terminal wealth requires the optimization of a concave function on a polygon, a numerical problem whose complexity grows exponentially in the number of considered assets. We demonstrate how this seemingly impractical numerical problem can be solved reliably and efficiently in order to prepare the model for practical use cases. To this end, we resort to a specifically designed factor construction for the Marshall–Olkin distribution that separates dependence parameters from idiosyncratic parameters, and we develop a tailor-made stochastic gradient descent algorithm with random constraint projections for the model's numerical implementation. Finally, we explain a new method to include transaction costs and apply the model in a real-world, high-dimensional example.

    Mathematics Subject Classification: Primary: 91B16, 91B70; Secondary: 49M05, 60J76.

    Citation:

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  • Figure 1.  The function $ K \mapsto -U_p(K) $ for different values of $ K $

    Figure 2.  Visualization of the SGD algorithm with step width $ \Delta_n = 1/\sqrt{n+1} $ for parameters $ d = 2 $, $ p = 0 $, $ \mathit{\boldsymbol{\eta}} = (0.038, 0.065) $, $ \mathit{\boldsymbol{\kappa}} = (0.3, 0.15) $, $ \lambda_{\{1\}} = 0.021 $, $ \lambda_{\{2\}} = 0.041 $, $ \lambda_{\{1, 2\}} = 0.009 $. The domain is specified by $ \mathit{\boldsymbol{\ell}} = (0, 0) $, $ \mathit{\boldsymbol{u}} = (0.35, 0.35) $, and $ \epsilon = 0.8 $. Here, the arithmetic average of the last $ m_A = 100 $ iterates is returned. Further, the stochastic gradient in each iteration is computed as the arithmetic average over $ m_S $ iid simulations of $ \mathit{\boldsymbol{F}}(\mathit{\boldsymbol{x}}^{(n)}, S) $ in step $ n $. Top left: $ \delta = \epsilon = 0.8 $, $ m_S = 1 $. Top right: $ \delta = 0.2 $, $ m_S = 1 $. Bottom left: $ \delta = 0.8 $, $ m_S = 20 $. Bottom right: $ \delta = 0.2 $, $ m_S = 20 $

    Figure 3.  Optimal portfolios computed via the SGD algorithm in an example with $ d = 2 $, $ p = -1 $, and $ \epsilon = 0.1 $. The model parameters are $ \mathit{\boldsymbol{\eta}} = (0.018, 0.025) $, $ \mathit{\boldsymbol{\kappa}} = (0.3, 0) $, $ \lambda_{\{1\}}\approx 0.0199 $, $ \lambda_{\{2\}} \approx 0.0099 $, $ \lambda_{\{1, 2\}} \approx 0.0201 $. In the cases with transaction costs it is assumed that $ \eta_{i, \pm} = \eta_i\, (1 \pm c/2) $ for $ c \in \{0, 0.025, 0.05, 0.075, 0.125, 0.25, 0.5\} $, as depicted in the plot. The contour lines correspond to the objective function without transaction costs

    Figure 4.  Left: Histogram of the values $ \mu_{i, \pm} = \eta_{i, \pm}-\Lambda_{\{i\}}\, (1-\kappa_i) $, $ i \in [d] $; but only such $ \mu_{i, +} \geq -2\% $ and only such $ \mu_{i, -} \leq 2\% $. Right: Portfolio weights of the top $ 15 $ holdings in the optimal portfolio $ \mathit{\boldsymbol{x}}_{\ast} $ for different risk aversion parameters $ p \in \{0, -2, -5\} $

    Figure 5.  Left: The top 15 holdings of for risk aversion $ p = -2 $ in the right plot of Figure 4, together with their weights in four further runs of the same SGD algorithm. Right: Runtime in minutes for the SGD algorithm with $ N = 100, 000 $, $ \epsilon = \delta = 0.6 $, $ m_A = m_S = 0 $, with increasing number of assets $ d \in [10, 410] $

    Table 1.  Enhanced parameters when transaction costs are taken into account

    old asset $ i $ new asset $ (i, +) $ new asset $ (i, -) $
    default intensity $ \Lambda_{\{i\}} $ $ \Lambda_{\{(i, +)\}}=\Lambda_{\{i\}} $ $ \Lambda_{\{(i, -)\}}=\Lambda_{\{i\}} $
    recovery rate $ \kappa_i $ $ \kappa_{i, +}=\kappa_i $ $ \kappa_{i, -}=\kappa_i $
    factor weight $ w_j^{(i)} $ $ w_j^{(i, +)}=w_j^{(i)} $ $ w_j^{(i, -)}=w_j^{(i)} $
    yield $ \eta_i $ $ \eta_{i, +}<\eta_i $ $ \eta_{i, -}>\eta_i $
    lower restriction $ \ell_i $ $ \ell_{i, +}=\max\{\ell_i, 0\}\geq 0 $ $ \ell_{i, -}=\min\{\ell_i, 0\} \leq 0 $
    upper restriction $ u_i $ $ u_{i, +}=\max\{u_i, 0\} \geq 0 $ $ u_{i, -}=\min\{u_i, 0\} \leq 0 $
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    Table 2.  One-parametric families of Laplace transforms $ \varphi $. The column "$ J \sim $" indicates how the jump distribution is simulated efficiently. We denote by $ U \sim \mathcal{U}(0, 1) $ a random variable that is uniform on $ (0, 1) $, and by $ E $ an independent standard exponential random variable

