American Institute of Mathematical Sciences

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June  2022, 1(2): 219-248. doi: 10.3934/fmf.2021009

Acceptability maximization

Gabriela Kováčová 1,, , Birgit Rudloff 1, and  Igor Cialenco 2,

1. 

Vienna University of Economics and Business, Institute for Statistics and Mathematics, Vienna A-1020, Austria
2. 

Illinois Institute of Technology, Department of Applied Mathematics, 10 W 32nd Str, Building RE, Room 220, Chicago, IL 60616, USA

* Corresponding author: Gabriela Kováčová

Received  September 2021 Revised  November 2021 Published  June 2022 Early access  May 2022

Fund Project: IC acknowledges partial support from the National Science Foundation (US) grant DMS-1907568

The aim of this paper is to study the optimal investment problem by using coherent acceptability indices (CAIs) as a tool to measure the portfolio performance. We call this problem the acceptability maximization. First, we study the one-period (static) case, and propose a numerical algorithm that approximates the original problem by a sequence of risk minimization problems. The results are applied to several important CAIs, such as the gain-to-loss ratio, the risk-adjusted return on capital and the tail-value-at-risk based CAI. In the second part of the paper we investigate the acceptability maximization in a discrete time dynamic setup. Using robust representations of CAIs in terms of a family of dynamic coherent risk measures (DCRMs), we establish an intriguing dichotomy: if the corresponding family of DCRMs is recursive (i.e. strongly time consistent) and assuming some recursive structure of the market model, then the acceptability maximization problem reduces to just a one period problem and the maximal acceptability is constant across all states and times. On the other hand, if the family of DCRMs is not recursive, which is often the case, then the acceptability maximization problem ordinarily is a time-inconsistent stochastic control problem, similar to the classical mean-variance criteria. To overcome this form of time-inconsistency, we adapt to our setup the set-valued Bellman's principle recently proposed in [23] applied to two particular dynamic CAIs - the dynamic risk-adjusted return on capital and the dynamic gain-to-loss ratio. The obtained theoretical results are illustrated via numerical examples that include, in particular, the computation of the intermediate mean-risk efficient frontiers.

Keywords: Acceptability index, acceptability maximization, optimal portfolio, gain-to-loss ratio, dynamic performance measures, tail-value-at-risk, set-valued Bellman principle.
Mathematics Subject Classification: 91G10, 93E20, 93E35, 49L20.
Citation: Gabriela Kováčová, Birgit Rudloff, Igor Cialenco. Acceptability maximization. Frontiers of Mathematical Finance, 2022, 1 (2) : 219-248. doi: 10.3934/fmf.2021009
References:
[1]

B. Acciaio, H. Föllmer and I. Penner, Risk assessment for uncertain cash flows: Model ambiguity, discounting ambiguity, and the role of bubbles, Finance and Stochastics, 16 (2012), 669-709.  doi: 10.1007/s00780-012-0176-1.

[2]

B. Acciaio and I. Penner, Dynamic risk measures, Advanced Mathematical Methods for Finance, Springer, Heidelberg, (2011), 1–34. doi: 10.1007/978-3-642-18412-3_1.

[3]

V. Agarwal and N. Y. Naik, Risks and portfolio decisions involving hedge funds, The Review of Financial Studies, 17 (2004), 63-98.  doi: 10.1093/rfs/hhg044.

[4]

P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.

[5]

F. Bellini and E. Di Bernardino, Risk management with expectiles, European Journal of Finance, 23 (2015), 487-506.  doi: 10.1080/1351847X.2015.1052150.

[6]

A. Bernardo and O. Ledoit, Gain, loss, and asset pricing, Journal of Political Economy, 108 (2000), 144-172.  doi: 10.1086/262114.

[7]

S. Biagini and J. Bion-Nadal, Dynamic quasi-concave performance measures, Journal of Mathematical Economics, 55 (2014), 143-153.  doi: 10.1016/j.jmateco.2014.02.007.

[8]

T. R. Bielecki, I. Cialenco and T. Chen, Dynamic conic finance via backward stochastic difference equations, SIAM J. Finan. Math., 6 (2015), 1068-1122.  doi: 10.1137/141002013.

[9]

T. R. Bielecki, I. Cialenco, S. Drapeau and M. Karliczek, Dynamic assessment indices, Stochastics, 88 (2016)), 1-44.  doi: 10.1080/17442508.2015.1026346.

[10]

T. R. Bielecki, I. Cialenco, I. Iyigunler and R. Rodriguez, Dynamic conic finance: Pricing and hedging via dynamic coherent acceptability indices with transaction costs, International Journal of Theoretical and Applied Finance, 16 (2103), 1350002, 36 pp. doi: 10.1142/S0219024913500027.

[11]

T. R. Bielecki, I. Cialenco and M. Pitera, A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective, Probability, Uncertainty and Quantitative Risk, 2 (2017), Paper No. 3, 52 pp. doi: 10.1186/s41546-017-0012-9.

[12]

T. R. Bielecki, I. Cialenco and M. Pitera, A unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time, Mathematics of Operations Research, 43 (2018), 204-221.  doi: 10.1287/moor.2017.0858.

[13]

T. R. Bielecki, I. Cialenco and Z. Zhang, Dynamic coherent acceptability indices and their applications to finance, Mathematical Finance, 24 (2014), 411-441.  doi: 10.1111/j.1467-9965.2012.00524.x.

[14]

A. Biglova, S. Ortobelli, S. T. Rachev and S. Stoyanov, Different approaches to risk estimation in portfolio theory, The Journal of Portfolio Management, 31 (2004), 103-112.  doi: 10.3905/jpm.2004.443328.

[15]

P. Cheridito and E. Kromer, Reward-risk ratio, Journal of Investment Strategies, 3 (2013), 1-16.  doi: 10.2139/ssrn.2144185.

[16]

P. Cheridito and M. Stadje, Time-inconsistency of VaR and time-consistent alternatives, Finance Research Letters, 6 (2009), 40-46.  doi: 10.1016/j.frl.2008.10.002.

[17]

A. Cherny and D. B. Madan, New measures for performance evaluation, The Review of Financial Studies, 22 (2009), 2571-2606. 

[18]

E. Eberlein and D. B. Madan, Hedge fund performance: sources and measures, Int. J. Theor. Appl. Finance, 12 (2009), 267-282.  doi: 10.1142/S0219024909005282.

