Robust Asset Allocationfor Institutional Investors

François Oustry

2

Complexity and uncertainty in finance

Morocco’s burgeoning automobile sector is driving the industrialisation of an economy still heavily dependent on agriculture. The country’s auto sector will help push GDP growth up 4.5 to 5 per cent in 2015-2016

3

Uncertainty in the work of Ibn Khaldun (1332-1406 – 350 years before Adam Smith!!)

االحتكارقوي عند اشتهر مما و

في التجربة و البصرالزرع احتكار أن األمصار

الغالء أوقات لتحينعلى. يعود أنه و مشؤم

الخسران و بالتلف فائدته

ابن مقدمةخلدون

4

Uncertainty by Ibn Khaldun (1332-1406)

Hoarding

Intelligent and experienced people in the cities know that it is inauspicious to hoard grain and to wait for high prices, and that the profit (expected) may be spoiled or lost through (hoarding)

Muqaddimah of Ibn Khaldun

Part of the uncertainty cannot be captured for prediction – yet can we act in a robust way knowing that we (including Mathematicians and Geeks!) do not know (everything)?

Ibn Khaldun, Franz Rosenthal, N. J. Dawood (1967), The Muqaddimah: An Introduction to History, pp. 5–6, Princeton University Press, ISBN 0-691-01754-9.(Joseph A. Schumpeter, History of Economic Analysis, edited from manuscript by Elizabeth B. Schumpeter and published after his death (New York: Oxford University Press, 1954), pp. 136, 788.

5

Robust and Resilient Structure Design & Construction since 1200 years!

Marginal risk driven dynamic system – absorbing waves via a semi-active system

6

Modern and efficient technology: semi-active tuned mass damper

Tuned mass dampers stabilize against violent motion caused by harmonic vibration. A tuned damper reduces the vibration of a system with a comparatively lightweight component so that the worst-case vibrations are less intense.

Transposition in Finance

L. El Ghaoui and F. Oustry and M. Oks. Worst-Case Value-At-Risk and Robust Portfolio Optimization. Operations Research, 2003.

7

Correlations: not intuitive, a source of complexityAn asset by itself may have a high volatility, but because it has a low or even negative correlation with other assets in the portfolio, adding some of the asset to the portfolio may reduce overall portfolio risk. http://www.systemic-risk-indices.com

8

Robust Marginal Risk Analyzis

From magnitude to specific sources of risk identification in a portfolio

From passive measurement to action: recommend trades that would reduce overall portfolio risk, including data and model uncertainties

Robust Marginal Risk Analyzis as a first step to Robust Portfolio Optimization

9

A key tool for risk allocation: Euler principle

F. Yang and Z. Wei. Generalized Euler identity for subdifferentials of homogeneous functions and applications. Mathematical Analysis and Applications, 337:516–523, 2008

10

Marginal Risk Decomposition (1)Standard Marginal Risk Decomposition

Structured Portfolio Perturbation

Generalized Marginal Risk Decomposition

Keel, Simon and Ardia, David, Generalized Marginal Risk (April 20, 2010). Journal of Asset Management, Vol 12, pp.123-131, June 2011.

11

Marginal Risk Decomposition (2)

Perturbation of a portfolio with no cash

Diagonal perturbation

Portfolio with cash/leverage

12

Regularizing Historical Value-at-Risk (1)Gourieroux, C. & Laurent, J. P. & Scaillet, O., 2000.Sensitivity analysis of Values at Risk, Journal of Empirical Finance, Elsevier, Elsevier, vol. 7(3-4), pages 225-245, NovemberScaillet , O., Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall . Mathematical Finance, Vol. 14, pp. 115-129, January 2004.

13

Regularizing Historical Value-at-Risk (2)

14

Why (strong) convexity is important CRUCIAL to ensure stability of optimal investment decisions?

f(x,u) and G(x,u) families of convex perturbed/uncertained functions (Pu)

Convex Active Risk Management/Asset Allocation

General primal-dual optimality conditions

Bonnans, J.F. and Shapiro, A., ``Optimization problems with perturbations, A guided tour", SIAM Review, vol. 40, pp. 228-264, 1998.

(P0)

15

Stability results under second-order growth conditions

C Lemaréchal, F Oustry, C Sagastizábal. The -Lagrangian of a convex function - 𝒰Transactions of the American Mathematical Society, 2000

Direct approach when L(.,.,u) is convex but not differentiable

16

A robust allocation model

• [0,100%] exposure on each position

• controls exposure to a « naive » momentum factor targeting to reach about 9% growth annual performance

• Monthly allocation from April 2007 to today

17

Numerical optimization: bundle methods

C. Lemaréchal, A. Nemirovskii, Y. Nesterov. New variants of bundle methods Mathematical Programming. 1995, Volume 69, Issue 1-3, pp 111-147

18

19

20

Kernel distribution analyzis: tail-risk reduction

21

Conclusion

Comments on the allocation process• Risk functions• Robust Optimization• Expected returns (e.g. generalization of Black-

Litterman model)

Questions? : ;را ك ش>َج<ز=يال: