@let@token Biases in Decomposing Holding Period Portfolio ...rcaf.dufe.edu.cn/upload/2014/1204/20141204090229774.pdf · A question Suppose that we have an equally-weighted (ew) portfolio

Biases in Decomposing Holding Period Portfolio Returns

Liu Weimin (Nottingham University Business School)

Norman Strong (Manchester Business School)

Dongbei University of Finance and Economics

December 12, 2014

Liu Weimin Decompose Portfolio Return December 12, 2014 1 / 21

A question

• Suppose that we

� have an equally-weighted (ew) portfolio containing N stocks

� hold the portfolio for m months (m > 1)

� know the monthly returns over the m-month holding period of each

stock in the portfolio

• What is the month-t (t = 1, 2, ..., m) return of the portfolio over

the m-month holding period?

� One guess is Rp,t =1

N

N∑i=1

Ri ,t

� Is the guess correct?


A question

• Suppose that we








N

N∑i=1

Ri ,t



A question

• Suppose that we








N

N∑i=1

Ri ,t



A question

• Suppose that we








N

N∑i=1

Ri ,t



A question

• Suppose that we








N

N∑i=1

Ri ,t



Example: Hold an ew two-stock portfolio for two months

• A, B, C, and D are non-dividend-paying stocks. Consider 2 scenarios:

� Scenario 1: the ew portfolio contains A and B

Price per share Rate of return

Month end 0 1 2 1 2

Stock A 5 6 3 0.2 −0.5Stock B 5 4 7 −0.2 0.75

Arithmetic average 0 0.125

� Scenario 2: the ew portfolio contains C and D


Month end 0 1 2 1 2

Stock C 5 6 9 0.2 0.5Stock D 5 4 1 −0.2 −0.75

Arithmetic average 0 −0.125






Month end 0 1 2 1 2

Stock A 5 6 3 0.2 −0.5Stock B 5 4 7 −0.2 0.75




Month end 0 1 2 1 2

Stock C 5 6 9 0.2 0.5Stock D 5 4 1 −0.2 −0.75







Month end 0 1 2 1 2

Stock A 5 6 3 0.2 −0.5Stock B 5 4 7 −0.2 0.75




Month end 0 1 2 1 2

Stock C 5 6 9 0.2 0.5Stock D 5 4 1 −0.2 −0.75







Month end 0 1 2 1 2

Stock A 5 6 3 0.2 −0.5Stock B 5 4 7 −0.2 0.75




Month end 0 1 2 1 2

Stock C 5 6 9 0.2 0.5Stock D 5 4 1 −0.2 −0.75







Month end 0 1 2 1 2

Stock A 5 6 3 0.2 −0.5Stock B 5 4 7 −0.2 0.75




Month end 0 1 2 1 2

Stock C 5 6 9 0.2 0.5Stock D 5 4 1 −0.2 −0.75







Month end 0 1 2 1 2

Stock A 5 6 3 0.2 −0.5Stock B 5 4 7 −0.2 0.75




Month end 0 1 2 1 2

Stock C 5 6 9 0.2 0.5Stock D 5 4 1 −0.2 −0.75







Month end 0 1 2 1 2

Stock A 5 6 3 0.2 −0.5Stock B 5 4 7 −0.2 0.75




Month end 0 1 2 1 2

Stock C 5 6 9 0.2 0.5Stock D 5 4 1 −0.2 −0.75







Month end 0 1 2 1 2

Stock A 5 6 3 0.2 −0.5Stock B 5 4 7 −0.2 0.75




Month end 0 1 2 1 2

Stock C 5 6 9 0.2 0.5Stock D 5 4 1 −0.2 −0.75







Month end 0 1 2 1 2

Stock A 5 6 3 0.2 −0.5Stock B 5 4 7 −0.2 0.75




Month end 0 1 2 1 2

Stock C 5 6 9 0.2 0.5Stock D 5 4 1 −0.2 −0.75







Month end 0 1 2 1 2

Stock A 5 6 3 0.2 −0.5Stock B 5 4 7 −0.2 0.75




Month end 0 1 2 1 2

Stock C 5 6 9 0.2 0.5Stock D 5 4 1 −0.2 −0.75



Outline

• Motivation of the study

• Decomposing holding period portfolio returns

� Decomposed buy-and-hold method

� Re-balanced method

• Biases from using the re-balanced method

• Data and sample

• Empirical evidence of the biases

• Conclusion

• This presentation is based on my RFS paper (2008, Vol. 21, pp.

2243–2274) with Norman Strong


Outline

• Motivation of the study

• Decomposing holding period portfolio returns

� Decomposed buy-and-hold method

� Re-balanced method

• Biases from using the re-balanced method

• Data and sample

• Empirical evidence of the biases

• Conclusion

• This presentation is based on my RFS paper (2008, Vol. 21, pp.

2243–2274) with Norman Strong


Motivation

• Decomposing a multiperiod portfolio return into single-period returns

is common. E.g., to test or estimate an asset pricing model, we often

decompose a multi-month portfolio return into monthly returns

• However, there is no explicit decomposition formula or description in

the literature, and some studies use a simplified method, which is

referred to as the “re-balanced” method in this paper

• The re-balanced method involves portfolios that

� nobody would seriously consider ex ante

� are poor investment vehicles due to transaction costs (TC)

� can lead to spurious statistical inferences


Motivation












Motivation












Motivation












Motivation












Motivation












Motivation (cont’d)

• Based on the precept that returns should measure the wealth effect

to the portfolio holder, we formalize the calculation by decomposing

the multiperiod buy-and-hold return into single-period returns

• We predict and find that re-balancing imparts an upward bias to the

size or value premium, and a downward bias to the momentum effect

� The bias in portfolio return can be over 8% per annum relative to

the decomposed buy-and-hold method

• We should be aware of the possible biases from using the re-balanced

method in research











method in research











method in research











method in research











method in research


Portfolio return over an m-month holding period

• The month-t return of stock i is

Rit =Pit + Dit

Pi ,t−1− 1 (1)

• Given monthly returns of each stock in a portfolio containing N

stocks, what is the portfolio’s m-month buy-and-hold return?

Rp, 1→m =N∑i=1

wi (1 + Ri1) · · · (1 + Rim)− 1 (2)

� (1 +Ri1) · · · (1 +Rim)− 1 = m-month buy-and-hold return of stock i

� wi is the portfolio weight of stock i and∑N

i=1 wi = 1

• Under the buy-and-hold assumption, equation (2) gives the actual

portfolio return over the m-month holding period (ignoring TC, etc.)




Rit =Pit + Dit

Pi ,t−1− 1 (1)



Rp, 1→m =N∑i=1

wi (1 + Ri1) · · · (1 + Rim)− 1 (2)



i=1 wi = 1






Rit =Pit + Dit

Pi ,t−1− 1 (1)



Rp, 1→m =N∑i=1

wi (1 + Ri1) · · · (1 + Rim)− 1 (2)



i=1 wi = 1






Rit =Pit + Dit

Pi ,t−1− 1 (1)



Rp, 1→m =N∑i=1

wi (1 + Ri1) · · · (1 + Rim)− 1 (2)



i=1 wi = 1






Rit =Pit + Dit

Pi ,t−1− 1 (1)



Rp, 1→m =N∑i=1

wi (1 + Ri1) · · · (1 + Rim)− 1 (2)



i=1 wi = 1




Decomposed buy-and-hold method

• We can decompose the m-month portfolio return given by equation

(2) into m monthly returns

Rpτ =N∑i=1

wi∏τ−1

t=1 (1 + Rit)∑Nj=1 wj

∏τ−1t=1 (1 + Rjt)

Riτ ∀ τ > 1; Rp1 =N∑i=1

wiRi1 (3)

• Equation (3) follows from

m∏τ=1

(1 + Rpτ )− 1 =N∑i=1

wi (1 + Ri1) · · · (1 + Rim)− 1

• Equation (3) shows: the month-τ portfolio return from an m-month

B&H strategy is the weighted average of individual stocks’ returns in

month-τ with each weight depending on prior holding period return





Rpτ =N∑i=1

wi∏τ−1

t=1 (1 + Rit)∑Nj=1 wj

∏τ−1t=1 (1 + Rjt)

Riτ ∀ τ > 1; Rp1 =N∑i=1

wiRi1 (3)


m∏τ=1

(1 + Rpτ )− 1 =N∑i=1

wi (1 + Ri1) · · · (1 + Rim)− 1








Rpτ =N∑i=1

wi∏τ−1

t=1 (1 + Rit)∑Nj=1 wj

∏τ−1t=1 (1 + Rjt)

Riτ ∀ τ > 1; Rp1 =N∑i=1

wiRi1 (3)


m∏τ=1

(1 + Rpτ )− 1 =N∑i=1

wi (1 + Ri1) · · · (1 + Rim)− 1








Rpτ =N∑i=1

wi∏τ−1

t=1 (1 + Rit)∑Nj=1 wj

∏τ−1t=1 (1 + Rjt)

Riτ ∀ τ > 1; Rp1 =N∑i=1

wiRi1 (3)


m∏τ=1

(1 + Rpτ )− 1 =N∑i=1

wi (1 + Ri1) · · · (1 + Rim)− 1





Two special cases of equation (3)

• An equally-weighted (ew) portfolio

� Equation (3) with wi = 1/N gives the month-τ return of an

equally-weighted portftfolio from an m-month holding period

Rewpτ =

N∑i=1

∏τ−1t=1 (1 + Rit)∑N

j=1

∏τ−1t=1 (1 + Rjt)

Riτ (4)

• A value-weighted (vw) portfolio

� Equation (3) with wi=MV i ,0/∑N

j=1MV j ,0 gives the month-τ return

of a value-weighted portfolio from an m-month holding period

Rvwpτ =

N∑i=1

MVi ,τ−1∑Nj=1MVj ,τ−1

Riτ (5)

? Eq. (5) assumes MV s are adjusted for capitalizations. Or, use (3)






Rewpτ =

N∑i=1

∏τ−1t=1 (1 + Rit)∑N

j=1

∏τ−1t=1 (1 + Rjt)

Riτ (4)





Rvwpτ =

N∑i=1


Riτ (5)







Rewpτ =

N∑i=1

∏τ−1t=1 (1 + Rit)∑N

j=1

∏τ−1t=1 (1 + Rjt)

Riτ (4)





Rvwpτ =

N∑i=1


Riτ (5)







Rewpτ =

N∑i=1

∏τ−1t=1 (1 + Rit)∑N

j=1

∏τ−1t=1 (1 + Rjt)

Riτ (4)





Rvwpτ =

N∑i=1


Riτ (5)







Rewpτ =

N∑i=1

∏τ−1t=1 (1 + Rit)∑N

j=1

∏τ−1t=1 (1 + Rjt)

Riτ (4)





Rvwpτ =

N∑i=1


Riτ (5)



Re-balanced method

• Some studies simplify the calculation of (3) to

R(rb)pτ =N∑i=1

wiRiτ , τ = 1, 2, ..., m (6)

� Equation (6) implies an investment strategy that re-balances the

weights each month to those determined at the beginning of the

holding period

• For an equally-weighted portfolio with the re-balanced method,

R(rb)ewpτ =1

N

N∑i=1

Riτ , τ = 1, 2, ..., m (7)

� The portfolio return in each holding-period month τ is the

arithmetic average of all stock returns in month τ


Re-balanced method


R(rb)pτ =N∑i=1

wiRiτ , τ = 1, 2, ..., m (6)



holding period


R(rb)ewpτ =1

N

N∑i=1

Riτ , τ = 1, 2, ..., m (7)




Re-balanced method


R(rb)pτ =N∑i=1

wiRiτ , τ = 1, 2, ..., m (6)



holding period


R(rb)ewpτ =1

N

N∑i=1

Riτ , τ = 1, 2, ..., m (7)




Re-balanced method


R(rb)pτ =N∑i=1

wiRiτ , τ = 1, 2, ..., m (6)



holding period


R(rb)ewpτ =1

N

N∑i=1

Riτ , τ = 1, 2, ..., m (7)




Re-balanced method


R(rb)pτ =N∑i=1

wiRiτ , τ = 1, 2, ..., m (6)



holding period


R(rb)ewpτ =1

N

N∑i=1

Riτ , τ = 1, 2, ..., m (7)




Evidence using the rebalanced method

• The problem is not a minor technical issue. Table 1 lists a number of

papers published in the top three finance journals over 2001–2005

• There were seven papers in JF in 2003–2005 (2+ papers per annum)

that used an incorrect decomposition; five in JFE

• For example, “hold this portfolio for 6 months. The portfolio is

re-balanced monthly to account for stocks that drop out of the

database.” (JFE 73, 2004, 525–565)—wrong method with wrong

reasoning

• There are many unclear cases in decomposing portfolio returns in the

literature










reasoning


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reasoning


literature


Issues with rebalanced method

• Rebalanced method contradicts the assumption of an m-month

holding period under examination

• Inaccurate in reflecting investor wealth, unless investors rebalance

their portfolios back to the initial weights at the beginning of every

holding-period month

• Rebalancing at regular intervals to keep initial portfolio weights fixed

is unrealistic

� Rebalancing may occur irregularly due to fund flows, new

information, liquidity requirements

� Rebalancing incurs extra transaction costs, making the rebalancing

strategy a poor investment vehicle

• Rebalanced method can lead to spurious inferencesLiu Weimin Decompose Portfolio Return December 12, 2014 12 / 21








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is unrealistic






Bias of the rebalanced method

• Based on equations (3) and (6), the bias in using the rebalanced

method relative to the decomposed buy-and-hold method is

Biasτ = E [R(rb)pτ − Rpτ ], τ = 1, 2, ..., m (8)

• Bias in the 2nd holding-period month with the ew portfolio is

Bias2 → Cov(r̃1, r̃2)− 1

N

N∑i=1

Cov(r̃i ,1, r̃i ,2) (9)

• Portfolio returns over a week/month are positively autocorrelated

• Individual returns over a week/month are negatively autocorrelated

due to microstructure effects, especially for small stocks

• Prediction: an upwards bias of the rebalanced method in finding a

size or value effect, and a downwards bias in momentum premiumLiu Weimin Decompose Portfolio Return December 12, 2014 13 / 21






Bias2 → Cov(r̃1, r̃2)− 1

N

N∑i=1

Cov(r̃i ,1, r̃i ,2) (9)











Bias2 → Cov(r̃1, r̃2)− 1

N

N∑i=1

Cov(r̃i ,1, r̃i ,2) (9)











Bias2 → Cov(r̃1, r̃2)− 1

N

N∑i=1

Cov(r̃i ,1, r̃i ,2) (9)











Bias2 → Cov(r̃1, r̃2)− 1

N

N∑i=1

Cov(r̃i ,1, r̃i ,2) (9)











Bias2 → Cov(r̃1, r̃2)− 1

N

N∑i=1

Cov(r̃i ,1, r̃i ,2) (9)






Data and research design

• NYSE/AMEX/NASDAQ common stocks over 1951–2003, the test

period starts at the beginning July 1951

• Data from CRSP/COMPUSTAT merged database

� Monthly return and MV

� Annual data for working out B/M

• Basic research design: sort stocks into portfolios

� MV deciles and B/M deciles are formed at the beginning of July

each year with NYSE breakpoints and held for 12 months

� We form r−6 deciles each month with a 6-month holding period

• We do not report the B/M results, which show similar magnitudes of

bias, but do not lead to spurious inferencesLiu Weimin Decompose Portfolio Return December 12, 2014 14 / 21









































































Empirical results of dc and rb strategies

L S L−SMV -based decile portfolios (1951–2003)

dc(%) 1.235 0.922 0.313 (1.65)

rb(%) 1.368 0.920 0.448 (2.34)

B/M-based decile portfolios (1951–2003)

dc(%) 1.554 0.847 0.708 (4.52)

rb(%) 1.748 0.843 0.905 (5.72)

R−6-based decile portfolios (1951–2003)

dc(%) 1.643 0.768 0.875 (4.91)

rb(%) 1.646 1.063 0.583 (3.01)

• For r−6 deciles over 1978–2003, dc L−S = 1.017% (t=3.66) per

month, but rb L−S = 0.569% (t=1.82) per monthLiu Weimin Decompose Portfolio Return December 12, 2014 15 / 21



dc(%) 1.235 0.922 0.313 (1.65)

rb(%) 1.368 0.920 0.448 (2.34)


dc(%) 1.554 0.847 0.708 (4.52)

rb(%) 1.748 0.843 0.905 (5.72)


dc(%) 1.643 0.768 0.875 (4.91)

rb(%) 1.646 1.063 0.583 (3.01)





dc(%) 1.235 0.922 0.313 (1.65)

rb(%) 1.368 0.920 0.448 (2.34)


dc(%) 1.554 0.847 0.708 (4.52)

rb(%) 1.748 0.843 0.905 (5.72)


dc(%) 1.643 0.768 0.875 (4.91)

rb(%) 1.646 1.063 0.583 (3.01)