[19]

E. Eberlein and D. B. Madan, Maximally acceptable portfolios, Inspired by Finance, Springer, Cham, (2014), 257–272. doi: 10.1007/978-3-319-02069-3_11.

[20]

W. N. Goetzmann, J. E. Ingersoll, M. I. Spiegel and I. Welch, Sharpening Sharpe ratios, NBER Working Paper No. 9116, (2002), 51 pp.

[21]

C. Karnam, J. Ma and J. Zhang, Dynamic approaches for some time-inconsistent optimization problems, Ann. Appl. Probab., 27 (2017), 3435-3477.  doi: 10.1214/17-AAP1284.

[22]

H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Science, 37 (1991), 519-531.  doi: 10.1287/mnsc.37.5.519.

[23]

G. Kováčová and B. Rudloff, Time consistency of the mean-risk problem, Operations Research, 69 (2021), 1100-1117.  doi: 10.1287/opre.2020.2002.

[24]

A. Löhne and B. Weißing, Equivalence between polyhedral projection, multiple objective linear programming and vector linear programming, Mathematical Methods of Operations Research, 84 (2016), 411-426.  doi: 10.1007/s00186-016-0554-0.

[25] D. Madan and W. Schoutens, Applied Conic Finance, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316585108.
[26]

R. D. Martin, S. Z. Rachev and F. Siboulet, Phi-alpha optimal portfolios and extreme risk management, The Best of Wilmott 1: Incorporating the Quantitative Finance Review, 1 (2003), 223. 

[27]

S. Ortobelli, A. Biglova, S. Stoyanov, S. Z. Rachev and F. Fabozzi, A comparison among performance measures in portfolio theory, IFAC Proceedings Volumes, 16th IFAC World Congress, 38 (2005), 1-5.  doi: 10.3182/20050703-6-CZ-1902.02236.

[28]

F. Riedel, Dynamic coherent risk measures, Stochastic Process. Appl., 112 (2004), 185-200.  doi: 10.1016/j.spa.2004.03.004.

[29]

E. Rosazza Gianin and E. Sgarra, Acceptability indexes via $g$-expectations: An application to liquidity risk, Mathematics and Financial Economics, 7 (2013), 457-475.  doi: 10.1007/s11579-013-0097-6.

[30]

H. Shalit and S. Yitzhaki, Mean-Gini, portfolio theory, and the pricing of risky assets, Journal of Finance, 39 (1984), 1449-1468.  doi: 10.1111/j.1540-6261.1984.tb04917.x.

[31]

W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442.  doi: 10.1111/j.1540-6261.1964.tb02865.x.

[32]

F. A. Sortino and S. Satchell, Managing Downside Risk in Financial Markets, Butterworth-Heinemann, 2001.

[33]

M. R. Young, A minimax portfolio selection rule with linear programming solution, Management Science, 44 (1998), 595-741.  doi: 10.1287/mnsc.44.5.673.

show all references

References:
[1]

B. Acciaio, H. Föllmer and I. Penner, Risk assessment for uncertain cash flows: Model ambiguity, discounting ambiguity, and the role of bubbles, Finance and Stochastics, 16 (2012), 669-709.  doi: 10.1007/s00780-012-0176-1.

[2]

B. Acciaio and I. Penner, Dynamic risk measures, Advanced Mathematical Methods for Finance, Springer, Heidelberg, (2011), 1–34. doi: 10.1007/978-3-642-18412-3_1.

[3]

V. Agarwal and N. Y. Naik, Risks and portfolio decisions involving hedge funds, The Review of Financial Studies, 17 (2004), 63-98.  doi: 10.1093/rfs/hhg044.

[4]

P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.

[5]

F. Bellini and E. Di Bernardino, Risk management with expectiles, European Journal of Finance, 23 (2015), 487-506.  doi: 10.1080/1351847X.2015.1052150.

[6]

A. Bernardo and O. Ledoit, Gain, loss, and asset pricing, Journal of Political Economy, 108 (2000), 144-172.  doi: 10.1086/262114.

[7]

S. Biagini and J. Bion-Nadal, Dynamic quasi-concave performance measures, Journal of Mathematical Economics, 55 (2014), 143-153.  doi: 10.1016/j.jmateco.2014.02.007.

[8]

T. R. Bielecki, I. Cialenco and T. Chen, Dynamic conic finance via backward stochastic difference equations, SIAM J. Finan. Math., 6 (2015), 1068-1122.  doi: 10.1137/141002013.

[9]

T. R. Bielecki, I. Cialenco, S. Drapeau and M. Karliczek, Dynamic assessment indices, Stochastics, 88 (2016)), 1-44.  doi: 10.1080/17442508.2015.1026346.

[10]

T. R. Bielecki, I. Cialenco, I. Iyigunler and R. Rodriguez, Dynamic conic finance: Pricing and hedging via dynamic coherent acceptability indices with transaction costs, International Journal of Theoretical and Applied Finance, 16 (2103), 1350002, 36 pp. doi: 10.1142/S0219024913500027.

[11]

T. R. Bielecki, I. Cialenco and M. Pitera, A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective, Probability, Uncertainty and Quantitative Risk, 2 (2017), Paper No. 3, 52 pp. doi: 10.1186/s41546-017-0012-9.

[12]

T. R. Bielecki, I. Cialenco and M. Pitera, A unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time, Mathematics of Operations Research, 43 (2018), 204-221.  doi: 10.1287/moor.2017.0858.

[13]

T. R. Bielecki, I. Cialenco and Z. Zhang, Dynamic coherent acceptability indices and their applications to finance, Mathematical Finance, 24 (2014), 411-441.  doi: 10.1111/j.1467-9965.2012.00524.x.

[14]

A. Biglova, S. Ortobelli, S. T. Rachev and S. Stoyanov, Different approaches to risk estimation in portfolio theory, The Journal of Portfolio Management, 31 (2004), 103-112.  doi: 10.3905/jpm.2004.443328.

[15]

P. Cheridito and E. Kromer, Reward-risk ratio, Journal of Investment Strategies, 3 (2013), 1-16.  doi: 10.2139/ssrn.2144185.

[16]

P. Cheridito and M. Stadje, Time-inconsistency of VaR and time-consistent alternatives, Finance Research Letters, 6 (2009), 40-46.  doi: 10.1016/j.frl.2008.10.002.

[17]

A. Cherny and D. B. Madan, New measures for performance evaluation, The Review of Financial Studies, 22 (2009), 2571-2606. 

