Risk Horizon and Expected Market Returns

Georges Hubner∗† Thomas Lejeune ‡

October 2012

Abstract

The paper proposes an equilibrium asset pricing model that accounts of the incompleteinformation on returns distribution and investors’ preferences. As the model defines anintuitive risk measure (risk horizon) which relates to a time horizon, the horizon of treasurysecurities can be exploited to calibrate the parameters of the model. As a by-product, time-varying estimates of the expected return on the market portfolio are extracted and theirrelevance for stock returns predictability is tested.

JEL Classification: G11; G12; C14

∗Deloitte Chaired Professor of Portfolio Management and Performance, HEC Management School - Uni-versity of Liege, Belgium; Associate Professor, School of Economics and Business, Maastricht University, theNetherlands; Founder & Chief Scientific Officer, Gambit Financial Solutions Ltd, Belgium. Correspondingaddress. University of Liege, HEC Management School, Rue Louvrex 14 - N1, B-4000 Liege, Belgium. Phone:(+32) 42327428. E-mail: g.hubner@ulg.ac.be†This research has been supported by the Dauphine-Amundi Chair in Asset Management. We wish to

thank Jan Annaert, Marie Briere and Michael Rockinger for helpful comments on previous partial versionsof the paper, participants to the 2012 annual workshop organized by the Amundi Asset Management Chairof Paris-Dauphine University, as well as seminar participants at HEC Montreal. The authors are grateful toDorothee Honhon for her invaluable contribution to an earlier version of this work. Financial support fromDeloitte (Belgium and Luxembourg) is gratefully acknowledged. All errors remaining are ours.‡FRS-FNRS Research Fellow, HEC Management School – University of Liege. E-mail:tlejeune@ulg.ac.be

1

1 Introduction

Asset pricing models provide insights into how the cross-sections of expected returns are

determined at equilibrium. Along the lines of the Capital Asset Pricing Model (CAPM)

introduced by Sharpe (1964), Lintner (1965) and Mossin (1966), individual security or port-

folio expected returns are often defined as a linear function to the expected return on a risky

fund, interpreted as the market portfolio. In the multiple use of traditional asset pricing

models, the expected market return becomes a key input in capital budgeting decisions, as-

set allocation and wealth projection. Consequently, the application of asset pricing formulas

for solving portfolio and corporate problems requires clean parameter values. Most model-

streat the expected market return as exogenous, and no clear guidelines are provided for its

estimation in practice.

In spite of the criticisms raised by Merton (1980), who suggests an exploratory method,

the estimation of expected returns based on the realized ones still remains the standard

(Mayfield, 2004). Realized returns are yet far from a perfect proxy as they are likely to

introduce large measurement error (Blume and Friend, 1973; Sharpe, 1978), to be influenced

by information surprises (Elton, 1999), to yield implausible implications for risk-aversion

(Mehra and Prescott, 1985), and to be biased due to potential learning effects (Lewellen and

Shanken, 2002). Another approach uses forward-looking estimates from analysts’ expecta-

tion measures or forecasts (Brav et al., 2005; Welch, 2000; Claus and Thomas, 2001). While

this approach seems more realistic due to its forward-looking aspect, it still relies upon the

assumption that analysts’ forecasts are representative of market-wide expectations, and do

not make use of market prices information.

Besides these reservations on the relevance of realized returns or forward-looking measures

for expected returns, these estimation approaches also lack of theoretical foundations. Our

paper overcomes these limits by tackling the issue of the expected market return estimation

inside an original asset pricing framework. The asset pricing model is based on an intuitive

risk measure, namely the notion of “risk horizon”, with reference to the time horizon required

for the convergence of a security’s mean return towards its expectation. Using standard re-

2

sults in the non-parametric statistical literature to characterize the risk horizon measure,

we relax many restrictive assumptions about return distributions or utility functions found

in traditional asset pricing models. The framework allows us to derive CAPM-like pricing

relationships. The model’s implications for market equilibrium provide the basis for the de-

termination of expected market portfolio returns. The originality of our approach lies in the

explicit inclusion of the term structure of interest rates within the equilibrium framework. As

the risk measure refers to a notion of time horizon, it can be matched with the risk horizon of

treasury securities whose expected returns are modelled by the yield curve. The framework

makes it then possible to use information contained in the term structure of interest rates

to calibrate the model parameters. Furthermore, through an arbitrage argument, an equi-

librium link between risk and return of Treasuries and the market portfolio can be exploited

and estimates of the expected return on the market portfolio can also be extracted.

This framework has the original advantage of providing an intuitive link between the

term structure of interest rates, the expected market portfolio and market-wide preferences

for asymmetric and fat-tail risks embedded in the risk horizon measure. Such a link is de-

sirable for several reasons. First, it is grounded on the theoretical basis of an asset pricing

framework that exploits the realistic view that entire return distribution and investors prefer-

ences are hardly observable, and only partial observation of moments of orders 2, 3 and 4 are

observed. To the extent that higher order moments are priced at equilibrium (Chunhachinda

et al., 1997; Harvey and Siddique, 2000; Fang and Lai, 1997; Dittmar, 2002), market percep-

tion of “moment risks” seems relevant to the estimation of expected market returns. Second,

term structure factors are shown to be good predictors of expected stock returns (Camp-

bell, 1987; Lettau and Wachter, 2009). Finally, if the market-wide attitudes towards risk is

a leading indicator of market sentiment, the ability to model the time-varying behavior of

their determinants will enable us to test new predictors of systemic market risks and assess

their relevance for pricing equity and interest-rate related securities.

In the empirical part of the paper, we propose applications of the risk horizon framework,

and use U.S. market data to calibrate the risk horizon parameters and the expected return

on the market portfolio. We find plausible parameters estimates with interesting cyclical

3

patterns in the time series of the expected return. The empirical relevance of these estimates

is examined with tests of statistical and economic predictive ability for stock excess returns.

The results provide some evidence on the added value of the parameter in out-of-sample

forecasts and asset allocation with respect to popular predictors found in the literature (see

a.o. Lettau and Ludvigson, 2001; Rapach and Wohar, 2006; Goyal and Welch, 2008).

The paper is structured as follows: Section 2 describes the model setup. Build on an

arbitrage argument, a system of equilibrium equations is derived in section 3, and relates

the market portfolio to the term structure of interest rates. The fourth section section

is dedicated to empirical applications, with parameters calibration on U.S. data and the

analysis of the predictive performance of the endogenously derived expected market return.

Section 5 concludes the article.

2 The Model

In this section, we describe the theoretical setup. We start with the set of assumptions re-

quired to construct the model around the risk horizon measure. A characterization of the

measure is then detailed with reference to standard results in the non-parametric statistical

literature.

2.1 Assumptions

Assume a set Ω of N + 1 risky assets indexed by i, and T periods, indexed by t. The first

set of assumptions that surrounds the model is identical to the CAPM framework: homoge-

nous assessment of returns distributions, lending and borrowing at a common continuously

compounded rate1, no transaction costs, all investors are price-takers and all securities are

infinitely divisible.

1The framework does not necessary require that this rate is riskless. Instead, a “least risky asset” (LRA),that is the asset whose horizon is lowest of all existing asset, can be used. Therefore, the use of risk horizon asrisk measure provide an interesting definition of what is traditionally meant as the risk-free rate, and allowsit to be stochastic.

4

The second set of assumptions differs from traditional framework, and weakens the strong

assumptions usually made on return distributions and utility functions. In the risk horizon

framework, they are replaced by the following assumptions.

Assumption 1: Periodic continuously compounded returns of each asset i for period t,

denoted Rit, are i.i.d. with unknown distribution. Only centered moments of orders 2 (µ2),

3 (µ3) and 4 (µ4) of the unconditional distribution of returns of any asset or portfolio i exist

and are known, and are denoted Vi, Si and Ki respectively.

This assumption is based on the realistic view that investors hardly observe the entire

probability distribution of asset returns, but might instead extract the information on the

riskiness of distribution with the use of measures of unconditional moments representing

volatility, asymmetry and fat-tailness. In this regard, Assumption 1 is therefore weaker than

traditional restrictions on returns distribution, as it only requires the existence2 and knowl-

edge of unconditional moments up to order 4. From period returns moments, it is easy to

derive the moments of the mean of a sequence of n drawings of these returns, Rn

i = 1n

∑nt=1 Rit.

They are equal to: V ni = Vi

n, Sni = Si

n2 , Kni = Ki

n3 +3(n−1)V 2

i

n3 .

