Equity Portfolio Optimization with Factor Tilts

Abstract

A variety of techniques are applied to improve upon passive capitalization weighted equity market portfolios via intelligent integration of the four equity market factors introduced by Fama, French and Carhart. In consideration of structural and regulatory constraints imposed upon most investment practitioners, long-only factor tilt portfolios are substituted for the traditional long-short factors, which facilitates simple implementation of techniques using liquid Exchange Traded Funds. Factor tilt portfolios are assembled using equal weight, equal volatility weight, risk parity and minimum variance optimizations, and simple filters are introduced to reduce turnover and commensurate trading frictions. Finally, a simple but prospective balanced portfolio framework is proposed.

Introduction

There has been a cluster of papers recently about factor allocations as an addition to the traditional asset allocation framework. We believe the market cap, small cap, and value equity factors described by Fama and French (1992), as well as the momentum factor identified by Jagadeesh and Titman (1993) and eventually specified by Carhart (1997), represent mildly interesting diversifiers for portfolios in certain contexts. However, many institutions are increasingly interested in how to intelligently allocate to these factors to improve the risk-adjusted performance of their equity portfolios. This led us to explore optimal allocation methods.

Sharpe introduced the first equity factor, sometimes called ‘market beta’, in 1964 when he described the Capital Asset Pricing Model (CAPM). The CAPM model was meant to explain the degree to which market returns to stocks were a function of non-diversifiable risk; stocks with higher non-diversifiable risk were theoretically presumed to possess a possess a higher required rate of return in order to compensate for the extra risk of owning them. Fama and French extended the model in 1992 to describe the excess returns observed in small capitalization stocks, and ‘value’ stocks, the latter which was defined by stocks’ price to book ratio. 

In Fama’s and French’s initial tests it was observed that stocks with low market capitalizations delivered higher returns than their large capitalization counterparts. It was also observed that stocks with high book values relative to their market values offered higher returns than stocks with low book to market ratios. Further, these performance anomalies could not be completely explained by higher market betas associated with the factor portfolios.

In 1993 Jagadeesh and Titman published research on the momentum factor, which they defined as a stocks’ 12 month historical return, with a lag of one month. They found that stocks that delivered high returns over the past 12 months (with a 1 month lag) tended to outperform over the next month; this worked in reverse as well, such that stocks with poor performance in the recent past tended to underperform over the next month.

While the low volatility anomaly was noted in the literature as far back as 1977 (Miller, 1977), Miller first published exclusively on the low volatility anomaly in 2001. Additionally, Eric Falkenstein submitted his dissertation on the phenomenon in 1994, but it seems the academic community was not ready to accept a theoretical framework that effectively repudiated the CAPM at the time. As a result, his findings were never published. However, the seminal work on this factor seems to belong to Ang, Hodrick, Xing and Zhang (2006) with their examination of U.S. stock markets, while Baker and Haugen (2012) confirmed Ang et al.’s results in all major international equity markets earlier this year.

Robeco provided a superb summary of the observed magnitude of the above anomalies in a 2011 paper, Strategic Allocation to Premiums in the Equity Market, from which we copied the table below.

Source: Robeco

These are not inconsiderable premiums considering that they represent simple, persistent, systematic techniques that anyone can apply. This is especially true now that there are liquid ETFs that effectively capture these factors that anyone can use in portfolios:

• Small-cap stocks: IWM (cap weighted) and EWRS (equal weight)

• Value stocks: IVE (S&P 500), IWD (Russell 1000), PRF (FTSE RAFI)

• Momentum stocks: PDP (all-cap), HMTM (large-cap), SMTM (small-cap)

• Low Volatility: SPLV (volatility weighted), LVOL (volatility weighted)

In contrast to most academic factor investigations, this paper will focus on the same long-only versions of the factors described in the Robeco paper, as most practitioners encounter structural or regulatory barriers to shorting. Further, we would argue that short factors should be disaggregated from long factors, as long and short factors often 'work' at different times, and long-only factors display more persistence in practice.

