Optimal Mean-Reversion Strategies

Scenario Description

Consider a financial asset whose price, Xt, follows a mean-reverting stochastic process. A common model for mean reversion is the Ornstein-Uhlenbeck (OU) process, defined by the stochastic differential equation (SDE):

Objective

The trader aims to maximize the expected cumulative profit from trading this asset over a finite horizon, subject to transaction costs. The trader’s control is the rate of buying or selling the asset, denoted by ut, at time t.

Hamilton-Jacobi-Bellman (HJB) Equation

To find the optimal trading strategy, we frame this as a stochastic control problem. The value function,V(t,Xt), represents the maximum expected profit from time t to the end of the trading horizon, given the current price level Xt. The HJB equation for this problem is:

where C(ut) represents the cost of trading, which can depend on the rate of trading ut. The term ut(Xt−C(ut)) captures the profit from trading, adjusted for transaction costs.

Solution Approach

Boundary and Terminal Conditions: Specify terminal conditions for V(T,XT), where T is the end of the trading horizon, and boundary conditions for V(t,Xt) based on the problem setup.

Solve the HJB Equation: The solution involves finding the function V(t,Xt) and the control policy ut∗ that maximizes the HJB equation. This typically requires numerical methods, especially for complex cost functions or when closed-form solutions are not feasible.

Interpret the Optimal Policy: The optimal control ut∗ derived from solving the HJB equation indicates the optimal rate of trading (buying or selling) at any time t and price level Xt, considering the mean-reverting nature of the price and the impact of transaction costs.

Key Insights

No-Trade Zones: The presence of transaction costs often leads to the creation of no-trade zones in the optimal policy, where the expected benefit from trading does not outweigh the costs.

Mean-Reversion Exploitation: The optimal strategy exploits mean reversion by adjusting the trading rate based on the deviation of the current price from the mean level, μ.

The Lipton & Lopez de Marcos Paper

“A Closed-form Solution for Optimal Mean-reverting Trading Strategies” contributes significantly to the literature on optimal trading strategies for mean-reverting instruments. The paper focuses on deriving optimal trading strategies that maximize the Sharpe Ratio by solving the Hamilton-Jacobi-Bellman equation associated with the problem. It outlines a method that relies on solving a Fredholm integral equation to determine the optimal trading levels, taking into account transaction costs.

The paper begins by discussing the relevance of mean-reverting trading strategies across various markets, particularly emphasizing the energy market’s suitability for such strategies. It acknowledges the practical challenges and limitations of previous analytical results, mainly asymptotic and applicable to perpetual trading strategies, and highlights the novelty of addressing finite maturity strategies.

A key contribution of the paper is the development of an explicit formula for the Sharpe ratio in terms of stop-loss and take-profit levels, which allows traders to deploy tactical execution algorithms for optimal strategy performance under different market regimes. The methodology involves calibrating the Ornstein-Uhlenbeck process to market prices and optimizing the Sharpe ratio with respect to the defined levels. The authors present numerical results that illustrate the Sharpe ratio as a function of these levels for various parameters and discuss the implications of their findings for liquidity providers and statistical arbitrage traders.

The paper also reviews traditional approaches to similar problems, including the use of renewal theory and linear transaction costs, and compares these with its analytical framework. It concludes that its method provides a valuable tool for liquidity providers and traders to optimally execute their strategies, with practical applications beyond theoretical interest.

The authors use the path integral method to understand the behavior of their solutions, providing an alternative treatment to linear transaction costs that results in a determination of critical boundaries for trading. This approach is distinct in its use of direct solving methods for the Fredholm equation and adjusting the trading thresholds through a numerical method until a matching condition is met.

This research not only advances the understanding of optimal trading rules for mean-reverting strategies but also offers practical guidance for traders and liquidity providers in implementing these strategies effectively.

