Portfolio Optimization Under Uncertainty

Portfolio Optimization Under Uncertainty

Guest LectureAdam Butler, CFA CAIA

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THE GOLDEN RULE OF INVESTMENT MANAGEMENT

Risk is the probability of not achieving financial objectives.

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For most investors, risk is defined as the probability of not meeting financial objectives.

Investing should have the exclusive objective of minimizing this risk.

What is the primary risk for most investors?

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The smooth geometric growth curve is a myth.

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Investing is a stochastic process; it is all about probabilities.

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The probability of any investment outcome is a function of expected return, and volatility around that expectation.

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Given an expected return and volatility we can quantify a range of outcomes at any investment horizon.

Source: Shiller, FRED (2013)

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STRUCTURAL DIVERSIFICATIONPortfolios must be robust to many possible market regimes.

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Traditional stock/bond portfolios are very sensitive to market regime.

Source: Deutsche Bank

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As a result, investors are vulnerable to an alarming range of outcomes, even over long horizons.

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Source: Shiller, FRED (2013)

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How can we make portfolios resilient to a wider range of regimes?

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• Stocks react favorably to accelerating economic growth and decelerating inflation.

• Treasuries respond favorably to decelerating growth and inflation

• Commodities respond favorably to accelerating inflation.• Gold responds well to some kinds of inflation, and to the

actions that authorities take to battle deflation.• Etc.

Financial theory offers clues about how assets will react in different environments.

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Portfolios with assets that thrive in each major regime are ‘structurally diversified’.

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One possible structurally diversified portfolio.

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RISK BASED OPTIMIZATIONCombining structural diversification with dynamic portfolio estimates.

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U.S. Stocks – VTI European Stocks - VGK

Japanese Stocks – EWJ Emerging Market Stocks - EEM

U.S. REITs - ICF International REITs - RWX

Commodities - DBC Gold - GLD

Intermediate Treasuries – IEF Long Treasuries - TLT

A simple structurally diversified universe for investigation.

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Equal weight resembles a traditional policy portfolio.

40% Equities20% Real Estate20% Alternatives20% Fixed Income

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Results: Equal Weight, Rebalanced Monthly

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Data source: Bloomberg

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Results: Equal Weight, Rebalanced Monthly

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• Assets included in the portfolio have wildly different ambient volatilities.

• Asset volatilities change profoundly over time.• Asset correlations change dramatically over time.

• As a result, asset risk contributions are highly unstable.

The simple policy portfolio framework has some challenges.

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Asset class volatilities are wildly unstable.

Ranges of Asset Class Volatility© Gestaltu


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• Examples of potential dynamic optimizations:– Naïve risk parity– Robust risk parity– Mean-variance optimization

• The following examples use short-term historical realized volatility and covariance as inputs for dynamic portfolio optimization.– Volatility estimate = 60 day historical observed volatility– Covariance estimate = 250 day historical observed covariance

Dynamic Asset Allocation applies dynamic parameter estimates to re-optimize portfolios at each rebalance period.

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In an equal weight portfolio, the lunatics run the asylum.Proportion of rolling 60-day historical volatility

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If we can estimate volatility, we can use these estimates to scale weights by inverse volatility: naïve risk parity.

Portfolio weights scaled by 1/rolling 60-day historical volatility

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Results: Naïve Risk Parity, Rebalanced Monthly

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Results: Naïve Risk Parity, Rebalanced Monthly


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Naïve risk parity assumes all assets have similar returns, and are similarly correlated. Is this a reasonable assumption?

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An asset contributes risk to a portfolio in proportion to its volatility AND its correlation to the portfolio itself.

37% Volatility Reduction

Negative MRC

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Robust risk parity seeks portfolio weights which equalize marginal risk contributions.

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Results: Robust Risk Parity (Equal Risk Contribution), Rebalanced Monthly

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Results: Robust Risk Parity (Equal Risk Contribution), Rebalanced Monthly


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MEAN VARIANCE OPTIMIZATIONMaximizing portfolio Sharpe ratio.

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Asset returns are even more unstable than volatility and covariance.

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Returns are more forecastable in the very short term, and the very long term. Not so much in the middle.

High frequency arms race

Momentum sweet spot

Value (long-term mean-reversion)


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Is it helpful to use long-term average return estimates with dynamic covariance estimates?

*Source: JPM Long Term Capital Market Return Assumptions, December 2013

Long-Term Returns*

DBC 3.8%

EEM 9.0%

EWJ 7.8%

GLD 4.3%

ICF 6.8%

IEF 4.3%

RWX 6.0%

TLT 3.3%

VGK 7.8%

VTI 7.5%

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Results: Long-term average returns with dynamic covariance estimates.

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Results: Mean variance using long-term average returns with dynamic covariance estimates.


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This intuition is sound because returns are empirically and theoretically proportional to risk.

Source: BridgewaterData source: Bloomberg

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• Mean variance optimization is implemented by maximizing the Sharpe Ratio.

• Maximum Diversification (Choueifaty, 2008) is mean-variance optimization where E(ri) = E(σi).

Are historical averages the only source of return estimates?


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Volatility is not the only measure of risk. And it’s worthwhile considering other simple methods like rank.

VolatilityDownside

Semivariance Drawdown Rank

DBC 12.5% 16.0% 60.3% 1

EEM 13.1% 19.4% 69.9% 5

EWJ 14.4% 19.5% 58.9% 5

GLD 9.0% 11.9% 38.8% 1

ICF 10.2% 16.6% 77.6% 4

IEF 4.2% 5.5% 11.4% 2

RWX 8.3% 13.0% 73.8% 4

TLT 6.9% 9.8% 26.6% 3

VGK 11.4% 16.7% 63.3% 5

VTI 10.1% 14.1% 55.5% 5


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Results: Heuristic mean-variance optimization with alternative return estimates.

max.drawdown

rank

downside.semi

volatility

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Results: Heuristic mean-variance optimization with alternative return estimates.


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REVISITING THE GOLDEN RULE

Did we achieve our goal of minimizing the risk of not achieving financial objectives?

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Thoughtful optimization can materially reduce the probability of not achieving financial objectives.

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Questions?

Thank you very much.

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Adam Butler416.572.5477

[email protected]

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Portfolio Optimization Under Uncertainty

Economy & Finance

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nave risk

alternative

equal risk

dynamic covariance

robust risk

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