This article will explore factors in a variety of asset allocation frameworks; we will introduce the factors individually and then see if we can create a better passive 'equity' basket, and extend this concept to create a better 'balanced' portfolio.

Factor Charts

To get us started, we have published below the charts and summary statistics for each of the major long-only factor tilts listed above. For all factors except the low volatility factor we sourced the data from Ken French's database. We used the S&P Low Volatility Index Total Return series for the low volatility anomaly. As the Low Vol data only goes back to 1991, the charts go back to 1992 in order to provide a year of 'priming' for the asset allocation overlays that we will introduce in later posts.

Large Cap Stocks

Source: Ken French database, 2012

Small Cap

 Source: Ken French database, 2012

Large Cap Value

Source: Ken French database, 2012

Large Cap Momentum

Source: Ken French database, 2012

Large Cap Low Volatility

Source: Standard and Poor’s, 2012

Equal Weight

The obvious next step in our exploration is to determine how well the factors work together in a portfolio. To answer, we first ran an equal weight factor tilt portfolio, rebalanced quarterly with data back to 1992:

Five Equity Factors, Equal Weight, Rabalanced Quarterly

Source: Ken French database, Standard & Poor’s, Yahoo Finance

Equal allocations to long-only factor tilts improve what is essentially a long-only beta portfolio (the Fama French Large Cap portfolio) by about 2.25% in terms of CAGR, and offer slight improvements to volatility, drawdowns, and the frequency of positive periods. Of course, these improvements come at the expense of extra trading.

Equal Volatility Weighting

We have explored volatility management techniques at length in many prior articles (see here, here, here, here, and here for a few examples). Equal volatility weighted portfolios are constructed to target an equal volatility contribution by all assets in the portfolio, such that the portfolio is always fully invested. This requires that volatility be estimated for each asset going into every rebalance period; we use the 60-day observed historical volatility for all estimates in this article to be consistent with our other articles, though there is no particular significance behind using this lookback horizon.

Five Equity Factors, Equal Volatility Weight, Rebalanced Quarterly

Source: Ken French database, Standard & Poor’s, Yahoo Finance

This technique doesn't offer much improvement over simple equal weighting, in all likelihood because the factor tilts are all so highly correlated during periods of market distress.

Position Size Volatility Limits

In this variation on the theme of volatility management, factors are granted equal volatility weighting in the portfolio up to a fixed volatility contribution limit, in this case 1% daily. When any asset exhibits volatility in excess of its 1% limit, exposure to that asset is scaled back in favour of cash in order to maintain our prescribed volatility limit. In this way, total portfolio exposure is less than 100% during periods where individual positions are highly volatile.

5 Equity Factors, Equal Volatility Budgets (1% daily), Rebalanced Quarterly

Source: Ken French database, Standard & Poor’s, Yahoo Finance

Setting position level volatility limits does improve risk-adjusted performance (see Sharpe ratio), in this case exclusively due to lower realized average portfolio volatility. More notably, drawdown is reduced by 40% because allocations are scaled back during the high volatility periods that are generally characterized by large drawdowns.

This technique does improve measurably with more active rebalancing, as evidenced by the results below based on a monthly rebalance schedule:

5 Equity Factors, Equal Weight Volatility Budgets (1% daily), Rebalanced Monthly

Source: Ken French database, Standard & Poor’s, Yahoo Finance

Observe: A realized Sharpe over 0.6 with no tactical overlay at all.

While some might object to the large number of trades, in this case the number of trades is deceiving because most trades are small and nuanced in reaction to small changes in volatility. Further, by setting range-based rebalancing targets of 25% (that is, when any allocation target changes by 25% or more relative to its current allocation in the portfolio, the whole portfolio is rebalanced to new target weights), we can reduce turnover by 70% with no loss in performance.