Assumption 2: All agents are rational risk averters with unknown utility functions.

However, they consider that the risk horizon Hi of a security is the shortest number of

periods such that

P[RHii ≤ Ei − λ

]≤ (1− Λ) + γP

[RHii ≥ Ei + λ

](1)

for constants λ ≥ 0, 0 ≤ γ ≤ 1, 0 ≤ Λ ≤ 1 and where Ei is the expected return of the

security3.

Assumption 2 totally relaxes the traditional restriction on investors’ utility functions. It

replaces it by indirect information about investors’ preferences that is inferred from their

2Although the existence of these moments is a strong assumption regarding some evidence from manyseries of returns (see Jondeau and Rockinger, 1999), it is commonly used for the derivation of asset pricingmodels.

3If Hi is not an integer, it is implicitly assumed that time intervals can be divided through a factor z sothat the returns are still i.i.d. and the product zHi is an integer.

5

assessment of the riskiness of a security. By the weak law of large numbers, they know that

the mean return of a security converges in probability towards its expected value over an

infinite horizon of investment. However, for a finite number of periods, investors could con-

sider this convergence to be sufficient to make the mean not significantly different from the

expectation, and therefore to consider the security as approximately riskless. This number of

periods can be defined as the risk horizon and is formalized by equation (1): the probability

of the mean return falling short from a lower bound Ei − λ must be smaller than a constant

(1 − Λ) plus a premium. Parameter λ determines the boundaries of the convergence inter-

val, and therefore brings information on the market-wide aversion for extreme risks. The

higher the λ, the larger the interval required by investors, and the smaller the risk horizon

Hi for any given asset i, ceteris paribus. Parameter Λ depicts the tolerance level assigned

by investors such that the security i is considered as approximately riskless over an horizon

of investment Hi. The premium is composed of the potential of exceeding the upper bound

Ei + λ multiplied by a proportionality coefficient γ ∈ [0, 1]. This latter parameter can be

viewed as the market trade-off coefficient between the downside and upside potential of the

asset. If γ is close to 0, investors only care about downside risk: the risk horizon of a security

is just the number of periods necessary to respect a Value-at-Risk constraint. On the other

hand, if γ is large, investors assign a very high weight to the upside return potential of the

security. The limiting case γ = 1 implies that the market participants assign the same weight

to returns above Ei + λ than to returns below Ei + λ. In particular, for a security with a

symmetric distribution of returns, P[RHii ≥ Ei + λ

]= P

[RHii ≤ Ei − λ

]and the constraint

given by (1) is always respected, meaning that the risk horizon of this security is null.

The calibration of the market-wide parameters Λ, λ and γ cannot be done arbitrarily,

but should be endogenously derived at equilibrium. This problem is discussed in section 3

when the information on the term structure of interest rates is introduced. At this point,

investors’ perception of the riskiness of a security is characterized by equation (1) that makes

use of the probability distribution P, which was assumed to be unknown by Assumption 1.

Therefore, what is currently missing is a way to link the risk horizon definition (1) with the

set of available information, that is, moments of order 2, 3 and 4. This link is performed in

the next section where an analytical expression for the risk horizon is derived.

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2.2 Characterization of the risk horizon

Equation (1) formulates the way risk horizon is used to measure the riskiness of any security

or portfolio. However, as the entire return distribution is assumed to be unknown, an estimate

of the probabilities for the mean returns to exceed a certain interval around the expectation

needs to be provided in order to use the risk horizon measure and derive associated pricing

relationships. Chebyshev inequalities already provide such an estimate under the form of an

upper bound for the cumulative distribution, but are “much too wide” and do not exploit

the information of moments higher than 2. Mallows (1956) generalized Chebyshev-type of

boundaries to moments of order 3 and 4, and provide an attractive approach to estimate

the probability of mean convergence by using only the information assumed to be known.

Indeed, information given in Assumption 1 allows to adapt this classical result to derive an

expression for the risk horizon (All proofs are gathered in the Appendix):

Proposition 2.1 If Si and Ki exist and are known, the risk horizon of a security i is givenby:

Hi = min H : Λ ≤ 1− πi(−λ,H) + γπi(λ,H)

where πi(x,H) =∆i

Q2i (x) + ∆i(1 + Hx2

Vi)

Qi(x) = −Hx2

Vi+Six

V 2i

+ 1

∆i =1

H

(Ki

V 2i

− S2i

V 3i

− 3

)+ 2

under the constraints ∆i > 0

λ >1

2

(SiV H

+

√S2i

V 2H2+ 4

)and

∂πi(x,H)

∂ | x |< 0 ∀x

∂πi(x,H)

∂H< 0 ∀ϕi ≤ H ≤ Φi

where ϕi = min

[(−x2S2

i + S2i − ViKi − 3V 3

i x2 + 3V 3

i + ViKix2)2

4V 3i x

2S2i

,x2S2

i

V 3i (1− x2)2

]Φi = max

[(−x2S2

i + S2i − ViKi − 3V 3

i x2 + 3V 3

i + ViKix2)2

4V 3i x

2S2i

,x2S2

i

V 3i (1− x2)2

]

7

The probability function πi(x,H) represents the upper bound of the corresponding prob-

abilities defined in equation (1) and is function of the unconditional moments. One can

rewrite πi(x,H) = P(RH

i − Ei > x)

for x > 0 and P(RH

i − Ei < x)

for x < 0. Note the

behavior of πi with respect to x and H. The higher the interval x around expectations, the

greater the probability mass inside it, and the lower π. Moreover, the higher the risk horizon

H, the more concentrated the distribution around expectation and the lower the π.

Proposition 2.1 does not relate the risk horizon to the centered moments of the distri-

bution in any particular manner: H is not necessarily increasing in variance, decreasing in

skewness or increasing in kurtosis.4 Therefore, the risk horizon framework complies with

(Brockett and Kahane, 1992) critique as it does not posit any moment preferences.

3 Equilibrium relations with the yield curve

The upper bound π described above enables us to link investors’ common assessment of as-

set riskiness defined by equation (1) with the information about returns distribution that is

available to investors: the unconditional moments of order 2, 3 and 4. Beyond its implica-

tions for traditional market equilibria, a major added value of the model rests on its potential

use of the term structure of interest rates. Instead of introducing superfluous complexity,

information contained in interest rates can be exploited in order to ensure the consistency

of the model with a non-constant term structure. Because the notion of risk is related to a

measure of time horizon, it can be matched with the risk horizon of a riskless zero-coupon

bond (i.e. a treasury security). The yield of this security as a function of maturity is precisely

modelled by the curve of the term structure.

As in the CAPM, considering a unique equilibrium market portfolio5, the expected return

4Nevertheless, if only the second moment is observed, Proposition 2.1 collapses to the application ofChebyschev’s inequality, and risk horizon increases monotonically with the variance. Traditional mean-variance analysis is therefore nested into this framework.

5This constitutes an additional assumption to our framework, as Simkowitz and Beedles (1978) and Mittonand Vorkink (2007) show that diversification is not necessarily desirable due to skewness. In this paper, weneed to restore this principle to provide a consistent equilibrium with the term structure of interest rates.

8

of any security derives from the rate of return of the risk-free rate and the expected return of

the market portfolio. In most partial equilibrium models, the latter input is taken as exoge-

nous. However, thanks to information contained in the term structure, we can retrieve the

expected rate of return of the market portfolio. Furthermore, Assumption 2 posits that the

notion of risk is captured by expression (1) whose input parameters λ, γ and Λ are unknown.

All we need for the calibration of these parameters is a sufficient number of points on the

term structure. As treasury securities are priced at equilibrium, they have to satisfy this

assumption. Nevertheless, they cannot be treated as risky as they are totally riskless if they

are held until maturity. Therefore, some very risk averse investors are likely to hold them

without combining them with the market portfolio at equilibrium. Thanks to an arbitrage ar-

gument, these treasury securities must yield the highest expected return among all securities

with the same horizon as theirs. If, for instance, the expected rate of return of the treasury

security was lower that the optimal combination of the market portfolio and the least risky

asset – say, the short term Treasury Bill, it would be possible for any investor with the same

risk horizon to sell treasury securities to increase her position in the optimal combination.

This would induce an increase in the price of the optimal combination and a decrease in the

price of the treasury security up to the point where expected returns become equal.

