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AQuantitativeApproachtoFaber'sTacticalAssetAllocation

ARTICLE·MARCH2012

DOI:10.2139/ssrn.1476225

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StefanoMarmi

ScuolaNormaleSuperiorediPisa

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ClaudioPacati

UniversitàdegliStudidiSiena

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WistonAdrianRisso

UniversityoftheRepublic,Uruguay

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RobertoRenò

UniversitàdegliStudidiSiena

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Retrievedon:23July2015

Electronic copy available at: http://ssrn.com/abstract=1476225

A quantitative approach

to Faber’s tactical asset allocation

Stefano Marmi∗ Claudio Pacati† Roberto Reno‡

Wiston Adrian Risso§

March 9, 2012

Abstract

Routinely, practictioners and academics alike propose the use of trading

strategies with an alleged improvement on the risk-return relation, tipically

entailing a considerably higher return for the given level of risk. A very

popular example is “A quantitative approach to tactical asset allocation” by

the fund manager M. Faber, a real hit in the SSRN online library. Is this paper

a counterexample to market efficiency? We reject this conclusion, showing

that a lot of caution should be used in this field, and we indicate a series of

bootstrapping experiments which can be easily implemented to evaluate the

performance of trading strategies.

∗Scuola Normale Superiore, Pisa (Italy), s.marmi@sns.it†Dipartimento di Economia Politica e Statistica, Universita di Siena (Italy), pacati@unisi.it‡Dipartimento di Economia Politica e Statistica, Universita di Siena (Italy), reno@unisi.it§Dipartimento di Economia Politica e Statistica, Universita di Siena (Italy), risso@unisi.it

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Electronic copy available at: http://ssrn.com/abstract=1476225

1 Introduction

In Faber (2007), a very simple trend-following strategy has been proposed and

“shown” to outperform the market. The simplicity of the tactical asset allocation

suggested and the remarkable performance of the strategy (no losing years from 1972

to 2007) have contributed to make it an outstanding editorial success. On January

20, 2012, the paper was still the most downloaded paper on SSRN (www.ssrn.com,

one of the largest online libraries of working papers) in the last 12 months and the

second most downloaded paper in total (surpassed only by Solove, 2008), reaching

the noteworthy figure of 84, 255 downloads. While using the number of downloads

to measure the scientific relevance of a paper is certainly questionable under many

respects, it is at least interesting to notice that the fourth paper in the list is Fama

(1998), authored by one of the recognized fathers of the doctrine of market efficiency.

In the literature, active portfolio strategies and “technical analysis” received

considerable attention (see, e.g., Lo and MacKinlay, 1988, Lo et al., 2000 and,

for a very recent discussion, Scholz and Walther, 2011 and the references therein).

Momentum strategies have been shown to have substantial power in explaining the

cross-section of stock returns (Jegadeesh and Titman, 1993), whereas the evidence

in favour of trend-following is much more controversial: early studies on the subject

suggested that moving averages trading rules had some predictive abilities in both

conditional means and variances (Brock et al., 1992) but Sullivan et al. (1999) and

Pukthuanthong et al. (2007) demonstrated that their forecasting performance had

disappeared in the last decade of the 20th century. More recently a simple theoretical

model for the profit and losses of a trend following strategy has also been proposed

and analytically solved (Potters and Bouchaud, 2011). Nonetheless, the overall

feelings of the academic community about these approaches tipically ranges from

indifference to scorn.

In this paper, we prefer to take an agnostic perspective on the matter. We start

from the consideration that, as the massive downloads of Faber’s paper show, techni-

2

cal analysis tend to receive more attention than what would be implied by academic

circles, and this attention tends to peak in periods of financial turmoil. However,

we also recognize that assessing whether a trading rule significantly outperforms the

market is not a difficult task, see e.g. Sullivan et al. (1999) for a thorough analysis of

this issue. In this paper we just cast this problem in a simple and theoretically sound

fashion, and we illustrate our technique on Faber’s trading strategy, also showing

that it is more likely due to data snooping than to actual predictive power.