September 23, 2018

Money Management – the Good, the Bad and the Ugly

The infatuation of futures traders with the subject of money management, (more aptly described as position sizing), is something of a puzzle for someone coming from a background in equities or forex. The idea is, simply, that one can improve one’s trading performance through the judicious use of leverage, increasing the size of a position at times and reducing it at others.

Perhaps the most widely known money management technique is the Martingale, where the size of the trade is doubled after every loss. It is easy to show mathematically that such a system must win eventually, provided that the bet size is unlimited. It is also easy to show that, small as it may be, there is a non-zero probability of a long string of losing trades that would bankrupt the trader before he was able to recoup all his losses. Still, the prospect offered by the Martingale strategy is an alluring one: the idea that, no matter what the underlying trading strategy, one can eventually be certain of winning. And so a virtual cottage industry of money management techniques has evolved.

One of the reasons why the money management concept is prevalent in the futures industry compared to, say, equities or f/x, is simply the trading mechanics. Doubling the size of a position in futures might mean trading an extra contract, or perhaps a ten-lot; doing the same in equities might mean scaling into and out of multiple positions comprising many thousands of shares. The execution risk and cost of trying to implement a money management program in equities has historically made the idea infeasible, although that is less true today, given the decline in commission rates and the arrival of smart execution algorithms. Still, money management is a concept that originated in the futures industry and will forever be associated with it.

Van Tharp on Position Sizing
I was recently recommended to read Van Tharp’s Definitive Guide to Position Sizing, which devotes several hundred pages to the subject. Leaving aside the great number of pages of simulation results, there is much to commend it. Van Tharp does a pretty good job of demolishing highly speculative and very dangerous “money management” techniques such as the Kelly Criterion and Ralph Vince’s Optimal f, which make unrealistic assumptions of one kind or another, such as, for example, that there are only two outcomes, rather than the multiple possibilities from a trading strategy, or considering only the outcome of a single trade, rather than a succession of trades (whose outcome may not be independent). Just as with the Martingale, these techniques will often produce unacceptably large drawdowns. In fact, as I have pointed out elsewhere, the use of leverage which many so-called money management techniques actually calls for increases in the risk in the original strategy, often reducing its risk-adjusted return.

As Van Tharp points out, mathematical literacy is not one of the strongest suits of futures traders in general and the money management strategy industry reflects that.

But Van Tharp himself is not immune to misunderstanding mathematical concepts. His central idea is that trading systems should be rated according to its System Quality Number, which he defines as:

SQN = (Expectancy / standard deviation of R) * square root of Number of Trades

R is a central concept of Van Tharp’s methodology, which he defines as how much you will lose per unit of your investment. So, for example, if you buy a stock today for $50 and plan to sell it if it reaches $40, your R is $10. In cases like this you have a clear definition of your R. But what if you don’t? Van Tharp sensibly recommends you use your average loss as an estimate of R.

Expectancy, as Van Tharp defines it, is just the expected profit per trade of the system expressed as a multiple of R. So

SQN = ( (Average Profit per Trade / R) / standard deviation (Average Profit per Trade / R) * square root of Number of Trades

Squaring both sides of the equation, we get:

SQN^2 = ( (Average Profit per Trade )^2 / R^2) / Variance (Average Profit per Trade / R) ) * Number of Trades

The R-squared terms cancel out, leaving the following:

SQN^2 = ((Average Profit per Trade ) ^ 2 / Variance (Average Profit per Trade)) * Number of Trades

Hence,

SQN = (Average Profit per Trade / Standard Deviation (Average Profit per Trade)) * square root of Number of Trades

There is another name by which this measure is more widely known in the investment community: the Sharpe Ratio.

On the “Optimal” Position Sizing Strategy
In my view, Van Tharp’s singular achievement has been to spawn a cottage industry out of restating a fact already widely known amongst investment professionals, i.e. that one should seek out strategies that maximize the Sharpe Ratio.

