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Multi-factor volatility and stock returns

Multi-factor volatility and stock

returns

Zhongzhi(Lawrence)He a,Jie Zhu b,?,Xiaoneng Zhu b

a Brock University,Ontario,Canada

b Shanghai University of Finance and Economics,and Shanghai Key Laboratory of Financial Information Technology,China

a r t i c l e i n f o

Article history:

Received13January2015 Accepted15September2015 Available online9October2015

JEL Classi?cation:

C58

G12

Keywords:

Multi-factor volatility

Cross-sectional returns

Out-of-sample predictability Asset allocation a b s t r a c t

In light of inconclusive evidence on the relation between market volatility and stock returns,this paper proposes a

model

and examines its impact on cross-sectional pricing.We also

evaluate and economic signi?cance of multi-factor volatility.We?nd

that the size and value dynamic factor earn signi?cant and positive variance

risk multi-factor volatility can signi?cantly improve the out-of-sample return

predictability with a positive economic gain in asset allocation.

1.Introduction

The question of whether volatility affects stock returns is an

enduring one in?nancial economics.Merton’s(1973)ICAPM inter-

prets the change in market volatility as the time-varying invest-

ment opportunity that characterizes a shift in the trade-off

between risk and return.In equilibrium,investors taking on addi-

tional risk should be compensated through higher expected return,

which implies a positive correlation in the volatility-return

relationship.However,empirical evidence on the volatility-return

relationship is still inconclusive within the single-factor volatility

framework.1The assumption of a single market volatility may lead

to model misspeci?cation if there exist additional sources of volatil-

ity risk that characterize a multi-dimensional change in the invest-

ment opportunity set.In this paper,we examine the effect of

multi-factor volatility on cross-sectional returns,and evaluate the

out-of-sample performance and economic signi?cance of multi-

factor volatility as compared to existing benchmarks.

Although?nancial theories do not specify the number and the

identity of volatility risk factors,the prominent Fama–French(FF)

three factors provide a natural guidance for our study on multi-

factor volatility.This is motivated by some empirical evidence that

the FF size and value factor are proxies for state variables that are

linked to fundamental risk in the economy(Liew and Vassalou,

2000;Vassalou,2003).We therefore investigate whether or not

the volatility of the size and value factor captures the macroeco-

nomic uncertainty risk by examining its impact on cross-

sectional returns,its out-of-sample performance,and economic

signi?cance.

To accomplish this,we develop a two-stage multi-factor volatil-

ity model using a dynamic Fama–French three-factor approach.

The?rst stage allows us to simultaneously estimate conditional

means(as an AR process)and conditional variances(as a GARCH

process)of the market,size,and value factor.The second stage

estimates variance risk premia of the three factors and tests the

cross-sectional pricing restriction.Non-arbitrage requires the exis-

tence of signi?cant variance risk premia that drive pricing errors

insigni?cant in the cross-section.In addition,we conduct out-of-

sample predictive regressions and asset allocation tests to ensure

that multi-factor volatility is non-elusive and carries a signi?cant

economic value.

https://www.doczj.com/doc/3d13886517.html,/10.1016/j.jbank?n.2015.09.013

?Corresponding author at:School of Finance,Shanghai University of Finance and

Economics,Guoding Road777,Shanghai200433,China.Tel.:+862165908403.

E-mail address:zhu.jie@https://www.doczj.com/doc/3d13886517.html,(J.Zhu).

1For time-series studies,Ghysels et al.(2005),Guo and Whitelaw(2006)and

Ludvigson and Ng(2007)?nd that aggregate volatility is positively related to market

expected returns,whereas Glosten et al.(1993),Brandt and Kang(2004)and

Christensen et al.(2010)document a negative relationship.In the cross-section of

stocks,Ang et al.(2006)and Adrian and Rosenberg(2008)?nd a negative price of

volatility risk,which is opposite to the prediction of the variance risk premium

literature(Bollerslev et al.,2009;Drechsler and Yaron,2011).

Our results on the multi-factor volatility model are https://www.doczj.com/doc/3d13886517.html,ing the Fama–French10?10size and book-to-market sorted portfolios as the primary test portfolios,2we?nd that sensi-tivities to the conditional variances of the three factors exhibit wide dispersions across the100portfolios.Furthermore,the conditional variances of the size and value factor carry highly signi?cant and positive risk premia,rendering the new model with a stronger asset-pricing performance than the competing models in terms of higher adjusted R-square,lower sum of squared errors,and lower pricing error statistics.To alleviate the potential model misspeci?ca-tion problem that might cause a spurious relationship,we test the pricing ability of multi-factor volatility using various asset-pricing models(i.e.,the CAPM,ICAPM,and CCAPM),and still?nd signi?cant results regardless of the models used.As for the out-of-sample fore-casting performance,the multi-factor volatility model beats the existing benchmarks with a wide margin.For example,the model produces lower forecasting errors for at least89out of the100port-folios in the forecast of one-period-ahead expected returns.This sug-gests that its superior pricing power is not caused by adding some random factors;instead,multi-factor volatility is an incremental source of pervasive risk factors that should be priced both in sample and out of sample.To gauge its economic signi?cance,our asset allo-cation results show that the multi-factor volatility model outper-forms other models by about1–3%annualized certainty equivalent returns(CER)in most cases,and the CER gap becomes even larger as investors exhibit more risk aversion.Our model also yields signif-icantly higher Sharpe ratios than most benchmark models.As for practical implications,the asset-allocation results suggest that by accounting for multi-factor volatility into portfolio decisions,asset managers can signi?cantly improve the risk-return trade-off of their portfolios.

The contribution of this paper is twofold.First,the paper sheds new light on the relationship between volatility risk and cross-sectional returns.In this literature,Ang et al.(2006)estimate a neg-ative price of volatility risk of approximatelyà1%per annum.3The idea is that since innovations in volatility are higher during reces-sions,stocks that co-vary with volatility are stocks that pay off in bad states,and these should require a smaller risk premium.In a sim-ilar vein,Adrian and Rosenberg(2008)decompose market volatility into the short-run and long-run component and?nd that both com-ponents constitute negative volatility risk premia.On the other hand, theoretical work of Bollerslev et al.(2009)and Drechsler and Yaron (2011)predict a positive variance risk premium that compensates for macroeconomic uncertainty risk.Han and Zhou(2011)take a cross-sectional pricing approach and?nd that stocks with higher var-iance risk premium earn higher expected returns.These studies esti-mate a positive price of volatility risk using options on the aggregate market and individual stocks.It is infeasible,however,to estimate multi-factor volatility using options https://www.doczj.com/doc/3d13886517.html,ing the cross-section of stock returns,rather than options data,allows us(1)to create portfo-lios of stocks that have different sensitivities to variations in multiple sources of volatility risk;(2)to control for a battery of other well-known risk-return effects.Our?nding of positive prices of multi-factor volatility risk supports the prediction of the variance risk premium literature.We show that each factor volatility is positively correlated with variance risk premium.This implies that multi-factor volatility can serve as empirical proxy for variance risk premium,but with a stronger pricing ability.Indeed,in the presence of highly sig-ni?cant and positive premia for the volatility of the size and value factor,the risk premium for the market volatility becomes insigni?-cant.This?nding is robust to in-sample pricing,out-of-sample fore-casting,alternative model and volatility speci?cations,and model test concerns raised by Lewellen et al.(2010).

Second,this paper contributes to the growing body of literature on out-of-sample return predictability by adding multi-factor vol-atility as a reliable set of predictors.In this strand of literature, Welch and Goyal(2008)and Simin(2008)argue that historical average returns forecast future returns better than predictive vari-ables for both the market and individual stocks.Campbell and Thompson(2008)show that imposing meaningful restrictions on coef?cients can largely improve the out-of-sample explanatory power of regressions.Rapach et al.(2010)recommend combining information from numerous economic variables to improve out-of-sample equity premium prediction.More recent studies propose technical indicators(Neely et al.,2014),time-varying coef?cients (Dangl and Halling,2012),sum-of-the-parts method(Ferreira and Santa-Clara,2011),and conditioning on market state(Henkel et al.,2011;Zhu and Zhu,2013;Zhu,forthcoming),among others, to predict stock market returns.In light of the literature,this paper argues that multi-factor volatility can signi?cantly improve the out-of-sample return predictability.The economic gains of multi-factor volatility are evaluated by a dynamic asset-allocation approach of Tu(2010).

The remainder of our paper is organized as follows.Section2 introduces the?rst stage of the multi-factor volatility model. Section3discusses the model estimation and estimated parame-ters.Section4conducts in-sample asset-pricing tests of the multi-factor volatility model as compared to the competing models.Section5performs out-of-sample forecasting tests of the multi-volatility model as compared to various benchmark models. Section6provides the asset allocation test to evaluate the economic value of the multi-factor volatility model.Section7 concludes the paper with future research directions.The appendix includes details of the estimation procedure and results from Monte Carlo simulations.

2.The multi-factor volatility model

The Fama–French three-factor model(1993,1996;FF3 hereafter)has been an empirical benchmark in the modern asset pricing literature.An open issue that has yet to be addressed in the literature is the relationship between the volatility of the Fama–French factors and expected stock returns.Fama and French(2012)explore the volatility of the market,size,and value premia of the three factors.Our paper formalizes this idea into con-ditional means and conditional variances of the three factors within a dynamic factor framework.

In the FF3model,three factors originate from the6portfolios formed on size and book-to-market ratio(BTM).He et al.(2010) (HHL hereafter)use the Kalman?lter to extract3latent dynamic factors from the6size and BTM portfolios by properly restricting factor loadings(betas).We adopt the HHL method and estimate the conditional means and conditional variances of the three fac-tors simultaneously.

Denote the6size and BTM portfolios as SL,SM,SH,BL,BM, and BH.Let R t??R SL;t R SM;t R SH;t R BL;t R BM;t R BH;t 0be a vector of demeaned excess returns on the6portfolios in month t.The latent MKT,SIZE,and BTM dynamic factors are given by D t?eD MKT;t D SIZE;t D BTM;tT0.With these notations,the dynamic factor model is represented in the state-space form,with the6observa-tion or measurement equations given by:

2Given the extreme returns in some corner portfolios,the100portfolios pose a

much greater challenge than the25portfolios(commonly used in the literature)in

terms of in-sample pricing and out-of-sample forecasting.In addition,using100

portfolios partially alleviates the model test problem raised by Lewellen et al.(2010).

3Another major?nding of Ang et al.(2006)is that the idiosyncratic volatility

relative to the Fama–French factors earns a signi?cantly negative risk premium,

which cannot be explained by the systematic market volatility.There is a contentious

debate between Ang et al.(2009),Bali and Cakici(2008)and Fu(2009)with regard to

the sign of idiosyncratic volatility risk premium.The asset-pricing role of idiosyn-

cratic volatility is beyond the scope of this paper.

Zhongzhi(Lawrence)He et al./Journal of Banking&Finance61(2015)S132–S149S133

R SL ;t

R SM ;t R SH ;t R BL ;t R BM ;t R BH ;t

666

64377777777777775?b SL b S b L

SM b S b M b SH

b S b H b BL

b B b L b BM

b B b M b BH

b B

b H

66666666666664377777777777775D MKT ;t D SIZE ;t D BTM ;t 26

66437775tw SL ;t

w SM ;t w SH ;t w BL ;t w BM ;t w BH ;t

6666666

643

77777777777775;e1T

where b s are the factor loadings on the unobserved factors and w t ??w SL ;t w SM ;t w SH ;t w BL ;t w BM ;t w BH ;t 0is the (6?1)vector of error terms.It is important to note that some equality restrictions must be imposed on the factor loading matrix so that the extracted factors have a predetermined interpretation of the market factor (D MKT ;t ),the size factor (D SIZE ;t ),and the BTM factor (D BTM ;t ).

The market factor is identi?ed from the 1st column of the factor loading matrix,with no restrictions on the factor loadings.Without imposing restrictions,the market factor captures common price dynamics across the 6portfolios consisting of market-wide assets.In this sense,the market factor can be interpreted as a composite index.

The size factor is used to capture the size-related variation in returns that is not captured by the market factor.The intuition for identifying the size factor is straightforward:assets within the same size group should have the same sensitivity to the size factor,whereas assets across size groups should have different sensitivities to the size factor.Hence,the equality restrictions imposed on the 2nd column of the factor loading matrix are:(1)the ?rst 3coef?cients in the 2nd column are all equal to b S so that it captures the small-size effect;(2)the last 3coef?cients in the 2nd column are all equal to b B so that it captures the large-size effect.

Similarly,the BTM factor can be identi?ed by imposing equal-ity restrictions on the 3rd column of the factor loading matrix.These restrictions re?ect the fact that assets within the same BTM group have the same sensitivity to the BTM factor and assets across BTM groups have different sensitivities to the BTM factor.Speci?cally,these equality restrictions are:(1)the ?rst and fourth coef?cients in the 3rd column are equal to b L so that it captures the low BTM effect;(2)the second and ?fth coef?cients in the 3rd column are equal to b M so that it captures the medium BTM effect;and (3)the third and sixth coef?cients in the 3rd column are equal to b H so that it captures the high BTM effect.

The three state equations are expressed as

D MKT ;t

D SIZ

E ;t D BTM ;t 2

64375?/MKT

0/SIZE 00

/BTM

64375D MKT ;t à1

D SIZ

E ;t à1D BTM ;t à1264375tv MKT ;t

v SIZE ;t v BTM ;t 2643

75;

e2T

where /s are the autoregressive coef?cients.With the conventional assumption of white noise errors,Eqs.(1)and (2)constitute the HHL dynamic factor model.To capture the conditional variance of the risk factors,we generalize the model and allow for conditional variance in risk factors.Let v t ??v MKT ;t v SIZE ;t v BTM ;t 0denote the (3?1)vector of state equation innovations,and

v t ?H 1=2

t g t ;

e3T

where H t is the conditional covariance matrix of v t .In addition,g t is the normalized residual,which is assumed to follow a standard nor-mal distribution.

We follow the convention and assume a diagonal H t ,where each of the diagonal elements follows a GARCH (1,1)process:

h MKT ;t

h SIZE ;t h BTM ;t

64375?x MKT x SIZE x BTM 264375ta MKT 000a SIZE 000a BTM 2

64375v 2MKT ;t à1

v 2SIZE ;t à1v 2BTM ;t à1

6643

775

tc MKT 000c SIZE 000c BTM 264375h MKT ;t à1h SIZE ;t à1h BTM ;t à1

75:e4T

The state-space model as in Eqs.(1)and (2)can be expressed in

a more succinct form:

R t ?BD t tw t ;e5TD t ?U D t à1tv t ;

e6T

where v t follows a GARCH process;B is a prespeci?ed constant factor-loading matrix whose columns have identifying restrictions on MKT,SIZE,and BTM;w t is a vector of idiosyncratic returns on the 6portfolios;and U is a diagonal autoregressive coef?cient matrix.Following the convention,we also assume that w t follows a normal distribution and w t is uncorrelated with v t .These restric-tions are given by:

E ?w t w 0

s ?

D ;for t ?s ;0;for t –s ;

e7TE ?w t v 0s ?0;for all t and s ;

e8T

where D ?diag ?r 2SL r 2SM r 2SH r 2BL r 2BM r 2

BH is a (6?6)diagonal covari-ance matrix of idiosyncratic risk for the observation equations.The conditional variance process can also be expressed in a more suc-cinct form:

h t j t à1?x tA v 2t à1tC h t à1;

e9T

where h t j t à1is the time-varying variance conditional on information set I t à1;x ??x MKT x SIZE x BTM 0is the (3?1)vector of unconditional variance;and A and C are diagonal coef?cient matrices.With these assumptions,Eqs.(5),(6),and (9)constitute the multi-factor vola-tility model.