5 Equity Factors, Equal Weight Volatility Budgets (1% daily), Rebalanced Monthly, 25% Filter

Source: Ken French database, Standard & Poor’s, Yahoo Finance

Risk Parity

The risk parity concept merges precepts from equal volatility weighting at the individual asset level, and fixed volatility budgeting at the portfolio level, with the idea that lower volatility assets can be levered up to provide a similar return contribution to the portfolio as more risky assets while better balancing risk across the portfolio.

Risk parity requires a volatility budget to be set at the portfolio level; in this case, we maintain the same 1% daily target for portfolio volatility using 60 day realized volatility as the estimate for each asset, as well as for the portfolio in aggregate.

Note from the chart below that it is not possible to reach our volatility target without the use of leverage; the realized volatility of the un-levered version is just 13%. 

5 Equity Factors, Risk Parity (1% daily), Rebalanced Monthly, 25% Filter

Source: Ken French database, Standard & Poor’s, Yahoo Finance

It is a simple thing to use traditional margin to reach our volatility target with up to 100% leverage (or a maximum portfolio exposure of 200%) when required during periods of low aggregate portfolio volatility.

Obviously this less constrained version provides better performance, adding 2% to annualized returns, with commensurately higher volatility but, perhaps surprisingly, almost no incremental boost in drawdown. The following simulation also includes a cost of margin equal to .5% above the t-bill rate, which is excruciatingly onerous, but we like to be conservative.

5 Equity Factors, Risk Parity (1% daily), Rebalanced Monthly, 25% Filter, Max 200% Exposure

Source: Ken French database, Standard & Poor’s, Yahoo Finance

Minimum Variance

Minimum variance algorithms strive to create optimal portfolios using the equations described by Modern Portfolio Theory, but with the objective of minimizing total portfolio variance rather than maximizing portfolio Sharpe. In contrast with standard mean-variance optimization therefore, minimum variance optimization does not require or use any return estimates, focusing instead exclusively on volatility and the covariance matrix.

The average correlation between momentum and value tilts over the past 20 years is 0.85; between value and low volatility, it is 0.82; and between momentum and low volatility it is 0.75. As a result, there is an opportunity to leverage the diversification between factors explicitly, a process for which minimum variance optimization is well suited.

5 Equity Factors, Minimum Variance, Rebalanced Monthly, 25% Filter

Source: Ken French database, Standard & Poor’s, Yahoo Finance

As with virtually all optimization procedures we have analyzed over the years, the minimum variance optimization is improved by overlaying a portfolio level volatility target. In this case we will use the same 1% daily target as in our risk parity example above.

5 Equity Factors, Minimum Variance, Rebalanced Monthly, 25% Filter, Portfolio Target Volatility (1%), Max 100% Exposure

Source: Ken French database, Standard & Poor’s, Yahoo Finance

This seems to be quite a powerful combination. We have achieved a Sharpe ratio of 0.84 with a pure beta portfolio. Returns increase by almost 4% per year and drawdowns are reduced by 45% while volatility drops by 40%.

Adventurous beta seekers might wish to lever up the minimum variance portfolio at opportune times in order to actually achieve the target 1% daily volatility. If we set a maximum leverage factor of 100%, we achieve the following profile:

5 Equity Factors, Minimum Variance, Rebalanced Monthly, 25% Filter, Portfolio Target Volatility (1% daily), Max 200% Exposure

Source: Ken French database, Standard & Poor’s, Yahoo Finance

A Better Balanced Fund

As a parting gift, we ran the minimum variance portfolio as the equity portion of a risk parity version of a traditional balanced portfolio, with a target volatility of 7% annualized (equal to the realized volatility of the 10 year Treasury over the same period), rebalanced quarterly.

5 Equity Factors (Minimum Variance) and 10-Year Treasuries, Risk Parity (7% annualized), Max 100% Exposure

Source: Ken French database, Standard & Poor’s, Yahoo Finance

Notice that, due to the lower volatility of the minimum variance factor tilt equity portfolio, and a low structural correlation between the factor portfolio and Treasuries, the realized volatility of this balanced fund is just 5.4% despite our 7% target. Of equal importance, the maximum drawdown for this portfolio is just 11.65% over the past 20 years!