Not that seeking to maximize the Sharpe Ratio is a bad idea – far from it. But then Van Tharp goes on to suggest that one should consider only strategies with a SQN of greater than 2, ideally much higher (he mentions SQNs of the order of 3-6).

But 95% or more of investable strategies have a Sharpe Ratio less than 2. In fact, in the world of investment management a Sharpe Ratio of 1.5 is considered very good. Barely a handful of funds have demonstrated an ability to maintain a Sharpe Ratio of greater than 2 over a sustained period (Jim Simon’s Renaissance Technologies being one of them). Only in the world of high frequency trading do strategies typically attain the kind of Sharpe Ratio (or SQN) that Van Tharp advocates. So while Van Tharp’s intentions are well meaning, his prescription is unrealistic, for the majority of investors.

One recommendation of Van Tharp’s that should be taken seriously is that there is no single “best” money management strategy that suits every investor. Instead, position sizing should be evolved through simulation, taking into account each trader or investor’s preferences in terms of risk and return. This makes complete sense: a trader looking to make 100% a year and willing to risk 50% of his capital is going to adopt a very different approach to money management, compared to an investor who will be satisfied with a 10% return, provided his risk of losing money is very low. Again, however, there is nothing new here: the problem of optimal allocation based on an investor’s aversion to risk has been thoroughly addressed in the literature for at least the last 50 years.

What about the Equity Curve Money Management strategy I discussed in a previous post? Isn’t that a kind of Martingale? Yes and no. Indeed, the strategy does require us to increase the original investment after a period of loss. But it does so, not after a single losing trade, but after a series of losses from which the strategy is showing evidence of recovering. Furthermore, the ECMM system caps the add-on investment at some specified level, rather than continuing to double the trade size after every loss, as in a Martingale.

But the critical difference between the ECMM and the standard Martingale lies in the assumptions about dependency in the returns of the underlying strategy. In the traditional Martingale, profits and losses are independent from one trade to the next. By contrast, scenarios where ECMM is likely to prove effective are ones where there is dependency in the underlying strategy, more specifically, negative autocorrelation in returns over some horizon. What that means is that periods of losses or lower returns tend to be followed by periods of gains, or higher returns. In other words, ECMM works when the underlying strategy has a tendency towards mean reversion.

CONCLUSION
The futures industry has spawned a myriad of position sizing strategies. Many are impractical, or positively dangerous, leading as they do to significant risk of catastrophic loss. Generally, investors should seek out strategies with higher Sharpe Ratios, and use money management techniques only to improve the risk-adjusted return. But there is no universal money management methodology that will suit every investor. Instead, money management should be conditioned on each individual investors risk preferences.

September 17, 2018

Creating Robust, High-Performance Stock Portfolios

Summary

In this article, I am going to look at how stock portfolios should be constructed that best meet investment objectives.

The theoretical and practical difficulties of the widely adopted Modern Portfolio Theory approach limits its usefulness as a tool for portfolio construction.

MPT portfolios typically produce disappointing out-of-sample results, and will often underperform a naïve, equally-weighted stock portfolio.

The article introduces the concept of robust portfolio construction, which leads to portfolios that have more stable performance characteristics, including during periods of high volatility or market corrections.

The benefits of this approach include risk-adjusted returns that substantially exceed those of traditional portfolios, together with much lower drawdowns and correlations.

Market Timing

In an earlier article, I discussed how investors can enhance returns through the strategic use of market timing techniques to step out of the market during difficult conditions.

To emphasize the impact of market timing on investment returns, I have summarized in the chart below how a $1,000 investment would have grown over the 25-year period from July 1990 to June 2014. In the baseline scenario, we assume that the investment is made in a fund that tracks the S&P 500 Index and held for the full term. In the second scenario, we look at the outcome if the investor had stepped out of the market during the market downturns from March 2000 to Feb 2003 and from Jan 2007 to Feb 2009.