The trading rule proposed by Faber (2007) for asset allocation among risky

assets and a risk-free asset is as simple as that: if the monthly closing price of

the risky asset is higher than its past ten-month average, buy the risky asset; else

buy the riskless asset. This “timing” model applied to each asset of a diversified

portfolio including United States stocks, the Morgan Stanley Capital International

EAFE Index (MSCI EAFE), Goldman Sachs Commodity Index (GSCI), National

Association of Real Estate Investment Trusts Index (NAREIT), and United States

government 10-year Treasury bonds leads to rather impressive results: Better risk-

adjusted performance than a reference equally-weighted yearly rebalanced passive

portfolio and, especially, a drastic reduction of the maximum drawdown. Published

by the author just before the 2007–2009 financial crises1 its growing popularity is

certainly due to the remarkable result of over thirty-five consecutive years of positive

performance.

What can we conclude from this result? Is the market risk-return relation vio-

lated? And, more far reaching: is the market efficient, at least in its weak form? Is

there non-linear dependence in stock market returns which can be exploited?

The answer to these questions may seem trivial or not, depending on the point

of view (academic or practioneer) adopted. Anyhow we think that Faber’s result

deserves some analysis. The point we would like to make here is that no empirical

results can be derived on the historical performance of financial markets if we do

1The electronic version of the paper on SSRN was updated on February 2009 to include out-of-sample results for the years 2006, 2007, and 2008.

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not estimate the statistical significance of the results. Thus, a possible correct and

well-posed question is: is Faber’s strategy violating the risk-return trade-off in a

statistically significant way?

The bibliography on this central question is certainly longer than this paper. We

try to keep it simple and we perform some bootstrap experiments. We analyze the

behaviour of each asset class considered, taken singularly and also as a component

of the diversified portfolio obtained by investing in each asset or in the risk-free asset

according to Faber’s simple moving average trading rule.

2 Bootstrapping experiments

We replicate Faber’s strategy on monthly S&P 500 returns and three-months Trea-

sury bill, using the latter as riskless asset, from January 1950 to June 2009, for a

total of 713 months.

Summary statistics are shown in Table 1. We confirm that Faber’s strategy

has a substantially higher mean and a lower standard deviation than the S&P 500

portfolio. The risk reduction is even more appealing if measured in terms of the

maximum drawdown of the portfolio since its inception: Faber’s strategy does not

avoid a significant drawdown (in correspondence of October 1987) but it avoids

the painful drawdowns of most prolonged bear markets. This is also graphically

illustrated in Figure 2.1, where the efficient frontier for the two assets is compared

with the mean and standard deviation obtained with Faber’s strategy. This is well

above the risk-return efficient frontier.

The question now is: how robust is the evidence of superior risk-adjusted per-

formance when examined in the light of the statistical variablity of returns? We

perform three very simple bootstrapping experiments, with increasing level of so-

phistication.

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Asset Mean Standard Maximumdeviation Drawdown

S&P 500 6.71% 14.60% −50%3M T-bills 4.99% 0.89%Faber 8.26% 10.54% −22%

Table 1: Summary Statistics (log returns – annualized figures)

2.1 Simple bootstrap

In the first experiment we replicate the S&P 500 returns by drawing (with replace-

ment) 500 simulated time series from the truly observed returns. Then, we compute

the mean and standard deviation of the portfolio obtained with Faber’s strategy in

each simulation. The result is displayed in Figure 2.2: the cloud of bootstrapped

Faber portfolios is obviously symmetrical with respect to the efficient frontier, as

it should be since deviations of returns from their mean must occur with the same

probability in both directions, coherently with the random walk assumption used

in the model. However, Faber’s original portfolio lies significantly to the left of the

cloud, indicating lower variance for the strategy when applied to actual historical

returns. The reason for this could be that this experiment assumes normal i.i.d. dis-

tributed returns for the S&P 500, an assumption well known to be counterfactual.