Many prospectus mutual funds can carry up to 125% exposure under their mandates. If we raise the maximum exposure to accommodate this limit, we can get closer to our 7% target, with commensurately juicier returns.

5 Equity Factors (Minimum Variance) and 10-Year Treasuries, Risk Parity (7% annualized), Max 125% Exposure

Source: Ken French database, Standard & Poor’s, Yahoo Finance

Conclusion

Long-only factor tilt portfolios are very accessible by average investors using highly liquid Exchange Traded Funds. Equity factors appear to provide some diversification benefits in a portfolio context, and algorithms that explicitly account for portfolio level volatility and factor covariance can provide a very substantial boost to absolute returns while reducing portfolio risk.

Indeed, investors might wish to explore these techniques as interesting complements to existing equity allocations, especially in the context of a diversified balanced portfolio.

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Histogram of Rolling 20-Year Real Total Returns to U.S. Stocks

The Permanent Portfolio Turns Japanese

Our last few articles (here and here) dealt with the Permanent Portfolio, a widely embraced static asset allocation concept proposed by Harry Browne in 1982. To review, the  simple Permanent Portfolio consists of equal weight allocations to cash (T-bills), Treasuries, stocks and gold to ward against the four major financial states of the world:

We can hear the chorus of protests about how cash is not a hedge against inflation. Tell it to the judge - by which we mean the only judge that counts - the evidence.

The following chart, taken from Ang's recent chapter on forecasting stock market returns, illustrates the correlation between inflation and the returns to cash and stocks over horizons from 1 month to 10 years. Note from the chart that equity returns in excess of cash (T-bill) yields are negatively correlated with inflation for at least 10 years after an inflation shock, while cash returns adapt quickly to changes in inflation.

Chart 1. Correlation of excess returns to cash and stocks with changes in inflation.

Source: Ang (2012)

The U.S. market has experienced at least three of the four states above since 1970. The 1970s was clearly a period of inflationary stagnation (stagflation), where commodities, gold and extremely short-duration bonds were the only sources of positive returns. This was followed by an intense and prolonged period of disinflationary growth from 1982 through 2000 which saw U.S. stocks compound at rates above 15% annualized for almost two decades. Long duration Treasury bonds also performed very well in this period as rates dropped from 15% in 1981 to 5% in 2001.

After 2000, aggressive central bank priming succeeded in creating a period of inflationary growth which ended in the Great Financial Crisis of 2008. This period was dominated by commodities, gold, and the markets of those emerging economies that benefited from high commodity prices.

We have no confidence whatsoever in defining the state of current markets, though we suspect we may experience a period of deflationary stagnation in the intermediate term before all is said and done.

Periods of deflationary stagnation are (so far) quite uncommon in the modern era, and especially among developed economies for which we have accurate and granular data. The one prominent standout is, of course, Japan which began its period of deflationary stagnation in 1990 after the coincident bursting of the Japanese banking, stock and property bubble.

Japan as a Deflationary Case Study

Deflation is a period of falling prices, and Japan offers plenty of examples of this phenomenon over the last 20 years or more. As seen in Chart 2 below, commercial real estate prices experienced the most profound bubble, rising by almost 4x from 1985 through 1990, but residential real estate also tripled in price over the same period. Over the subsequent 22 years, commercial property collapsed to half the value it went for in 1985, and residential real estate currently goes for about the same price it went for almost 30 years ago.

Chart 2. Japanese Real Estate Prices, 1985 - 2010

Source: japaninvestor.net

The story of the Japanese stock market is now very well known. Japanese stocks (Nikkei) dropped almost 80% from 1990 through their interim bottom in 2003, and then dropped further to their bottom in early 2009. Total returns to Japanese stocks over the period were -5.29% annualized, for a total return of -71% through June 29, 2012.