Fig. 1: Value of $1,000 Jul 1990-Jun 2014 – S&P 500 Index with and without Market Timing

Source: Yahoo Finance, 2014

After 25 years, the investment under the second scenario would have been worth approximately 5x as much as in the baseline scenario. Of course, perfect market timing is unlikely to be achievable. The best an investor can do is employ some kind of market timing indicator, such as the CBOE VIX index, as described in the previous article.

Equity Long Short

For those who mistrust the concept of market timing or who wish to remain invested in the market over the long term regardless of short-term market conditions, an alternative exists that bears consideration.

The equity long/short strategy, in which the investor buys certain stocks while shorting others, is a concept that reputedly originated with Alfred Jones in the 1940s. A long/short equity portfolio seeks to reduce overall market exposure, while profiting from stock gains in the long positions and price declines in the short positions. The idea is that the investor’s equity investments in the long positions are hedged to some degree against a general market decline by the offsetting short positions, from which the concept of a hedge fund is derived.

There are many variations on the long/short theme. Where the long and short positions are individually matched, the strategy is referred to as pairs trading. When the portfolio composition is structured in a way that the overall market exposure on the short side equates to that of the long side, leaving zero net market exposure, the strategy is typically referred to as market-neutral. Variations include dollar-neutral, where the dollar value of aggregate long and short positions is equalized, and beta-neutral, where the portfolio is structured in a way to yield a net zero overall market beta. But in the great majority of cases, such as, for example, in 130/30 strategies, there is a residual net long exposure to the market. Consequently, for the most part, long/short strategies are correlated with the overall market, but they will tend to outperform long-only strategies during market declines, while underperforming during strong market rallies.

Modern Portfolio Theory

Theories abound as to the best way to construct equity portfolios. The most commonly used approach is mean-variance optimization, a concept developed in the 1950s by Harry Markovitz (other more modern approaches include, for example, factor models or CVAR – conditional value at risk).

If we plot the risk and expected return of the assets under consideration, in what is referred to as the investment opportunity set, we see a characteristic “bullet” shape, the upper edge of which is called the efficient frontier (See Fig. 2). Assets on the efficient frontier produce the highest level of expected return for a given level of risk. Equivalently, a portfolio lying on the efficient frontier represents the combination offering the best possible expected return for a given risk level. It transpires that for efficient portfolios, the weights to be assigned to individual assets depend only on the volatilities of the individual assets and the correlation between them, and can be determined by simple linear programming. The inclusion of a riskless asset (such as US T-bills) allows us to construct the Capital Market Line, shown in the figure, which is tangent to the efficient frontier at the portfolio with the highest Sharpe Ratio, which is consequently referred to as the Tangency or Optimal Portfolio.

Fig. 2: Investment Opportunity Set and Efficient Frontier

Source: Wikipedia

Paradise Lost

Elegant as it is, MPT is open to challenge as a suitable basis for constructing investment portfolios. The Sharpe Ratio is often an inadequate representation of the investor’s utility function – for example, a strategy may have a high Sharpe Ratio but suffer from large drawdowns, behavior unlikely to be appealing to many investors. Of greater concern is the assumption of constant correlation between the assets in the investment universe. In fact, expected returns, volatilities and correlations fluctuate all the time, inducing changes in the shape of the efficient frontier and the composition of the optimal portfolio, which may be substantial. Not only is the composition of the optimal portfolio unstable, during times of financial crisis, all assets tend to become positively correlated and move down together. The supposed diversification benefit of MPT breaks down when it is needed the most.

I want to spend a little time on these critical issues before introducing a new methodology for portfolio construction. I will illustrate the procedure using a limited investment universe consisting of the dozen stocks listed below. This is, of course, a much more restricted universe than would typically apply in practice, but it does provide a span of different sectors and industries sufficient for our purpose.