2.2 Historical simulation

In order to incorporate the most salient feature of the data, that is heteroskedas-

ticity, we apply the well known historical simulation approach (Barone-Adesi et al.,

1999, 1998; Boudoukh et al., 1998). To this end, we now bootstrap standardized

residuals form a GARCH(1,1) process (Bollerslev, 1986). A simple GARCH(1,1)

model is indeed able to remove from standardized residuals the AR and ARCH ef-

fects displayed by the original series, as can be formally tested with, for example, a

Ljung-Box-Pierce’s Q-test or an Engle’s ARCH test. Results, shown in Figure 2.3,

confirm the above result, with the Faber original portfolio now being well inside the

5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

0.09Efficient Frontier and Faber Portfolio

Risk(Standard Deviation)

Exp

ecte

d R

etur

n

Efficient FrontierFaber

Figure 2.1: Efficient frontier and Faber portfolio applied to S&P500-Tbills.

simulated cloud. It is interesting to remark that, with respect to the first experiment

(simple bootstrap), the means of the simulated Faber portfolios remain more or less

the same, whereas their variance becomes more dispersed. This effect is clearly due

to the simulated returns heteroskedasticity, a crucial feature of the data which is

now included in the simulation model.

2.3 Bivariate historical simulation

The first two experiments where performed by bootstrapping the S&P 500 returns

and leaving the T-bill rates unchanged. Even if we estimate an unsignificant negative

correlation of -2.67% (between S&P returns and T-bill increments), one could argue

that the correlation structure might matter in certain periods. Thus, we perform

a third experiment in which we estimate a GARCH(1,1) also on the T-bill rate

increments. Then we bootstrapped the pairs of standardized residuals of the two

series, as prescribed by the bivariate version of the historical simulation methodology.

The result is shown in Figure 2.4 and confirms the previous findings, with the

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−0.05

0

0.05

0.1

0.15Normal Bootsrap

Risk(Standard Deviation)

Exp

ecte

d R

etur

n

Efficient FrontierFaberReplicated Faber portfolios

Figure 2.2: Efficient frontier, Faber original portfolio and Faber bootstrapped port-folios.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−0.05

0

0.05

0.1

0.15Bootsrap by Historical Simulation

Risk(Standard Deviation)

Exp

ecte

d R

etur

n

Efficient FrontierFaberReplicated Faber Portfolios

Figure 2.3: Efficient frontier, Faber original portfolio and bootstrapped Faber port-folios from an historical simulation (2 points outside the axes range removed forvisual improvement).

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−0.05

0

0.05

0.1

0.15Bootsrap by Bivariate Historical Simulation

Risk(Standard Deviation)

Exp

ecte

d R

etur

n

Efficient FrontierFaberReplicated Faber Portfolios

Figure 2.4: Efficient frontier, Faber original portfolio and bootstrapped Faber port-folios from a bivariate historical simulation (4 points outside the axes range removedfor visual improvement).

original Faber portfolio significantly inside the simulated cloud. These findings are

confirmed by the comparison of the maximum drawdown of the simulated S&P500

with the timing strategy shown in Figure 2.5

2.4 Multivariate analysis

We can repeat the very same analysis on a set of multiple assets, as suggested in

Faber (2007), and we use the very same assets used in his analysis: S&P500, EAFE,

GSCI, NAREIT, TBOND, 3M T-bills. We use monthly returns from January 1973

to June 2009; summary statistics are reported in Table 2. The window for the

moving-average is again equal to 10 months. Figure 2.6 shows the starting point.

The efficient frontier is, as it should be, above the single assets, and the Faber

portfolio is, also in the multivariate case, well above the efficient frontier. Again, by

looking at Figure 2.6 alone, one may be tempted to conclude that Faber’s strategy

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−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Scatter plot

S&P500 Max Draw Down

Fab

er P

ortfo

lio M

ax D

raw

Dow

n

Replicated PortfoliosActual Portfolio

Figure 2.5: Maximum drawdowns of simulated S&P 500 and Faber portfolios.