Chart 3. Nikkei total return, 1990 - 2012

Source: Data from Bloomberg

Despite the best efforts of the Bank of Japan to debase their currency (and re-inflate asset prices) through extensive quantitative easing over the past 20+ years, the Yen has appreciated strongly, moving from almost 160 yen to 1 U.S. dollar in 1990 to under 80 currently (i.e. one dollar can purchase half as many yen today as in 1990, so the yen has doubled in value vs. the dollar).

Chart 4. Number of Japanese yen to 1 U.S. dollar

Source: FRED database

The strength of the yen has complicated efforts to diversify into alternative global asset classes, which are mostly denominated in dollars. The following chart (Chart 5.) of gold in yen highlights this challenge for Japanese investors. While gold's strength since 2000 has swamped yen strength, investors endured a 57% drawdown over the first 10 years, and did not break even on their investment until 2005!

Chart 5. Gold priced in Japanese Yen

Source: Data from Bloomberg

The Permanent Portfolio Turns Japanese

Intuitively, given the poor performance of Japanese stocks; extremely low short term interest rates over much of the last 15 years, and; yen strength eating into dollar denominated assets like gold, the Japanese Permanent Portfolio was not nearly as fruitful as the U.S. version.

Note: The daily total return data for Japanese asset classes that we use in this analysis deviate somewhat from other similar studies of the Japanese Permanent Portfolio that we've seen. While we have done our best to source and, in the case of interest rate indices, transform the data appropriately, the absolute performance numbers of the strategy are, in our opinion, less important. Rather, we are much more interested in validating (or not) the risk management techniques we proposed in our earlier Permanent Portfolio Shakedown Part II, in the context of a Japan's very different market environment.

Chart 6. Simple Permanent Portfolio Japan

Source: Bloomberg

Clearly the Japanese Permanent Portfolio (PP) delivered much better total returns over the past 20 years than the Japanese stock market on its own, and indeed performed better than the traditional Japanese 50/50 stock bond portfolio that we explored in great detail in Rebalancing Japan. 

Volatility Management

In examining the impact of volatility management on the Japanese Permanent Portfolio, we will follow the same investigative trajectory that we followed in our original Permanent Portfolio Shakedown Part II for the next series of case studies. Note that, while the original Japanese PP exhibited an average volatility of about 8% annualized, we will stick with 7% portfolio volatility targets in order to be consistent with the original U.S. based articles.

As a reminder, we target portfolio volatility by measuring the volatility of the portfolio at each rebalance period, and adjust total portfolio exposure lower if observed volatility is too high by substituting t-bills. The first case maintains equal portfolio exposure to all four assets, but reduces pro-rata exposure in response to expanding portfolio volatility.

Chart 7. Simple Permanent Portfolio Japan, 7% target volatility, rebalanced monthly, 1992 - 2012

Source: Bloomberg

In our experience, the primary utility of managing portfolio volatility to a target is a reduction in drawdowns and increase in Sharpe ratio, with a very small sacrifice to absolute returns. The Japanese PP outcome is consistent with this observation, as Sharpe ratio increases slightly while drawdown drops substantially to 13% from 22%.

Risk Parity

The next step is to migrate to an equal volatility allocation instead of the equal capital allocation example in Chart 7. 

Adherents to the risk parity philosophy aim to create portfolios where each asset class contributes an equal amount of volatility to the portfolio rather than an equal amount of capital. Chart 5. approximates a risk parity approach using stocks, gold and Treasuries, with a 7% risk target. The T-bill allocation is dynamic where cash expands and contracts in the portfolio in order to keep the portfolio volatility close to our 7% target. 

