How to build a cross-asset value factor

Since the seminal paper “Value and Momentum Everywhere” by Asness et al. (2013), the investment community’s understanding of hedge-fund strategies has progressed enormously. What was previously seen as pure alpha suddenly became commoditised systematic beta, as most asset classes seem to exhibit the same double patterns. First, assets outperforming and underperforming in the past tend to continue doing so in the coming weeks, forming a momentum effect. Second, assets showing elevated (respectively low) prices in comparison with their historical trend tend to generate negative (respectively positive) returns, forming a value effect. Both effects correlate negatively.

If both effects are now well known, forming a value strategy remains a source of complexity for a variety of reasons. There is a lack of consensus within the financial community on how to gauge the valuation of an asset. Using financial metrics such as rates or dividend yields shows a lack of consistency across asset classes. In addition, within an asset class, common valuation metrics can strongly depend on sectors. For instance, price-to-book ratios make sense for a majority of sectors, but much less so for financials. Finally, some of these financial metrics rely on expectation variables, such as the price-earnings ratio: practitioners have a tendency to use analyst forecasts instead of past realised values, adding a final layer of complication. Asness et al. (2013) proposed an interesting solution to all of these issues: using asset prices (or futures) and dividing these prices by their value five years ago. By doing so, the value analyst obtains a mean-reverting ratio across all asset classes, at the cost of assuming that value trends last for about a 5-year period.

The link between ’long term’ past returns and valuation is well established and was first documented by DeBondt and Thaler (1985) on individual stocks. Similarly, Fama and French (1996) found that portfolios constructed using book-to-market measures are highly correlated with portfolios constructed with a 5-year return metric. Black (1986) posited the Value premium as the counterpart of the Momentum premium. The author postulated that market trends are created by relatively uninformed trades that push asset prices far from the unobserved fundamental value of an asset, until the spread becomes so wide that mean reversion will occur. Measuring valuation as long-term reversal thus appears highly appealing because it provides an approach free of economic models that is purely based on price data.

The question, however, remains about the choice of the econometric specification that will deliver such a metric. One of the several issues with the 5-year window is its ad hoc feature. A genuine value measure should produce mean-reverting and stationary signals: the 5-year window ratio happens to deliver some of that, but not in a controlled manner.

This paper¹ aims to build on the intuition that the easiest measure to gauge valuation is the price of assets, consistent with the economic theory. Instead of a lookback period of our own choosing, we propose to perform a classical trend-cycle decomposition of the log of asset prices. The ‘cycle’ part of this decomposition should resemble what Asness et al. (2013) attempted to exploit, but leverage a theoretically sound timeseries model.

Several methods could be used to extract such a value signal: non-parametric methods such as the Hodrick-Prescott or the Baxter-King filters would be natural candidates, yet they are unstable in estimating the decomposition on the extremities of samples due to edge effects. The right edge – meaning the most recent points in the sample –  is of an utmost interest to us, given that is where the investment signal will sit.

Recently, Hamilton (2018) proposed an appealing contender to this filtering approach, namely an econometric filter. That band-pass filter produces cycle components that are stationary by construction. Our experiment shows how this cycle element is also mean-reverting, with a half-life which is data-driven instead of being explicitly controlled by a model.

Therefore, our analysis explores using Hamilton’s trend/cycle decomposition in order to gauge if it can also serve to derive value measures that are employed later to build a cross-asset value factor using consistent metrics across all asset classes.

This article unfolds as follows: the next section presents the methodology we employ to build the value signal in a general context. An empirical application is considered in section 3 to show the benefits of the approach. Section 4 concludes with an overview.  

Building the valuation signal

Let  represent the price of a given financial asset (including e.g. the value of an index). Let   denote the log of the asset price. That variable is typically an integrated variable, exhibiting a stochastic trend or a persistence so near to random-walk that it is not stationary. The index  here typically refers to days, for we intend to use daily observations with potentially overlapping samples. Relying on quarterly observations could be another possibility but would mean using a lower number of observations.²

Hamilton (2018) proposes a method to filter a trend out of this timeseries by running the following regression:

with   a Gaussian disturbance that is identically distributed but not necessarily independent,  the value of  quarters before and  the maximum number of lags considered in the regression. Hamilton (2018) recommends 4 lags. Obviously, such a regression equation applies if is sampled at a quarterly frequency or not. The  coefficients are -valued parameters that can either be estimated via Ordinary Least Squares or Maximum Likelihood.

Let us denote the estimated value of , that is:

 is the ’trend’ part of this cycle/trend decomposition. The estimated value of ,  , is the cyclical component, from which we intend on deriving a valuation metric.

An interesting property of this valuation approach is that although the fitted residuals  should be non-integrated, they are still capable of displaying some form of persistence, similarly to most mispricing measures. Fitting an Autoregressive process to the estimated  will allow us to explicitly measure the persistence and half-life of the value anomaly. This offers an interesting by-product to Hamilton’s approach, coming alongside stationarity.