Another identi?cation issue is associated with the GARCH pro-cess.Indeed,v t is unobservable,so the estimation of the model via the Kalman ?lter is not operable.Following Kim and Nelson (1999,Chapter 6)and Harvey et al.(1992),we circumvent the unobservable error issue by normalizing the parameters and impose the following restriction:

x j ?1àa j àc j ;

e10T

where j ?eMKT ;SIZE ;BTM T.

3.Model estimation

Under the assumptions in Eqs.(3),(7),and (8),the Kalman ?lter

is a statistically optimal procedure to extract the unobserved fac-tors from a ?nite set of observed returns.While using the Kalman ?lter,we simultaneously run the GARCH process according to Eq.(4)to estimate the conditional variance,h t ,with the normalization condition Eq.(10).The parameters in Eqs.(1),(2),and (4)are esti-mated via the Kalman ?lter and the GARCH procedure,with the state vector and variance being initialized by their unconditional means and variances of D t .The Kalman ?lter and the GARCH pro-cedure is then iterated for t ?1;...;T to recursively extract the conditional expectations of the dynamic factors,D t j t à1and D t j t ,and their conditional variance,h t j t à1.The procedure for extracting the factors and factor variances via the Kalman ?lter and the GARCH procedure is brie?y described in Appendix A .

S134Zhongzhi (Lawrence)He et al./Journal of Banking &Finance 61(2015)S132–S149

3.1.Data and estimation methodology

We obtain monthly returns on the6portfolios formed by sort-

ing on size and BTM from Kenneth French’s data library.4To com-

pute excess returns on the6portfolios,we subtract the1-month T-

bill rate.The sample used in this study ranges from January1961to

December2010,covering600months(50years).

After computing the excess returns on the6portfolios,we?rst

estimate26model parameters in the observation equations,the

state equations,and the GARCH process using the maximum

likelihood method.To this end,we run the Kalman?lter combined

with the GARCH model,as described in Kim and Nelson(1999),

with iterations until all parameters converge according to a

prespeci?ed precision.After obtaining these point estimates for

parameters,we run the Kalman?lter and the GARCH model

again to recursively extract the ex ante(forecast at month tà1) factors D t j tà1?eD MKT;t j tà1;D SIZE;t j tà1;D BTM;t j tà1T0,the ex post (updated at month t)factors D t j t?eD MKT;t j t;D SIZE;t j t;D BTM;t j tT0,as well as the conditional variance H t j tà1?eh MKT;t j tà1;h SIZE;t j tà1;h BTM;t j tà1T0, for t?1961=01—2010=12.

3.2.Empirical results

3.2.1.Descriptive Statistics

Panel A of Table1presents the mean and variance of the excess

returns(over the1-month T-bill rate,in percentage)for the6size-

and BTM-sorted portfolios.Note that the mean returns increase

monotonically with the BTM within each of the2size groups,

and the results are consistent with the well-known value effect.

For the portfolio of small and growth stocks(SL),we?nd that it

has the highest variance with only the second smallest mean

returns.This inverted risk-return feature for the SL group remains

a challenge for asset pricing.

The point estimates of the model parameters from the maxi-

mum likelihood method are presented in Panel B of Table1.Most

of the parameter estimates are statistically signi?cant at the1%

level,indicating that excess returns on the6portfolios conform

to the three-dimensional factor structure.

Speci?cally,the loadings on the market factor,D MKT,in the

small-stock group(b SL;b SM,and b SH)are signi?cantly larger than those in the big-stock group(b BL;b BM,and b BH).Since small stocks are relatively riskier than big stocks,this result implies that the

dynamic factor,D MKT,captures the market risk well.As for the load-

ings on the size factor,D SIZE,the magnitude of loadings in the big-

stock group(b B)is roughly twice as large as that of the small-stock

group(b S),which indicates that returns are more sensitive to the

size factor in the big-stock group.We also?nd that low(L)BTM

stocks have a signi?cantly negative loading(à1.291)on the BTM

factor,D BTM,while the high(H)BTM stocks have a signi?cantly

positive loading(0.856),providing evidence of the growth and

value effects.In contrast,the loading on the factor in the medium

BTM group(b M)is much smaller and insigni?cant.This is a desir-

able feature since it implies that in a portfolio with a medium level

of BTM,stock returns are not sensitive to the BTM factor.All

GARCH parameters(a j s and c j s)are signi?cant at any conventional level.Furthermore,the stationary condition is satis?ed:a jtc j<1 for any j,where j?eMKT;SIZE;BTMT.The results con?rm that

there exists the GARCH effect in the dynamic process of factor

variances.

Regarding the AR(1)coef?cients on the3factors,we?nd that both/

MKT

and/

BTM

are signi?cant at the1%level,while/

SIZE

is mar-ginally signi?cant around the level of10%.The result indicates that a change in the factor realizations by1standard deviation in the current month will affect the unobserved factor realizations by 7–30%in magnitude in the following month.To visualize the level of time variation in investors’optimal forecasts about the reward/ risk ratio,we draw a graph of the maximum ex ante squared Sharpe Ratio(SSR t j tà1)based on Eq.(11)over the sample period (1961/01–2010/12).

SSR t j tà1?D0t j tà1Pà1t j tà1D t j tà1;e11T

where D t j tà1?eD MKT;t j tà1;D SIZE;t j tà1;D BTM;t j tà1T0is the(3?1)vector of the ex ante dynamic factors,and P t j tà1is a(3?3)matrix including the mean squared error(MSE)of D t j tà1.

Fig.1illustrates that substantial variation exists in investors’forecast about their risk-return trade-off.The average value of the ex ante SSR is0.10,which is comparable to that of the FF3fac-tors(0.07).The ratio is the highest(0.632)in March2000,when the S&P500index reached its peak in our sample period.

We report in Table2descriptive statistics and correlation coef-?cients for the ex post factors D t j t?eD MKT;t j t;D SIZE;t j t;D BTM;t j tT0,the ex ante factors D t j tà1?eD MKT;t j tà1;D SIZE;t j tà1;D BTM;t j tà1T0,and the conditional variance H t j tà1?er2MTK;t j tà1;r2SIZE;t j tà1;r2BTM;t j tà1T0.For

notational simplicity,we denoteeD1

MKT;

;D1

SIZE

;D1

BTM

T0and

eD0

MKT;

;D0

SIZE

;D0

BTM

T0as the time series collection of D t j t and D t j tà1, respectively;ander2MTK;r2SIZE;r2BTMT0as the conditional variance estimated at time tà1.For comparison purposes,we also report the statistics for the FF3factors and the excess return on the CRSP value-weighted market(VWM)index.

The?rst column of Table2illustrates that the means of the ex post and ex ante estimates of the dynamic factors are all zero by construction.The means of the conditional variance are close to one for r2MTK and r2SIZE,but the mean of r2BTM is larger than one. The magnitude of these mean values is consistent with the vari-ance(STD squared)of the ex post dynamic factors

eD1

MKT;

;D1

SIZE

;D1

BTM

T0.The middle part of the table presents the corre-lation coef?cients between the dynamic factors,conditional vari-ance,and the FF3factors.We can see from the table that the FF 3factors are correlated with each other:the absolute values of the correlation coef?cients range from24%to31%.However,the ex post and the ex ante factors are weakly correlated with each other within each of the two groups:the absolute values of the cor-relation coef?cients are1–10%.This implies that each of D MKT;D SIZE, and D BTM independently captures speci?c aspects in return varia-tions,whereas MKT,SMB,and HML tend to jointly explain return variations.The correlation of conditional variance is positive and relatively high with each other within the group:the values of the correlation coef?cients are from58%to62%,which implies that the variance of the dynamic factors tends to move together.

As for the cross-correlation coef?cients between conditional variances,dynamic factors,and FF3factors,we note that condi-tional variances of the dynamic factors are weakly correlated with other variables,including the ex ante and ex post dynamic factors, and FF3factors.The absolute values of the correlation coef?cients range from1%to19%,and are less than10%in most cases.The weak correlation between conditional variances and other factors implies that the dynamics of conditional variances contain addi-tional information that is not captured by either dynamic factors or FF3factors.We aim to examine whether or not and to what extent this additional information about return dynamics is impor-tant in asset pricing and out-of-sample forecasting.

4The6portfolios are the original assets used by Fama and French(1993)to

construct the FF3factors.For robustness,we also use5?5portfolios sorted on size

and book-to-market ratio to extract dynamic factors.We?nd that dynamic factors

extracted from25portfolios are highly correlated with those from6portfolios.The

pricing performance is also very similar.

Zhongzhi(Lawrence)He et al./Journal of Banking&Finance61(2015)S132–S149S135

We also

plot the conditional volatility process

er MKT ;t j t à1;r SIZE ;t j t à1;r BTM ;t j t à1Tfor the three factors,as given by Fig.2.The ?gure illustrates that all three series of factor volatility exhibit a large time variation.Speci?cally,the market volatility peaked during the years of 1974,1987,2000,and 2007,which cor-responds to the Oil Crisis,the Black Monday stock market crash,the Dotcom Bubble burst,and the Subprime Crisis,respectively,indicating that market uncertainty or risk increased signi?cantly during these crisis periods.In contrast,the size factor volatility ?uctuates less than the market factor volatility in most time peri-ods.However,it increased sharply when the Dotcom Bubble https://www.doczj.com/doc/3d13886517.html,stly,the dynamic of the BTM factor volatility exhibits a large time-variation along with the market volatility.

3.2.2.Relation with variance risk premium

In the variance risk premium literature,Bollerslev et al.(2009)and Drechsler and Yaron (2011)predict a positive variance risk premium.Bollerslev et al.(2009)relate this result to the volatility of the variance of consumption which should be priced in an equilibrium model with recursive preference.In this subsection,we demonstrate the empirical relationship between variance risk premium and our dynamic volatility.Following Bollerslev et al.(2009),we de?ne variance risk premium as the implied volatility (ImV)minus the realized volatility (RV)on S&P 500index:

VRP t ?Im V t àRV t ;

e12T

in which RV t is calculated as follows

Table 1

Summary statistics and parameter estimates.Item

SM SH BL BM BH Panel A:descriptive statistics about the 6portfolios returns (in percent)formed on size and BTM Mean 0:4490:8651:0450:4330:5070:685Variance 48:0629:8931:8822:3119:10

22:14

Panel B:maximum likelihood estimates of the parameters of the model Parameter Estimated value Parameter Estimated value Parameter Estimated value b SL 6:297???

b M 0:192/SIZE à0:0684(0.439)(0.226)(0.0549)b SM 5:152???b H

0:856???/BTM

0:296???(0.357)(0.280)(0.0578)b SH 5:218???r 2SL 1:134???a MKT 0:0932???(0.363)(0.219)(0.0314)b BL 3:052???r 2SM 0:353???c MKT 0:827???(0.332)(0.0653)(0.0548)b BM 2:856???r 2SH 0:906???a SIZE 0:196???(0.309)(0.119)(0.0506)b BH 3:208???r 2BL 1:689???c SIZE 0:735???(0.317)(0.229)(0.0681)b S 1:680???r 2BM 1:387???a BTM 0:228???(0.477)(0.134)(0.0511)b B 3:144???r 2BH

1:757???c BTM

0:738???(0.464)(0.172)(0.0544)

b L

à1:291???/MKT

0:222???(0.346)

(0.0470)

Panel A of Table 1reports the means and variance of excess returns (over the 1-month T-bill rate,in percent)on the 6size and book-to-market (BTM)sorted portfolios over

600months (50years,1961/01–2010/12).Panel B reports the maximum likelihood estimates of the parameters for the model obtained using the Kalman ?lter and GARCH procedure.Standard errors are reported in the parentheses.The 6size-and BTM-sorted portfolios are notated as SL,SM,SH,BL,BM,and BH,where S and B denote ‘‘small”and ‘‘big”in ?rm size,respectively,and L,M,and H denote ‘‘low”,‘‘medium”,and ‘‘high”in the BTM ratio,respectively.R t ??R SL ;t R SM ;t R SH ;t R BL ;t R BM ;t R BH ;t 0is a e6?1Tvector of demeaned excess returns on the 6portfolios at month t ;and D t ??D MKT ;t D SIZE ;t D BTM ;t 0is a vector of zero-mean unobserved state vector at month t .The model is speci?ed as

(i)observation equations

R SL ;t R SM ;t R SH ;t R BL ;t R BM ;t R BH ;t 2

6666666643777777775?b SL b S b L

b SM b S b M b SH b S b H b BL b B b L b BM b B b M b BH

b B

b H

266666666437777777

MKT ;t

D SIZ

E ;t D BTM ;t 26

4375tw SL ;t w SM ;t w SH ;t w BL ;t w BM ;t w BH ;t 26666666643777777775;(ii)state equations

D MKT ;t D SIZ

E ;t D BTM ;t

4375?/MKT

000/SIZE 00

/BTM

264375D MKT ;t à1D SIZE ;t à1D BTM ;t à1

64375tv MKT ;t v SIZE ;t v BTM ;t

26437

5;and (iii)conditional variance equations

v t

?H 1=2

t g t ;h MKT ;t

h SIZE ;t h BTM ;t

64375?x MKT x SIZE x BTM 2643

75ta MKT 0

a SIZE

a BTM 2

64375v 2

MKT ;t à1v 2SIZE ;t à1v 2

BTM ;t à126643775tc

MKT 00

c SIZE 0

c BTM 264375h MKT ;t à1h SIZE ;t à1h BTM ;t à1

643

75:???

indicates coef?cients signi?cantly different from zero at the 1%level.

S136

Zhongzhi (Lawrence)He et al./Journal of Banking &Finance 61(2015)S132–S149

RV t?

X m

i?1r2

;e13T

where m is the number of return observations available during month t.The implied volatility is downloaded from Hao Zhou’s website,5which starts from January1990,and we extend the series to December2010.

Table3presents the correlation and regression results between variance risk premium and dynamic volatility.From Panel A we can see that all three dynamic volatility(h MKT;t;h SIZE;t,and h BTM;t) are positively correlated with variance risk premium.The correla-tion coef?cient between h MKT;t and variance risk premium,0.285,is the highest among the three.This is not surprising since we have no data on implied volatility of the SMB and HML portfolio,and thus it is impossible to construct variance risk premium on the SMB and HML portfolio.In addition,the regression results in Panel B shows that the estimated parameters for all three dynamic vola-tility(h MKT;t;h SIZE;t,and h BTM;t)are positive and signi?cant at the con-?dence level of1%.The result further con?rms that the dynamic volatility is positively correlated with variance risk premium.We thus conclude that our multi-factor dynamic volatility may serve as an empirical proxy for theoretically motivated variance risk premium.

Furthermore,our?nding of a positive relationship between var-iance risk premium and three-dimensional dynamic volatility empirically justi?es the theoretical foundation for using the vari-ance of the Fama–French factors as additional pricing factors (Chabi-Yo,2011).In a variance risk premium framework similar to Bollerslev et al.(2009),Chabi-Yo(2011)shows that stock returns are positively correlated with the variance of the Fama–French factors.This further shows that dynamic volatility proxies for variance risk premium.

4.Testing the multi-factor volatility model

4.1.Formulating the asset pricing tests

Our multi-factor volatility model assumes a factor structure for the excess return on a given asset or portfolio,R i(i?1;...;N) with the true factors identi?ed as D t??D MKT;t D SIZE;t D BTM;t 0whose prior(ex ante)expectations are extracted as D t j tà1??D MKT;t j tà1D SIZE;t j tà1D BTM;t j tà1 0,and posterior(ex post) expectations are extracted as D t j t??D MKT;t j t D SIZE;t j t D BTM;t j t 0using the Kalman?lter.By construction,both D t j t and D t j tà1have zero unconditional means.

Conditional on the information set,I tà1,the factor structure for the excess return at time t is

R i;t?E?R i;t j I tà1 tB1ieD tàD t j tà1Tt i;t;e14T

where B1

??b1

MKT;i

SIZE;i

BTM;i

is a(1?3)row vector of constant factor loadings on factor innovations for asset i.To be empirically testable,we also assume that Ee i;t j I tà1T?0and Ee i;t D t j I tà1T?0.

Table2

Descriptive statistics and correlations of the factors and volatility.

Correlation coef?cients

Item Mean STD D1

MKT

D1SIZE D1BTM D0MKT D0SIZE D0BTM r2MKT r2SIZE r2BTM MKT SMB HML VWM D1MKT0:001:021:00

D1SIZE0:000:94à0:011:00

BTM

0:001:12à0:080:101:00

D0MKT0:000:230:17à0:040:061:00

SIZE

0:000:082à0:130:11à0:080:011:00

D0BTM0:000:33à0:03à0:010:19à0:09à0:101:00

r2 MKT 1:030:430:050:030:090:010:080:061:00

r2 SIZE 0:991:04à0:040:100:15à0:01à0:060:050:621:00

BTM

1:401:770:010:010:190:050:030:190:580:611:00

MKT0:454:530:750:61à0:190:09à0:02à0:050:02à0:02à0:041:00

SMB0:273:170:84à0:53à0:120:17à0:16à0:020:04à0:080:010:311:00

HML0:402:94à0:230:090:970:03à0:060:150:070:120:15à0:30à0:241:00

VWM0:884:360:660:71à0:170:0730:01à0:040:02à0:02à0:020:990:17à0:271:00 This table reports means,standard deviations,and correlations of the estimates of the dynamic factors,conditional variances,and the FF3factors over the sample period (1961/01–2010/12).The variables are de?ned as follows:D t j t?eD1MKT;D1SIZE;D1BTMT0,a vector of the ex post(updated at month t)dynamic factors;D t j tà1?eD0M;D0S;D0BT0,a vector of ex ante(forecast at month tà1)dynamic factors;H t j tà1?er2MTK;r2SIZE;r2BTMT,the factor variance conditioned on month tà1;andeMKT;SMB;HMLT0,a vector of the FF3 factors.VWM is the CRSP value weighted monthly return series for the S&P500index.

5https://www.doczj.com/doc/3d13886517.html,/pubs/feds/2007/200711/200711abs.html.

Zhongzhi(Lawrence)He et al./Journal of Banking&Finance61(2015)S132–S149S137

Ang et al.(2006)and Adrian and Rosenberg(2008)examine the cross-sectional relationship between market volatility and stock returns.We generalize this line of research from a single market volatility to multi-factor volatility.Hence,we assume that E?R i;t j I tà1 is not only linearly related to D t j tà1,but also linearly related to h t j tà1??h MKT;t h SIZE;t h BTM;t 0.Thus,we have:

E?R i;t j I tà1 ?R itB0i D t j tà1tB h i h t j tà1;e15T

where i is the unconditional mean of R i;t,and B0

i ??b0

MKT;i

SIZE;i

BTM;i

and B h

i ??b h

MKT;i

b h

SIZE;i

b h

BTM;i

are respectively(1?3)row vector of

constant factor loadings on D t j tà1and h t j tà1.

As in HHL(2010),Eqs.(14)and(15)imply the following asset pricing test model:

R i;t?R itB?i D t j tà1tB1i D t j ttB h i h t j tà1t i;t;e16T

where B?

i ?B0

àB1

??eb0

MKT;i

àb1

MKT;i

Teb0

SIZE;i

àb1

SIZE;i

Teb0

BTM;i

àb1

BTM;i

T ?

?b?

MKT;i b?

SIZE;i

BTM;i

.In the sense that B?

captures the portion of B0

that

is not explained by B1

i ,B?

is labeled as the residual factor loadings.

Eq.(16)extends the dynamic factor pricing model(DFPM)of HHL (2010)by including conditional variances of the dynamic factors, so is named the multi-factor volatility pricing model(MFVPM).

The MFVPM implies the following relationship for the cross-sectional excess returns on assets:

R i?B?i k0tB1i k1tB h i k h;e17T

where k0??k0MKT k0SIZE k0BTM 0is a(3?1)vector of ex ante factor risk premia,k1??k1MKT k1SIZE k1BTM 0is a(3?1)vector of ex post factor risk premia,and k h??k h MKT k h SIZE k h BTM 0is a(3?1)vector of variance risk premia.We aim to test if k h?0in cross-sectional asset pricing.

4.2.Model tests with100size and book-to-market sorted portfolios

We use the two-pass regression procedure(Cochrane,2005, Chapter12;Brennan et al.,2004)to test the MFVPM.Test assets are the100size-and BTM-sorted portfolios.Two reasons account for the selection of the100portfolios instead of the frequently used 25size-and BTM-sorted portfolios.First,from a statistical perspec-tive,because the MFVPM includes9factors,the test based on the 100portfolios is more powerful than the test based on the25port-

Table3

Correlations and regression results of variance risk premium on dynamic factor

volatility.

h MKT;t h SIZE;t h BTM;t

Panel A:Correlation

VRP0:2850:1660:159

Panel B:Regression results

h MKT;t h SIZE;t h BTM;t Constant R2

VRP10:73???àà16:86???0:08

VRPà2:17???à25:20???0:03

VRPàà1:23???2:49???0:03

This table reports the correlation and regression results between variance risk

premium and corresponding dynamic factor volatility.Variance risk premium is

calculated as the implied volatility minus the realized volatility.Panel A reports the

correlation between variance risk premium and corresponding dynamic factor

volatility.Panel B reports the regression results of variance risk premium on cor-

responding dynamic factor volatility.The estimation period is from1990=01to

2010=12.???indicates signi?cance level at1%.

S138Zhongzhi(Lawrence)He et al./Journal of Banking&Finance61(2015)S132–S149

folios.Secondly,from the empirical test perspective,100portfolios contain return dynamics that are much harder to explain than the 25portfolios.So if the MFVPM can explain the returns of the 100portfolios signi?cantly better than other competing models,it validates the empirical power of the MFVPM.6The use of 100port-folios may also partially alleviate the concern of Lewellen et al.(2010)that model test results based on the traditional 25size-and BTM-sorted portfolios may be misleading.

4.2.1.Time-series regressions

In the ?rst pass,we specify a multiple time-series regression that provides estimates of the loadings with respect to the ex ante factors,the ex post factors,and the variance factors.More speci?-cally,the time-series regression is as follows:

R i ;t ?a i tX 3j ?1

b 1i ;j D j ;t j t tX 3j ?1

b ?

i ;j D j ;t j t à1tX 3j ?1

b h i ;j h j ;t j t à1te i ;t ;

e18T

where R i ;t is the excess return on portfolio i (i ?1;...;100)and

D j ;t j t ;D j ;t j t à1,and h j ;t j t à1are ex post factors,ex ante factors,and the variance factor j ;j ?eMKT ;SIZ

E ;BTM T,respectively.

Tables 4–6contain the estimates of the variance factor loadings (b h i ;j s)(i ?1;...;100)from the time-series regression for the 100portfolios.

The values of the variance loadings exhibit a wide range for all the three variance factors.For the variance of the market factor,the loadings range from à1:150to 1:535.For the variance of the size factor,the loadings range from à0:616to 0:539.For the variance of the BTM factor,the loadings range from à0:432to 0:213.The wide range of values indicates that the loadings on factor variances indeed capture a signi?cant portion of time-series variation in the portfolio returns (i.e.,the average adjusted-R 2for the time-series regressions is 0:85).For comparison,the average adjusted-R 2for the time-series regressions from the FF3is 0:78.

Note that some of the loadings on factor variances are statisti-cally insigni?cant at the conventional level,indicating that these loadings are imprecisely estimated.To make sure that the imprecisely estimated variance loadings are not random factors,we follow Petkova (2006)and perform a Monte Carlo experiment.

We also perform the same testing procedure on the usual 25portfolios.The sign of the estimates is the same.However,we lose statistical power due to the ?nite sample problem.The results are available upon request.

Table 4

Time-series regression results for loadings on volatility of the dynamic market factor.Item

b h MKT 1(low)

2345678910(high)Size

1(small)0:299à0:192à0:3500:225à0:350à0:566?à0:589??0:003à0:113à0:01220:722?à0:682??

0:002à0:698??à0:074à0:015à0:493?à0:3590:2400:11730:707??0:351à0:1430:0197à0:3030:0572à0:0520:1420:191à0:15940:3580:1220:900???

0:0529à0:059à0:300à0:2300:1150:1290:6265à0:089à0:528?0:0630:4510:337à0:0480:0930:409à0:0080:2986à0:222à0:3240:1470:2150:0030:4140:3800:959???

0:303à0:1227à0:272à0:2630:0480:2520:0530:837???

à0:0160:4250:3910:3548à0:635??0:1560:1880:492?0:542?0:3690:1500:2620:2711:535???

à0:3000:429?0:3510:0430:238à0:235à0:1270:514?0:677?0:36210(big)

à0:536??

à0:238

0:123

à0:251

0:089

à0:124

0:133

0:285

à0:581

à1:150?

This table reports the estimated loadings on variance of the dynamic market factor (h MKT ;t j t à1)for 100portfolios sorted on size and BTM ratio from the time-series regression.

R i ;t ?a i t

X 3j ?1

b 1i ;j D j ;t j t tX 3j ?1

b ?i ;j D j ;t j t à1tX 3j ?1

b h i ;j h j ;t j t à1tu i ;t ;where R i ;t is the excess return (in percentage)on portfolio i (i ?1;...;100)and D j ;t j t ;D j ;t j t à1,and h j ;t j t à1are ex post factors,ex ante factors,and the volatility factor j

(j ?MKT ;SIZE ;BTM ),respectively.?;??,and ???denote signi?cance at 10%,5%,and 1%levels,respectively.

Table 5

Time-series regression results for loadings on volatility of the dynamic size factor.Item

b h SIZE 1(low)

234

910(high)Size

1(small)à0:496??à0:138à0:211à0:1120:293??0:1820:0040:032à0:025à0:331???

2à0:0470:539???0:0600:330??0:210?à0:1350:390???0:362???à0:141à0:2223à0:198à0:337??

à0:101à0:0610:234??à0:458???

à0:196?à0:110à0:091à0:0764à0:0890:0390:032à0:0390:1440:1320:263??0:2310:011à0:369?5à0:0860:308??0:1350:1240:1110:0060:223?0:1200:490???à0:21860:0490:0460:2030:060à0:1070:303??à0:225?0:0290:405???0:26170:077à0:0210:240?0:0370:099à0:104à0:024à0:076à0:1710:37880:111à0:089à0:011à0:035à0:252?0:0920:1380:121à0:305??

à0:3859

à0:063à0:005à0:1510:067à0:0490:0870:218?0:0670:158à0:01110(big)

0:286???

0:139

à0:098

à0:338???

à0:130

à0:306?

0:049

à0:616???

à0:235

0:181

This table reports the estimated loadings on variance of the dynamic size factor (h SIZE ;t j t à1)for the 100portfolios sorted on size and BTM ratio from the time-series regression.

R i ;t ?a i t

X 3j ?1

b 1i ;j D j ;t j t tX 3j ?1

b ?i ;j D j ;t j t à1tX 3j ?1

b h i ;j h j ;t j t à1tu i ;t ;where R i ;t is the excess return (in percentage)on portfolio i (i ?1;...;100)and D j ;t j t ;D j ;t j t à1,and h j ;t j t à1are ex post factors,ex ante factors,and variance factor j

(j ?MKT ;SIZE ;BTM ),respectively.?;??,and ???denote signi?cance at the 10%,5%,and 1%levels,respectively.

Zhongzhi (Lawrence)He et al./Journal of Banking &Finance 61(2015)S132–S149

S139

This experiment describes the small sample empirical distributions of the different parameters of interest(Appendix B describes the procedure and reports the results).We conclude from the simulation results that the estimates of the factor loadings are unbiased.Their insigni?cance is due to the large standard errors.

4.2.2.Cross-sectional regressions

The second step of the two-pass regression procedure involves relating the average excess returns of all of the assets to their expo-sure to the risk factors in the model.Hence,we regress the sample mean of monthly excess returns on the factor loadings in the cross-section as follows:

R i?

j?1k1

^b1

i;j

j?1

^b?

i;j

j?1

k h

^b h

i;j

te i;e19T

where R i is the mean excess return on portfolio i(i?1;...;100),^b1i;j is the estimated loading on ex post factor j;j?eMKT;SIZE;BTMT;^b?i;j is the residual factor loading on ex ante factor j,and^b h i;j is the esti-

mated loading on the variance factors.k1

j ;k0

,and k h

represent the

prices of risk for the corresponding factors.If the cross-sectional variation in factor loadings is important in pricing the assets,then the prices of risk should be signi?cant.e i is the residual term that measures the pricing error of the MFVPM for portfolios i.

The estimation results for the factor risk premia and pricing errors for the DFPM(without the factor variance term),the MFVPM (with the factor variance term),as well as the FF3,are presented in Table7.Following Shanken(1992),we adjust for errors-in-variables and report the adjusted statistics.

Lewellen et al.(2010)point out that traditional asset pricing tests can be highly misleading.The apparently strong explanatory power(high cross-sectional R2and small pricing errors)provides weak support for the model.They suggest reporting the GLS R2 instead of the OLS R2to address this issue.Thus we also report the GLS R2for the cross-sectional regression.

For the MFVPM estimates,the sign and magnitude of the risk premia for the ex ante and ex post dynamic factors are similar to those from the DFPM.In addition,the levels of the risk premia for the three variance terms areà0:059;0:281and0.395,respec-tively,which are much larger than the risk premia of the dynamic factors.Furthermore,the risk premia for the variance of the size and BTM factor are signi?cant at the1%level,implying that the variance of the two factors are important pricing factors.By adding factor variances,the variations in the cross-sectional returns are now largely captured.The pricing errors statistics,including the

OLS R2eR2OLST,the sum of squared errors(SSE),and the quadratic term e0Rà1e,con?rm our conjecture.For example,the R2OLS for the MFVPM is67%,while it is57%for the DFPM and52%for the FF3.

The SSE and e0Rà1e of pricing errors are signi?cantly smaller for the MFVPM than for the DFPM and FF3.Not surprisingly,the GLS R2eR2GLSTis lower than the corresponding OLS R2for all three mod-

els,but R2

GLS

from MFVPM(18%)is still higher than that of the DFPM (10%)and FF3(6%).Thus,by adding factor variances,the MFVPM can explain an additional10%cross-sectional variations in expected returns.