Denote yτ the continuously compounded rate of return of a treasury security maturing

at τ . As for the risk-free rate Rf , the current yield-to-maturity of this security is equal to

its expected return over the next period, denoted Eτ . Its periodic future rate of return

is risky6 with its own moments Vτ , Sτ and Kτ . The treasury security is totally riskless if

held until maturity, but it can nevertheless be considered as approximately riskless for an

horizon of investment shorter than τ . Therefore, the risk horizon of such a security must

be equal to some time horizon Hτ = kτ with 0 ≤ k ≤ 1. In other words, any investment

in a riskless bullet bond whose horizon exceeds the one of the least risky asset will entail

a positive risk horizon, as for any other risky investment. With the help of a non-constant

term structure of interest rates, it is possible to endogenously derive the parameters k, λ, γ

and Λ, together with the expected return on the market portfolio (hereafter denoted Em),

6Of course, the pull-to-par phenomenon induces that debt returns are not stationary. Therefore, theargument only holds for a rolled-over portfolio of debt with a constant maturity.

9

following an arbitrage argument. This is done through a new fund separation theorem.

Proposition 3.1 If y = 1 and my ≡ m and if the term structure of interest rates obeys

a n-dimensional yield-factor model corresponding to maturities τ 1, ..., τn and risk horizons

Hτ1 , ..., Hτn, then the parameters Λ, γ, λ, k and the expected return on the market portfolio

Em are determined at equilibrium by solving the following system of equations:πτ j(−λ, kτ j)− γπτ j(λ, kτ j) = 1− Λ j = 1, ..., nπpj(−λ, kτ j)− γπpj(λ, kτ j) = 1− Λ j = 1, ..., nατ jEm + (1− ατ j)Rf = Eτ j j = 1, ..., n

(2)

where

Rpjt = ατ jRmt + (1− ατ j)Rft (3)

with m being the risky fund held at equilibrium. The equilibrium entails separation between

one risky fund and n treasury securities.

Note that the principle underlying this proposition is not easily transposable to other

asset pricing models. In our setup, risk is measured through a time period, which induces

that treasury securities are totally riskless for a given horizon (their maturity), and “ap-

proximately riskless” for their risk horizon. Therefore, they do not belong to the set of risky

securities. If another measure of risk were used (e.g. the variance of returns) these securities

could not be held as riskless for a given period of time as their returns would display some

variability.

Before we bring system (2) to the data, it is useful to examine its implications for the

endogenous expected market return. A closer examination of the bottom equations of the

system allows us to re-express the expected excess return on the market portfolio as follows:

Em − Rf =Eτj−Rfατj

, with this latter equation being required to hold for all Treasuries. This

decomposition emphasizes the presence of two components of Em in the model. The first one

is the expected return on Treasuries in excess of the risk-free asset (Eτ j − Rf ). It reflects

information contained in the term structure. The other component is the weight assigned by

investors on the market portfolio in the optimal risky portfolio. It is related to stock market

data, as it is influenced by the changes in statistical risk (moments) of the risky portfolio,

10

through the required matching between the risk horizon of this portfolio and the risk horizon

of treasury securities. As a result, periods of high riskiness (i.e. highly unfavourable Vm,

Sm and Km) apply a downward pressure to weights assigned to the market portfolio because

of this matching constraint. If the term structure is upward sloping, reduction in market

weights goes along with an increase in the expected excess return. This can be interpreted

as if investors perceive the high riskiness of the market but expect higher expected return

in the future, reflecting the anticipation of recovery from the information contained in the

positively sloped yield curve. In case of downward sloping curve (negative Eτ j − Rf ), the

expected excess return on the market portfolio decreases with lower weights (as we now have

Em − Rf < 0). The expected return Em might even turn negative if Eτ j − Rf is sufficiently

negative or in case of sufficiently low weights ατ j . This is consistent with the view that neg-

atively sloped yield curve indicates market anticipation of economic downturn (see a.o. Ang

et al., 2006) and investors expect negative market returns. On the other hand, if the mo-

ments riskiness is low, higher weights are allocated to the market portfolio, and the required

premium is closer to the treasury premium (Eτ j −Rf ). Our theoretical framework therefore

provides an intuitive link between the expected risk premium, the information contained

in the term structure, and “moments risks”. The exploitation of this link to calibrate risk

horizon parameters and the expected market return as well as its potential power for stock

return predictability is examined in the empirical application presented below.

4 Empirical application

Proposition 3.1 provides a framework that enables us to endogenously calibrate the param-

eters underlying the risk horizon measures and the expected return on the market portfolio.

In this section, we perform this calibration on U.S. market data. The framework has the

original characteristic of simultaneously linking market-wide risk premium estimates to the

term structure of interest rates which naturally refers to a notion of horizon risk, the aversion

for extreme risks, which is embedded in our measure of risk horizon through parameters λ

and Λ, and the market-wide aversion for asymmetric risks (parameter γ). This link is desir-

able for at least two reasons. First, it will enable us to discriminate the relative importance

11

of asymmetric and fat-tail risks in market-wide risk premia, in a broader context than under

multi-moment models that only look at equity markets. Second, if the market-wide attitudes

towards risk is a leading indicator of market sentiment, the ability to model the time-varying

behavior of their determinants enables us to test new predictors of systemic market risks. In

particular, the relevance of information contained in the estimated expected return is verified

with tests of stock return forecasting ability and asset allocation applications. Our objective

is to analyse whether the endogenously calibrated risk premium reveals able to add statisti-

cally and economically significant value to forecasts from a model that already contains the

popular predictors found in the literature. We begin with the description of the dataset and

the methodology applied to these tests.

4.1 Data

We collect a sample of US monthly term structure factors from Gurkaynak et al. (2007) for

the period beginning in January 1980 until May 2010.We treat this sample with the Nel-

son and Siegel (1987) curve fitting model augmented by Svensson (1994) (hereafter NSS)

to retrieve US spot rates of required maturities. Table 1 reports some descriptive statistics

on NSS factors and US treasury rates and returns. Monthly returns of 1, 3, 5 and 10-year

treasury securities allow us to estimate Vτ , Sτ and Kτ needed to obtain the four functions

πτ in equation (2)7. Moments are computed on a 36 monthly return window8. Continuously

compounded returns of 3-month treasury security and logarithmic monthly total returns (in-

cluding reinvested dividends) of the S&P 500 index provide our estimates for Rf and Rm

respectively. Prices of the S&P 500 total return index are from the updated database of Goyal

and Welch (2008). It consists of month-end values from Center for Research and Security

Press (CRSP). Horizon Hτ is set equal to kDτ where Dτ is the Macaulay duration of US

Treasuries expressed in months and k a multiplier allowing for the possibility that Treasuries

might be considered as approximately riskless for an horizon lower than their maturities.

7As robustness check a system with only three treasury securities has also been implemented. Calibrationresults do not change significantly, except from the presence of more outliers, indicating less precise estimatesof our parameters.

8Similar results are obtained when longer calibration windows are considered in the model simulations,but the length of the estimation period is reduced. The accuracy of some parameter estimates appears todiminish (that is, they touch boundaries of the range of authorized values more often) when shorter calibrationwindows are used.

12

The expected return Eτ of each treasury security equals its corresponding NSS yield. The

time series of these yields are represented in figure 1.

Insert Table 1 and Figure 1 about here

In order to examine the relevance of the risk premium estimates, we implement some

in-sample and out-of-sample stock return forecasting tests and perform an asset allocation

application. In this analysis, the predictive power of our estimates is compared to other

popular predictors of stock returns. Most of the popular economic variables are collected

from the updated dataset of Goyal and Welch (2008)9:

1. Book-to-market ratio (btm): ratio of book value to market value for the Dow Jones

Industrial Average index.

2. Dividend payout ratio (de): logarithm of the ratio of a twelve-month moving sum

of dividends and a twelve-month moving sum of earnings.

3. Default spread (dfy): difference between the yields of BAA-rated and AAA-rated

corporate bonds.

4. Default returns (dfr): difference between the returns of BAA-rated and AAA-rated

corporate bonds.

5. Earnings-Price ratio (ep): logarithm of the ratio between a twelve-month moving

sum of earnings and S&P stock price.

6. Inflation (infl): log returns of the CPI (all urban consumers). Taking account of

delay in the release of the CPI index, we follow Goyal and Welch (2008) and use the

lagged series of this variable.

7. Long-term yield (lty): yield on long-term treasury bonds.

8. Long-term return (ltr): return on long-term treasury bonds.

9The description of the data and their sources are provided in Goyal and Welch (2008).