 

Building the cross-asset value factor

We now detail how we obtain the composition of the proposed cross-asset value factor building on Asness et al. (2013.). Let  be the estimated residuals for asset  at time  in a set of  assets  . As in our model,  can be interpreted as the deviation from the ‘long term’ value of the price series, a positive   means an expensive asset, while a negative one means a cheap asset. We thus propose to construct our valuation metrics as:

where denotes the volatility of the residuals. The negative sign yields a positive signal for cheap assets and a negative one for expensive assets, while the volatility rescaling is necessary to have comparable metrics across different assets and asset classes. Let   be the portfolio weight for asset  at time . It is obtained from the below formulaic expression:

  ,

where rank(.) is the increasing-rank function. The weights across assets within the investment universe made of the assets sum to zero, representing a Dollar-neutral long-short portfolio.  is a scaling factor such that the overall portfolio is scaled to one dollar long and one dollar short, as in Asness et al. (2013).

Let  be the return on asset  between  and . The factor portfolio performance is given by:

.

As noted in Asness et al. (2013), using such a ranking function instead of creating a simple spread portfolio leads to a more diversified exposure and less extreme weights. We note that the long-short setup we have adopted for literature-consistency reasons can easily be modified to obtain a directional value factor by using weights computed as:

,

with  now acting as a control variable for risk and the overall leverage that the factor can reach. In the next section we present empirical applications showcasing how this value factor construction can be used in different settings.

In this section, we propose different applications of the proposed methodology in order to gauge its ability to create a value strategy simply from the price or index level of financial assets.

 

A first illustration

To start with an illustrative application, we compare a classic measure of valuation – the dividend yield ratio – to our approach in the case of equities. The popularity of the dividend yield to assess the valuation of equities stems from two factors: first, its simple computation; second, the fact that it can be compared across most sectors and with long term yield for relative valuation assessment. It also has limitations, such as being difficult to apply to growth companies. Also, in the context of building a cross asset value factor, a similar kind of metric needs to be defined for assets such as commodities, which is not straightforward.

Figure 1 compares the price-to-dividend ratio of the S&P500 index and the valuation measure we introduced in section 2. The ratio has been obtained from 5-year rolling regressions with , therefore using four different quarters as recommended in Hamilton (2018). This analogy is not quantitative, merely a visual comparison between what is widely accepted as a valuation measure and what we propose in this paper. This comparison also echoes the 5-year return and book-to-market relationship (see Gerakos and Linnainmaa, 2018). As clearly shown in figure 1, both time series show a strong covariation. Indeed, this pure price-based measure explains 78% of the dividend yield variation through time.  

 

Figure 1. Price/dividend ratio of the MSCI World vs. Hamilton (2022)’s valuation signal

Source: Bloomberg, LOIM. For illustrative purposes only. As of January 2024.

 

Investigating the cross-asset value factor across samples

Next, we created four different datasets composed of liquid futures only, similarly to Moskowitz et al. (2012). These futures relate to four markets: bonds, equities, currencies and commodities and are listed in the Appendix. Our database run from 2 January 2000 to 31 January 2024, encompassing slightly more than 24 years of daily prices. Our value measure is computed using five years of rolling overlapping daily data and using four quarterly lags, that is .

Our empirical strategy unfolded as follows: first, we created value portfolios at a single asset class level in order to assess if our approach could generate a value factor within asset classes. Then we created cross-asset value portfolios by progressively combining asset classes, starting with bonds and equities, then adding commodities and finally currencies. In addition, we ran two kinds of factor compositions: one targeting an ex-ante volatility of 15% using an exponentially moving average with parameters .94 – equivalent to a 11-day half-life – , and another not targeting volatility, for which the 1 dollar long 1 dollar short legs drive the risk of the portfolio.

With these settings, we obtained the results shown in table 1 and the indexed performances shown in figure 2. From a single asset class perspective, each value factor sees a positive performance, whether in the volatility targeted version or not. Sharpe ratios fluctuate between .11 and 1.12 depending on the asset class and the type of implementation. The skewness of these value returns has a tendency to be positive: large positive returns are more frequent than large negative returns. This is consistent with what one can expect from a value factor:  unless they are value traps, value trades have a tendency to deliver small negative returns (given their short carry nature) and large positives when the value anomaly rapidly gets repriced.

Now turning to the cross-asset setup, the results are relatively similar: Sharpe ratios are in the region of .2 to .4, that is to say in line with the historical Sharpe ratio of standard risk premia (see Illmanen et al., 2014). These numbers go down in the case of a volatility targeted portfolio to .14 to .26. In this set-up too, skewness is positive in most cases (five out of six) – an expected feature for valuation strategies.

In summary, the proposed approach seems to deliver appealing results. We now turn to the statistical properties of our value factor and our valuation signals. In particular, we investigate the relationship between our factor and existing valuation and momentum strategies. We then present empirical test results for the non-stationarity of prices, the stationarity to the value signal and finally investigating its half-life, emphasising the sound econometric features of our methodology.

 

Table 1. Performance statistics of the single asset class and cross asset class value factors

Source: Bloomberg, LOIM. In the cross asset columns, B stands for bonds, E for equities, C commodities and F forex.  For illustrative purposes only. CAGR stands for compound annual growth rate. MaxDD stands for maximum drawdown.

