# Asset allocation and multi-asset arbitrage

J.-G. Attali

The objective of this course is to introduce some of the tools needed to construct a diversified portfolio across the major asset classes, namely equities, bonds, credit, currencies and commodities. Once justified the existence of Objectives: In this course, tools are introduced that allow the evaluation of certain risks by taking into account the characteristics and history of the insured. They thus make it possible to take into account the heterogeneity of a portfolio, the evolution of a risk in the course of time, and to refine thus the techniques of tarification.

## Course outline

I. Generalized linear model in a priori pricing

- Formulation and estimation of parameters
- Confidence intervals and fit
- Examples and illustrations in pricing

II. Credibility theory, a posteriori pricing

- Bayesian credibility
- Linear credibility
- Hierarchical credibility
- Taking into account a time drift

III. Bonus-Malus

- Generalities on Markov chains
- Construction of a bonus-malus system

IV. Technical provisions

- Deterministic methods
- Mack’s method and extensions
- Factor methods
- Use of generalized linear models
- Measuring uncertainty in provisions

## In order to determine the long-term risk premium for each asset class, a number of quantitative techniques for determining risk factors, such as principal component analysis and factor analysis, will be presented. We will be careful to specify the underlying assumptions of each technique in order to determine its scope. Certain assets that do not respect traditional assumptions, such as credit, will naturally lead us to present the most recent academic developments, such as independent component analysis.

These factorial approaches will, in particular, allow us to construct efficient indices aiming at capturing the traditional beta at best, as most of the major benchmarks are not optimal in a risk-return sense. They will also allow us to demonstrate some alternative management strategies used by certain hedge funds and based on exposure to alternative factors such as the size of the company in the equity world or the slope of rates in the bond world, strategies that can be made market-neutral by cancelling out exposure to the “market” factor. We will speak here of alternative beta in the sense that it is the search for indexation to a source of recurrent and stable performance under certain macroeconomic conditions. These approaches will finally make it possible to address the question of the optimal hedging of a portfolio against an event that could affect it (rise in rates for a bond portfolio, etc.). We will then deal with the creation of value through the search for alpha by arbitrage, i.e. the financial exploitation of temporary imbalances. This part will provide an opportunity to discuss econometric methods for valuing assets in order to arbitrage within a class (platinum vs. gold, euro vs. dollar, etc.) or between classes, such as the famous Fed model which establishes a long-term arbitrage relationship between stocks and bonds. The search for beta, whether traditional or alternative, is a long-term approach known as “strategic”, while the search for alpha is a medium- to short-term approach known as “tactical”. In order to determine an optimal asset allocation, we will finally detail the Black and Litterman portfolio optimisation method, which is in line with the Markowitz approach, and which makes it possible to integrate tactical signals into a strategic allocation.

Prerequisites: CAPM, Markowitz optimization, principal component analysis, linear regression.

## References

- Black, F. and Litterman, R. (1990).
*Asset Allocation: Combining Investors Views with Market Equilibrium. Fixed Income*Research, Goldman, Sachs & Company, September. - Black, F., R. Litterman (1991)
*Global asset allocation with equities, bonds and currencies, Fixed Income Research*, Goldman, Sachs & Company, October. - Black, F., R. Litterman (1992)
*Global portfolio optimization*, Financial Analysts Journal, September/October, 28-43. - Boulier, J.-F., Hartpence, M.L. *Fundamental-driven and Tactical Asset Allocation : what really matters ? *Banque & Marchés n° 73 novembre-décembre 2004
- Campbell, John Y., Andrew W. Lo and A. Craig MacKinlay (1997)
*The Econometrics of Financial Markets*, Princeton University Press. - Campbell, John Y. and Robert J. Shiller, (1988a),
*Stock prices, earnings and expected dividends*, Journal of Finance,{ 43}, pp.661-676. - Campbell, John Y. and Robert J. Shiller, (1988b),
*The dividend-price ratio and expectations of future dividends and discount factors*, Review of Financial Studies, { 1}(3), pp.195-228. - Cardoso, J.F. (1998),
*Blind signal separation : statistical principles*, Proc IEEE 86, 2009-2025. - Durieu, C. & Kieffer, M. (2003),
*Analyse en composantes indépendantes pour la séparation aveugle de sources*, - - - Colloque sur l’enseignement des technologies et des sciences de l’information et des systèmes, CETSIS-EEA, Toulouse, pp. 143-146. - Fama, E & French, K. (1993),
*Common risk factors in the returns of stock and bonds*, Journal of Financial Economics, vol 33, pp. 3-56. - Hamilton, James D. (1994).
*Time Series Analysis*, Princeton University Press. - Litterman, R. & Scheinkman, J., (1988),
*Common Factors affecting bond returns*. Financial strategies group, Goldman Sachs. - Engle, R.F. and Granger, C.W.J. (1987),
*Co-integration and Error Correction : Representation, Estimation and Testing*, Econometrica 55, 251-76. - Johansen, S. (1991),
*Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models*, Econometrica, 59, pp. 1551-1580. - Phillips, P. C. and Loretan, M. (1991),
*Estimating Long-Run Economic Equilibria*, Review of Economic Studies, Vol. 58, 407-36.