[18]

E. Eberlein and D. B. Madan, Hedge fund performance: sources and measures, Int. J. Theor. Appl. Finance, 12 (2009), 267-282.  doi: 10.1142/S0219024909005282.

[19]

E. Eberlein and D. B. Madan, Maximally acceptable portfolios, Inspired by Finance, Springer, Cham, (2014), 257–272. doi: 10.1007/978-3-319-02069-3_11.

[20]

W. N. Goetzmann, J. E. Ingersoll, M. I. Spiegel and I. Welch, Sharpening Sharpe ratios, NBER Working Paper No. 9116, (2002), 51 pp.

[21]

C. Karnam, J. Ma and J. Zhang, Dynamic approaches for some time-inconsistent optimization problems, Ann. Appl. Probab., 27 (2017), 3435-3477.  doi: 10.1214/17-AAP1284.

[22]

H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Science, 37 (1991), 519-531.  doi: 10.1287/mnsc.37.5.519.

[23]

G. Kováčová and B. Rudloff, Time consistency of the mean-risk problem, Operations Research, 69 (2021), 1100-1117.  doi: 10.1287/opre.2020.2002.

[24]

A. Löhne and B. Weißing, Equivalence between polyhedral projection, multiple objective linear programming and vector linear programming, Mathematical Methods of Operations Research, 84 (2016), 411-426.  doi: 10.1007/s00186-016-0554-0.

[25] D. Madan and W. Schoutens, Applied Conic Finance, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316585108.
[26]

R. D. Martin, S. Z. Rachev and F. Siboulet, Phi-alpha optimal portfolios and extreme risk management, The Best of Wilmott 1: Incorporating the Quantitative Finance Review, 1 (2003), 223. 

[27]

S. Ortobelli, A. Biglova, S. Stoyanov, S. Z. Rachev and F. Fabozzi, A comparison among performance measures in portfolio theory, IFAC Proceedings Volumes, 16th IFAC World Congress, 38 (2005), 1-5.  doi: 10.3182/20050703-6-CZ-1902.02236.

[28]

F. Riedel, Dynamic coherent risk measures, Stochastic Process. Appl., 112 (2004), 185-200.  doi: 10.1016/j.spa.2004.03.004.

[29]

E. Rosazza Gianin and E. Sgarra, Acceptability indexes via $g$-expectations: An application to liquidity risk, Mathematics and Financial Economics, 7 (2013), 457-475.  doi: 10.1007/s11579-013-0097-6.

[30]

H. Shalit and S. Yitzhaki, Mean-Gini, portfolio theory, and the pricing of risky assets, Journal of Finance, 39 (1984), 1449-1468.  doi: 10.1111/j.1540-6261.1984.tb04917.x.

[31]

W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442.  doi: 10.1111/j.1540-6261.1964.tb02865.x.

[32]

F. A. Sortino and S. Satchell, Managing Downside Risk in Financial Markets, Butterworth-Heinemann, 2001.

[33]

M. R. Young, A minimax portfolio selection rule with linear programming solution, Management Science, 44 (1998), 595-741.  doi: 10.1287/mnsc.44.5.673.

Figure 1.  Efficient frontiers (black) of the mean-risk problems and elements with the highest mean-to-risk ratio (green). All frontiers are depicted in the $ (\rho, \mathbb{E}) $ plane for the returns $ v_T - v_t $ with $ v_t = 1 $
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Figure 2.  Efficient frontiers for returns over time. The mean-risk profiles and the corresponding values of dRAROC are depicted for three trading strategies: the time consistent mean-risk strategy in one state $ \omega $ (yellow triangle), the switching strategy (red diamond) and the myopic strategy (magenta square). The element of the frontier with the highest dRAROC is also depicted at each time (green circle)
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Figure 3.  Efficient frontiers (black) of the problems (17) depicted for wealth $ V_t = 0. $ All frontiers are depicted in the $ (\mathbb{E}_t(V_T^-), \mathbb{E}_t (V_T)) $ plane. The corresponding highest value of $ dGLR $ (the slope of the frontier) is given
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Algorithm 1.  Approximating maximal acceptability α* via risk minimization
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Table 1.  Algorithm 1 for AIT, GLR and RAROC in a toy market model
Panel A: Return matrix R in the toy market model (two assets and four states of the world).
AIT GLR RAROC
Iter $x_L$ $x_U$ $x$ $p(x)$ Iter $x_L$ $x_U$ $x$ $p(x)$ Iter $x_L$ $x_U$ $x$ $p(x)$
Step 1 Step 1 Step 1
$1$ $0$ $\infty$ $2$ + $1$ $0$ $\infty$ $2$ $-$ $1$ $0$ $\infty$ $2$ +
$2$ $0$ $2$ $1$ + $2$ $2$ $\infty$ $4$ + $2$ $0$ $2$ $1$ +
$3$ $0$ $1$ $0.5$ $-$ Step 2 $3$ $0$ $1$ $0.5$ $-$
Step 2 $1$ $2$ $4$ $3$ $-$ Step 2
$1$ $0.5$ $1$ $0.75$ $-$ $2$ $3$ $4$ $3.5$ + $1$ $0.5$ $1$ $0.75$ $-$
$2$ $0.75$ $1$ $0.875$ + $3$ $3$ $3.5$ $3.25$ + $2$ $0.75$ $1$ $0.875$ +
$3$ $0.75$ $0.875$ $0.8125$ + $4$ $3$ $3.25$ $3.125$ $-$ $3$ $0.75$ $0.875$ $0.8125$ $-$
$4$ $0.75$ $0.8125$ $0.7813$ + $5$ $3.125$ $3.25$ $3.1875$ + $4$ $0.8125$ $0.875$ $0.8438$ +
$5$ $0.75$ $0.7813$ $0.7656$ + $6$ $3.125$ $3.1875$ $3.1563$ + $5$ $0.8125$ $0.8438$ $0.8281$ +
$6$ $0.75$ $0.7656$ $0.7578$ $-$ $7$ $3.125$ $3.1563$ $3.1406$ $-$ $6$ $0.8125$ $0.8281$ $0.8203$ $-$
$7$ $0.7578$ $0.7656$ $0.7617$ $-$ $8$ $3.1406$ $3.1563$ $3.1484$ + $7$ $0.8203$ $0.8281$ $0.8242$ +
$8$ $0.7617$ $0.7656$ $0.7637$ $-$ $9$ $3.1406$ $3.1484$ $3.1445$ + $8$ $0.8203$ $0.8242$ $0.8223$ +
$9$ $0.7637$ $0.7656$ $0.7647$ $-$ $10$ $3.1406$ $3.1445$ $3.1426$ $-$ $9$ $0.8203$ $0.8223$ $0.8213$ $-$
$10$ $0.7647$ $0.7656$ $0.7651$ $-$ $11$ $3.1426$ $3.1445$ $3.1436$ + $10$ $0.8213$ $0.8223$ $0.8218$ +
$11$ $0.7651$ $0.7656$ $0.7654$ + $12$ $3.1426$ $3.1436$ $3.1431$ + $11$ $0.8213$ $0.8218$ $0.8215$ +
$12$ $0.7651$ $0.7654$ $0.7653$ $-$ $13$ $3.1426$ $3.1431$ $3.1428$ $-$ $12$ $0.8213$ $0.8215$ $0.8214$ $-$
$13$ $0.7653$ $0.7654$ $0.7653$ $-$ $14$ $3.1428$ $3.1431$ $3.1429$ + $13$ $0.8214$ $0.8215$ $0.8215$ +
$0.76532$ $0.76538$ $15$ $3.1428$ $3.1429$ $3.1429$ + $0.82141$ $0.82147$
$h^\epsilon = (55.17\%, 44.83\%)$ $3.14282$ $3.14288$ $h^\epsilon = (93.75\%, 6.25\%)$
$h^\epsilon = (73.33\%, 26.67\%)$
Panel B: Iterations of Algorithm 1 with input parameters $x_0 = 2, \epsilon = 10^{-4}$ and $\bar{M} = 15$. The last two rows give, respectively, the bounds $x_L$ and $x_U$ on the maximal acceptability, and an $\text{V@R}epsilon$-optimal portfolio.
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Panel A: Return matrix R in the toy market model (two assets and four states of the world).
AIT GLR RAROC
Iter $x_L$ $x_U$ $x$ $p(x)$ Iter $x_L$ $x_U$ $x$ $p(x)$ Iter $x_L$ $x_U$ $x$ $p(x)$
Step 1 Step 1 Step 1
$1$ $0$ $\infty$ $2$ + $1$ $0$ $\infty$ $2$ $-$ $1$ $0$ $\infty$ $2$ +
$2$ $0$ $2$ $1$ + $2$ $2$ $\infty$ $4$ + $2$ $0$ $2$ $1$ +
$3$ $0$ $1$ $0.5$ $-$ Step 2 $3$ $0$ $1$ $0.5$ $-$
Step 2 $1$ $2$ $4$ $3$ $-$ Step 2
$1$ $0.5$ $1$ $0.75$ $-$ $2$ $3$ $4$ $3.5$ + $1$ $0.5$ $1$ $0.75$ $-$
$2$ $0.75$ $1$ $0.875$ + $3$ $3$ $3.5$ $3.25$ + $2$ $0.75$ $1$ $0.875$ +
$3$ $0.75$ $0.875$ $0.8125$ + $4$ $3$ $3.25$ $3.125$ $-$ $3$ $0.75$ $0.875$ $0.8125$ $-$