Chart 8. Simple Permanent Portfolio Japan, Risk Parity, 7% target volatility, 1992 - 2012

Source: Bloomberg

The Risk Parity overlay delivers a substantial uptick in performance and portfolio stability, as evidenced by the more consistently upward sloping equity line. Absolute portfolio performance increases substantially, as does the Sharpe ratio, which is double the Sharpe of the original portfolio. Higher returns are accompanied by a lower drawdown as well, under 12%. Risk Parity also improves the percentage of positive rolling 12-month periods, which we take very seriously as a measure of consistency. 

The large improvement from Risk Parity is clearly the result of the fact that risk parity approaches tend to greatly overweight bond positions, given the low volatility of bonds relative to stocks and gold. Japanese bonds were the best performing of the four asset classes over the period.

Tactical Approaches

To repeat from our original article:

The potential problem, as we see it, with any static asset allocation, including the permanent portfolio's permanent equal weight allocations to stocks, bonds, gold and cash, is that sometimes everything is expensive all at once, and returns to all asset classes have the potential to be low or negative in tandem. The current environment may represent one of these periods, where certainly bonds and cash are more expensive than they have ever been, with yields on Treasuries lower than at any other time in the last 220 years (source: Bank of America). Cash yields essentially zero. We think stocks are expensive as well (see here andhere), and gold is at best a wildcard, having rallied by 500% or more from its lows in 2001. 

If we are right, and the permanent portfolio is vulnerable to synchronized losses, then it makes sense to explore some tactical or dynamic overlays to help avoid investing in asset classes in sustained downtrends. 

Mebane Faber is credited with bringing moving averages to the masses with his Quantitative Approach to Tactical Asset Allocation whitepaper in 2005. In it, he describes an approach that applies a 10-month moving average to basket of 5 asset classes: stocks, Treasuries, commodities, REITs and international stocks. While there is nothing magical about the 10-month moving average, this approach is ubiquitously cited elsewhere, and in our testing we observed no material difference with other moving averages. The following simulations apply monthly rebalancing.

Chart 9. Simple Permanent Portfolio Japan, 10-Month MA, 1992 - 2012

Source: Bloomberg

The 10-month moving average overlay does improve absoute returns and volatility, but at the expense of a larger drawdown, which is higher even than the 22% drawdown experienced by the original Japanese PP.

The last approach marries Faber's tactical overlay with a portfolio risk target of 7%. This approach delivered the best risk-adjusted performance for the U.S. model, so we were quite interested in how it would work in Japan.

Chart 10. Simple Permanent Portfolio Japan, 10-Month MA, 7% Target Volatility, 1992 - 2012

Source: Bloomberg

The target volatility technique performs similar magic in Japan as it did for the U.S. PP, though the absolute numbers are still not nearly as good. Returns of 5% annualized are 40% higher than the original Japanese PP portfolio, and risk-adjusted returns are more than double, with a Sharpe of 0.77 compared with the original portfolio's .37. Drawdown drops to a tolerable 13.7% from 22%.

Conclusion

The Japanese experience has clearly been very different from the U.S. experience over the past 20 years, as Japan has endured a persistent period of deflationary stagnation while the U.S. has so far experienced growth of both the deflationary and inflationary variety. This has had a profound impact on Japanese stocks and bonds, as well as the yen, which in turn impacted returns to Japanese investors from foreign assets like gold.

That said, the traditionally implemented Permanent Portfolio would have served Japanese investors much better than stocks on their own, or even a balanced stock/bond portfolio over the past 20 years, with double the Sharpe ratio, higher returns, and much lower drawdowns. 

Consistent with our findings in the U.S., techniques that manage the volatility of the individual portfolio holdings and/or target a total portfolio volatility delivered similar or better returns, and for the most part much lower drawdowns. The Risk Parity and Tactical approach with target volatility delivered the highest, most consistent returns with the lowest drawdowns.

Overall, passive Japanese investors were better served by the Permanent Portfolio than other traditional asset allocation models over the past 20 years, but even the best tactical and risk management overlays could only produce returns of about 5% per year. U.S. and other international investors might wish to consider this potential outcome in the context of retirement or institutional funding needs.