Adobe Systems Inc. (NASDAQ:ADBE)

E. I. du Pont de Nemours and Company (NYSE:DD)

The Dow Chemical Company (NYSE:DOW)

Emerson Electric Co. (NYSE:EMR)

Honeywell International Inc. (NYSE:HON)

International Business Machines Corporation (NYSE:IBM)

McDonald’s Corp. (NYSE:MCD)

Oracle Corporation (NYSE:ORCL)

The Procter & Gamble Company (NYSE:PG)

Texas Instruments Inc. (NASDAQ:TXN)

Wells Fargo & Company (NYSE:WFC)

Williams Companies, Inc. (NYSE:WMB)

If we follow the procedure outlined in the preceding section, we arrive at the following depiction of the investment opportunity set and efficient frontier. Note that in the following, the S&P 500 index is used as a proxy for the market portfolio, while the equal portfolio designates a portfolio comprising identical dollar amounts invested in each stock.

Fig. 3: Investment Opportunity Set and Efficient Frontiers for the 12-Stock Portfolio

Source: MathWorks Inc.

As you can see, we have derived not one, but two, efficient frontiers. The first is the frontier for standard portfolios that are constrained to be long-only and without use of leverage. The second represents the frontier for 130/30 long-short portfolios, in which we permit leverage of 30%, so that long positions are overweight by a total of 30%, offset by a 30% short allocation. It turns out that in either case, the optimal portfolio yields an average annual return of around 13%, with annual volatility of around 17%, producing a Sharpe ratio of 0.75.

So far so good, but here, of course, we are estimating the optimal portfolio using the entire data set. In practice, we will need to estimate the optimal portfolio with available historical data and rebalance on a regular basis over time. Let’s assume that, starting in July 1995 and rolling forward month by month, we use the latest 60 months of available data to construct the efficient frontier and optimal portfolio.

Fig. 4 below illustrates the enormous variation in the shape of the efficient frontier over time, and in the risk/return profile of the optimal long-only portfolio, shown as the white line traversing the frontier surface.

Fig. 4: Time Evolution of the Efficient Frontier and Optimal Portfolio

Source: MathWorks Inc.

We see in Fig. 5 that the outcome of using the MPT approach is hardly very encouraging: the optimal long-only portfolio underperforms the market both in aggregate, over the entire back-test period, and consistently during the period from 2000-2011. The results for a 130/30 portfolio (not shown) are hardly an improvement, as the use of leverage, if anything, has a tendency to exacerbate portfolio turnover and other undesirable performance characteristics.

Fig. 5: Value of $1,000: Optimal Portfolio vs. S&P 500 Index, Jul 1995-Jun 2014

Source: MathWorks Inc.

Part of the reason for the poor performance of the optimal portfolio lies with the assumption of constant correlation. In fact, as illustrated in Fig 6, the average correlation between the monthly returns in the twelve stocks in our universe has fluctuated very substantially over the last twenty years, ranging from a low of just over 20% to a high in excess of 50%, with an annual volatility of 38%. Clearly, the assumption of constant correlation is unsafe.

Fig. 6: Average Correlation, Jul 1995-Jun 2014

Source: Yahoo Finance, 2014

To add to the difficulties, researchers have found that the out of sample performance of the naïve portfolio, in which equal dollar value is invested in each stock, is typically no worse than that of portfolios constructed using techniques such as mean-variance optimization or factor models¹. Due to the difficulty of accurately estimating asset correlations, it would require an estimation window of 3,000 months of historical data for a portfolio of only 25 assets to produce a mean-variance strategy that would outperform an equally-weighted portfolio!

Without piling on the agony with additional concerns about the MPT methodology, such as the assumption of Normality in asset returns, it is already clear that there are significant shortcomings to the approach.

Robust Portfolios

Many attempts have been made by its supporters to address the practical limitations of MPT, while other researchers have focused attention on alternative methodologies. In practice, however, it remains a challenge for any of the common techniques in use today to produce portfolios that will consistently outperform a naïve, equally-weighted portfolio. The approach discussed here represents a radical departure from standard methods, both in its objectives and in its methodology. I will discuss the general procedure without getting into all of the details, some of which are proprietary.

Let us revert for a moment to the initial discussion of market timing at the start of this article. We showed that if only we could time the market and step aside during major market declines, the outcome for the market portfolio would be a five-fold improvement in performance over the period from Aug 1990 to Jun 2014. In one sense, it would not take “much” to produce a substantial uplift in performance: what is needed is simply the ability to avoid the most extreme market drawdowns. We can identify this as a feature of what might be described as a “robust” portfolio, i.e. one with a limited tendency to participate in major market corrections. Focusing now on the general concept of “robustness”, what other characteristics might we want our ideal portfolio to have? We might consider, for example, some or all of the following:

Ratio of total returns to max drawdown
Percentage of profitable days
Number of drawdowns and average length of drawdowns
Sortino ratio
Correlation to perfect equity curve
Profit factor (ratio of gross profit to gross loss)
Variability in average correlation

The list is by no means exhaustive or prescriptive. But these factors relate to a common theme, which we may characterize as robustness. A portfolio or strategy constructed with these criteria in mind is likely to have a very different composition and set of performance characteristics when compared to an optimal portfolio in the mean-variance sense. Furthermore, it is by no means the case that the robustness of such a portfolio must come at the expense of lower expected returns. As we have seen, a portfolio which only produces a zero return during major market declines has far higher overall returns than one that is correlated with the market. If the portfolio can be constructed in a way that will tend to produce positive returns during market downturns, so much the better. In other words, what we are describing is a long/short portfolio whose correlation to the market adapts to market conditions, having a tendency to become negative when markets are in decline and positive when they are rising.

The first insight of this approach, then, is that we use different criteria, often multi-dimensional, to define optimality. These criteria have a tendency to produce portfolios that behave robustly, performing well during market declines or periods of high volatility, as well as during market rallies.

The second insight from the robust portfolio approach arises from the observation that, ideally, we would want to see much greater consistency in the correlations between assets in the investment universe than is typically the case for stock portfolios. Now, stock correlations are what they are and fluctuate as they will – there is not much one can do about that, at least directly. One solution might be to include other assets, such as commodities, into the mix, in an attempt to reduce and stabilize average asset correlations. But not only is this often undesirable, it is unnecessary – one can, in fact, reduce average correlation levels, while remaining entirely with the equity universe.

The solution to this apparent paradox is simple, albeit entirely at odds with the MPT approach. Instead of creating our portfolio on the basis of combining a group of stocks in some weighting scheme, we are first going to develop investment strategies for each of the stocks individually, before combining them into a portfolio. The strategies for each stock are designed according to several of the criteria of robustness we identified earlier. When combined together, these individual strategies will merge to become a portfolio, with allocations to each stock, just as in any other weighting scheme. And as with any other portfolio, we can set limits on allocations, turnover, or leverage. In this case, however, the resulting portfolio will, like its constituent strategies, display many of the desired characteristics of robustness.

Let’s take a look at how this works out for our sample universe of twelve stocks. I will begin by focusing on the results from the two critical periods from March 2000 to Feb 2003 and from Jan 2007 to Feb 2009.

Fig. 7: Robust Equity Long/Short vs. S&P 500 index, Mar 2000-Feb 2003

Source: Yahoo Finance, 2014

Fig. 8: Robust Equity Long/Short vs. S&P 500 index, Jan 2007-Feb 2009

Source: Yahoo Finance, 2014

As might be imagined, given its performance during these critical periods, the overall performance of the robust portfolio dominates the market portfolio over the entire period from 1990:

Fig. 9: Robust Equity Long/Short vs. S&P 500 index, Aug 1990-Jun 2014

Source: Yahoo Finance, 2014

It is worth pointing out that even during benign market conditions, such as those prevailing from, say, the end of 2012, the robust portfolio outperforms the market portfolio on a risk-adjusted basis: while the returns are comparable for both, around 36% in total, the annual volatility of the robust portfolio is only 4.8%, compared to 8.4% for the S&P 500 index.

A significant benefit to the robust portfolio derives from the much lower and more stable average correlation between its constituent strategies, compared to the average correlation between the individual equities, which we considered before. As can be seen from Fig. 10, average correlation levels remained under 10% for the robust portfolio, compared to around 25% for the mean-variance optimal portfolio until 2008, rising only to a maximum value of around 15% in 2009. Thereafter, average correlation levels have drifted consistently in the downward direction, and are now very close to zero. Overall, average correlations are much more stable for the constituents in the robust portfolio than for those in the traditional portfolio: annual volatility at 12.2% is less than one-third of the annual volatility of the latter, 38.1%.

Fig. 10: Average Correlations Robust Equity Long/Short vs. S&P 500 index, Aug 1990-Jun 2014

Source: Yahoo Finance, 2014

The much lower average correlation levels mean that it is possible to construct fully diversified portfolios in the robust portfolio framework with fewer assets than in the traditional MPT framework. Put another way, a robust portfolio with a small number of assets will typically produce higher returns with lower volatility than a traditional, optimal portfolio (in the MPT sense) constructed using the same underlying assets.

In terms of correlation of the portfolio itself, we find that over the period from Aug 1990 to June 2014, the robust portfolio exhibits close to zero net correlation with the market. However, the summary result disguises yet another important advantage of the robust portfolio. From the scatterplot shown in Fig. 11, we can see that, in fact, the robust portfolio has a tendency to adjust its correlation according to market conditions. When the market is moving positively, the robust portfolio tends to have a positive correlation, while during periods when the market is in decline, the robust portfolio tends to have a negative correlation.

Fig. 11: Correlation between Robust Equity Long/Short vs. S&P 500 index, Aug 1990-Jun 2014

Source: Yahoo Finance, 2014

Optimal Robust Portfolios

The robust portfolio referenced in our discussion hitherto is a naïve portfolio with equal dollar allocations to each individual equity strategy. What happens if we apply MPT to the equity strategy constituents and construct an “optimal” (in the mean-variance sense) robust portfolio?

The results from this procedure are summarized in Fig. 12, which shows the evolution of the efficient frontier, traversed by the risk/return path of the optimal robust portfolio. Both show considerable variability. In fact, however, both the frontier and optimal portfolio are far more stable than their equivalents for the traditional MPT strategy.

Fig. 12: Time Evolution of the Efficient Frontier and Optimal Robust Portfolio

Source: MathWorks Inc.

Fig. 13 compares the performance of the naïve robust portfolio and optimal robust portfolio. The optimal portfolio does demonstrate a small, material improvement in risk-adjusted returns, but at the cost of an increase in the maximum drawdown. It is an open question as to whether the modest improvement in performance is sufficient to justify the additional portfolio turnover and commensurate trading cost and operational risk. The incremental benefits are relatively minor, because the equally weighted portfolio is already well-diversified due to the low average correlation in its constituent strategies.

Fig. 13: Naïve vs. Optimal Robust Portfolio Performance Aug 1990-Jun 2014

Source: Yahoo Finance, 2014

Conclusion

The limitations of MPT in terms of its underlying assumptions and implementation challenges limits its usefulness as a practical tool for investors looking to construct equity portfolios that will enable them to achieve their investment objectives. Rather than seeking to optimize risk-adjusted returns in the traditional way, investors may be better served by identifying important characteristics of strategy robustness and using these to create strategies for individual equities that perform robustly across a wide range of market conditions. By constructing portfolios composed of such strategies, rather than using the underlying equities, investors may achieve higher, more stable returns under a broad range of market conditions, including periods of high volatility or market drawdown.

¹ Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?, Victor DeMiguel, Lorenzo Garlappi and Raman Uppal, The Review of Financial Studies, Vol. 22, Issue 5, 2007.