$4$ $0.75$ $0.8125$ $0.7813$ + $5$ $3.125$ $3.25$ $3.1875$ + $4$ $0.8125$ $0.875$ $0.8438$ +
$5$ $0.75$ $0.7813$ $0.7656$ + $6$ $3.125$ $3.1875$ $3.1563$ + $5$ $0.8125$ $0.8438$ $0.8281$ +
$6$ $0.75$ $0.7656$ $0.7578$ $-$ $7$ $3.125$ $3.1563$ $3.1406$ $-$ $6$ $0.8125$ $0.8281$ $0.8203$ $-$
$7$ $0.7578$ $0.7656$ $0.7617$ $-$ $8$ $3.1406$ $3.1563$ $3.1484$ + $7$ $0.8203$ $0.8281$ $0.8242$ +
$8$ $0.7617$ $0.7656$ $0.7637$ $-$ $9$ $3.1406$ $3.1484$ $3.1445$ + $8$ $0.8203$ $0.8242$ $0.8223$ +
$9$ $0.7637$ $0.7656$ $0.7647$ $-$ $10$ $3.1406$ $3.1445$ $3.1426$ $-$ $9$ $0.8203$ $0.8223$ $0.8213$ $-$
$10$ $0.7647$ $0.7656$ $0.7651$ $-$ $11$ $3.1426$ $3.1445$ $3.1436$ + $10$ $0.8213$ $0.8223$ $0.8218$ +
$11$ $0.7651$ $0.7656$ $0.7654$ + $12$ $3.1426$ $3.1436$ $3.1431$ + $11$ $0.8213$ $0.8218$ $0.8215$ +
$12$ $0.7651$ $0.7654$ $0.7653$ $-$ $13$ $3.1426$ $3.1431$ $3.1428$ $-$ $12$ $0.8213$ $0.8215$ $0.8214$ $-$
$13$ $0.7653$ $0.7654$ $0.7653$ $-$ $14$ $3.1428$ $3.1431$ $3.1429$ + $13$ $0.8214$ $0.8215$ $0.8215$ +
$0.76532$ $0.76538$ $15$ $3.1428$ $3.1429$ $3.1429$ + $0.82141$ $0.82147$
$h^\epsilon = (55.17\%, 44.83\%)$ $3.14282$ $3.14288$ $h^\epsilon = (93.75\%, 6.25\%)$
$h^\epsilon = (73.33\%, 26.67\%)$
Panel B: Iterations of Algorithm 1 with input parameters $x_0 = 2, \epsilon = 10^{-4}$ and $\bar{M} = 15$. The last two rows give, respectively, the bounds $x_L$ and $x_U$ on the maximal acceptability, and an $\text{V@R}epsilon$-optimal portfolio.
Table 2.  Iterations of the modified, the mixed and the zero-level version of Algorithm 1 for $\text{GLR}$ in the market model from Table 1, Panel A ($ \alpha^* = 3.1428 $) with the tolerance $ \epsilon = 10^{-4} $. In the modified version the bisection is performed on the parameter $ q = \frac{1}{2+x} \in [0, 0.5] $ after verifying the signs of $ p(0) $ and $ p(\infty) $. The termination criterion is set on the parameter $ x $ to guarantee an $ \epsilon $-solution is obtained. With the termination criterion on the parameter $ q $ the algorithm would finish after $ 13 $ iteration of Step 2, however, the interval for maximal acceptability would have length 1.6e-03. The mixed version switches to a bisection on the parameter $ x $ as soon as a finite upper bound $ x_U $ is obtained. -the zero-level version computes after each iteration the level $ y $ for which the portfolio solving the risk minimization problem has zero risk. This level is used as a lower bound. The algorithm is run with initial parameters $ x_0 = 2, \epsilon = 10^{-4} $ and $ \bar{M} = 15 $
Modified algorithm for GLR Mixed algorithm for GLR Zero-level algorithm for GLR
Iter $q_L$ $q_U$ $q$ $x$ $p(x)$ Iter $q_L$ $q_U$ $q$ $x$ $p(x)$ Iter $x_L$ $x_U$ $x$ $y$ $p(x)$
Step 1 Step 1 Step 1
$0$ $\infty$ + $0$ $\infty$ + $1$ $0$ $\infty$ $2$ $3.1429$ $-$
$0.5$ $0$ $-$ $0.5$ $0$ $-$ $2$ $3.1429$ $\infty$ $6.2857$ $3.1429$ +
Step 2 Step 2 Step 2
$1$ $0$ $0.5$ $0.25$ $2$ $-$ $1$ $0$ $0.5$ $0.25$ $2$ $-$ $1$ $3.1429$ $6.2857$ $4.7143$ $3.1429$ +
$2$ $0$ $0.25$ $0.125$ $6$ + $2$ $0$ $0.25$ $0.125$ $6$ + $2$ $3.1429$ $4.7143$ $3.9286$ $3.1429$ +
$3$ $0.125$ $0.25$ $0.1875$ $3.3333$ + Iter $x_L$ $x_U$ $x$ $p(x)$ $3$ $3.1429$ $3.9286$ $3.5357$ $3.1429$ +
$4$ $0.1875$ $0.25$ $0.2188$ $2.5714$ $-$ $3$ $2$ $6$ $4$ + $4$ $3.1429$ $3.5357$ $3.3393$ $3.1429$ +
$5$ $0.1875$ $0.2188$ $0.2031$ $2.9231$ $-$ $4$ $2$ $4$ $3$ $-$ $5$ $3.1429$ $3.3393$ $3.2411$ $3.1429$ +
$6$ $0.1875$ $0.2031$ $0.1953$ $3.1200$ $-$ $5$ $3$ $4$ $3.5$ + $6$ $3.1429$ $3.2411$ $3.1920$ $3.1429$ +
$7$ $0.1875$ $0.1953$ $0.1914$ $3.2245$ + $6$ $3$ $3.5$ $3.25$ + $7$ $3.1429$ $3.1920$ $3.1674$ $3.1429$ +
$8$ $0.1914$ $0.1953$ $0.1934$ $3.1717$ + $7$ $3$ $3.25$ $3.125$ $-$ $8$ $3.1429$ $3.1674$ $3.1551$ $3.1429$ +
$9$ $0.1934$ $0.1953$ $0.1943$ $3.1457$ + $8$ $3.125$ $3.25$ $3.1875$ + $9$ $3.1429$ $3.1551$ $3.1490$ $3.1429$ +
$10$ $0.1943$ $0.1953$ $0.1948$ $3.1328$ $-$ $9$ $3.125$ $3.1875$ $3.1563$ + $10$ $3.1429$ $3.1490$ $3.1459$ $3.1429$ +
$11$ $0.1943$ $0.1948$ $0.1946$ $3.1393$ $-$ $10$ $3.125$ $3.1563$ $3.1406$ $-$ $11$ $3.1429$ $3.1459$ $3.1444$ $3.1429$ +
$12$ $0.1943$ $0.1946$ $0.1945$ $3.1425$ $-$ $11$ $3.1406$ $3.1563$ $3.1484$ + $12$ $3.1429$ $3.1444$ $3.1436$ $3.1429$ +