    Name $ \varphi(x) $ $ \theta \in $ $ \varphi^{-1}(y) $ $ J \sim $
    constant $ e^{-\theta\, x} $ $ (0, \infty) $ $ -\log(y)/\theta $ $ \theta $
    exponential $ \frac{\theta}{x+\theta} $ $ (0, \infty) $ $ \theta\, \frac{1-y}{y} $ $ E/\theta $
    stable $ e^{-x^{\theta}} $ $ (0, 1) $ $ (-\log(y))^{\frac{1}{\theta}} $ $ \sin( \theta\, \pi\, U)\, \Big( \frac{\cos\big(\pi\, U\, (1-\theta)-\pi/2\big)}{E\, \cos\big(\pi(U-1/2)\big)^{1/(1-\theta)}}\Big)^{\frac{1-\theta}{\theta}} $
    gamma $ (1+x)^{-\theta} $ $ (0, \infty) $ $ y^{-\frac{1}{\theta}}-1 $ $ \Gamma(\theta, 1) $, see [21, p. 240 ff]
    inv. Gauss 1 $ e^{-\theta\, (\sqrt{2\, x+1}-1)} $ $ (0, \infty) $ $ \frac{1}{2}\, \Big\{\Big( 1-\frac{\log(y)}{\theta}\Big)^2-1 \Big\} $ IG$ (\theta, 1) $, see [21, p. 245]
    inv. Gauss 2 $ e^{-(\sqrt{2\, x+\theta^2}-\theta)} $ $ (0, \infty) $ $ \frac{1}{2}\, \big\{\big( \theta-\log(y)\big)^2-\theta^2 \big\} $ IG$ (1, \theta) $, see [21, p. 245]
     | Show Table
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  • [1] N. AkutsuM. Kijima and K. Komoribayashi, A portfolio optimization model for corporate bonds subject to credit risk, Journal of Risk, 6 (2004), 31-48. 
    [2] F. AnderssonH. MausserD. Rosen and S. Uryasev, Credit risk optimization with conditional value-at-risk criterion, Mathematical Programming, 89 (2001), 273-291.  doi: 10.1007/PL00011399.
    [3] D. P. Bertsekas, Projected Newton methods for optimization problems with simple constraints, SIAM Journal of Control and Optimization, 20 (1982), 221-246.  doi: 10.1137/0320018.
    [4] Z. I. Botev, P. L'Ecuyer, R. Simard and B. Tuffin, Static network reliability estimation under the Marshall–Olkin copula, ACM Transactions on Modeling and Computer Simulation 26 (2016) article number 14, 1-28. doi: 10.1145/2775106.
    [5] D. BrigoJ.-F. Mai and M. Scherer, Markov multi-variate survival indicators for default simulation as a new characterization of the Marshall–Olkin law, Statistics and Probability Letters, 114 (2016), 60-66.  doi: 10.1016/j.spl.2016.03.013.
    [6] A. Capponi and J. E. Figueroa-Lopez, Dynamic portfolio optimization with a defaultable security and regime-switching, Mathematical Finance, 24 (2014), 207-249.  doi: 10.1111/j.1467-9965.2012.00522.x.
    [7] A. CapponiJ. E. Figueroa-Lopez and A. Pascucci, Dynamic credit investment in partially observed markets, Finance and Stochastics, 19 (2015), 891-939.  doi: 10.1007/s00780-015-0272-0.
    [8] Y. Elouerkhaoui, Pricing and hedging in a dynamic credit model, International Journal of Theoretical and Applied Finance, 10 (2007), 703-731.  doi: 10.1142/S0219024907004408.
    [9] P. Embrechts, F. Lindskog and A. J. McNeil, Modelling dependence with copulas and applications to risk management, in Handbook of Heavy Tailed Distributions in Finance, ed. S. Rachev, Elsevier/ North-Holland, Amsterdam, (2003), 329-384. doi: 10.1016/B978-044450896-6.50010-8.
    [10] K. Giesecke, A simple exponential model for dependent defaults, Journal of Fixed Income, 13 (2003), 74-83. 
    [11] K. GieseckeB. KimJ. Kim and G. Tsoukalas, Optimal credit swap portfolios, Management Science, 60 (2014), 2291-2307.  doi: 10.1287/mnsc.2013.1890.
    [12] T. Goll and L. Rüschendorf, Minimax and minimal distance martingale measures and their relationship to portfolio optimization, Finance and Stochastics, 5 (2001), 557-581.  doi: 10.1007/s007800100052.
    [13] H. Joe, Multivariate Models and Dependence Concepts, Monogr. Statist. Appl. Probab., 73 Chapman & Hall, London, 1997. doi: 10.1201/b13150.
