Equity Portfolio Optimization with Factor Tilts

In Asset Allocation, Correlation, Factors, Institutional, Minimum Variance, Optimization, Risk Parity on


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.
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 hereherehere, 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
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.