13

9. Net equity expansion (ntis): ratio of a twelve-month moving sum of net equity

issues by NYSE-listed stocks to the total end-of-year market capitalization of NYSE

stocks.

10. Stock variance (svar): monthly sum of squared daily returns on the S&P 500 index.

11. Term spread (tms): difference in yields between a long-term treasury bond and a

3-month t-bill.

In addition, we add the following variables as candidate predictors:

12. Business conditions (bus): the Aruoba-Diebold-Scotti indicator (from Aruoba et al.

(2009)), designed to track business conditions at high frequency and available on the

Philadelphia Fed website10. It mixes the low and high frequency seasonally adjusted

economic indicators whose release are closely watched by investors: weekly initial job-

less claims, monthly payroll employment, industrial production, personal income less

transfer payments, manufacturing and trade sales, and quarterly real GDP. Our vari-

able is centered around mean value so that positive values indicate better-than-average

business conditions, while negative values shows relatively worse-than-average condi-

tions.

13. Federal funds rate (ffr): the monetary policy rate collected on the website of the

Federal Reserve.

14. Market liquidity conditions (liqu): Pastor and Stambaugh (2003)’s liquidity fac-

tor is used for our liquidity component. Their aggregate measure is computed from

individual daily stock data and rely on the principle that order flows induce more im-

portant return reversals when liquidity is lower. Indicator values are typically negative

while larger in absolute magnitude when liquidity is lower.

In forecasting regressions, the dependent variable is the logarithmic total return on the

S&P 500 minus the logarithmic return on the three-month Treasury Bill. The summary

statistics of candidate predictors are described in Table 2. When needed, first differences or

detrending adjustments are applied to ensure stationarity.

10The Aruoba-Diebold-Scotti indicator is available at http://www.philadelphiafed.org/research-and-data/real-time-center/business-conditions-index/ .

14

Insert Table 2 about here

4.2 Calibration

Information contained in the term structure of interest rates allow us to endogenously extract

parameters k, λ, Λ, γ, as well as the expected market return Em and portfolio weights ατ .

To calibrate these parameters, the system of 12 equations with 9 unknowns described in

(2) is solved numerically11. Motivated by the theory, and to avoid unrealistic estimates, the

following set of constraints is applied:0 ≤ k ≤ 10 < λ ≤ 0.050 ≤ Λ ≤ 0.990 ≤ γ ≤ 10 ≤ ατ j ≤ 1

The upper bound for Λ of 0.99 means that investors require 99% of confidence in the

convergence of mean returns towards their expected value to consider an asset as approxi-

mately riskless. Constraints on parameter λ are rather lax with (monthly) asset mean returns

required to not falling out of a lower bound of 0.05 from expectations. Weights ατ are con-

strained between 0 and 1 so that no short sales are allowed12.

Due to the high non-linearities that characterize system (2), the simultaneous optimiza-

tion of the parameters is arduous. To overcome this difficulty, we apply a two-step iteration

approach to find the optimal values of the parameters at each date of our sample period. In

the first step, the risk horizon parameters (k, λ, Λ and γ) are kept fixed while only the weights

(ατ j) of the market portfolio and its expected return (Em) are allowed to vary. The weights

are then fixed to the optimized value found at the first step, and risk horizon parameters are

optimized in a second step. The expected return of the market portfolio (Em) is optimized

at each step. We then iterate between the two steps until the calibrated values converge.

At each date t, starting values comes from calibrated values in t − 113. Calibrated parame-

ters are reported in table 3. The time evolution of the estimates are displayed in figures 2 to 5.

11The SQP, Quasi-Newton algorithm implemented via the fmincon solver in Matlab is used in this task.12Results are robust to simulations with short sales allowed. Detailed results are available upon request.13This approach has shown to provide smoother estimates of the parameters compared to an approach

where all parameters are estimated simultaneously.

15

Insert Table 3 and Figures 2 till 6 about here

We start by considering the estimates of the risk horizon parameters (rows 1-4 in table

3). The estimates of the duration multiplier k are very close to 1, indicating that US Trea-

suries are perceived as approximately risk-free for an horizon close to their maturities. The

trade-off coefficient γ between downside and upside potential has a mean of 0.11 and display

infrequent jumps during the sample period. Estimates of λ imply that investors require, on

average, the distribution of mean returns to concentrate in the interval [−1%,+1%] around

their expectation on a monthly basis. The width of this interval is somehow compensated

by the calibrated values of Λ, which closely stick to the upper bound imposed. These results

derive from the matching of the risky portfolio with the risk horizon of Treasuries in system

(2). The difference in riskiness between Treasuries and the optimal portfolio is such that,

to respect this constraints, the model has to allow large intervals around expectations. The

recent financial crisis period (mid-2007 - 2010) provides extreme estimates. The sharp in-

crease in the riskiness of the market (reflected by its statistical moments) induces the model

to provide high λ and γ and reduce the weights of the market portfolio. However there is

a trade-off with multiplier k: as λ and γ reach high levels, the risk horizon constraint on

Treasuries becomes relatively lower and the model leads to choose a smaller risk horizon H

to reflect it, thereby reducing k. This explains the sharp decrease in k along with high levels

of λ and γ during this period. Such a behavior can be interpreted as the translation of the

“flight-to-quality” phenomenon in the model: a lower value of k corresponds to government

bonds being considered safe earlier.

The time series of the estimated weights are highly cyclical as depicted by figure 5.

Periods of small weights correspond to economic envrionments characterized by relatively

lower interest rates and higher uncertainty (higher volatility, more negative skewness and

higher kurtosis) on the stock market. During these periods, the model rests more heavily

on the risk-free asset to achieve low-yield, less risky objectives. As stressed in the discussion

of Proposition 3.1, the information conveyed by the weights influence the expected market

return Em. The estimates of Em have an annualized sample mean of 7.52% with an annualized

16

standard deviation of 0.77%. Figure 6 shows the time series evolution of Em and the 10-year

treasury yield. The difference between the expected value on the risky asset and the risk-

free treasury (denoted Em y10) is represented in green. This premium is characterized by

relatively low values in normal times (low premium over the 10-year yield) and high values

in crisis periods. Indeed, as discussed in the previous section, the expected excess return

on the market is closer to the treasury premium when market weights (α) are high, which

correspond to periods with relatively low market risk (favourable environment). In contrast,

periods of higher market risk are characterized by lower weights. Moreover, the yield curve

tends to flatten or inverse before important market downturns (see figure 1). This explains

the reduction and negative values of the the premium Em − y10 before NBER recessions

(represented in grey in figure 6). In the aftermath of downturns, we observe higher values for

Em as the yield curve returns to an upward-sloping shape and the weights stay at relatively

low values. The predictive power of this cyclical behavior in the estimated premium Em−y10

for stock returns is analyzed in the next section.

4.3 Predictive performance

We now investigate the predictive power of our premium estimates. In-sample and out-of-

sample tests are performed along the lines of the work of Rapach and Wohar (2006) and

Goyal and Welch (2008). In regressions, the dependent variable is the log total return on

the S&P 500 index minus the log return on a risk-free Treasury Bill. Using the financial

variables described above, we examine the predictive power of multivariate regressions for

stock returns. This analysis aims to produce forecasts from a model with and without

the premium estimate (Em − y10), and to investigate the statistical as well as economic

significance of this new variable.

4.3.1 Statistical predictive performance

The predictive ability is assessed by traditional statistics: root mean squared error of the

prediction (RMSE), cumulative squared prediction errors, Theil’s U and the Diebold and

Mariano (1995) and West (1996) statistic, adjusted by Clark and West (2007), for comparing

nested model forecasts (MSPE). This last statistic is used to test the null hypothesis that

an unrestricted model forecast is equivalent to a restricted model forecast, against the one-

17

sided (upper tail) alternative that the unrestricted model forecasts better than the restricted

model. Clark and West (2007) show that the traditional statistic needs to be adjusted to take

account of the noise in forecasts of the unrestricted model introduced by coefficients whose

population values are zero. They also show that their adjusted statistic has an asymptotic

distribution well approximated by the standard normal, so that one-sided critical values from

the standard normal distribution can be used (i.e. 1.282 for significance at 10% or 1.645 for

5%). The in-sample and out-of-sample forecasts are compared to the unconditional historical

mean. The period from Jan 1983 to Dec 1991 is used as the starting calibration period for

out-of-sample forecasts. Once a forecasts has been made in t + 1, the calibration period is

extended with the information available on t + 1 to produce the forecasts of period t + 2,

and so on. Out-of-sample forecasts of the unconditional model are generated using the mean

that prevails over each calibration period.