    [14] J. Kallsen, Optimal portfolios for exponential Lévy processes, Mathematical Methods of Operations Research, 51 (2000), 357-374.  doi: 10.1007/s001860000048.
    [15] H. Kraft and M. Steffensen, How to invest optimally in corporate bonds: A reduced-form approach, Journal of Economic Dynamics and Control, 32 (2008), 348-385.  doi: 10.1016/j.jedc.2007.02.001.
    [16] H. Kraft and M. Steffensen, Asset allocation with contagion and explicit bankruptcy procedures, Journal of Mathematical Economics, 45 (2009), 147-167.  doi: 10.1016/j.jmateco.2008.08.006.
    [17] F. Lindskog and A. J. McNeil, Common poisson shock models: Applications to insurance and credit risk modelling, ASTIN Bulletin, 33 (2003), 209-238.  doi: 10.2143/AST.33.2.503691.
    [18] J.-F. Mai, Correction to "Portfolio opimization for credit-risky assets under Marshall–Olkin dependence", Applied Mathematical Finance, 28 (2021), 96-99.  doi: 10.1080/1350486X.2021.1956708.
    [19] J.-F. Mai, Portfolio opimization for credit-risky assets under Marshall–Olkin dependence, Applied Mathematical Finance, 26 (2020), 598-618.  doi: 10.1080/1350486X.2020.1727755.
    [20] J.-F. Mai and M. Scherer, Reparameterizing Marshall–Olkin copulas with applications to sampling, Journal of Statistical Computation and Simulation, 81 (2011), 59-78.  doi: 10.1080/00949650903185961.
    [21] J.-F. Mai and M. Scherer, Simulating Copulas, 2$^{nd}$ edition, World Scientific Press, Singapore, 2017.
    [22] H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. 
    [23] H. Markowitz, Portfolio selection: Efficient diversification of investments, Wiley, New York, (1959).
    [24] A. W. Marshall and I. Olkin, A multivariate exponential distribution, Journal of the American Statistical Association, 62 (1967), 30-44.  doi: 10.1080/01621459.1967.10482885.
    [25] O. MatusJ. BarreraE. Moreno and G. Rubino, On the Marshall–Olkin copula model for network reliability under dependent failures, IEEE Transactions on Reliability, 68 (2019), 451-461.  doi: 10.1109/TR.2018.2865707.
    [26] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560.
    [27] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X.
    [28] M. Nutz, Power utility maximization in constrained exponential Lévy models, Mathematical Finance, 22 (2012), 690-709.  doi: 10.1111/j.1467-9965.2011.00480.x.
    [29] P. Pasricha, D. Selvamuthu, G. D'Amico and R. Manca, Portfolio optimization of credit risky bonds: A semi-Markov process approach, Financial Innovation, 6 (2020), 14 pp. doi: 10.1186/s40854-020-00186-1.
    [30] D. SaundersC. Xiouros and S. Zenios, Credit risk optimization using factor models, Annals of Operations Research, 152 (2007), 49-77.  doi: 10.1007/s10479-006-0136-2.
    [31] S. Shalev-Shwartz and S. Ben-David, Understanding Machine Learning: From Theory to Algorithms, Cambridge University Press, 2014. doi: 10.1017/CBO9781107298019.
    [32] J. A. SirignanoG. Tsoukalas and K. Giesecke, Large-scale loan portfolio selection, Operations Research, 64 (2016), 1239-1255.  doi: 10.1287/opre.2016.1537.
    [33] Y. SunR. Mendoza-Arriaga and V. Linetsky, Marshall–Olkin distributions, subordinators, efficient simulation, and applications to credit risk, Advances in Applied Probability, 49 (2017), 481-514.  doi: 10.1017/apr.2017.10.
    [34] M. Wang and D. P. Bertsekas, Stochastic first-order methods with random constraint projection, SIAM Journal of Optimization, 26 (2016), 681-717.  doi: 10.1137/130931278.
    [35] M. B. Wise and V. Bhansali, Portfolio allocation to corporate bonds with correlated defaults, Journal of Risk, 5 (2002), 39-58. 
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