4.3.Robustness check

In this section,we conduct further robustness tests to address several concerns that might lead to model misspeci?cations and spurious estimation results.7

4.3.1.Correlations between factor volatility

The diagonal assumption on H t is restrictive.It is possible that volatility of individual factors covaries with each other.8To address the issue,we allow the off-diagonal elements in H t to be non-zero and consider the following dynamic conditional correlation(DCC) multivariate GARCH model of Engle(2002)as an alternative to Eq.(4).9

H t?A t R t A t;e20Twhere A t is a diagonal matrix consisting of

????????????

h MKT;t

;

???????????

h SIZE;t

;

????????????

h BTM;t

Tand R t is the conditional correlation matrix as follows

Table6

Time-series regression results for loadings on volatility of the dynamic btm factor.

Item b h

SIZE

1(low)2345678910(high) Size

1(small)0:1320:0780:164?à0:007à0:0590:0170:107?à0:044à0:0440:025 2à0:0930:0300:1300:183??à0:0100:164???0:087à0:0210:018à0:026 30:0850:131?0:0860:062à0:0860:138??0:068à0:084à0:027à0:059 40:0020:035à0:0750:0360:036à0:010à0:213???à0:104à0:166??0:026 5à0:0890:133?0:021à0:087à0:048à0:018à0:161???à0:053à0:0830:162 6à0:0440:053à0:056à0:0280:021à0:162??à0:013à0:110à0:235???à0:085 70:191??0:0610:0140:037à0:058à0:0240:0950:067à0:053à0:432???80:185???0:116?0:1060:131?0:096à0:018à0:0920:0490:0200:007 90:173???0:141??0:052à0:0680:115?0:019à0:077à0:096à0:221???à0:064 10(big)à0:035à0:006à0:0060:213???0:143?0:008à0:015à0:0900:114à0:152 This table reports the estimated loadings on variance of the dynamic BTM factor(h BTM;t j tà1)for the100portfolios sorted on size and BTM ratio from the time-series regression.

R i;t?a itX3

j?1

i;j

D j;t j tt

j?1

i;j

D j;t j tà1t

j?1

b h

i;j

h j;t j tà1tu i;t;

where R i;t is the excess return(in percentage)on portfolio i(i?1;...;100)and D j;t j t;D j;t j tà1,and h j;t j tà1are ex post factors,ex ante factors,and variance factor j (j?MKT;SIZE;BTM),respectively.?;??,and???denote signi?cance at10%,5%,and1%levels,respectively.

7Lewellen et al.(2010)offer a number of suggestions to alleviate the problem of

model misleading.Our robustness checks are consistent with some approaches

recommended by them.

8We thank one referee for pointing out the issue.

9It is well known that multivariate models face the problem of curse of

dimensionality.The DCC-GARCH model can reduce parameters signi?cantly while

keeping the?exibility in constructing the conditional correlation between variables. S140Zhongzhi(Lawrence)He et al./Journal of Banking&Finance61(2015)S132–S149

R t ?1

q 12;t q 13;t q 12;t

1q 23;t q 13;t

q 23;t 1

643

75:e21T

Let h ij ;t ???????h i ;t p ??????

h j ;t p q ij ;t ;i ;j ?eMKT ;SIZE ;BTM Tbe the conditional covariance between h i ;t and h j ;t .In addition to use Eq.(4)to describe the dynamics of factor volatility,we assume h ij ;t also follows a GARCH process:

h ij ;t ?h ij ta ij v i ;t à1v j ;t à1tb ij h ij ;t à1:

e22T

Panel A of Table 8reports correlation estimates between factor volatility obtained with and without conditional correlation.It is clear that the two sets of h t s are highly correlated.For example,the correlation between volatility of the dynamic market factor with and without off-diagonal restriction is 0.81.The ?gure is even higher for the dynamic size and BTM factor.Thus,the obtained h t with these two speci?cations are quite similar.

Panel B of Table 8reports the estimation results from the two-state asset pricing tests by using h t obtained from https://www.doczj.com/doc/3d13886517.html,paring with the corresponding results in Table 7,we ?nd that the additional gain from allowing h t non-diagonal is marginal.The

overall model performance (R 2OLS ;R 2GLS ;SSE ,and e 0R 1

e )is robust,i.e.,the sign and signi?cance for all risk premia remain the same as in the previous case.Thus we may conclude that keeping h t diagonal does not materially alter our results.

4.3.2.Alternative testing assets

Lo and MacKinlay (1990)raise an issue of possible data-snooping biases when drawing inferences from samples of characteristic-sorted data.Furthermore,it is well-known that the FF3cannot explain the momentum effect.To alleviate the potential problem and to see if the MFVPM can capture the momentum effect,we now extend the scope of the test assets by adding 10momentum portfolios,which are also obtained from Professor Kenneth French’s website.The 10momentum portfolios are con-structed monthly (for month t portfolio,it is formed on month t à1)using NYSE prior (for month t ,the past returns from month t à2to month t à12are included)return decile breakpoints.

We again run the two-pass regressions for the extended 110portfolios.For brevity purposes,the estimation results from the time-series regression,which are quite similar to those for the ori-ginal 100portfolios,are not reported.In the following Table 9,we

Table 8

Estimation results from the DCC-GARCH model.

h MKT ;t

h SIZE ;t h BTM ;t Panel A:correlation h 1

MKT ;t

0:81--h 1SIZE ;t -0:97-h 1

BTM ;t -

0:95

Panel B:MVFPM with H 1t Risk premium k 1MKT

k 1SIZE

k 1BTM

k 0MKT k 0SIZE k 0BTM k h MKT

k h SIZE

k h BTM

0:230???

0:0280:322???0:0120:0050:072?à0:0890:275???0:368??

Pricing errors R 2

OLS

R 2GLS SSE e 0R à1e 0:687

0:187

2:02

129:18

Panel A of the table reports the correlation between h t with and without conditional correlation assumption,i.e.from Eq.(4)and Eq.(22),respectively.h j ;t refers to dynamic

volatility obtained without conditional correlation,while h 1

j ;t refers to dynamic volatility with conditional correlation,j ?eMKT ;SIZE ,BTM T.Panel B reports the results for the

asset pricing tests by using h t with conditional correlation as pricing factors.H 1t ?eh 1MKT ;t ;h 1SIZE ;t ;h 1BTM ;t Trefers to the factor volatility with conditional correlation.?,??

,and ???denote signi?cance at the level of 10%,5%,and 1%,respectively.

Table 7

Cross-sectional regression results for the MFVPM,DFPM,and FF3model.MFVPM

Risk premium k 1MKT

k 1SIZE k 1BTM

k 0MKT k 0SIZE k 0BTM k h MKT

k h SIZE

k h BTM

0:130???0:0510:233???0:0080:0020:062?à0:0590:281???0:395??

Pricing errors R 2OLS R 2GLS SSE e 0R à1e 0:6720:1822:07133:25DFPM

Risk premium k 1MKT

k 1SIZE k 1BTM

k 0MKT

k 0SIZE k 0BTM 0:134???0:0390:216???à0:0050:0010:045

Pricing errors R 2OLS R 2GLS SSE e 0R à1e 0:5670:0962:52180:30

FF3

Risk premium k MKT

k SIZE

k BTM

0:428??0:261??0:468???Pricing errors

R 2OLS R 2GLS SSE e 0R à1e 0:520

0:059

2:88

190:85

This

table contains the results from the cross-sectional regressions for the 100portfolios sorted on size and BTM ratio.The regression is R i ?P 3j ?1k 1j ^b 1i ;j tP 3j ?1k 0j ^b ?i ;j tP 3j ?1k h j ^b h i ;j te i ,where R i is the mean excess return on portfolio i (i ?1;...;100),^b 1i ;j is the estimated loading on ex post factor j (j ?MKT ;SIZE ;BTM ),^b ?i ;j and ^b h i ;j are the estimated factor loadings on variances.k 1j ;k 0j ,and k h j are the prices of risk.SSE is the sum of squared errors,and e 0R à1e is the quadratic test statistic,which follows a v 2eN àk Tdistribution under the null hypothesis that the pricing errors are jointly zero,where N is the number of assets and k is the number of

pricing factors.?,??,and ???denote signi?cance at 10%,5%,and 1%level,respectively.The test statistics are Shanken (1992)corrected.The R 2OLS and R 2GLS refer to the OLS R 2

and

GLS R 2,respectively.The R 2GLS is calculated as in Lewellen et al.(2010).The data are from 1961=01to 2010=12.

Zhongzhi (Lawrence)He et al./Journal of Banking &Finance 61(2015)S132–S149

S141

report the estimates of the factor risk premia and pricing errors for the MFVPM,DFPM,and FF3.

The results in Table9are largely consistent with those in

Table7.The estimates for all k h s are positive,and k h

SIZE and k h

BTM

remain signi?cant.The pricing errors statistics show that the MFVPM again out-performs both the DFPM and https://www.doczj.com/doc/3d13886517.html,paring

the R2

OLS

in Tables7and9,we note that for the FF3the value is low-ered from52%in Table7to only38%in Table9.For the DFPM,it is

lowered from57%to47%.R2

OLS

for the MFVPM is also deceased to 58%,but still higher than that of FF3and DFPM.If we compare

GLS

,it is more than obvious that MFVPM out-performs the other

two.The R2

GLS

for MFVPM is now17%,which is1%lower than in

Table7.However,the R2

GLS

for DFPM and FF3is lowered by2%or

more.

Hou et al.(2015)argue that the FF3fails to account for a wide array of asset pricing anomalies.They construct a new investment-based model and?nd that many of the anomalies that prove challenging for the FF3model can be captured by the new model.Following their spirit,we consider alternative testing port-folios that may present a challenge for the FF3and dynamic fac-tors.These testing portfolios are constructed based on?ve well-known anomalies,including deciles sorted on idiosyncratic volatil-ity(IVOL,Ang et al.,2006),net stock issues(NSI,Pontiff and Woodgate,2008),composite issues(CEI,Daniel and Titman, 2006),abnormal corporate investment(ACI,Titman et al.,2004), and total accruals(TA,Richardson et al.,2005).Please refer to Hou et al.(2015)for more details of the portfolio construction. Due to data availability,monthly returns from1972:01to 2010:12are collected for the testing purpose.

The cross-sectional regression results are reported in Table10. It is not surprising that the FF3cannot explain the anomalies at all.The R2from OLS is only0.162,the R2from GLS is even worse,

only0.024.The results from DFPM are better,the R2

OLS

is0.378,

and R2

GLS

reaches0.108.Again,the MFVPM performs the best among

the three models,the R2

OLS is0.478,and the R2

GLS

is0.159,signi?-

cantly higher than that of DFPM and FF3.More importantly,the

risk premia on volatility of the dynamic size and BTM factor,k h

SIZE

and k h

BTM

,remain signi?cant.It con?rms that the good performance

of MFVPM is not achieved by testing the model on speci?c portfo-

lios.It can explain anomalies much better than the competing

benchmark models.

4.3.3.Spurious relationship

Model misspeci?cation is a more serious issue that might lead

to spurious?ndings of variance risk premium.Given that factor

variances are extracted from the dynamic factor structure,it is a

major concern that the positive premia of the variance risk can

only be found within the MFVPM framework.What about the per-

formance of multi-factor volatility that does not rely on the

MFVPM speci?cation?To alleviate such a model misspeci?cation

problem,we consider various asset-pricing models including the

CAPM,ICAPM,and CCAPM,and test the sign and signi?cance of

multi-factor volatility under these alternative models.If variance

risk is correlated with other known risk factors,then its pricing

effect should be subsumed by other factors and become insigni?-

cant in the cross-sectional test.On the other hand,if variance risk

premium tends out to be still signi?cantly positive when tested

with various pricing models,then the concern of model misspeci-

?cation is largely relieved.The results in Table11show that our

variance based pricing factors are robust to different model speci-

?cations.When tested under the CAPM,ICAPM(Petkova,2006),

and CCAPM(Kang et al.,2011),risk premia of the size and BTM fac-

tor variances remain positive and signi?cant.This implies that,

regardless of model speci?cations,the variances of the size and

BTM factor are additional sources of priced risks above and beyond

traditional risk factors in ICAPM and CCAPM.

The results discussed so far have been cast in variance form.

However,the volatility,as a nonlinear monotone transforms of

the variance,is often used as an alternative measure of risk.Fol-

lowing Bollerslev et al.(2009),we take the square root of our con-

ditional variances of the market,size,and book-to-market factor,

MFVPM

Risk premium k1

MKT k1

SIZE

BTM

MKT

SIZE

BTM

k h

MKT

k h

SIZE

k h

BTM

0:113???0:0280:295???à0:094???à0:0160:461???0:2250:428???0:678???Pricing errors R2

OLS

R2GLS SSE e0Rà1e

0:5750:1703:41142:68

DFPM

Risk premium k1

MKT k1

SIZE

BTM

MKT

SIZE

BTM

0:099??0:0620:249???à0:0520:058??0:298???

Pricing errors R2

OLS

R2GLS SSE e0Rà1e

0:4690:0563:93193:83

FF3

Risk premium k MKT k SIZE k BTM

0:0480:521???0:713???

Pricing errors R2

OLS

R2GLS SSE e0Rà1e

0:3780:0324:74239:3

This table contains the results from the cross-sectional regression for the100portfolios sorted on size and BTM ratio plus the10momentum portfolios.The sample mean of monthly excess return(in percentage)is regressed on the estimated factor loadings obtained from the time-series regression as in the equation.

i?X3

j?1

^b1

i;j

j?1

^b?

i;j

j?1

k h

^b h

i;j

te i;

where i is the mean excess return on portfolio i(i?1;...;110),^b1i;j is the estimated loading on ex post factor j(j?MKT;SIZE;BTM),^b?i;j is the residual factor loading on ex ante factor j,and^b h i;j is the estimated loading on variance factors.k1j;k0j,and k h j represent the prices of risk for corresponding pricing factors.adj R2is the adjusted R2.SSE is the sum of squared errors,and e0Rà1e is the quadratic test statistic,which follows a v2eNàkTdistribution under the null hypothesis that the pricing errors are jointly zero,where N is

the number of assets and k is the number of pricing factors.??,and???denote signi?cance at the5%,and1%level,respectively.The test statistics are Shanken(1992)corrected. The R2OLS and R2GLS refer to the OLS R2and GLS R2,respectively.The R2GLS is calculated as in Lewellen et al.(2010).The data are from1961=01to2010=12.

and check the pricing effects of the volatility measures.Table 11indicates little change in the sign and signi?cance of volatility risk premia.Actually the test statistics con?rm that using standard deviation instead of variance as the proxy for volatility brings an even better result (higher R 2s and smaller pricing errors).So our results are robust to different volatility measures.

Finally,to check if the sample size affects the results,we divide the sample into two equal subsamples (thus each subsample contains 300observations),and re-do the two-pass estimation procedure.10We ?nd that the variation in sample size does not change the sign and signi?cance for most estimates,especially for those of factor vari-ances,indicating that the MFVPM is consistent for various samples.