Several models are used to test the predictive ability of our estimate. First we consider in-

dividual regressions, where only the lags of the premium estimate (Em − y10) are introduced

as regressors. We then include these lags in multivariate regressions along with popular

financial predictors, and examine the predictive added value of (Em − y10). An intercept

is included in all regressions. The predictive power of first differences of our estimates is

also tested in separate models. Two types of lag structure are considered. In a first step,

regressors enter regressions with one lag. Regressions with this lag structure are reported

as models of type A. In a second analysis, a recursive model selection similar to the one

described in Pesaran and Timmermann (1995) is performed to retain the lag structure with

the best predictive power. This lag selection approach yields “type B”-models. To obtain

a relatively parsimonious specification and avoid potential over-fitting issues, we follow a

general-to-specific approach. Starting with 12 lags for each variables (i.e. 1 year), in-sample

criteria (adjusted R2, AIC and BIC statistics) are used to select the most relevant lag struc-

ture of each variable. The joint irrelevance of the dropped lags and variables are controlled

with F-tests. This model selection is recursive, as it is applied at each forecasting date. It

is therefore consistent with the use of information available in real time. For information,

regressions performed on the overall period are reported in table 4. Model B1 only include

traditional predictors and reports an adjusted R2 of 9.6%. Models B2 to B5 consider different

18

lags of the estimated (Em − y10) and display an adjusted R2 around 11.5%. For comparison

purposes, we also include model B6 and B7 that contain only lags of (Em − y10).

Insert Table 4 about here

Table 5 reports the different models along with their lag structure and their forecasting

performance. The first three columns provide information on the lag structure of the finan-

cial variables (first column), the premium (second column) and its difference (third column)

respectively. The next columns summarize information on in-sample and out-of-sample fore-

casts. The last two columns summarize the comparative performance of the specifications.

They report the MSPE test statistic for a comparison of the models and the unconditional

regression (before last column) and the models of the same type with financial variables only

(last column). The upper panel of the table reports the forecasting statistics of the uncondi-

tional model (historical mean only), and the other two panels are dedicated to the regression

models of type A (medium panel) and B (lower panel).

Insert Table 5 about here

In-sample tests are conducted with forecasts computed over the overall sample period,

from Jan 1983 to May 2010. All type A models have in-sample performance very close to the

unconditional model. Only model A3 and A5 marginally reduce the RMSE statistics with

respect to the historical mean predictions. If we select optimal lags for the financial variables

(model B1), the in-sample RMSE criterion is slightly better. Adding the optimal lags of our

premium estimates or its first difference brings a marginal improvement over other popular

predictors. However, significant differences can be observed in out-of-sample forecasts. While

forecasts of type A models do not statistically differ from the predictions of the unconditional

model, models A4 and A5, that only includes first lag of (Em − y10) or its first differences,

reach statistically better accuracy than their peers. This is underlined by lower RMSE and

significant MSPE statistic from the test of better forecasting ability with respect to model

A1.

19

When optimal lags are selected, the forecasts are improved when compared to the un-

conditional model, as can be observed from statistics of type B models. The MSPE test

statistics indicate that all the models inside this class perform statistically better than the

unconditional model in terms of out-of-sample predictions. Figure 7 reports the cumulative

sum of the squared error of out-of-sample predictions from prevailing historical mean minus

the cumulative sum of squared errors of model B1. An increase in this difference indicates

better performance of the latter. The cumulative difference indicates that the overall out-

of-sample improvements of popular predictors is especially important in periods of market

downturns, indicated in grey on the figure. Models of type B that include the optimal lags

of our premium estimates seem to deliver more accurate forecasts than model B1, as the

significant MSPE statistics in the last column of table 5 reports. The premium estimate

(Em − y10) seems to deliver some predictive ability that is not contained in popular predictors

found in the literature, and is therefore complementary to those predictors. Figures 8 and 9

display the cumulative performance for model B2 and B3 respectively.

Insert Figures 7 till 10 about here

In particular, the parsimonious models with the optimal lags of (Em − y10), namely (B6)

or its first difference (B7) alone, also display a positive and significant MSPE statistics de-

spite higher out-of-sample RMSE than model B1. Figure 10 shows that this higher RMSE

comes mostly from less accurate predictions during the recent financial downturn period.

While model B6 seems to exhibit more accurate forecasts during periods of normal times or

expansion, its performance during the 2007-08 financial crisis brings the cumulative difference

towards negative area. This observation is confirmed in sub-period analyses performed in

tables 6 and 7. These tables report the in-sample and out-of-sample forecasting performance

of the different models in periods of normal or bullish markets (table 6) and during periods

of bearish markets (table 7). Bearish markets periods are measured as the top 3 drawdown

periods in the price evolution of the S&P 500 total return index, and covers recent finan-

cial crises (LTCM crisis in July and August 1998, internet bubble from September 2000 to

February 2003 and recent subprime turmoil from December 2007 to February 2009). During

normal and bullish times, models of type-B that includes our premium estimates seems to

20

provide better forecasts with statistically significant MSPE statistics over both the uncon-

ditional model and models with popular predictors only. Over bearish market periods, the

complementarity of our premium estimates is less obvious, with no statistically significant

MSPE statistic when compared to a traditional model with popular predictors.

Insert Tables 6 and 7 about here

4.3.2 Economic predictive performance

In addition to the tests of statistical significance of the premium forecasts, we also conduct

an asset allocation application to examine their economic significance. In this application,

a self-financing portfolio is built and re-balanced according to premium out-of-sample fore-

casts. The portfolio consists in a market portfolio invested in the S&P 500 index (Rm) and

a risk-free asset (return on a Treasury Bill, Rf ). The monthly rebalancing of this portfolio

starts in Jan 1992 and ends in May 2010, and is based on the following trading rule: when-

ever the model predicts a significant negative equity premium, the portfolio takes a short

position in Rm and a long position in Rf . Otherwise, the portfolio stays long in (Rm −Rf ).

From our simulations, we test different thresholds to determine the significance of negative

forecasts. Only forecasts lower than these thresholds are considered as (economically) sig-

nificantly negative, and imply a rebalancing. The actual returns generated by this strategy

are then computed and compared to the performance of a passive strategy that stays long

in (Rm −Rf ) during the period. The results are also compared with returns generated by

forecasts from other models, and reported in table 8. Results are reported for a threshold

of 0bps and −30bps. Two Jobson and Korkie (1981) statistics for significant differences in

Sharpe ratio are also reported14. The first statistic tests the relative performance between an

active and a passive strategy, while the second compares the performance between models

14The formula for Jobson and Korkie (1981) statistic for testing the Sharpe ratio difference between twodifferent securities or portfolios is as follows:

zJK =s1 − s2√

θ

where

si =µi

σi

21

with or without the premium (Em − y10) (i.e. model B1 versus model B2).

Insert Table 8 about here

Over the entire out-of-sample period (upper panel in table 8), the portfolio performance

under active strategies outperforms the passive strategy results in terms of cumulative total

returns. These results are confirmed from the graph of cumulative returns of the different

strategies (figure 11). The early 2000’s and the 2007-08 financial turmoil is characterized by

a sharp decrease in the passive strategy, while strategies based on out-of-sample forecasts of

models of type B generate stable or even increasing returns during these periods. This better

performance comes mostly from the opportunity to rebalance the portfolio during downturn

periods. The active strategies display positive returns and significantly improve the Sharpe

ratio during these periods. The out-performance is reinforced once we select a lower threshold

to determine significant negative predictions. Moreover, forecasts from model B2 with esti-

mates of (Em − y10) improve the cumulative performance in each sub-sample period, when

compared to model B1 where the premium is excluded. The difference in Sharpe ratios turns

out to be statistically significant over both normal or bullish markets and periods of market

downturn.

Insert Figure 11 about here

To sum up, the tests of predictive performance reveal some forecasting ability of our

endogenously derived premium (Em − y10). This ability appears to bring some added value

with respect to other popular candidate predictors presented in the literature, with significant

out-performance when the premium is added to a complete model. The statistical out-

performance comes from more accurate out-of-sample predictions of the equity premium. An

for i = 1, 2 is the estimated Sharpe ratio and

θ = T−1[2− 2ρ1,2 +

1

2

(s21 + s22 − 2s1s2ρ

21,2

)]is the estimated variance of the difference in Sharpe ratios. The correlation between returns of portfolio 1and 2 is denoted by ρ1,2.

22

asset allocation application indicates better cumulative performance and Sharpe ratios when

these forecasts are used to rebalance the positions in a portfolio composed of a market index

and a risk-free asset.

23

5 Conclusion

The paper analyses the link between market expected returns, the information contained

term structure of interest rates and market-wide preferences for volatility, asymmetric and

fat-tail risks. We adopt an intuitive, non-parametric approach to associate the notion of risk

to a holding period that is necessary for the mean returns of a portfolio to be considered as

riskless. As the risk measure (risk horizon) provided by this model refers to a time horizon,

it is possible to characterize a system of equilibrium equations used to calibrate model pa-

rameters and the expected market return.

When the model is taken to the U.S. market data, we find economically significant es-

timates of the risk horizon parameters and the expected market return. Using forecasting

tests and asset allocation simulation, we show that the estimated expected return exhibits

some predictive power for the equity premium, and add value to popular predictors of the

equity premium found in the literature.

The extraction of such an assessment of expected market returns has interesting implica-

tions for future research. For instance, a general multi-moment asset pricing model can be

developed and tested with our estimates of expected market returns. Again, the risk hori-

zon framework seems appropriate as it does not rely on traditional restrictive assumptions

on either the returs distribution or utility functions and complies with both Simkowitz and

Beedles (1978) and Brockett and Kahane (1992) critiques. Another potential application

concerns portfolio allocation application. The assessment of portfolio optimal allocations ex-

tracted from the optimization based on investor’s desire to maximize expected return under

risk horizon constraints is possible within our framework. Several risk profiles can be covered

by varying the model parameters, and allocation could be compared to traditional objective

functions.

24

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6 Appendix

6.1 Proofs

Proof of Proposition 2.1

See Mallows (1956) using the moments of RH

i . The signs of the derivative with respect

to | x | is straightforward. The sign of the second derivative can be obtained by using the

variable transformation y =√Hx√Vi

and noting that dπi(y,H)dH

= ∂πi(y,H)∂y

dydH

+ ∂πi(y,H)∂H

. The first

term is always strictly negative and the second term is nonpositive for ϕi ≤ H ≤ Φi, which

completes the proof.

Proof of Proposition 3.1

The first line of the system corresponds to the definition of the risk horizon applied to

each treasury security. The second and third set represent the horizon risk and expected

return of a portfolio constructed only with a combination of the least risky asset and the

market portfolio. The second set of equation entails that each portfolio has the same risk

as the corresponding treasury security, while the thrid set of equation equates expected re-

turns. As each portfolio pk is mean-horizon efficient, by arbitrage, each treasury security is

mean-horizon efficient as well. The calibration of λ, γ, Λ and Em follows from solving the

nonlinear system of equations.

27

6.2 Tables

Table 1: Summary statistics of US term structure factors and zero Treasuries for the periodJan 1980 to May 2010.

NSS param. Mean Median min Max Std

Level (in %) 6,12 4,43 0,00 25,00 5,79Slope (in %) -0,71 0,10 -20,13 7,78 4,88Curvature 1 -1,61 0,00 -6,31 0,93 2,75Curvature 2 1,67 0,15 -1,00 6,27 2,70λ1 (1st hump) 0,03 0,02 0,00 0,24 0,03λ2 (2nd hump) 0,09 0,09 0,00 0,99 0,08

Mkt Data (in %) Mean Median Std Skewn. Ex. Kurt.

y1 5,91 5,56 3,41 0,63 0,24y3 6,40 5,88 3,23 0,59 -0,16y5 6,73 6,16 3,05 0,64 -0,29y10 7,27 6,58 2,74 0,74 -0,36R1 6,57 5,64 1,99 2,58 16,30R3 7,90 7,16 4,69 0,87 8,42R5 8,82 8,54 6,98 0,33 3,31R10 10,35 10,52 12,49 0,35 2,28Rf 5,79 5,40 1,09 1,46 3,93Rm 10,43 15,95 15,69 -0,94 3,36

Notes: The Nelson-Siegel-Svensson model (NSS) is used to retrieve the US spot rates using factors

from Gurkaynak et al. (2007). Level, Slope, Curvature 1 and 2 are NSS factors β0, β1, β2 and

β3, respectively in Gurkaynak et al. (2007). Factors λ1 and λ2 are parameters that determine

the location of the two humps in the term structure maturity span. Prices of hypothetical zero

Treasuries are computed according to the following formula: Pm,t = 100e−ym,tm12 where ym,t is the

US spot rate with maturity m expressed in annual terms. Logarithmic monthly returns (represented

by R in the table) are computed from these prices: Rt = ln(Pm−1,t

Pm,t

). Rm is the logarithmic period

return of the S&P 500 total return index. Rf is the continuously compounded return on 3-month

US T-bill. All yields and returns are here reported in annual terms.

28

Table 2: Descriptive statistics of the candidate predictors, period from Jan 1983 to May2010.

Overall period Expansions Recessions

Mean Std. Dev. Mean Std. Dev. Mean Std. Dev.btm 0,353 0,170 0,373 0,172 0,229 0,083bus -0,090 0,760 0,072 0,562 -1,065 1,026de -0,730 0,398 -0,753 0,382 -0,588 0,462

dfy 0,010 0,004 0,010 0,004 0,013 0,007dfr -0,0002 0,015 0,0002 0,013 -0,003 0,023dp -3,756 0,394 -3,706 0,385 -4,059 0,305ep -3,026 0,432 -2,952 0,406 -3,471 0,293ffr 4,986 2,591 5,351 2,524 2,798 1,807

infl 0,002 0,003 0,003 0,003 0,001 0,005liqu -0,026 0,069 -0,019 0,060 -0,068 0,100lty 0,070 0,022 0,073 0,022 0,051 0,007ltr 0,008 0,030 0,008 0,028 0,010 0,039

ntis 0,006 0,022 0,008 0,019 -0,004 0,030svar 0,003 0,005 0,002 0,004 0,007 0,010tms 0,024 0,013 0,023 0,013 0,027 0,013

Notes: Most of the dataset is collected from Goyal and Welch (2008). Refer to the text for names

and description of the different variables. Variables bus (Business conditions indicator), ffr (Federal

funds rate) and liqu (Pastor and Stambaugh (2003) liquidity measure) have been added. The overall

period considered extends from Jan 1983 to May 2010 and corresponds to the estimation period

of risk horizon parameters. Recessions dates are 1998:07-08, 2000:09-2003:02, and 2007:12-2009:02

and correspond to the top 3 drawdown periods in the S&P 500 index.

29

Table 3: Summary statistics of the risk horizon parameter estimates over the period startingin Jan 1983 to May 2010.

Mean Median min Max Std. Skewn. Ex. kurt.

k 0.924 1.000 0.012 1.000 0.237 -3.323 9.441λ 0.011 0.008 0.004 0.050 0.008 3.070 10.743Λ 0.990 0.990 0.990 0.990 0.000 -1.005 -2.012γ 0.111 0.000 0.000 0.974 0.253 2.211 3.552

ατ1 0.239 0.219 0.000 0.502 0.113 0.289 -0.712ατ3 0.413 0.383 0.117 0.726 0.173 0.224 -1.239ατ5 0.531 0.496 0.172 0.931 0.216 0.206 -1.253ατ10 0.712 0.705 0.273 1.000 0.247 -0.112 -1.453

Em (in %) 7.515 7.177 -1.544 13.708 0.767 -0.039 -0.034Em −Rf (in %) 2.762 3.117 -9.075 12.397 1.393 1.748 -0.303Em − y1 (in %) 2.323 1.698 -6.278 12.320 0.835 1.266 2.446Em − y10 (in %) 1.044 1.025 -6.921 8.682 0.929 3.083 -0.083

Correlations

Em y1 y10 Em − y1 Em − y101 -0.204 0.237 0.839 0.886 Em

1 0.731 -0.704 -0.552 y11 -0.234 -0.240 y10

1 0.950 Em − y11 Em − y10

Notes: Risk Horizon parameters are represented in the top 4 rows of the table: k, λ,Λ and γ. ατ iis the weight associated to the market in the portfolio with same risk horizon and expected return

that a Treasury with maturity τ i. Estimates of expected return Em and yields information are

expressed in annual terms.

30

Table 4: Regression results for models of type B, with an optimal lag structure selected basedon in-sample criteria. Standard errors are in parentheses. * indicates significance at the 10percent level. ** indicates significance at the 5 percent level. *** indicates significance atthe 1 percent level.

B1 B2 B3 B4 B5 B6 B7

const -0.0123* -0.0116* -0.0112* -0.0130* -0.0114* 0.00367 0.00424(0.00655) (0.00679) (0.00672) (0.00725) (0.00656) (0.00304) (0.00267)

dfy(-5) 6.32** 7.12** 7.11** 7.20** 7.25**(2.37) (2.11) (2.11) (1.95) (2.11)

dfy(-6) -4.38** -5.22** -5.28** -5.11** -5.39**(2.07) (1.80) (1.80) (1.79) (1.87)

ep(-2) 0.0713** 0.0781** 0.0751** 0.0728** 0.0758**(0.0309) (0.0317) (0.0306) (0.0310) (0.0289)

ep(-7) -0.0657** -0.0675** -0.0724** -0.0695** -0.0717**(0.0234) (0.0246) (0.0217) (0.0243) (0.0225)

ffr(-9) 0.0220** 0.0209** 0.0204** 0.0210** 0.0217**(0.00952) (0.0101) (0.00992) (0.0100) (0.0103)

lty(-1) -0.724** -0.762** -0.748** -0.719** -0.736**(0.280) (0.291) (0.284) (0.292) (0.290)

ntis(-2) -1.29** -1.35** -1.36** -1.20** -1.26**(0.576) (0.593) (0.595) (0.581) (0.592)

ntis(-3) 1.46** 1.52** 1.52** 1.39** 1.41**(0.586) (0.603) (0.607) (0.594) (0.596)

svar(-1) -1.42** -1.49** -1.48** -1.53** -1.51**(0.301) (0.281) (0.279) (0.278) (0.275)

(Em − y10)(−10) 6.10** 6.36** 6.09**(2.18) (2.19) (2.03)

(Em − y10)(−11) -6.83** -6.85** -4.89**(1.77) (1.73) (2.16)

d(Em − y10)(−10) 6.46** 5.84** 5.55**(1.80) (1.89) (1.89)

d(Em − y10)(−11) -2.44(1.67)

(Em − y10)(−1) -2.32(2.03)

(Em − y10)(−2) 1.53(3.04)

d(Em − y10)(−1) -1.70(1.99)

d(Em − y10)(−2) 0.528(2.24)

n 319 318 318 318 317 318 318Adj.R2 0.096 0.116 0.118 0.113 0.115 0.014 0.015

lnL 556 558 558 559 557 536 536

31

Tab

le5:

Sta

tist

ical

pre

dic

tive

per

form

ance

test

s:re

sult

sof

in-s

ample

and

out-

of-s

ample

fore

cast

ing

test

s.

Lag

stru

ctu

reIn

-sam

ple

perf

orm

an

ce

Ou

t-of-

sam

ple

perf

orm

an

ce

Fin

anci

als

Em−y 1

0d

iff(E

m−y 1

0)

RM

SE

Th

eil’

sU

RM

SE

Th

eil’

sU

MS

PE

MS

PE

stat.

vs

Ust

at.

vs

Fin

.Uncond.Model

Un

on

on

o44

8.19

1.00

431.

921.

00

TypeA

A1

firs

tla

gn

on

o44

9.62

1.00

470.

771.

090.6

6A

2fi

rst

lag

firs

tla

gn

o44

8.78

1.00

475.

241.

10

0.5

9-1

.53

A3

firs

tla

gn

ofi

rst

lag

447.

961.

0047

1.45

1.09

0.6

40.0

7A

4n

ofi

rst

lag

no

448.

741.

0043

5.82

1.01

-1.0

93.37

A5

no

no

firs

tla

g44

8.04

1.00

433.

351.

00

-0.8

83.34

TypeB

B1

opti

m.

lags

no

no

427.

080.

9541

3.05

0.96

2.34

B2

opti

m.

lags

10,1

1n

o42

2.13

0.94

408.

490.

952.75

1.94

B3

opti

m.

lags

no

1042

2.14

0.94

406.

480.

94

2.82

2.01

B4

opti

m.

lags

1,2,

10,1

1n

o42

1.86

0.94

413.

410.

962.52

1.2

7B

5op

tim

.la

gsn

o1,

2,10

,11

421.

850.

9441

1.86

0.95

2.57

1.3

7B

6n

o10

,11

no

448.

391.

0043

0.97

1.00

1.4

01.4

2B

7n

on

o10

448.

591.

0042

8.82

0.99

1.5

61.4

6

Notes:

In-s

amp

lep

erio

dco

rres

pon

ds

toth

eov

eral

lca

lib

rati

onp

erio

d(J

an19

83to

May

2010

).O

ut-

of-s

am

ple

fore

cast

sare

ob

tain

for

per

iod

from

Jan

1992

toM

ay20

10.

Mod

els

ofty

peA

use

only

the

info

rmat

ion

onth

efi

rst

lag

ofca

nd

idate

pre

dic

tors

.T

yp

eB

use

sa

mod

else

lect

ion

app

roac

hb

ased

onin

-sam

ple

crit

eria

(ad

just

edR

2,AIC

,BIC

).MSPE

stat

isti

csin

bold

corr

esp

on

dto

a10%

sign

ifica

nce

level

,in

bol

dan

dit

alic

toa

5%le

vel.

32

Tab

le6:

Sta

tist

ical

pre

dic

tive

per

form

ance

test

s:re

sult

sof

the

in-s

ample

and

out-

of-s

ample

fore

cast

ing

test

sduri

ng

per

iods

ofnor

mal

and

bullis

hm

arke

ts.

Lag

stru

ctu

reIn

-sam

ple

perf

orm

an

ce

Ou

t-of-

sam

ple

perf

orm

an

ce

Fin

anci

als

Em−y 1

0d

iff(E

m−y 1

0)

RM

SE

Th

eil’

sU

RM

SE

Th

eil’

sU

MS

PE

MS

PE

stat.

vs

Ust

at.

vs

Fin

.Uncond.Model

Un

on

on

o39

5.17

1.00

327.

031.

00

TypeA

A1

firs

tla

gn

on

o39

4.64

0.99

938

3.32

1.17

-0.9

1A

2fi

rst

lag

firs

tla

gn

o39

4.12

1.00

388.

341.

19

-1.0

2-1

.33

A3

firs

tla

gn

ofi

rst

lag

392.

660.

9938

1.79

1.17

-0.8

90.8

9A

4n

ofi

rst

lag

no

395.

421.

0033

3.99

1.02

-1.8

43.29

A5

no

no

firs

tla

g39

5.84

1.00

327.

471.

00

-0.0

83.14

TypeB

B1

opti

m.

lags

no

no

388.

220.

9833

5.83

1.03

0.8

4B

2op

tim

.la

gs10

,11

no

382.

330.

9732

9.39

1.01

1.4

81.66

B3

opti

m.

lags

no

1038

1.88

0.97

325.

280.

99

1.82

2.10

B4

opti

m.

lags

1,2,

10,1

1n

o38

1.32

0.96

334.

441.

021.0

31.1

4B

5op

tim

.la

gsn

o1,

2,10

,11

381.

230.

9632

7.95

1.00

1.5

62.03

B6

no

10,1

1n

o39

4.05

1.00

324.

040.

99

1.4

62.77

B7

no

no

1039

4.05

1.00

320.

230.

982.22

2.98

Notes:

Nor

mal

and

bu

llis

hm

arke

tp

erio

ds

are

defi

ned

by

excl

ud

ing

the

top

3d

raw

dow

np

erio

ds

mea

sure

din

the

pri

ceev

olu

tion

of

the

S&

P50

0to

tal

retu

rnin

dex

from

the

over

all

sam

ple

per

iod

.It

ther

efor

eex

clu

des

Ju

lyan

dA

ug

1998,

the

per

iod

start

ing

from

Sep

t20

00to

Feb

2003

,an

dfr

omD

ec20

07to

Feb

2009

.M

od

els

ofty

peA

use

only

the

info

rmat

ion

onth

efi

rst

lag

of

can

did

ate

pre

dic

tors

.T

yp

eB

use

sa

mod

else

lect

ion

app

roac

hb

ased

onin

-sam

ple

crit

eria

(ad

just

edR

2,AIC

,BIC

).MSPE

stati

stic

sin

bold

corr

esp

ond

toa

10%

sign

ifica

nce

level

,in

bol

dan

dit

alic

toa

5%le

vel.

33

Tab

le7:

Sta

tist

ical

pre

dic

tive

per

form

ance

test

s:re

sult

sof

the

in-s

ample

and

out-

of-s

ample

fore

cast

ing

test

sduri

ng

per

iods

ofb

eari

shm

arke

ts.

Lag

stru

ctu

reIn

-sam

ple

perf

orm

an

ce

Ou

t-of-

sam

ple

perf

orm

an

ce

Fin

anci

als

Em−y 1

0d

iff(E

m−y 1

0)

RM

SE

Th

eil’

sU

RM

SE

Th

eil’

sU

MS

PE

MS

PE

stat.

vs

Ust

at.

vs

Fin

.Uncond.Model

Un

on

on

o68

4.14

1.00

693.

731.

00

TypeA

A1

firs

tla

gn

on

o60

7.39

0.89

705.

801.

021.1

2A

2fi

rst

lag

firs

tla

gn

o60

8.86

0.89

709.

711.

02

1.1

2-0

.75

A3

firs

tla

gn

ofi

rst

lag

608.

070.

8971

0.96

1.02

1.0

6-0

.80

A4

no

firs

tla

gn

o68

4.63

1.00

692.

931.

000.3

21.69

A5

no

no

firs

tla

g67

9.95

0.99

697.

161.

00

-1.2

71.65

TypeB

B1

opti

m.

lags

no

no

587.

870.

8662

0.24

0.89

2.26

B2

opti

m.

lags

10,1

1n

o58

3.47

0.85

618.

820.

892.46

1.0

1B

3op

tim

.la

gsn

o10

585.

540.

8662

0.66

0.89

2.39

0.5

5B

4op

tim

.la

gs1,

2,10

,11

no

584.

360.

8562

4.12

0.90

2.38

0.6

3B

5op

tim

.la

gsn

o1,

2,10

,11

584.

310.

8563

2.03

0.91

2.23

-0.0

1B

6n

o10

,11

no

680.

911.

0069

6.14

1.00

0.5

4-0

.12

B7

no

no

1068

2.22

1.00

696.

421.

00

0.3

2-0

.12

Notes:

Bea

rish

mar

ket

per

iod

sco

rres

pon

dto

the

top

3d

raw

dow

np

erio

ds

mea

sure

din

the

pri

ceev

olu

tion

of

the

S&

P500

tota

lre

turn

ind

ex.

Itin

clu

des

Ju

lyan

dA

ug

1998

(LT

CM

coll

apse

),th

ep

erio

dst

arti

ng

from

Sep

t20

00to

Feb

2003

(inte

rnet

bu

bb

le),

an

dfr

omD

ec20

07to

Feb

2009

(rec

ent

fin

anci

alcr

ash).

Mod

els

ofty

peA

use

only

the

info

rmat

ion

onth

efi

rst

lag

of

can

did

ate

pre

dic

tors

.T

yp

eB

use

sa

mod

else

lect

ion

app

roac

hb

ased

onin

-sam

ple

crit

eria

(ad

just

edR

2,AIC

,BIC

).MSPE

stati

stic

sin

bold

corr

esp

ond

toa

10%

sign

ifica

nce

level

,in

bol

dan

dit

alic

toa

5%le

vel.

34

Table 8: Economic predictive performance tests: result of the asset allocation application.

Strategy Passive B1 B2 B1 B2

threshold = 0 bps threshold = -30 bpsOverall

Cumul. Ret. 74,03% 137,96% 178,15% 213,74% 252,23%Mean 4,02% 7,49% 9,67% 11,61% 13,70%

Std 14,94% 14,83% 14,73% 14,61% 14,46%Skewness -0,93 -0,08 -0,12 -0,15 -0,18

Excess Kurt. 1,92 1,72 1,82 1,92 2,07Sharpe R 0,27 0,50 0,65 0,79 0,94

JK stat vs passive 2,68 4,25 6,34 7,87JK stat vs active 3,02 2,85

Turnover 0,00 3,26 3,58 2,17 2,39Normal/bullish

Cumul. Ret. 220,49% 136,21% 155,11% 195,34% 225,56%Mean 15,21% 9,39% 10,70% 13,47% 15,56%

Std 11,13% 11,65% 11,56% 11,31% 11,08%Skewness -0,10 -0,13 -0,18 -0,09 -0,05

Excess Kurt. -0,12 -0,14 -0,06 -0,15 -0,18Sharpe R 1,36 0,80 0,92 1,19 1,40

JK stat vs passive -4,75 -3,58 -1,73 0,35JK stat vs active 1,57 2,38

Turnover 0,00 3,10 3,38 1,79 2,07Bearish

Cumul. Ret. -146,46% 1,75% 23,04% 18,40% 26,67%Mean -37,40% 0,45% 5,88% 4,70% 6,81%

Std 20,42% 23,15% 23,09% 23,11% 23,07%Skewness -0,27 0,17 0,04 0,06 0,01

Excess Kurt. -0,04 0,29 0,28 0,28 0,29Sharpe R -1,80 0,02 0,25 0,20 0,29

JK stat vs passive 6,68* 7,60* 7,50* 7,77*JK stat vs active 3,39 1,82

Turnover 0,00 3,83 4,34 3,57 3,57

Notes: The overall period for this analysis starts from Jan 1992 to May 2010. Bearish market

periods correspond to the top 3 drawdown periods measured in the price evolution of the S&P

500 total return index. It includes July and Aug 1998 (LTCM collapse), the period starting from

Sept 2000 to Feb 2003 (internet bubble), and from Dec 2007 to Feb 2009 (recent financial crash).

Mean returns and standard deviations are in annual terms. Turnover is the number of portfolio

rebalancing per year. *due to negative Sharpe ratio for the passive strategy during these periods,

the statistic cannot be interpreted.

35

6.3 Figures

Figure 1: Sample of US treasury yields of maturity 1, 3, 5 and 10 years. Yields are generatedusing the Nelson-Siegel-Svensson model and factors from Gurkaynak et al. (2007). NBERrecessions are in grey.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

1985 1990 1995 2000 2005 2010

y1y3y5

y10

36

Figure 2: Estimates of parameter k.NBER recessions are in grey.

0

0.2

0.4

0.6

0.8

1

1985 1990 1995 2000 2005 2010

k

Figure 3: Estimates of parameter λ.

0

0.01

0.02

0.03

0.04

0.05

1985 1990 1995 2000 2005 2010

lam

bda

Figure 4: Estimates of parameter γ.

0

0.2

0.4

0.6

0.8

1

1985 1990 1995 2000 2005 2010

gam

ma

Figure 5: Estimates of weights ατ ofthe market portfolio.

0

0.2

0.4

0.6

0.8

1

1985 1990 1995 2000 2005 2010

alpha1alpha3alpha5

alpha 10

Figure 6: Annualized estimates of the expected return on the market portfolio Em, the10-year interest rate and the premium (Em − y10). NBER recessions are in grey.

-0.1

-0.05

0

0.05

0.1

0.15

1985 1990 1995 2000 2005 2010

Emy10

(Em - y10)

37

Figure 7: Cumulative difference of squared errors of out-of-sample prediction between a modelwith optimal lags of financial variables (B1) and the historical mean predictions. Periods ofmarket downturns are indicated in grey.

-0.02

-0.01

0

0.01

0.02

0.03

0.04

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010

Cum

ulat

ive

SS

E d

iffer

ence

Figure 8: Cumulative difference of squared errors of out-of-sample prediction for model B2.Periods of market downturns are indicated in grey.

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010

versus Uversus B1

38

Figure 9: Cumulative difference of squared errors of out-of-sample prediction for model B3.Periods of market downturns are indicated in grey.

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010

versus Uversus B1

Figure 10: Cumulative difference of squared errors of out-of-sample prediction for model(B6). Periods of market downturns are indicated in grey.

-0.02

-0.01

0

0.01

0.02

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010

versus Uversus B1

39

Figure 11: Cumulative returns of asset allocation strategies based on model out-of-sample forecasts.

The self-financing portfolio is rebalanced when models predict significant negative returns according

to a threshold of 30bps. Periods of market downturns are indicated in grey.

-0.5

0

0.5

1

1.5

2

2.5

3

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010

PassiveB1B2

40