$13$ $0.1943$ $0.1945$ $0.1944$ $3.1441$ + $12$ $3.1406$ $3.1484$ $3.1445$ + $13$ $3.1429$ $3.1436$ $3.1432$ $3.1429$ +
$14$ $0.1944$ $0.1945$ $0.1944$ $3.1433$ + $13$ $3.1406$ $3.1445$ $3.1426$ $-$ $14$ $3.1429$ $3.1432$ $3.1430$ $3.1429$ +
$15$ $0.1944$ $0.1945$ $0.1944$ $3.1429$ + $14$ $3.1426$ $3.1445$ $3.1436$ + $15$ $3.1429$ $3.1430$ $3.1430$ $3.1429$ +
$16$ $0.1944$ $0.1945$ $0.1945$ $3.1427$ $-$ $15$ $3.1426$ $3.1436$ $3.1431$ + $(x_L, x_U) = (3.14286, 3.14295)$
$17$ $0.1944$ $0.1945$ $0.1944$ $3.1428$ $-$ $16$ $3.1426$ $3.1431$ $3.1428$ $-$ $h^\epsilon = (73.33\%, 26.67\%)$
$18$ $0.1944$ $0.1944$ $0.1944$ $3.1429$ $-$ $17$ $3.1428$ $3.1431$ $3.1429$ +
$(x_L, x_U) = (3.14285, 3.14290)$ $18$ $3.1428$ $3.1429$ $3.1429$ +
$q_U - q_L =$ 1.9e-06, $x_U - x_L =$ 5.0e-05 $(x_L, x_U) = (3.14282, 3.14288)$
$h^\epsilon = (73.33\%, 26.67\%)$ $h^\epsilon = (73.33\%, 26.67\%)$
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Modified algorithm for GLR Mixed algorithm for GLR Zero-level algorithm for GLR
Iter $q_L$ $q_U$ $q$ $x$ $p(x)$ Iter $q_L$ $q_U$ $q$ $x$ $p(x)$ Iter $x_L$ $x_U$ $x$ $y$ $p(x)$
Step 1 Step 1 Step 1
$0$ $\infty$ + $0$ $\infty$ + $1$ $0$ $\infty$ $2$ $3.1429$ $-$
$0.5$ $0$ $-$ $0.5$ $0$ $-$ $2$ $3.1429$ $\infty$ $6.2857$ $3.1429$ +
Step 2 Step 2 Step 2
$1$ $0$ $0.5$ $0.25$ $2$ $-$ $1$ $0$ $0.5$ $0.25$ $2$ $-$ $1$ $3.1429$ $6.2857$ $4.7143$ $3.1429$ +
$2$ $0$ $0.25$ $0.125$ $6$ + $2$ $0$ $0.25$ $0.125$ $6$ + $2$ $3.1429$ $4.7143$ $3.9286$ $3.1429$ +
$3$ $0.125$ $0.25$ $0.1875$ $3.3333$ + Iter $x_L$ $x_U$ $x$ $p(x)$ $3$ $3.1429$ $3.9286$ $3.5357$ $3.1429$ +
$4$ $0.1875$ $0.25$ $0.2188$ $2.5714$ $-$ $3$ $2$ $6$ $4$ + $4$ $3.1429$ $3.5357$ $3.3393$ $3.1429$ +
$5$ $0.1875$ $0.2188$ $0.2031$ $2.9231$ $-$ $4$ $2$ $4$ $3$ $-$ $5$ $3.1429$ $3.3393$ $3.2411$ $3.1429$ +
$6$ $0.1875$ $0.2031$ $0.1953$ $3.1200$ $-$ $5$ $3$ $4$ $3.5$ + $6$ $3.1429$ $3.2411$ $3.1920$ $3.1429$ +
$7$ $0.1875$ $0.1953$ $0.1914$ $3.2245$ + $6$ $3$ $3.5$ $3.25$ + $7$ $3.1429$ $3.1920$ $3.1674$ $3.1429$ +
$8$ $0.1914$ $0.1953$ $0.1934$ $3.1717$ + $7$ $3$ $3.25$ $3.125$ $-$ $8$ $3.1429$ $3.1674$ $3.1551$ $3.1429$ +
$9$ $0.1934$ $0.1953$ $0.1943$ $3.1457$ + $8$ $3.125$ $3.25$ $3.1875$ + $9$ $3.1429$ $3.1551$ $3.1490$ $3.1429$ +
$10$ $0.1943$ $0.1953$ $0.1948$ $3.1328$ $-$ $9$ $3.125$ $3.1875$ $3.1563$ + $10$ $3.1429$ $3.1490$ $3.1459$ $3.1429$ +
$11$ $0.1943$ $0.1948$ $0.1946$ $3.1393$ $-$ $10$ $3.125$ $3.1563$ $3.1406$ $-$ $11$ $3.1429$ $3.1459$ $3.1444$ $3.1429$ +
$12$ $0.1943$ $0.1946$ $0.1945$ $3.1425$ $-$ $11$ $3.1406$ $3.1563$ $3.1484$ + $12$ $3.1429$ $3.1444$ $3.1436$ $3.1429$ +
$13$ $0.1943$ $0.1945$ $0.1944$ $3.1441$ + $12$ $3.1406$ $3.1484$ $3.1445$ + $13$ $3.1429$ $3.1436$ $3.1432$ $3.1429$ +
$14$ $0.1944$ $0.1945$ $0.1944$ $3.1433$ + $13$ $3.1406$ $3.1445$ $3.1426$ $-$ $14$ $3.1429$ $3.1432$ $3.1430$ $3.1429$ +
$15$ $0.1944$ $0.1945$ $0.1944$ $3.1429$ + $14$ $3.1426$ $3.1445$ $3.1436$ + $15$ $3.1429$ $3.1430$ $3.1430$ $3.1429$ +
$16$ $0.1944$ $0.1945$ $0.1945$ $3.1427$ $-$ $15$ $3.1426$ $3.1436$ $3.1431$ + $(x_L, x_U) = (3.14286, 3.14295)$
$17$ $0.1944$ $0.1945$ $0.1944$ $3.1428$ $-$ $16$ $3.1426$ $3.1431$ $3.1428$ $-$ $h^\epsilon = (73.33\%, 26.67\%)$
$18$ $0.1944$ $0.1944$ $0.1944$ $3.1429$ $-$ $17$ $3.1428$ $3.1431$ $3.1429$ +
$(x_L, x_U) = (3.14285, 3.14290)$ $18$ $3.1428$ $3.1429$ $3.1429$ +
$q_U - q_L =$ 1.9e-06, $x_U - x_L =$ 5.0e-05 $(x_L, x_U) = (3.14282, 3.14288)$
$h^\epsilon = (73.33\%, 26.67\%)$ $h^\epsilon = (73.33\%, 26.67\%)$
Table 3.  The behavior of Algorithm 1 for various input parameters in a market model with $ d = 10 $ assets with short-selling constraints
Panel A: $\text{AIT}$, maximal acceptability $\alpha^* = 25.45$.
$x_0$ $\epsilon$ $M$ Step 1 Step 2 Run time
Iter $[x_L, x_U]$ Iter $x_U - x_L$ (s)
$2$ $10^{-4}$ $15$ 5 $[16, 32]$ 18 6.1e-05 3.78
$20$ $10^{-4}$ $15$ 2 $[20, 40]$ 18 7.6e-05 3.40
$200$ $10^{-4}$ $15$ 4 $[25, 50]$ 18 9.5e-05 3.56
$2$ $10^{-8}$ $15$ 5 $[16, 32]$ 31 7.5e-09 6.22
$2^{20}$ $10^{-4}$ $15$ 15 $[0, 64]$ no Step 2 1.88
$2^{20}$ $10^{-4}$ $30$ 17 $[16, 32]$ 18 6.1e-05 4.67