5.Out-of-sample forecasting performance

We have illustrated that the conditional variance of the dynamic factors plays a key role in explaining the variations in cross-sectional asset returns,and the MFVPM performs signi?-cantly better than the DFPM and the FF3in pricing assets.How-ever,these results are obtained in the context of in-sample tests.A more relevant question for researchers and practitioners is whether the model can provide more accurate forecasts in a vari-ety of out-of-sample settings.Our out-of-sample tests are related to a growing body of literature that examines out-of-sample pre-dictability of stock returns (Campbell and Thompson,2008;Welch and Goyal,2008;Simin,2008;Rapach et al.,2010;He et al.,2015;and others).In particular,Simin (2008)argues that while conditional asset pricing models out-perform unconditional models in the in-sample tests,they perform poorly in the out-of-sample tests.He also documents that simply using the historical average return gives the best performance in the one-step-ahead forecasts for most of his testing assets,compared to other sophis-ticated models.In this section,we focus on the performance of the MFVPM in the out-of-sample context.Speci?cally,we make pair-wise comparisons of the accuracy in one-step-ahead forecasts.We consider the following ?ve competing models:(i)CAPM;(ii)the time-series average of historical monthly returns for the train-ing period (to be de?ned later)for each portfolio (MEAN);(iii)the

conditional FF3model (CFF3);(iv)the dynamic factor pricing model (DFPM);and (v)the multi-factor volatility pricing model (MFVPM).We include the low cost forecast (historical average return)and the CAPM in the analysis for the purpose of comparison with the results of Simin (2008).We consider the CFF3model given the concern that the better performance of the MFVPM may be achieved by simply adding more factors.The CFF3model is a conditional beta version of the FF3,where each of the loadings on the FF3s is modeled as a linear function of the lagged one-month T-bill rate.Finally,we include the DFPM to compare the forecasting performance.

We again use the 100size-and BTM-sorted portfolios as our test assets.The one-step-ahead out-of-sample forecasts of the asset returns (in real terms)are performed for the last 240obser-vations for each model,according to the period of 1991/01–2010/12(i.e.the last 20years).For the construction of each fore-cast at month t ,the model is re-estimated using data up to month t à1(the training period),and the estimated parameters are then used to construct the predicting return at month t .The squared forecast error for model i at t ;u i ;t ,is de?ned as:

u 2i ;t ?eR t àb R i ;t T2;

e23T

where R t is the realized return at t and b R

i ;t is the forecast of the return for model i at t .For pair-wise comparisons between compet-ing models,we compute the average of differences in the mean squared forecast errors (MSFEs),,as in Simin (2008):

d ?1T X T t ?1

eu 21;t àu 22;t T;e24T

where u 21;t and u 22;t are squared forecast errors for models 1and 2,respectively,as computed in Eq.(23),and T is the total number of forecasts eT ?240T.Following Simin (2008),we compute DM-STAT based on Diebold and Mariano (1995)to test the null hypoth-

esis of H 0:d ?0.The test statistic is de?ned as:

DM àSTAT ?d

????????????2p b f d e0T

r ;

e25T

where b f d e0Tis a consistent estimator of the spectral density of

?u 2

1;t àu 22;t at frequency 0and 2p =T is the length of time required

To save space,the results are not presented but available upon request.

MFVPM

Risk premium k 1MKT k 1SIZE

k 1BTM k 0MKT k 0SIZE

k 0BTM

k h MKT k h SIZE

k h BTM

0:07180:173???0:248?0:01250:0406?à0:061?

0:05130:114??0:483??

Pricing errors R 2OLS R 2GLS SSE e 0R à1e 0:4780:1590:7475:75DFPM

Risk premium k 1MKT k 1SIZE

k 1BTM

k 0MKT k 0SIZE

k 0BTM

0:06200:176???0:249???0:01700:0419???à0:0529

Pricing errors R 2OLS R 2GLS SSE e 0R à1e 0:3780:1080:7982:56

FF3

Risk premium k MKT

k SIZE

k BTM

0:537???à0:369?0:415??Pricing errors

R 2OLS R 2GLS SSE e 0R à1e 0:162

0:024

1:11

127:56

This table contains the results from the cross-sectional regressions for the 5deciles (50portfolios in total)sorted on various anomalies,including NSI,IVOL,CEI,ACI,and TA.

The regression is R i ?P 3j ?1k 1j

1i ;j

tP 3j ?1k 0j

^b ?i ;j

tP 3j ?1k h j

^b h i ;j

te i ,where R i is the mean excess return on portfolio i (i ?1;...;50),^b 1i ;j

is the estimated loading on ex post factor j (j ?MKT ;SIZE ;BTM ),^b ?i ;j and ^b h i ;j are the estimated factor loadings on variances.k 1j ;k 0j ,and k h j are the prices of risk.SSE is the sum of squared errors,and e 0R à1e is the quadratic test statistic,which follows a v 2eN àk Tdistribution under the null hypothesis that the pricing errors are jointly zero,where N is the number of assets and k is the number of

pricing factors.?,??,and ???denote signi?cance at 10%,5%,and 1%level,respectively.The test statistics are Shanken (1992)corrected.The R 2OLS and R 2GLS refer to the OLS R 2

and

GLS R 2,respectively.The R 2GLS is calculated as in Lewellen et al.(2010).The data are from 1972=01to 2010=12.

for the process to complete a full cycle.To examine if the average of is 0,we also conduct the overall t -test for each of the pairs using d s obtained from the 100portfolios.

Table 12reports the results for an evenly-spaced subsample of 100portfolios,with the interval to be 4.Thus,the reported portfo-lios are Portfolios 1;5;9;...;97,which includes 25portfolios in total.For completion,we also report several statistics for the whole 100portfolios.These statistics include the number of positive d s,the number of negative the number of signi?cantly positive d s,the number of signi?cantly negative d s,as well as the mean of for the 100portfolios.

A ?rst glance of Table 12indicates that the MFVPM beats the existing benchmarks by a wide margin.The MFVPM produces lower forecasting errors for at least 89out of the 100portfolios in the forecast of one-period-ahead expected returns.Take an example of the comparative results between the CAPM and the MFVPM presented in the second column.The number of positive and negative d s shows that 96d s are positive and 4d s are negative,meaning that the MFVPM provides lower forecasting errors than

the CAPM in 96out of 100portfolios.The number of signi?cant positive and negative d s indicates that when the CAPM performs worse than the MFVPM in forecasting,it generates signi?cantly lar-ger forecasting errors.On the other hand,when the MFVPM per-forms worse,it generates insigni?cant forecasting errors compared to the CAPM.In fact,none of those negative s is signif-icant.This is true not only for the reported 25portfolios,but also for all the 100portfolios.In contrast,the number of signi?cantly positive d s is 37.The mean of d s is also positive and signi?cant.All of these statistics indicate that,when compared to the CAPM,the MFVPM provides more accurate forecasts.

The forecasting performance for the MFVPM,as compared to other models,follows a similar pattern.It is worth noting that the perfor-mance of the CFF3is inferior to that of the MFVPM.Thus,the better predictive power of the MFVPM is not driven by using additional fac-tors in the model.In addition,from the mean of d s,we see that the sta-tistic between DFPM and MFVPM is the smallest,but is still signi?cantly positive.This result implies that the DFPM is better in forecasting when compared to the CAPM,the historical mean,and the CFF3model;this is not the case,however,for comparison with

Cross-sectional regression results for various models.CAPM +volatility Risk premium k MKT k H MKT k H SIZE

k H BTM 0:438

???0:09980:540??1:67???Pricing errors R 2OLS R 2GLS SSE e 0R à1e 0:2570:0474:41101:92ICAPM +volatility Risk premium k MKT k U DIV

k U TERM k U DEF k U RF

0:392

???

à2:80?10à4

5:78?10à4

5:09?10à5

à0:00168???Risk premium k U SIZE

k U BTM

k H MKT

k H SIZE

k H BTM

0:256???0:522???à0:07300:215???0:462???

Pricing errors R 2OLS R 2GLS SSE e 0R à1e 0:6400:1332:00123:08CCAPM +volatility Risk premium k cons k H MKT

k H SIZE

k H BTM

0:531

0:572??0:614??1:300???Pricing errors

R 2OLS R 2GLS SSE e 0R à1e 0:195

0:0584:98159:76MFVPM with standard deviation Risk premium k 1MKT

k 1SIZE

k 1BTM

k 0MKT k 0SIZE

0:130???

0:05160:230???0:005à0:001

Risk premium k 0BTM

k ??h p MKT

k ??h p SIZE

k ??h p BTM

0:0550

à0:02800:112???0:105??Pricing errors

R 2OLS

R 2GLS SSE e 0R à1e 0:685

0:174

1:99

130:48

This table contains the results from the cross-sectional regressions for the 100portfolios sorted on size and BTM ratio.

1.The ?rst regression (CAPM):R i ?k MKT ^b i ;MKT tP 3j ?1k h j ^b h i ;j te i ,where R i is the mean excess return on portfolio i (i ?1;...;100),^b i ;MKT is the estimated loading on market

returns,^b

h i ;j

are the estimated factor loadings on variances.k MKT and k h j

are the prices of risk.2.The second regression (ICAPM):

The factors are chosen as by Petkova (2006).We run the following VAR regression:

DIV t

TERM t DEF t RF t R HML ;t R SMB ;t

B B B B B B B B @1

C C C C C C C C A ?A DIV t à1TERM t à1DEF t à1RF t à1R HML ;t à1R SMB ;t à1

0B B B B B B B B @1

C C C C C C C C A tu t And the residual vector u t is collected as factors.3.The third regression (CCAPM):

We follow Kang et al.(2011)and use the consumption growth rate as the factor.4.The fourth regression (MFVPM with standard deviation).

We use the standard deviation instead of the conditional variance as the proxy for volatility in the two-pass regression.

SSE is the sum of squared errors,and e 0R à1e is the quadratic test statistic,which follows a v 2eN àk Tdistribution under the null hypothesis that the pricing errors are jointly zero,where N is the number of assets and k is the number of pricing factors.??and ???denote signi?cance at 5%and 1%level,respectively.The test statistics are Shanken

(1992)corrected.The R 2OLS and R 2GLS refer to the OLS R 2and GLS R 2,respectively.The R 2

GLS is calculated as in Lewellen et al.(2010).The data are from 1961=01to 2010=12.

the MFVPM.Finally,we note that using the historical mean provides similar forecasting performance when compared to the CAPM and the CFF3model,which is consistent with the results in Simin (2008).In summary,the MFVPM out-performs the competing

models in the out-of-sample tests.By adding multi-factor volatility risk in the return generating process,the model has largely increased its predictive power.The better forecasting performance is not driven by using additional factors.Therefore,multi-factor volatility risk is advocated as a reliable set of predictors in the literature on out-of-sample return predictability.

6.Asset allocation

In the previous section,we show that the MFVPM has positive out-of-sample forecasting power for portfolio returns.However,the statistical predictive power does not mechanically imply eco-

nomic signi?cance.We assess the economic value of the predictive power of the MFVPM using conditional mean–variance analysis,which underlies most common measures of portfolio management.Speci?cally,we investigate the implications of the MFVPM for a manager who makes dynamic portfolio decisions.

Following Tu (2010),we denote the ?rst and second moments of the returns obtained from the different models as in Section 5by E t eR t t1Tand Var t eR t t1T,respectively,where R t t1?er 1;t t1;r 2;t t1;ááá;r N ;t t1;N ?100T,the optimal portfolio solves:

max x x 0E t eR t t1Tà12

cx 0

Var t eR t t1Tx

;e26T

where c is the coef?cient of risk aversion.

We apply the following constraints for the optimal weight vector x :

Portfolio 930:3471:1580:4470:0357Portfolio

0:3851:0230:5060:131Mean of d

0:628???3:276???0:794???0:238???Number of positive ds 9610010089Number of negative ds

40011Number of signi?cantly positive ds 37546127Number of signi?cantly negative 0

This table presents the pair-wise comparison of the accuracy in one-step-ahead forecasts from the competing models using the 100size-and BTM-sorted portfolios.The competing models are as follows:(i)the Capital Asset Pricing Model,CAPM;(ii)the historical mean,which is the time-series average of monthly returns for the training period;(iii)the Conditional FF3model,CFF3;(iv)the dynamic factor pricing model,DFPM;v)the multi-factor volatility pricing model,MFVPM.The one-step-ahead out-of-sample forecasts of asset returns (in real terms)are performed for the last 240observations for each model,according to the period of 1991/01–2010/12,i.e.the last 20years in the sample.For the construction of each forecast at month t ,the model is re-estimated using data up to month t à1(the training period),and the estimated parameters are then used to construct the predicted return at month t .The average of differentials in the mean squared forecast errors (MSFEs),is de?ned as.

d ?

1T X

T t ?1eu 21;t

àu 22;t T;where u 21;t and u 2

2;t are squared forecast errors for model 1and 2,respectively,as computed in Eq.(23),and T is the total number of the forecasts eT ?240T.The DM-STAT to test the null hypothesis of H 0:d ?0is given as:

DM àSTAT ?d

????????????2p b f d e0T

r ;

where b f d e0Tis a consistent estimator of the spectral density of ?u 21;t àu 2

2;t at frequency 0and 2p =T is the length of time required for the process to complete a full cycle.To examine if the average of is 0,we also conduct the overall t -test for each of the pairs using obtained from the 100portfolios.To save space,only s for evenly-selected 25portfolios are reported.In addition,the number of signi?cantly positive s,the number of signi?cantly negative as well as the mean of for the all 100portfolios are also reported.?;??,and ???denote signi?cance at 10%,5%,and 1%level,respectively.

1.P N i ?1x i ?1,which is the budget constraint.

2.à0:26x i 60:2,or 06x i 60:2.11

The objective function from Eq.(26)de?nes the ?rst perfor-mance measure for asset allocation,which is termed the certainty equivalent returns (CERs)for the various models.The second mea-sure is the Sharpe ratio associated with the optimal weight:

SR ?x 0E t eR t t1T

???????????????????????????????x 0Var t eR t t1Tx

p :

e27T

The sample period for the performance measure is from 1991/01to 2010/12,for a total of 240months.For each month t ,we estimate the conditional expected return and conditional vari-ance from the forecasted returns for a rolling window of a length of

60months for each model.The expected returns and variance are used to obtain the optimal weight via Eq.(26).The obtained opti-mal weight is then applied to the realized returns to calculate the CERs and SR,Eqs.(26)and (27),to compare the performance for the different models.The results are given in Table 13.

Panel A of Table 13presents the results without the short-selling constraint (à0:26x i 60:2).Following Tu (2010)and other literature,we choose the coef?cient of risk aversion c to be 1;5,and 10.The values in the table measure the performance difference (CERs or SRs)between MFVPM and its competing models.The posi-tive value means that MFVPM out-performs the competing model in terms of higher certainty equivalent return or higher Sharpe ratio of the resulting portfolio.We ?nd that,for both CER and SR measures,the MFVPM yields superior performance at the 1%sig-ni?cance level for all other models except for the DFPM that is marginally (c ?5)or insigni?cantly (c ?10)better in the CER measure.For example,the ?rst column of the CER panel indicates that an investor with c ?1can obtain an annualized excess return of 3%if he switches from CAPM to MFVPM.Note that with the

The ?rst inequality implies that short-selling is allowed,while the leverage ratio is restricted to be 20%.The second inequality restricts short-selling.In both cases,the maximum weight for each asset in the portfolio is 20%so that no single asset will dominate in the portfolio.These constraints are often applied in investment practices.

Asset allocation results for out-of-sample forecasts:model comparison.Panel A:results without short-sell constraints

Comparison vs.MFVPM for CERs

CAPM Historical mean CFF3DFPM c ?13:00???0:87???1:15???0:61???c ?521:85??9:44???0:72???à0:54?c ?10

46:20???16:81???2:25???à0:00

Comparison vs.MFVPM for sharpe ratio CAPM

Historical mean CFF3DFPM c ?10:098???

0:063???

0:070???

0:058???c ?50:185???0:151???0:067???0:066???c ?10

0:221???

0:174???

0:076???

0:082???

