generalized modern portfolio theory · portfolio optimization in an uncertain world marielle de...

Portfolio optimization in an uncertain world

Marielle de Jong Head of Fixed-Income Quant Research at Amundi

 Stowe, March 2017

 Presented at the 29th Annual Research Conference of Northfield in Stowe, Vermont

Generalized Modern Portfolio Theory

Generalized MPT - de Jong

In this speech we

1. generalize the Markowitz optimization problem

2. solve it

3. link it to risk parity and similar methods

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Modern Portfolio Theory  

Generalized MPT - de Jong * 3

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Generalize Modern Portfolio Theory We relax the certainty hypothesis,

asset returns don’t strictly obey to predefined laws accord the definition of utility,

investors seek to minimize loss due to unfavourable price shocks, which can be

foreseen – within the sphere defined by the Hessian unforeseen – unframed and accord the optimization objective

minimize loss due to

foreseeable shocks – via variance minimization unforeseeable shocks – via diversification maximization


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Diversification  


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Generalized Modern Portfolio Theory  


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Solution to the generalized problem  

Generalized MPT - de Jong Proof by Roncalli (2013) 7

diversification

variance

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Risk parity is optimal

in an uncertain world By admitting uncertainty one acknowledges the need to protect against the

unknown.

Portfolio is optimal, i.e. loss is minimized, when portfolio variance and diversification optimized.

Risk parity when variance-over-diversification at maximum.

Risk parity solution not unique.

frontier of solutions depending on θ entropy (disorder/uncertainty) several measures of diversification


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Optimal solutions depending on θ variance-diversification frontier

High entropy – optimal portfolio 1/N

Intermediate – optimal portfolio 1/ risk contribution

Low entropy – optimal portfolio Markowitz Generalized MPT - de Jong 9

diversification

variance

ϑ → ∞

ϑ = 0

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Choice of diversification measure  


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Bayesian optimization is optimal  


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Diversification measure based on risk estimates  


portfolio volatility

asset volatility

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Recap  

Generalized MPT - de Jong 14

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Investment example - equity-bond allocation problem  

Generalized MPT - de Jong Example put forward by Qian (2011) 15

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Strategies coherent with generalized MPT

1. Top-down approach* (not full optimization) As covariance more predictable within asset classes than between,

Markowitz optimization is deployed within- and risk budgeting between.

2. Bridgewater® All Weather Funds Four plausible economic scenarios are weighted by their respective volatility levels. 3. Global Macro Asset class weights set inversely proportional to volatility levels

Generalized MPT - de Jong * Discussed by Scherer (2007) Portfolio Construction & Risk Budgeting

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A bit of philosophy  


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Risk model that defines order

Whether to adopt a risk factor, assess if it more damaging to ignore it – so that risk efficiency will be foregone to adopt it – so that diversification will be foregone

Statistical significance may not be an effective criterion. We explore two models

Model 1. one notch down from darkness only volatility levels predictable

Model 2. two notches down from darkness: Capital Asset Pricing Model


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Risk model that defines order  

Generalized MPT - de Jong * Clarke et al. (2013) derive the near closed-form solution 19

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Back-test Model 2bis on corporate bonds

Testbed – Region: Eurozone - Universe: Barclays Euro Corporate – Period: 2007 to 2016 - Monthly data frequency – Industries: Barclays level 3 sectors aggregated to ten groups

Test setup

– Portfolio rebalanced at regular intervals (no foresight) – Ex post performance over test period compared with standard benchmark Generalized MPT - de Jong 20

Consumer Discretionary

Consumer Staples

Utility & Energy

Finance_SENIOR

Finance_SUB

Insurance

Basic Industry

Captial Goods

Communication

IT

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Parity portfolio

Ten sectors contribute equally to overall portfolio risk (DTS) Within sectors firms (bond issuers) contribute equally

Generalized MPT - de Jong As of June 2016 21

0%

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Euro Corporate Bond Benchmark

Parity portfolio

Back-test: return

Generalized MPT - de Jong Performance calculations based on monthly total returns data provided by Barclays.

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Amundi Quant Research

Parity portfolio in line with benchmark and resilient during crises.

Underlying figures on next page

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Key figures

Generalized MPT - de Jong All figures are annual if not specified 23

benchmark Parity index Parity portfolio return volatility Sharpe ratio max drawdown TE # bonds # issuers turnover per month # weight

4.8% 4.1% 1.2 -6.3% - 1843 400 30 0.3%

5.4% 3.4% 1.6 -4.3% 1.64% 1843 400 30 -

5.4% 3.4% 1.6 -4.1% 0.11% (from target) < 233 < 127 10 1.6%

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Convex risk profile

Risk parity amortizes drawdowns

Generalized MPT - de Jong As of June 2016 24

-3%

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-4% -2% 0% 2% 4%

QQ plot market vs parity

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Induced quality bias The parity index has a structural bias towards low-indebted firms As measured by two debt ratios. Debt to assets Debt to equity


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Market indexParity index 0,0

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Market index

Parity index

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Induced size bias The parity index has a structural bias towards small firms as measured by two firm fundamentals, sales revenues and book value Sales Assets


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Market indexParity index

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Marketindex

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Conclusion

Incorporating uncertainty and consequently generalizing the Markowitz optimization problem makes sense. It corresponds to investment practice, with a list a portfolio construction methods with alternative philosophies, e.g. regret theory Incorporating uncertainty doesn’t suit active high-conviction investment.


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generalized modern portfolio theory · portfolio optimization in an uncertain world marielle de...

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