$2^{-10}$ $10^{-4}$ $15$ 15 $[16, \infty]$ no Step 2 4.61
$2^{-10}$ $10^{-4}$ $30$ 16 $[16, 32]$ 18 6.1e-05 7.15
Panel B: $\text{GLR}$, maximal acceptability $\alpha^* = 279.62$.
$x_0$ $\epsilon$ $M$ Step 1 Step 2 Run time
Iter $[x_L, x_U]$ Iter $x_U - x_L$ (s)
$2$ $10^{-4}$ $15$ 9 $[256,512]$ 22 6.1e-05 21.53
$20$ $10^{-4}$ $15$ 5 $[160,320]$ 21 7.6e-05 17.27
$200$ $10^{-4}$ $15$ 2 $[200,400]$ 21 9.5e-05 15.74
$2$ $10^{-8}$ $15$ 9 $[256,512]$ 35 7.5e-09 30.40
$2^{25}$ $10^{-4}$ $15$ 15 $[0, 2048]$ no Step 2 6.50
$2^{25}$ $10^{-4}$ $30$ 18 $[0.5,1]$ 22 6.1e-05 23.91
$2^{-10}$ $10^{-4}$ $15$ 15 $[16, \infty]$ no Step 2 13.41
$2^{-10}$ $10^{-4}$ $30$ 20 $[256,512]$ 22 6.1e-05 30.27
Panel C: $\text{RAROC}$, maximal acceptability $\alpha^* = 279.62$.
$x_0$ $\epsilon$ $M$ Step 1 Step 2 Run time
Iter $[x_L, x_U]$ Iter $x_U - x_L$ (s)
$2$ $10^{-4}$ $15$ 2 $[2, 4]$ 15 6.1e-05 7.20
$20$ $10^{-4}$ $15$ 4 $[2.5, 5]$ 15 7.6e-05 9.41
$200$ $10^{-4}$ $15$ 8 $[1.56, 3.13]$ 14 9.4e-05 12.84
$2$ $10^{-8}$ $15$ 2 $[2, 4]$ 28 7.5e-09 11.07
$2^{20}$ $10^{-4}$ $15$ 15 $[0,64]$ no Step 2 10.55
$2^{20}$ $10^{-4}$ $30$ 20 $[2, 4]$ 15 6.1e-05 19.59
$2^{-15}$ $10^{-4}$ $15$ 15 $[0.5, \infty]$ no Step 2 7.04
$2^{-15}$ $10^{-4}$ $30$ 18 $[2, 4]$ 15 6.1e-05 13.41
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Panel A: $\text{AIT}$, maximal acceptability $\alpha^* = 25.45$.
$x_0$ $\epsilon$ $M$ Step 1 Step 2 Run time
Iter $[x_L, x_U]$ Iter $x_U - x_L$ (s)
$2$ $10^{-4}$ $15$ 5 $[16, 32]$ 18 6.1e-05 3.78
$20$ $10^{-4}$ $15$ 2 $[20, 40]$ 18 7.6e-05 3.40
$200$ $10^{-4}$ $15$ 4 $[25, 50]$ 18 9.5e-05 3.56
$2$ $10^{-8}$ $15$ 5 $[16, 32]$ 31 7.5e-09 6.22
$2^{20}$ $10^{-4}$ $15$ 15 $[0, 64]$ no Step 2 1.88
$2^{20}$ $10^{-4}$ $30$ 17 $[16, 32]$ 18 6.1e-05 4.67
$2^{-10}$ $10^{-4}$ $15$ 15 $[16, \infty]$ no Step 2 4.61
$2^{-10}$ $10^{-4}$ $30$ 16 $[16, 32]$ 18 6.1e-05 7.15
Panel B: $\text{GLR}$, maximal acceptability $\alpha^* = 279.62$.
$x_0$ $\epsilon$ $M$ Step 1 Step 2 Run time
Iter $[x_L, x_U]$ Iter $x_U - x_L$ (s)
$2$ $10^{-4}$ $15$ 9 $[256,512]$ 22 6.1e-05 21.53
$20$ $10^{-4}$ $15$ 5 $[160,320]$ 21 7.6e-05 17.27
$200$ $10^{-4}$ $15$ 2 $[200,400]$ 21 9.5e-05 15.74
$2$ $10^{-8}$ $15$ 9 $[256,512]$ 35 7.5e-09 30.40
$2^{25}$ $10^{-4}$ $15$ 15 $[0, 2048]$ no Step 2 6.50
$2^{25}$ $10^{-4}$ $30$ 18 $[0.5,1]$ 22 6.1e-05 23.91
$2^{-10}$ $10^{-4}$ $15$ 15 $[16, \infty]$ no Step 2 13.41
$2^{-10}$ $10^{-4}$ $30$ 20 $[256,512]$ 22 6.1e-05 30.27
Panel C: $\text{RAROC}$, maximal acceptability $\alpha^* = 279.62$.
$x_0$ $\epsilon$ $M$ Step 1 Step 2 Run time
Iter $[x_L, x_U]$ Iter $x_U - x_L$ (s)
$2$ $10^{-4}$ $15$ 2 $[2, 4]$ 15 6.1e-05 7.20
$20$ $10^{-4}$ $15$ 4 $[2.5, 5]$ 15 7.6e-05 9.41
$200$ $10^{-4}$ $15$ 8 $[1.56, 3.13]$ 14 9.4e-05 12.84
$2$ $10^{-8}$ $15$ 2 $[2, 4]$ 28 7.5e-09 11.07
$2^{20}$ $10^{-4}$ $15$ 15 $[0,64]$ no Step 2 10.55
$2^{20}$ $10^{-4}$ $30$ 20 $[2, 4]$ 15 6.1e-05 19.59
$2^{-15}$ $10^{-4}$ $15$ 15 $[0.5, \infty]$ no Step 2 7.04
$2^{-15}$ $10^{-4}$ $30$ 18 $[2, 4]$ 15 6.1e-05 13.41
Table 4.  A comparison of the different versions of the algorithm in a market with $ d = 10 $ assets and $ \vert \Omega \vert = 1000 $ states of the world both with and without short-selling. A tolerance $ \epsilon = 10^{-4} $ is used for all algorithms, the original and zero-level version use $ x_0 = 2 $ and $ \bar{M} = 15 $. Obtaining the final approximation $ [x_L, x_U] $ is denoted in the table by $ \alpha^* $, values are listed to two decimal places
Panel A: $\text{AIT}$, the maximal acceptability with short-selling constraints ($h \geq 0$) is $\alpha^* = 25.45,$ without short-selling constraints ($h$ free) it is $\alpha^* =25.72$.
lgorithm Step 1 Bisection on $q$ Bisection on $x$ $x_U - x_L$ Run time
Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ (s)
$h \geq 0$ Original 5 $[16, 32]$ 18 $\alpha^*$ 6.1e-05 3.32
Modified 2 $[0, \infty]$ 23 $\alpha^*$ 8.3e-05 3.67
Mixed 2 $[0, \infty]$ 5 $[15, 31]$ 18 $\alpha^*$ 6.1e-05 3.90
Zero level 3 $[23.42, 46.84]$ 18 $\alpha^*$ 5.9e-05 2.96
$h$ free Original 5 $[16, 32]$ 18 $\alpha^*$ 6.1e-05 4.89
Modified 2 $[0, \infty]$ 23 $\alpha^*$ 8.5e-05 5.11
Mixed 2 $[0, \infty]$ 5 $[15, 31]$ 18 $\alpha^*$ 6.1e-05 5.09
Zero level 3 $[23.45, 46.89]$ 18 $\alpha^*$ 5.1e-05 4.36
Panel B: $\text{GLR}$, the maximal acceptability with short-selling constraints ($h \geq 0$) is $\alpha^* = 279.62,$ without short-selling constraints ($h$ free) it is $\alpha^* =288.88$.
Algorithm Step 1 Bisection on $q$ Bisection on $x$ $x_U - x_L$ Run time
Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ (s)
$h \geq 0$ Original 9 $[256,512]$ 22 $\alpha^*$ 6.1e-05 19.45
Modified 2 $[0, \infty]$ 29 $\alpha^*$ 7.4e-05 18.42
Mixed 2 $[0, \infty]$ 8 $[254,510]$ 22 $\alpha^*$ 6.1e-05 19.75
Zero level 3 $[279.62,559.24]$ 22 $\alpha^*$ 6.7e-05 14.54
$h$ free Original 9 $[256,512]$ 22 $\alpha^*$ 6.1e-05 39.56
Modified 2 $[0, \infty]$ 29 $\alpha^*$ 7.9e-05 40.85
Mixed 2 $[0, \infty]$ 8 $[254,510]$ 22 $\alpha^*$ 6.1e-05 41.66
Zero level 3 $[288.88,577.76]$ 22 $\alpha^*$ 6.8e-05 32.17
Panel C: $\text{RAROC}$, the maximal acceptability with short-selling constraints ($h \geq 0$) is $\alpha^* = 2.98,$ without short-selling constraints ($h$ free) it is $\alpha^* =3.08$.
Algorithm Step 1 Bisection on $q$ Bisection on $x$ $x_U - x_L$ Run time
Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ (s)
$h \geq 0$ Original 2 $[2, 4]$ 15 $\alpha^*$ 6.1e-05 5.88
Modified 2 $[0, \infty]$ 18 $\alpha^*$ 6.0e-05 5.88
Mixed 2 $[0, \infty]$ 2 $[1, 3]$ 15 $\alpha^*$ 6.1e-05 5.62
Zero level 2 $[2.98, 5.96]$ 15 $\alpha^*$ 9.1e-05 5.53
$h$ free Original 2 $[2, 4]$ 15 $\alpha^*$ 6.1e-05 6.91
Modified 2 $[0, \infty]$ 18 $\alpha^*$ 6.3e-05 8.00
Mixed 2 $[0, \infty]$ 3 $[3, 7]$ 18 $\alpha^*$ 6.1e-05 8.12
Zero level 2 $[3.08, 6.15]$ 15 $\alpha^*$ 9.4e-05 7.04
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Panel A: $\text{AIT}$, the maximal acceptability with short-selling constraints ($h \geq 0$) is $\alpha^* = 25.45,$ without short-selling constraints ($h$ free) it is $\alpha^* =25.72$.
lgorithm Step 1 Bisection on $q$ Bisection on $x$ $x_U - x_L$ Run time
Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ (s)
$h \geq 0$ Original 5 $[16, 32]$ 18 $\alpha^*$ 6.1e-05 3.32
Modified 2 $[0, \infty]$ 23 $\alpha^*$ 8.3e-05 3.67
Mixed 2 $[0, \infty]$ 5 $[15, 31]$ 18 $\alpha^*$ 6.1e-05 3.90
Zero level 3 $[23.42, 46.84]$ 18 $\alpha^*$ 5.9e-05 2.96
$h$ free Original 5 $[16, 32]$ 18 $\alpha^*$ 6.1e-05 4.89
Modified 2 $[0, \infty]$ 23 $\alpha^*$ 8.5e-05 5.11
Mixed 2 $[0, \infty]$ 5 $[15, 31]$ 18 $\alpha^*$ 6.1e-05 5.09
Zero level 3 $[23.45, 46.89]$ 18 $\alpha^*$ 5.1e-05 4.36
Panel B: $\text{GLR}$, the maximal acceptability with short-selling constraints ($h \geq 0$) is $\alpha^* = 279.62,$ without short-selling constraints ($h$ free) it is $\alpha^* =288.88$.
Algorithm Step 1 Bisection on $q$ Bisection on $x$ $x_U - x_L$ Run time
Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ (s)
$h \geq 0$ Original 9 $[256,512]$ 22 $\alpha^*$ 6.1e-05 19.45
Modified 2 $[0, \infty]$ 29 $\alpha^*$ 7.4e-05 18.42
Mixed 2 $[0, \infty]$ 8 $[254,510]$ 22 $\alpha^*$ 6.1e-05 19.75
Zero level 3 $[279.62,559.24]$ 22 $\alpha^*$ 6.7e-05 14.54
$h$ free Original 9 $[256,512]$ 22 $\alpha^*$ 6.1e-05 39.56
Modified 2 $[0, \infty]$ 29 $\alpha^*$ 7.9e-05 40.85
Mixed 2 $[0, \infty]$ 8 $[254,510]$ 22 $\alpha^*$ 6.1e-05 41.66
Zero level 3 $[288.88,577.76]$ 22 $\alpha^*$ 6.8e-05 32.17
Panel C: $\text{RAROC}$, the maximal acceptability with short-selling constraints ($h \geq 0$) is $\alpha^* = 2.98,$ without short-selling constraints ($h$ free) it is $\alpha^* =3.08$.
Algorithm Step 1 Bisection on $q$ Bisection on $x$ $x_U - x_L$ Run time
Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ Iter $[x_L, x_U]$ (s)
$h \geq 0$ Original 2 $[2, 4]$ 15 $\alpha^*$ 6.1e-05 5.88
Modified 2 $[0, \infty]$ 18 $\alpha^*$ 6.0e-05 5.88
Mixed 2 $[0, \infty]$ 2 $[1, 3]$ 15 $\alpha^*$ 6.1e-05 5.62
Zero level 2 $[2.98, 5.96]$ 15 $\alpha^*$ 9.1e-05 5.53
$h$ free Original 2 $[2, 4]$ 15 $\alpha^*$ 6.1e-05 6.91
Modified 2 $[0, \infty]$ 18 $\alpha^*$ 6.3e-05 8.00
Mixed 2 $[0, \infty]$ 3 $[3, 7]$ 18 $\alpha^*$ 6.1e-05 8.12
Zero level 2 $[3.08, 6.15]$ 15 $\alpha^*$ 9.4e-05 7.04
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