Panel B:Results with short-sell constraints

Comparison vs.MFVPM for CERs

CAPM

Historical mean CFF3DFPM c ?10:10???0:02???0:02???0:01???c ?50:48???0:34???0:06???0:17???c ?100:97???

0:91???

0:00

0:35???

Comparison vs.MFVPM for sharpe ratio CAPM

Historical Mean CFF3DFPM c ?10:006??0:0020:008???0:010???c ?50:012???0:0050:0010:007??c ?100:019???

0:008??

à0:002

0:009???

This table presents comparison results between various models.The optimal portfolio solves the following function:

max x x 0E eR t t1j I t Tà1

cx 0VAR eR t t1j I t Tx :Alternatively,the Sharpe Ratio of the optimal portfolio is given by.

SR ?x 0E eR t t1j I t T???????????????????????????????????x 0VAR eR t t1j I t Tx

p ;

subject to:

X N i ?1

x i ?1;and

à0:26x i 60:2for Panel A ;and 0

6x i 60:2for Panel B ;where

x i ?

V i

:V i is the dollar amount invested in asset i ,and V is the dollar amount of the portfolio.The above weight inequality restricts the minimum and maximum weight an individual asset can take in the portfolio,note that short-selling is allowed for the results in Panel A,where the leverage is allowed to less than 20%.Short-selling is not allowed for the results in Panel B.In both cases,the maximum weight for an individual asset is 20%.After the optimal weight is obtained via maximizing the CER Eq.(26),it is applied to the realized returns at time t ,and the CER and Sharpe ratio are used as two measures for asset allocation.The difference between MFVPM and other models are then reported in Panel A and B.The risk aversion coef?cient c ?1;5;10.?;??,and ???indicate signi?cance level at 10%,5%,and 1%,respectively.The data are from 1991/01to 2010/12.

increased value of c,the gain from MFVPM is more apparent.With c?10,the investor can obtain an excess return of46%annually with MFVPM over CAPM.Flemming et al.(2001)argue that under the CER framework as in Eq.(26),the change in volatility plays a much more important role than the change in expected returns. Since MFVPM can capture the dynamic risk in volatility,it can pro-vide higher CERs when c is higher,in which case volatility indi-cates a more important position.Furthermore,The SR differences are signi?cant regardless of the value of c.For example,when c?5,the MFVPM can earn an excess return of0:2%annually per unit of risk compared to CAPM.

The results with the short-selling constraint(06x i60:2)are provided in Panel B of Table13.The CERs for MFVPM are now bet-ter than any other models,regardless of the value of c,though eco-nomic gains look smaller with additional constraints.For example, when c?10,MFVPM can earn an excess return of0.97%annually over CAPM.And the SR for MFVPM beats other models in most cases.The asterisks in Table13con?rm that the performance dif-ference is signi?cant.

In summary,we conclude from the above results that(1)there are signi?cant economic gains of the MFVPM in asset allocation; and(2)multi-factor volatility is important for investment practi-tioners making asset allocation decisions.

7.Concluding remarks

In light of inconclusive evidence on the volatility-return rela-tionship in the cross-section of stocks,this paper examines the in-sample pricing and out-of-sample forecasting effects of the con-ditional variances of the market,size,and value factor using a multi-factor volatility approach.We demonstrate that the condi-tional variances of the size and value factor earn highly signi?cant and positive risk premia that add additional pricing power to the existing asset-pricing models.In addition,the multi-factor volatil-ity model can forecast expected returns more accurately than the existing benchmark models and signi?cantly enhance out-of-sample asset-allocation gains.The superior in-sample pricing power and out-of-sample predictive ability of the multi-factor vol-atility model come from the conditional variances of the size and value dynamic factor.In the presence of highly signi?cant and positive risk premia of the size and value factor variances,the risk premium of the market volatility vanishes.Our?ndings are robust to various speci?cations and model test concerns.

Our paper generalizes a single market volatility risk(Ang et al., 2006;Adrian and Rosenberg,2008)to multi-factor volatility risk in cross-sectional pricing.The?ndings of signi?cant and positive var-iance risk premia support the theoretical prediction of the variance risk premium literature(Bollerslev et al.,2009;Drechsler and Yaron,2011).Furthermore,the paper adds multi-factor volatility as a new set of reliable predictors to the growing body of literature on out-of-sample return predictability(Campbell and Thompson, 2008;Welch and Goyal,2008;Simin,2008;Rapach et al.,2010 and others).The strong in-sample and out-of-sample performance of multi-factor volatility calls for future research on its relation to a multi-dimensional source of macroeconomic uncertainty risk.

Acknowledgements

We would like to thank Carol Alexander(the managing editor), Morten Nielsen and Paresh Narayan(guest editors)and two anon-ymous referees for providing valuable comments that helped to improve the paper.We are also grateful to Dennis Alba,Bent Jesper Christensen,Bing Han,Sun Hwee Huat,Guangzhong Li,Rahman Shahidur,Frank Stephan,Timo Terasvirta,Neng Wang,Harold Zhang,participants at the1st Conference on Recent Developments in Financial Econometrics and Applications at Deakin University, and seminar participants at Aarhus University,Central University of Finance and Economics,Shanghai University of Finance and Eco-nomics,and Southwestern University of Finance and Economics for helpful comments.He gratefully acknowledges the support from the Eastern Scholarship of Shanghai and the support from the Social Sciences and Humanities Research Council of Canada (SSHRC).Xiaoneng Zhu acknowledges the?nancial support from the Natural Science Foundation of China(Grant No.71473281). Appendix A.The Kalman?lter and the GARCH procedure

At the beginning of month t,investors make prior assessments about the conditional means and variances of the unobserved fac-tors(D t)based on the information set I tà1.Thus,with the condi-tional distribution D t j I tà1$NeD t j tà1;P t j tà1T,and from Eq.(2),we have the following properties:

D t j tà1?U D tà1j tà1;eA:1TP t j tà1?U P tà1j tà1U0th t;eA:2Twhere h t follows the GARCH process equation:

h t j tà1?xtA v2tà1tC h tà1:

Table B.1

Monte Carlo experiment for the loading of volatility on the market factor.

b h

MKT

Null 2.510509097.5 Portfolio10:299à0:612à0:2890:2920:9001:210 Portfolio5à0:350à0:933à0:739à0:3560:0300:229 Portfolio9à0:113à0:566à0:415à0:1110:1880:348 Portfolio130:002à0:700à0:4580:0010:4660:710 Portfolio17à0:493à1:040à0:856à0:494à0:1370:059 Portfolio210:707à0:0030:2510:7081:1741:420 Portfolio25à0:303à0:813à0:642à0:3040:0310:210 Portfolio290:191à0:384à0:1850:1930:5760:778 Portfolio330:9000:3160:5110:9001:2871:487 Portfolio37à0:230à0:774à0:583à0:2330:1270:316 Portfolio41à0:089à0:679à0:475à0:0890:2970:495 Portfolio450:337à0:203à0:0220:3380:6920:886 Portfolio49à0:008à0:596à0:394à0:0040:3810:581 Portfolio530:147à0:467à0:2560:1440:5450:747 Portfolio570:380à0:1980:0120:3820:7430:935 Portfolio61à0:272à0:904à0:684à0:2760:1410:358 Portfolio650:053à0:515à0:3120:0480:4170:625 Portfolio690:391à0:305à0:0590:3870:8471:090 Portfolio730:188à0:421à0:2060:1890:5880:794 Portfolio770:150à0:447à0:2390:1450:5390:754 Portfolio81à0:300à0:788à0:621à0:2990:02390:201 Portfolio850:238à0:301à0:1140:2360:5810:765 Portfolio890:677à0:0080:2310:6731:1221:360 Portfolio930:123à0:358à0:1890:1210:4370:599 Portfolio970:133à0:491à0:2710:1340:5440:758 This table and the following two tables present results from a Monte Carlo exper-iment,designed as follows.First,factor loadings for100size-and BTM-sorted portfolios are estimated from time-series regressions.The risk premia associated with the loadings are determined in cross-sectional regressions.The OLS estimates from this two-pass procedure are taken as given,i.e.,the null hypothesis is that the estimated model is correct.Then,10,000time series of returns are generated under the estimated model.The simulated returns are used to estimate a new set of factor loadings for each asset,as well as factor risk premia and adjusted R2from the cross-sectional regressions.In this manner,the small sample distributions of the betas, the cross-sectional risk premia,and the adjusted R2s are generated.The distribution of the betas with respect to variance of dynamic market factor is reported in this table.The column‘‘null”implies the values under the null hypothesis.The columns of2.5,10,50,90,97.5refer to the2.5%,the10%,the50%,the97.5%critical value, respectively.

Using the Kalman?lter,we update the inference about D t based on I t.In a conditional distribution D t j I t$NeD t j t;P t j tT,the ex post expectations of the true factors(D t)are given by:

D t j t?D t j tà1tK t e t j tà1;eA:3Twhere K t?P t j tà1B0Rà1t j tà1is the Kalman gain matrix,and R t j tà1?eBP t j tà1B0tDTis the variance–covariance matrix of the forecast error,e t j tà1?R tàR t j tà1.The conditional expectations of R t on I tà1;R t j tà1are:

R t j tà1?BD t j tà1:eA:4TThe updated variances of D t;P t j t,is given by

P t j t?P t j tà1àK t BP t j tà1:eA:5TGiven these speci?cations,the maximum likelihood method can be applied to estimate the parameters.Let h be the parameter vec-tor,that is,h?eB;U;D;A;CT;26parameters in total.By assump-tion,the forecast errors follow a normal distribution,thus,the log-likelihood function is as follows:

lneLeR t j hTT?àT

lne2pTà

1R T

t?1

lnedeteV tTTt

;eA:6T

where V t is the conditional covariance matrix of R t:

V t?BP t j tà1B0tD:eA:7TThe parameter vector h is chosen to maximize the log-likelihood function from Eq.(A.6).

Appendix B.Monte Carlo simulation

The Monte Carlo exercise is designed as follows.First,given the estimated factor loadings b b s obtained from the time-series regres-sions with the extracted dynamic factors and factor variance,we simulate10;000time series of returns under the estimated model as follows:

i;t

?a it

j?1

b b1

i;j

D j;t j tt

j?1

b b0

i;j

D j;t j tà1t

j?1

b b h

i;j

h j;t j tà1te?i;t:eB:1T

where i?1;...;100,j?eMKT;SIZE;BTMT,and e?i;t stands for the bootstrap residual.e?i;t is generated from the sample distribution. The simulated returns are then used to estimate a new set of factor loadings for each asset:

i;t

?b a it

j?1

^b1

i;j

D j;t j tt

j?1

^b0

i;j

D j;t j tà1t

j?1

^b h

i;j

h j;t j tà1te i;eB:2T

where^b0i;j;^b1i;j,and^b h i;j are the estimated factor loadings using the simulated returns.i?1;...;100,and j?eMKT;SIZE;BTMT.In this way,the small sample distributions of the betas are generated.

We obtain the?nite distribution of the betas on the factor var-iance from the time-series regressions equation(B.1).The null hypothesis is that the model in Eq.(B.1)is correct.From the simu-lation results,as reported in Tables B.1,B.2,B.3,we?nd that,on one hand,the loadings on the variance of the dynamic factors are unbiased:the50%critical value of each distribution is very close to the value under the null hypothesis.On the other hand,the betas are very dispersed.For example,b b h MKT for the smallest size and lowest BTM portfolio has a value ofà0.612with a2.5%critical value and1.21with a97.5%critical value.This implies that the estimates of the factor loadings are unbiased,and the imprecise results are due to large standard errors.

The results indicate that,despite the large standard errors,the cross-sectional stage of the MFVPM captures the effect of signi?-cant risk premiums associated with the variance of the dynamic factors.The Monte Carlo experiment con?rms that the variance

Table B.2

Monte Carlo experiment for the loadings of volatility on the size factor.

b h

SIZE

Null 2.510509097.5 Portfolio1à0.496à0.905à0.770à0.494à0.220à0.072 Portfolio50.2930.0220.1170.2960.4720.567 Portfolio9à0.025à0.243à0.166à0.0270.1130.185 Portfolio130.060à0.266à0.1530.0560.2710.378 Portfolio170.3900.1390.2280.3890.5570.650 Portfolio21à0.198à0.523à0.411à0.2010.0130.130 Portfolio250.234à0.0010.0760.2340.3910.471 Portfolio29à0.091à0.361à0.267à0.0930.0820.176 Portfolio330.032à0.249à0.1510.0310.2130.301 Portfolio370.2630.0150.0970.2640.4340.519 Portfolio41à0.086à0.359à0.268à0.0860.0940.189 Portfolio450.111à0.140à0.0520.1110.2750.363 Portfolio490.4900.2150.3110.4880.6680.763 Portfolio530.203à0.0760.0190.2040.3890.482 Portfolio57à0.225à0.487à0.398à0.223à0.0510.032 Portfolio610.077à0.223à0.1210.0790.2700.374 Portfolio650.099à0.160à0.0720.1010.2700.360 Portfolio69à0.171à0.496à0.384à0.1700.0410.156 Portfolio73à0.011à0.290à0.195à0.0100.1720.270 Portfolio770.138à0.141à0.04610.1370.3220.412 Portfolio81à0.063à0.292à0.215à0.0630.0860.165 Portfolio85à0.050à0.300à0.209à0.0510.1100.195 Portfolio890.158à0.158à0.0490.1590.3640.476 Portfolio93à0.098à0.330à0.248à0.1000.0490.130 Portfolio970.049à0.244à0.1450.0500.2380.342 This table reports the distribution of the betas with respect to the variance of dynamic size factor.The column‘‘null”implies the values under the null hypoth-esis.The columns of2.5,10,50,90,97.5refer to the2.5%,the10%,the50%,the 97.5%critical value,respectively.Table B.3

Monte Carlo experiment for the loadings of volatility on the BTM factor.

b h

BTM

Null 2.510509097.5 Portfolio10.132à0.099à0.0180.1300.2850.363 Portfolio5à0.059à0.208à0.156à0.0600.0370.089 Portfolio9à0.044à0.158à0.119à0.0440.0310.072 Portfolio130.130à0.0460.0160.1320.2470.307 Portfolio170.087à0.049à0.0020.0870.1780.225 Portfolio210.085à0.092à0.0310.0860.2030.263 Portfolio25à0.086à0.215à0.170à0.0860.0010.046 Portfolio29à0.027à0.173à0.123à0.0260.0700.120 Portfolio33à0.075à0.224à0.173à0.0740.0230.076 Portfolio37à0.213à0.349à0.303à0.213à0.124à0.078 Portfolio41à0.089à0.236à0.186à0.0890.0090.061 Portfolio45à0.048à0.185à0.138à0.0490.0400.088 Portfolio49à0.083à0.231à0.181à0.0850.0140.067 Portfolio53à0.056à0.207à0.155à0.0580.0420.096 Portfolio57à0.013à0.160à0.108à0.0120.0810.129 Portfolio610.1910.0300.0860.1900.2960.350 Portfolio65à0.058à0.200à0.152à0.0590.0340.083 Portfolio69à0.053à0.231à0.169à0.0530.0610.123 Portfolio730.106à0.0440.0080.1060.2040.259 Portfolio77à0.092à0.242à0.190à0.0930.0060.061 Portfolio810.1730.0460.0910.1740.2550.299 Portfolio850.115à0.0160.0290.1160.2010.249 Portfolio89à0.221à0.393à0.334à0.220à0.107à0.049 Portfolio93à0.006à0.127à0.085à0.0050.0750.116 Portfolio97à0.015à0.174à0.118à0.0170.0870.141 This table reports the distribution of the betas with respect to the variance of dynamic BTM factor.The column‘‘null”implies the values under the null hypoth-esis.The columns of2.5,10,50,90,97.5refer to the2.5%,the10%,the50%,the 97.5%critical value,respectively.

of the dynamic factors plays a critical role in explaining the varia-tions in the cross-sectional returns.