 

Figure 2. Indexed performance across datasets

Source: Bloomberg, LOIM. For illustrative purposes only. As of January 2024.

 

Correlation to momentum and existing cross-asset value

To further explore the characteristics of our approach, we now present a correlation analysis between our value factor and (1) a momentum strategy and (2) the Asness et al. (2013) value factor. The momentum strategy is built according to the methodology presented in Moskowitz et al.  (2012) and uses the past 12 months of returns to measure trends across assets. The value factor from Asness and Pedersen relies on the ranked comparison between asset prices at time t and five years before.

Figure 3 summarises these findings. The factors obtained with our approach are negatively correlated with their momentum counterparts, as could be expected. These correlations range from -0.18 for equities to +0.64 for forex in the case of single asset datasets. In the case of cross-asset samples, that correlation is globally about -0.6. Hence, the correlations obtained with cross-asset universes are stronger than those with single asset classes.

The correlations between our value factors and Asness et al. (2012)’s are all positive. In the cross-asset case, they are also larger in scale than in the case of single asset classes.

Both empirical results are, therefore, consistent with intuition.

 

Figure 3. Correlation of the cross-asset factors with the momentum and Moskowitz value factor

Source: Bloomberg, LOIM. For illustrative purposes only.

 

Half-life and stationarity

One benefit of the proposed approach arises from its ties to econometrics. Hamilton (2018)’s approach is close in spirit to cointegration, as the price series are non-stationary variables while the residuals are stationary. Furthermore, this cyclical component should exhibit persistence that can be measured through an AR(1) model, once stationarity conditions have been validated. Such an estimate will provide us with information regarding the half-life of the value anomaly, which could be used to further fine tune the strategy.

To test the stationary of the value signal, given its suspected autocorrelation, an Augmented Dickey-Fuller test is appropriate. In a nutshell our tests all rejected the null hypothesis of integrated process, both in and out of sample. All of our value signals are therefore found to be stationary. We ran a similar test on Asness et al (2013)’s value measure and failed to reject the null hypothesis in all cases, too.

Second, once we ensured our residuals were stationary, we estimated for each of them an AR(1) model from which we derived a half-life. The results are presented in figure 4. The average half-life is 153 days, which is about seven months. The half-life of commodities (160 days), currencies (168 days) and equities (154 days) is marginally higher than the half-life for bonds (118 days). This is an advantage of our approach compared to other valuation methodologies.

 

Figure 4. Half-life of the value signal per asset class

Source: Bloomberg, LOIM. For illustrative purposes only.

 

Robustness checks

This final section presents a set of robustness checks for the value factor we propose.

First, in figure 5, we present the breakeven t-cost for each value factors, that is the average t-costs necessary to make the performance of each factor equal to zero. This breakeven t-cost ranges from 3 basis points  in the case of forex to 104 bps in the case of equities. Individual asset classes, therefore, show a large disparity in terms of t-costs, while cross-asset factors show a more stable level for these of around 10 bps. Given the liquidity of the instruments we use here, these t-costs numbers are higher than usual market conventions, validating the meaningfulness of such a value strategy.

 

Figure 5. Breakeven t-costs per type of investment universe

Source: Bloomberg, LOIM. For illustrative purposes only.

This paper proposes an econometric, model-free approach to building a cross-asset value factor. This simple strategy solves the difficulty facing analysts when comparing the valuation of different asset classes and proposes a unified way to compute such a value measure. 

Across single and multi-asset investment universes, our approach consistently yields a factor that (1) delivers a positive Sharpe ratio, (2) is negatively correlated to a momentum factor while (3) being positively correlated to other known measures of value. As a by-product, our signals are stationary in- and out-of-sample, therefore, enabling their half-life to be quantified explicitly. 
 

Asness, C. S., Moskowitz, T. J., & Pedersen, L. H. (2013). Value and momentum everywhere. The Journal of Finance, 68(3), 929-985.

Black, F, (1986). Noise. The Journal of Finance, 41(3), 529-543.

De Bondt, W. F., & Thaler, R. (1985). Does the stock market overreact?. The Journal of Finance, 40(3), 793-805.

Fama, E. F., & French, K. R. (1996). Multifactor explanations of asset pricing anomalies. The Journal of Finance, 51(1), 55-84.

Hamilton, J. D. (2018). Why you should never use the Hodrick-Prescott filter. Review of Economics and Statistics, 100(5), 831-843.

Ilmanen, A., Maloney, T., & Ross, A. (2014). Exploring macroeconomic sensitivities: How investments respond to different economic environments. The Journal of Portfolio Management, 40(3), 87-99.

Gerakos, J., & Linnainmaa, J. T. (2018). Decomposing value. The Review of Financial Studies, 31(5), 1825-1854.

Moskowitz, T. J., Ooi, Y. H., & Pedersen, L. H. (2012). Time series momentum. Journal of Financial Economics, 104(2), 228-250.

 

Appendix

Table 2. List of futures and forwards used in the research

Source: LOIM. For illustrative purposes only