Asset Mean Standarddeviation

S&P 500 5.77% 15.73%EAFE 7.53% 15.00%GSCI 9.05% 20.49%NAREIT 7.85% 18.21%TBOND 7.89% 8.89%3M T-bills 6.00% 0.94%Faber 10.36% 6.77%

Table 2: Multivariate Analysis Summary Statistics (log returns – annualized figures)

provides a substantially super-efficient approach to asset allocation. This is however

jumping to the conclusion, and forgetting the usual adagio that past-performance is

no guarantee of future results.

In order to be able to assess to what extent the past can be representative of the

future when the uncertainty and the variability of risk and return are taken into ac-

count we then perform the very same experiment of the univariate bootstrap. We fit

a GARCH(1,1) model for each of the series, and sample the residuals as suggested in

Barone-Adesi et al. (1999). Figure 2.7 shows the “cloud” of obtained random port-

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.05

0.06

0.07

0.08

0.09

0.1

0.11Efficient Frontier and Faber Portfolio

Risk(Standard Deviation)

Exp

ecte

d R

etur

n

Efficient FrontierFaberAssets

Figure 2.6: Efficient frontier, Faber portfolio and the assets used in the multivariateanalysis.

folios. The boostrapped Faber portfolios form a sort of thickened efficient frontier2

and, again, the Faber portfolio is inside the cloud, showing that its overperformance

is not statistically significant. This is further shown in the exercise in the following

section.

2.5 Economic significance

We can evaluate the significance of the success of the Faber portfolio through an

economic metric. We assume that the utility function of the investor is quadratic

(thus of the IARA form) and given by:

u(x) = x−1

2γx2, 0 ≤ x ≤

1

γ,

2Note also the high level of symmetry of the cloud with respect to the frontier

10

0.05 0.1 0.15 0.2 0.25−0.05

0

0.05

0.1

0.15

0.2

0.25

Bootstrap by Multivariate Historical Simulation

Risk(Standard Deviation)

Exp

ecte

d R

etur

n

Efficient FrontierFaberReplicated Faber Portfolios

Figure 2.7: Efficient frontier, Faber portfolio and bootstrapped Faber portfoliosfrom a multivariate historical simulation (4 points outside the axes range removedfor visual improvement).

where γ > 0 is a constant coefficient of risk aversion. Quadratic utility insures

that the mean-variance approach used in this paper is exact. Table 3 reports, for

different values of γ, the percentage of bootstrapped portfolios (in the multivariate

case) which have a higher utility of the Faber portfolio. Such a percentage can

be interpreted as the p-value of the test that the Faber portfolio outperforms the

efficient frontier, and it is never less than 5%. The value γ = 0 corresponds to a risk-

neutral investor, and provides the higher p-value. We consider the cases 0 ≤ γ ≤ 5.

Notice that an agent with a risk aversion coefficient γ > 4 has a very high risk

aversion: e.g. she does not accept an investment with expected return 10% and risk

(standard deviation) 20%, that is she would never invest in the stock market.

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γ p-value0.0 14.2%1.0 12.4%2.0 11.4%3.0 9.8%4.0 8.1%5.0 7.3%

Table 3: Percentage of portfolios which have a higher utility of the Faber portfolioas a function of the coefficient of risk aversion

3 Conclusions

Our results on the timing strategy used in Faber’s A Quantitative Approach to

Tactical Asset Allocation applied to a set of portfolio indeces suggest that the his-

torical outperformance of the strategy is, actually, not incompatible with the statis-

tical variability of the historical returns. In our statistical analysis, which we per-

formed with bootstrapping techniques, we just implemented a very simple model,

the GARCH(1,1). Multivariate GARCH should not be considered as an attempt

to provide a precise description of markets returns, which is still a fertile ongoing

research issue, but only to describe parsimoniously some of their salient features.

It could however well be that Faber’s results are suggestive of an alternative

hypotheses on the behavior of market returns, such as dynamic nonlinearities or the

fact that market crashes are more likely during bear markets than vice versa. These

hypothesis are intriguing and constitute a possible research agenda. Our paper just

shows that a lot of caution should be used in evaluating trading strategies, and

suggests a simple way to test them.

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