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建设工程经济计算公式汇总

一级建造师《建设工程经济》计算公式汇总 1、等额支付系列的终值、现值、资金回收和偿债基金计算等额支付系列现金流量序列是连续的，且数额相等，即：），，，，常数（n t A A t 321 ①终值计算（即已知A 求F ） i i A F n 11 ）（ ②现值计算（即已知A 求P ） n n n i i i A i F P ）（）（）（ 1111 ③资金回收计算（已知P 求A ） 111 n n i i i P A ）（）（ ④偿债基金计算（已知F 求A ） 1 1 n i i F A ）（ 2、有效利率的计算包括计息周期有效利率和年有效利率两种情况。（2）年有效利率，即年实际利率。年初资金P ，名义利率为r ，一年内计息m 次，则计息周期利率为 m r i 。根据一次支付终值公式可得该年的本利和F ，即： m m r P F 1 根据利息的定义可得该年的利息I 为： 111m m m r P P m r P I 再根据利率的定义可得该年的实际利率，即有效利率i eFF 为： 11i eff m m r P I 3、财务净现值 t c t n t i CO CI FNPV 10 式中 FNPV ——财务净现值；（CI-CO ）t ——第t 年的净现金流量（应注意“＋”、“-”号）； i c ——基准收益率； n ——方案计算期。 4、财务内部收益率（FIRR ——Financial lnternaI Rate oF Return ）其实质就是使投资方案在计算期内各年净现金流量的现值累计等于零时的折现率。其数学表达式为：

t t n t FIRR CO CI FIRR FNPV 10 式中 FIRR ——财务内部收益率。 5、投资收益率指标的计算是投资方案达到设计生产能力后一个正常生产年份的年净收益总额（不是年销售收入）与方案投资总额（包括建设投资、建设期贷款利息、流动资金等）的比率： %100 I A R 式中 R ——投资收益率； A ——年净收益额或年平均净收益额； I ——总投资 6、总投资收益率总投资收益率（ROI ）表示总投资的盈利水平 %100 TI EBIT ROI 式中 EBIT-----技术方案正常年份的年息税前利润或运营期内平均息税前利润； TI------技术方案总投资包括建设投资、建设期利息和全部流动资金。 7、资本金净利润率（ROE ）技术方案资本金净利润率（ROE ）表示技术方案盈利水平 %100 EC NP ROE 式中 NP----技术方案正常年份的年净利润或运营期内年平均净利润，净利润＝利润总额-所得税 EC----技术方案资本金 8、静态投资回收期 ·当项目建成投产后各年的净收益（即净现金流量）均相同时，静态投资回收期计算： A I P t 式中 I ——总投资； A ——每年的净收益。 ·当项目建成投产后各年的净收益不相同时，静态投资回收期计算：流量出现正值年份的净现金的绝对值上一年累计净现金流量现正值的年份数累计净现金流量开始出 1－ t P 9、借款偿还期余额盈余当年可用于还款的盈余当年应偿还借款额的年份数借款偿还开始出现盈余 1－d P 10、利息备付率利息备付率＝息税前利润/计入总成本费用的应付利息。式中：息税前利润——即利润总额与计入总成本费用的利息费用之和（不含折旧、摊销费 11、偿债备付率偿债备付率＝（息税前利润加折旧和摊销-企业所得税）/应还本付息的金额式中：应还本付息的资金——包括当期还贷款本金额及计入总成本费用的全部利息；息税前利润加折旧和摊销-企业所得税＝净利润+折旧+摊销+利息 12、总成本 C ＝C F ＋C u ×Q C ：总成本；C F ：固定成本；C u ：单位产品变动成本；Q ：产销量量本利模型

建筑工程工程量计算公式

、平整场地：建筑物场地厚度在±30cm 以内的挖、填、运、找平。 1、平整场地计算规则 (1)清单规则：按设计图示尺寸以建筑物首层面积计算。 (2 )定额规则：按设计图示尺寸以建筑物外墙外边线每边各加2 米以平方米面积计算。 2、平整场地计算公式 S= (A+4 ) X ( B+4 ) =S 底+2L 外+16 式中：S———平整场地工程量；A———建筑物长度方向外墙外边线长度；B———建筑物宽度方向外墙外边线长度；S 底———建筑物底层建筑面积；L 外———建筑物外墙外边线周长。该公式适用于任何由矩形组成的建筑物或构筑物的场地平整工程量计算。二、基础土方开挖计算开挖土方计算规则 ( 1 )、清单规则：挖基础土方按设计图示尺寸以基础垫层底面积乘挖土深度计算。 ( 2)、定额规则：人工或机械挖土方的体积应按槽底面积乘以挖土深度计算。槽底面积应以槽底的长乘以槽底的宽，槽底长和宽是指基础底宽外加工作面，当需要放坡时，应将放坡的土方量合并于总土方量中。 2 、开挖土方计算公式： (1) 、清单计算挖土方的体积：土方体积=挖土方的底面积X挖土深度。 (2) --------------------------------------------------------------------------------------------- 、定额规则：基槽开挖：V= (A+2C+X H) HXL。式中：V --------------------------------------------------------- 基槽土方量；A ----------- 槽底宽度；C———工作面宽度；H———基槽深度；L———基槽长度。. 其中外墙基槽长度以外墙中心线计算，内墙基槽长度以内墙净长计算，交接重合出不予扣除。基坑体积；A—基坑开挖：V=1/6H[A X B+a X b+(A+a) x(B+b)+a xb]。式中：V 基坑上口长度；B———基坑上口宽度；a———基坑底面长度；b———基坑底面宽度。

建设项目工程经济计算公式汇总

.. 一级建造师《建设工程经济》计算公式汇总 1、等额支付系列的终值、现值、资金回收和偿债基金计算等额支付系列现金流量序列是连续的，且数额相等，即：），，，，常数（n t A A t 321①终值计算（即已知 A 求F ） i i A F n 1 1 ）（②现值计算（即已知 A 求P ） n n n i i i A i F P ）（）（）（1111 ③资金回收计算（已知 P 求A ） 1 1 1 n n i i i P A ）（）（④偿债基金计算（已知 F 求A ） 1 1 n i i F A ）（2、有效利率的计算包括计息周期有效利率和年有效利率两种情况。（2）年有效利率，即年实际利率。年初资金P ，名义利率为r ，一年内计息m 次，则计息周期利率为 m r i 。根据一次支付终值公式可得该年的本利和F ，即： m m r P F 1 根据利息的定义可得该年的利息I 为： 1 1 1 m m m r P P m r P I 再根据利率的定义可得该年的实际利率，即有效利率 i eFF 为： 1 1 i eff m m r P I 3、财务净现值 t c t n t i CO CI FNPV 1 式中 FNPV ——财务净现值；（CI-CO ）t ——第t 年的净现金流量（应注意“＋” 、“-”号）； i c ——基准收益率；n ——方案计算期。

.. 4、财务内部收益率（FIRR ——Financial lnternaI Rate oF Return ）其实质就是使投资方案在计算期内各年净现金流量的现值累计等于零时的折现率。其数学表达式为： t t n t FIRR CO CI FIRR FNPV 10 式中 FIRR ——财务内部收益率。 5、投资收益率指标的计算是投资方案达到设计生产能力后一个正常生产年份的年净收益总额（不是年销售收入）与方案投资总额（包括建设投资、建设期贷款利息、流动资金等）的比率： % 100I A R 式中 R ——投资收益率； A ——年净收益额或年平均净收益额；I ——总投资 6、总投资收益率总投资收益率（ROI ）表示总投资的盈利水平 % 100TI EBIT ROI 式中 EBIT-----技术方案正常年份的年息税前利润或运营期内平均息税前利润； TI------技术方案总投资包括建设投资、建设期利息和全部流动资金。7、资本金净利润率（ ROE ）技术方案资本金净利润率（ ROE ）表示技术方案盈利水平 % 100EC NP ROE 式中 NP----技术方案正常年份的年净利润或运营期内年平均净利润，净利润＝利润总额-所得税 EC----技术方案资本金 8、静态投资回收期 ·当项目建成投产后各年的净收益（即净现金流量）均相同时，静态投资回收期计算： A I P t 式中 I ——总投资；A ——每年的净收益。 ·当项目建成投产后各年的净收益不相同时，静态投资回收期计算：流量出现正值年份的净现金的绝对值上一年累计净现金流量现正值的年份数累计净现金流量开始出 1 －t P 9、借款偿还期余额盈余当年可用于还款的盈余当年应偿还借款额的年份数借款偿还开始出现盈余 1 －d P 10、利息备付率利息备付率＝息税前利润 /计入总成本费用的应付利息。式中：息税前利润——即利润总额与计入总成本费用的利息费用之和（不含折旧、摊销费 11、偿债备付率偿债备付率＝（息税前利润加折旧和摊销-企业所得税）/应还本付息的金额式中：应还本付息的资金——包括当期还贷款本金额及计入总成本费用的全部利息；息税前利润加折旧和摊销 -企业所得税＝净利润 +折旧+摊销+利息

土建工程量计算公式大全

土建全套工程量计箅砌筑砂浆采用你说M强度等级，抹灰砂浆设计都是直接注明采用**:**:**，比如1：1：6在混合砂浆。就不拿你说在M20说事了，这么高在强度等级在砌筑砂浆人还没有用过，比如就水泥砂浆1：3，1斤水泥三斤砂，一方约2.25吨，面积乘以厚度就得到了，根据比例一算就可以了。计算砂浆的体积，比如你这里是2CM厚，4038.34平方，那么体积就是： 4038.34*2/100=80.8立方米。那么需要砂的数量就是80.8立方米。如果按重量算，就是80.8*1.4=113吨。水泥是用来填充砂子的空隙的，不必计算体积。20M砂浆中，水泥：砂=1：5 可以算的。 1M3砌体砂浆的净用量（标准砖）=1-0.24x0.115x0.053x标准砖的净用量 1M3标准砖的净用量=1/[砌体厚X（标准砖长+灰缝厚）X（标准砖厚+灰缝厚）]x2x 砌体厚度的砖数。砌体厚半砖取0.11M 一砖取0.24M 一砖半取0.365M。砌体厚度的砖数半砖取0.5 一砖取1 一砖半取1.5。灰缝厚度一般取0.01M。砂浆有水灰比，可以根据这个比例算出沙子和水泥的用量，再根据工程量计算出要使用多少砂浆即可。基础部分工程量计算一、平整场地：建筑物场地厚度在&plus mn;30cm以内的挖、填、运、找平。 1、平整场地计算规则（1）清单规则：按设计图示尺寸以建筑物首层面积计算。（2）定额规则：按设计图示尺寸以建筑物外墙外边线每边各加2米以平方米面积计算。2、平整场地计算公式 S=（A+4）×（B+4）=S底+2L外+16 式中：S―――平整场地工程量；A―――建筑物长度方向外墙外边线长度；B―――建筑物宽度方向外墙外边线长度；S底―――建筑物底层建筑面积；L外―――建筑物外墙外边线周长。

建筑工程工程量计算公式大全

?工程量计算规则公式汇总土建工程工程量计算规则公式汇总平整场地: 建筑物场地厚度在±30cm以的挖、填、运、找平. 1、平整场地计算规则（1）清单规则：按设计图示尺寸以建筑物首层面积计算。（2）定额规则：按设计图示尺寸以建筑物首层面积计算。 2、平整场地计算法（1）清单规则的平整场地面积：清单规则的平整场地面积＝首层建筑面积（2）定额规则的平整场地面积：定额规则的平整场地面积＝首层建筑面积 3、注意事项（1）、有的地区定额规则的平整场地面积：按外墙外皮线外放2米计算。计算时按外墙外边线外放2米的图形分块计算，然后与底层建筑面积合并计算；或者按“外放2米的中心线×2=外放2米面积” 与底层建筑面积合并计算。这样的话计算时会出现如下难点： ①、划分块比较麻烦，弧线部分不好处理，容易出现误差。 ②、2米的中心线计算起来较麻烦，不好计算。 ③、外放2米后可能出现重叠部分，到底应该扣除多少不好计算。（2）、清单环境下投标人报价时候可能需要根据现场的实际情况计算平整场地的工程量，每边外放的长度不一样。大开挖土 1、开挖土计算规则（1）、清单规则：挖基础土按设计图示尺寸以基础垫层底面积乘挖土深度计算。（2）、定额规则：人工或机械挖土的体积应按槽底面积乘以挖土深度计算。槽底面积应以槽底的长乘以槽底的宽，槽底长和宽是指混凝土垫层外边线加工作面，如有排水沟者应算至排水沟外边线。排水沟的体积应纳入总土量。当需要放坡时，应将放坡的土量合并于总土量中。 2、开挖土计算法（1）、清单规则： ①、计算挖土底面积：法一、利用底层的建筑面积+外墙外皮到垫层外皮的面积。外墙外边线到垫层外边线的面积计算（按外墙外边线外放图形分块计算或者按“外放图形的中心线×外放长度”计算。）法二、分块计算垫层外边线的面积（同分块计算建筑面积）。 ②、计算挖土的体积：土体积＝挖土的底面积*挖土深度。（2）、定额规则： ①、利用棱台体积公式计算挖土的上下底面积。 V=1/6×H×(S上+ 4×S中+ S下)计算土体积（其中，S上为上底面积，S中为中截面面积，S下为下底面面积）。如下图 S下＝底层的建筑面积+外墙外皮到挖土底边线的面积（包括工作面、排水沟、放坡等）。用同样的法计算S中和S下 3、挖土计算的难点 ⑴、计算挖土上中下底面积时候需要计算“各自边线到外墙外边线图”部分的中心线，中心线计算起来比较麻烦（同平整场地）。

建筑工程工程量计算规则公式汇总

建筑工程量计算规则公式汇总一、平整场地建筑物场地厚度在±30cm以内的挖、填、运、找平。 1、平整场地计算规则（1）清单规则：按设计图示尺寸以建筑物首层面积计算。（2）定额规则：按设计图示尺寸以建筑物首层面积计算。 2、平整场地计算方法（1）清单规则的平整场地面积：清单规则的平整场地面积＝首层建筑面积（2）定额规则的平整场地面积：定额规则的平整场地面积＝首层建筑面积 3、注意事项（1）有的地区定额规则的平整场地面积：按外墙外皮线外放2米计算。计算时按外墙外边线外放2米的图形分块计算，然后与底层建筑面积合并计算；或者按“外放2米的中心线×2=外放2米面积” 与底层建筑面积合并计算。这样的话计算时会出现如下难点： ①划分块比较麻烦，弧线部分不好处理，容易出现误差。 ②2米的中心线计算起来较麻烦，不好计算。 ③外放2米后可能出现重叠部分，到底应该扣除多少不好计算。（2）清单环境下投标人报价时候可能需要根据现场的实际情况计算平整场地的工程量，每边外放的长度不一样。二、大开挖土方 1、开挖土方计算规则（1）清单规则：挖基础土方按设计图示尺寸以基础垫层底面积乘挖土深度计算。（2）定额规则：人工或机械挖土方的体积应按槽底面积乘以挖土深度计算。槽底面积应以槽底的长乘以槽底的宽，槽底长和宽是指混凝土垫层外边线加工作面，如有排水沟者应算至排水沟外边线。排水沟的体积应纳入总土方量内。当需要放坡时，应将放坡的土方量合并于总土方量中。 2、开挖土方计算方法（1）清单规则： ①计算挖土方底面积：

方法一、利用底层的建筑面积+外墙外皮到垫层外皮的面积。外墙外边线到垫层外边线的面积计算（按外墙外边线外放图形分块计算或者按“外放图形的中心线×外放长度”计算。）方法二、分块计算垫层外边线的面积（同分块计算建筑面积）。 ②计算挖土方的体积：土方体积＝挖土方的底面积*挖土深度。（2）定额规则： ①利用棱台体积公式计算挖土方的上下底面积。 V=1/6×H×(S上+ 4×S中+ S下)计算土方体积（其中，S上为上底面积，S中为中截面面积，S下为下底面面积）。如下图 S下＝底层的建筑面积+外墙外皮到挖土底边线的面积（包括工作面、排水沟、放坡等）。用同样的方法计算S中和S下 3、挖土方计算的难点（1）计算挖土方上中下底面积时候需要计算“各自边线到外墙外边线图”部分的中心线，中心线计算起来比较麻烦（同平整场地）。（2）中截面面积不好计算。（3）重叠地方不好处理（同平整场地）。（4）如果出现某些边放坡系数不一致，难以处理。 4、大开挖与基槽开挖、基坑开挖的关系槽底宽度在3m以内且长度是宽度三倍以外者或槽底面积在20m2以内者为地槽，其余为挖土方。三、满堂基础垫层 1、满堂基础垫层工程量：如图所示，（1）素土垫层的体积；（2）灰土垫层的体积；（3）砼垫层的体积；（3）垫层模板。 2、满堂基础垫层工程量计算方法（1）素土垫层体积的计算：利用棱台的计算公式：素土垫层体积=1/6×H×(S上+ 4×S中+ S下)计算土方体积（其中，S上为上底面积，S中为中截面面积，S下为下底面面积）。

建筑工程经济公式汇总

建筑工程经济公式汇总 1、单利计算单i P I t ?= 式中 I t ——代表第t 计息周期的利息额；P ——代表本金；i 单——计息周期单利利率。 2、复利计算 1-?=t t F i I 式中 i ——计息周期复利利率；F t-1——表示第（t －1）期末复利本利和。而第t 期末复利本利和的表达式如下： ) 1(1i F F t t +?=- 3、一次支付的终值和现值计算 ①终值计算（已知P 求F 即本利和） n i P F )1(+= ②现值计算（已知F 求P ） n n i F i F P -+=+= )1() 1( 4、等额支付系列的终值、现值、资金回收和偿债基金计算等额支付系列现金流量序列是连续的，且数额相等，即：），，，，常数（n t A A t 321=== ①终值计算（即已知A 求F ） i i A F n 11-+=）（ ②现值计算（即已知A 求P ） n n i i i A i F P ）（）（）（+-+=+=-11 11 ③资金回收计算（已知P 求A ） 111-++=n n i i i P A ）（）（ ④偿债基金计算（已知F 求A ） 11-+=n i i F A ）（ 5、名义利率r 是指计息周期利率：乘以一年内的计息周期数m 所得的年利率。即：m i r ?= 6、有效利率的计算包括计息周期有效利率和年有效利率两种情况。（1）计息周期有效利率，即计息周期利率i ，由式（1Z101021）可知： m r I = （1Z101022-1）（2）年有效利率，即年实际利率。年初资金P ，名义利率为r ，一年内计息m 次，则计息周期利率为 m r i =。根据一次支付终值公式可得该年的本利和F ，即： m m r P F ?? ? ?? +=1

建筑工程常用计算公式

一、平整场地(建筑物场地厚度在±30cm以内的挖、填、运、找平) 1、平整场地计算规则 (1)清单规则：按设计图示尺寸以建筑物首层面积计算。 (2)定额规则：按设计图示尺寸以建筑物外墙外边线每边各加2米以平方米面积计算。 2、平整场地计算公式 S=(A+4)×(B+4)=S底+2L外+16 式中：S——平整场地工程量；A——建筑物长度方向外墙外边线长度；B——建筑物宽度方向外墙外边线长度；S底——建筑物底层建筑面积；L外——建筑物外墙外边线周长。该公式适用于任何由矩形组成的建筑物或构筑物的场地平整工程量计算。二、基础土方开挖计算 1、开挖土方计算规则

(1)清单规则：挖基础土方按设计图示尺寸以基础垫层底面积乘挖土深度计算。 (2)定额规则：人工或机械挖土方的体积应按槽底面积乘以挖土深度计算。槽底面积应以槽底的长乘以槽底的宽，槽底长和宽是指基础底宽外加工作面，当需要放坡时，应将放坡的土方量合并于总土方量中。 2、开挖土方计算公式 (1)清单计算挖土方的体积：土方体积＝挖土方的底面积×挖土深度。 (2)定额规则：基槽开挖：V=(A+2C+K×H)H×L。式中：V——基槽土方量；A——槽底宽度；C——工作面宽度；H——基槽深度；L——基槽长度。其中外墙基槽长度以外墙中心线计算，内墙基槽长度以内墙净长计算，交接重合出不予扣除。基坑开挖：V=1/6H[A×B+a×b+(A+a)×(B+b)+a×b]。式中：V——基坑体积；A——基坑上口长度；B——基坑上口宽度；a——基坑底面长度；b——基坑底面宽度。三、回填土工程量计算规则及公式

1、基槽、基坑回填土体积=基槽(坑)挖土体积-设计室外地坪以下建(构)筑物被埋置部分的体积。式中室外地坪以下建(构)筑物被埋置部分的体积一般包括垫层、墙基础、柱基础、以及地下建筑物、构筑物等所占体积 2、室内回填土体积=主墙间净面积×回填土厚度-各种沟道所占体积主墙间净面积=S底-(L中×墙厚+L内×墙厚) 式中：底——底层建筑面积；L中——外墙中心线长度；L内——内墙净长线长度。回填土厚度指室内外高差减去地面垫层、找平层、面层的总厚度。四、运土方计算规则及公式运土是指把开挖后的多余土运至指定地点，或是在回填土不足时从指定地点取土回填。土方运输应按不同的运输方式和运距分别以立方米计算。运土工程量=挖土总体积-回填土总体积

建筑工程量计算公式及计算方法大全

建筑工程量计算公式及计算方法大全一、平整场地：建筑物场地厚度在±30cm以内的挖、填、运、找平。 1、平整场地计算规则（1）清单规则：按设计图示尺寸以建筑物首层面积计算。（2）定额规则：按设计图示尺寸以建筑物外墙外边线每边各加2米以平方米面积计算。 2、平整场地计算公式 S=（A+4）×（B+4）=S底+2L外+16 式中：S———平整场地工程量；A———建筑物长度方向外墙外边线长度；B———建筑物宽度方向外墙外边线长度；S底———建筑物底层建筑面积；L外———建筑物外墙外边线周长。该公式适用于任何由矩形组成的建筑物或构筑物的场地平整工程量计算。二、基础土方开挖计算开挖土方计算规则（1）、清单规则：挖基础土方按设计图示尺寸以基础垫层底面积乘挖土深度计算。（2）、定额规则：人工或机械挖土方的体积应按槽底面积乘以挖土深度计算。槽底面积应以槽底的长乘以槽底的宽，槽底长和宽是指基础底宽外加工作面，当需要放坡时，应将放坡的土方量合并于总土方量中。 2、开挖土方计算公式：（1）、清单计算挖土方的体积：土方体积＝挖土方的底面积×挖土深度。（2）、定额规则：基槽开挖：V=（A+2C+K×H）H×L。式中：V———基槽土方量；A———槽底宽度；C———工作面宽度；H———基槽深度；L———基槽长度。. 其中外墙基槽长度以外墙中心线计算，内墙基槽长度以内墙净长计算，交接重合出不予扣除。基坑开挖：V=1/6H[A×B+a×b+(A+a)×(B+b)+a×b]。式中：V———基坑体积；A—基坑上口长度；B———基坑上口宽度；a———基坑底面长度；b———基坑底面宽度。三、回填土工程量计算规则及公式 1、基槽、基坑回填土体积=基槽（坑）挖土体积-设计室外地坪以下建（构）筑物被埋置部

建筑工程工程量计算公式大全

工程量计算规则公式汇总 ~ 土建工程工程量计算规则公式汇总平整场地: 建筑物场地厚度在±30cm以内的挖、填、运、找平. 1、平整场地计算规则（1）清单规则：按设计图示尺寸以建筑物首层面积计算。（2）定额规则：按设计图示尺寸以建筑物首层面积计算。 2、平整场地计算方法（1）清单规则的平整场地面积：清单规则的平整场地面积＝首层建筑面积：（2）定额规则的平整场地面积：定额规则的平整场地面积＝首层建筑面积 3、注意事项（1）、有的地区定额规则的平整场地面积：按外墙外皮线外放2米计算。计算时按外墙外边线外放2米的图形分块计算，然后与底层建筑面积合并计算；或者按“外放2米的中心线×2=外放2米面积” 与底层建筑面积合并计算。这样的话计算时会出现如下难点： ①、划分块比较麻烦，弧线部分不好处理，容易出现误差。 ②、2米的中心线计算起来较麻烦，不好计算。 ③、外放2米后可能出现重叠部分，到底应该扣除多少不好计算。（2）、清单环境下投标人报价时候可能需要根据现场的实际情况计算平整场地的工程量，每边外放的长度不一样。大开挖土方《 1、开挖土方计算规则（1）、清单规则：挖基础土方按设计图示尺寸以基础垫层底面积乘挖土深度计算。（2）、定额规则：人工或机械挖土方的体积应按槽底面积乘以挖土深度计算。槽底面积应以槽底的长乘以槽底的宽，槽底长和宽是指混凝土垫层外边线加工作面，如有排水沟者应算至排水沟外边线。排水沟的体积应纳入总土方量内。当需要放坡时，应将放坡的土方量合并于总土方量中。 2、开挖土方计算方法（1）、清单规则： ①、计算挖土方底面积：方法一、利用底层的建筑面积+外墙外皮到垫层外皮的面积。外墙外边线到垫层外边线的面积计算（按外墙外边线外放图形分块计算或者按“外放图形的中心线×外放长度”计算。）方法二、分块计算垫层外边线的面积（同分块计算建筑面积）。 ②、计算挖土方的体积：土方体积＝挖土方的底面积*挖土深度。（2）、定额规则： ①、利用棱台体积公式计算挖土方的上下底面积。 V=1/6×H×(S上+ 4×S中+ S下)计算土方体积（其中，S上为上底面积，S中为中截面面积，S 下为下底面面积）。如下图 S下＝底层的建筑面积+外墙外皮到挖土底边线的面积（包括工作面、排水沟、放坡等）。用同样的方法计算S中和S下

建筑工程经济公式大全

建筑工程经济公式汇总 1、单利计算式中 I t ——代表第t 计息周期的利息额；P ——代表本金；i 单——计息周期单利利率。 2、复利计算 1-?=t t F i I 式中 i ——计息周期复利利率；F t-1——表示第（t －1）期末复利本利和。而第t 期末复利本利和的表达式如下： ) 1(1i F F t t +?=- 3、一次支付的终值和现值计算 ①终值计算（已知P 求F 即本利和） n i P F )1(+= ②现值计算（已知F 求P ） n n i F i F P -+=+= )1() 1( 4、等额支付系列的终值、现值、资金回收和偿债基金计算等额支付系列现金流量序列是连续的，且数额相等，即：），，，，常数（n t A A t 321=== ①终值计算（即已知A 求F ） i i A F n 11-+=）（ ②现值计算（即已知A 求P ） n n n i i i A i F P ）（）（）（+-+=+=-1111 ③资金回收计算（已知P 求A ） 111-++=n n i i i P A ）（）（ ④偿债基金计算（已知F 求A ） 1 1-+=n i i F A ）（ 5、名义利率r 是指计息周期利率：乘以一年内的计息周期数m 所得的年利率。即：m i r ?= 6、有效利率的计算包括计息周期有效利率和年有效利率两种情况。（1）计息周期有效利率，即计息周期利率i ，由式（1Z101021）可知：

m r I = （1Z101022-1）（2）年有效利率，即年实际利率。年初资金P ，名义利率为r ，一年内计息m 次，则计息周期利率为 m r i =。根据一次支付终值公式可得该年的本利和F ，即： m m r P F ?? ? ?? +=1 根据利息的定义可得该年的利息I 为： ??? ?????-??? ??+=-??? ?? +=111m m m r P P m r P I 再根据利率的定义可得该年的实际利率，即有效利率i eFF 为： 7、财务净现值 ()()t c t n t i CO CI FNPV -=+-= ∑10 （1Z101035）式中 FNPV ——财务净现值；（CI-CO ）t ——第t 年的净现金流量（应注意“＋”、“-”号）； i c ——基准收益率； n ——方案计算期。 8、财务内部收益率（FIRR ——Financial lnternaI Rate oF Return ）其实质就是使投资方案在计算期内各年净现金流量的现值累计等于零时的折现率。其数学表达式为： ()()()t t n t FIRR CO CI FIRR FNPV -=+-= ∑10 （1Z101036-2）式中 FIRR ——财务内部收益率。 9、财务净现值率（FNPVR ——Financial Net Present Value Rate ）其经济含义是单位投资现值所能带来的财务净现值，是一个考察项目单位投资盈利能力的指标，考察投资的利用效率，作为财务净现值的辅助评价指标。计算式如下： p I FNPV FNPVR = （1Z101037-1） () t i F P I I c k t t p ,,/0 ∑== （1Z101037-2）式中 I p ——投资现值； I t ——第t 年投资额； k ——投资年数；

工程项目造价常用计算公式大全

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黄铜板重量（公斤）=0.0085×厚×宽×长度铝板重量（公斤）=0.00171×厚×宽×长度园紫铜管重量（公斤）=0.028×壁厚×（外径-壁厚）×长度园黄铜管重量（公斤）=0.0267×壁厚×（外径-壁厚）×长度园铝管重量（公斤）=0.00879×壁厚×（外径-壁厚）×长度园钢重量（公斤）=0.00617×直径×直径×长度注：公式中长度单位为米，面积单位为平方米，其余单位均为毫米长方形的周长=（长+宽）×2 正方形的周长=边长×4 长方形的面积=长×宽正方形的面积=边长×边长三角形的面积=底×高÷2 平行四边形的面积=底×高梯形的面积=（上底+下底）×高÷2 直径=半径×2 半径=直径÷2 圆的周长=圆周率×直径=圆周率×半径×2 圆的面积=圆周率×半径×半径长方体的表面积= （长×宽+长×高＋宽×高）×2 长方体的体积=长×宽×高正方体的表面积=棱长×棱长×6 正方体的体积=棱长×棱长×棱长

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(2)带形基础 (3)独立基础 1、独立基础和条形基础 (1)独立基础:V＝a’×b’×厚度＋棱台体积 (2)条形基础:V＝断面面积×沟槽长度 (1)砖基础断面计算砖基础多为大放脚形式,大放脚有等高与不等高两种.等高大放脚是以墙厚为基础,每挑宽1/4砖,挑出砖厚为2皮砖.不等高大放脚,每挑宽1/4砖,挑出砖厚为1皮与2皮相间(见图10-18).

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