Empirical evidence of the biases

1/3 lowest-price sample 1/3 highest-price sample

L(%) S(%) L−S L(%) S(%) L−SBias in MV -based decile portfolio returns per month

Raw 0.590 −0.002 0.592 −0.038 0.002 −0.040

α̂3F 0.530 −0.141 0.671 −0.062 −0.034 −0.028

Bias in B/M-based decile portfolio returns per month

Raw 0.373 0.200 0.173 0.001 −0.123 0.124

α̂3F 0.283 0.102 0.181 −0.019 −0.203 0.183

Bias in R−6-based decile portfolio returns per month

Raw 0.106 0.694 −0.588 −0.028 0.037 −0.065

α̂3F 0.072 0.594 −0.523 −0.053 0.001 −0.054

• Figures are return differences per month between rb and dc, and blue

figures are statistically significantLiu Weimin Decompose Portfolio Return December 12, 2014 16 / 21




Raw 0.590 −0.002 0.592 −0.038 0.002 −0.040

α̂3F 0.530 −0.141 0.671 −0.062 −0.034 −0.028


Raw 0.373 0.200 0.173 0.001 −0.123 0.124

α̂3F 0.283 0.102 0.181 −0.019 −0.203 0.183


Raw 0.106 0.694 −0.588 −0.028 0.037 −0.065

α̂3F 0.072 0.594 −0.523 −0.053 0.001 −0.054






Raw 0.590 −0.002 0.592 −0.038 0.002 −0.040

α̂3F 0.530 −0.141 0.671 −0.062 −0.034 −0.028


Raw 0.373 0.200 0.173 0.001 −0.123 0.124

α̂3F 0.283 0.102 0.181 −0.019 −0.203 0.183


Raw 0.106 0.694 −0.588 −0.028 0.037 −0.065

α̂3F 0.072 0.594 −0.523 −0.053 0.001 −0.054



Examples of spurious inferences in the literature

• Mean return pm of MV deciles (NYSE/AMEX/NASDAQ, 7/63–6/94)

Small B ig S−B

Replication of Barber and Lyon (1997, JF 52, 875–883)

dc(%) 1.396 0.990 0.407 (1.38)

rb(%) 1.638 1.012 0.626 (2.04)

Results of Barber and Lyon (1997)

rb(%) 1.68 1.03 0.65 (2.02)

• Mean return pm of R−6 deciles (NYSE/AMEX, 7/1926–12/1994)

W inner Loser W−L

Replication of Chordia and Shivakumar (2002, JF 57, 985–1019)

dc(%) 1.507 0.989 0.518 (2.31)

rb(%) 1.579 1.315 0.264 (1.12)

Results of Chordia and Shivakumar (2002)

rb(%) 1.61 1.34 0.27 (1.10)




Small B ig S−B


dc(%) 1.396 0.990 0.407 (1.38)

rb(%) 1.638 1.012 0.626 (2.04)


rb(%) 1.68 1.03 0.65 (2.02)


W inner Loser W−L


dc(%) 1.507 0.989 0.518 (2.31)

rb(%) 1.579 1.315 0.264 (1.12)


rb(%) 1.61 1.34 0.27 (1.10)




Small B ig S−B


dc(%) 1.396 0.990 0.407 (1.38)

rb(%) 1.638 1.012 0.626 (2.04)


rb(%) 1.68 1.03 0.65 (2.02)


W inner Loser W−L


dc(%) 1.507 0.989 0.518 (2.31)

rb(%) 1.579 1.315 0.264 (1.12)


rb(%) 1.61 1.34 0.27 (1.10)




Small B ig S−B


dc(%) 1.396 0.990 0.407 (1.38)

rb(%) 1.638 1.012 0.626 (2.04)


rb(%) 1.68 1.03 0.65 (2.02)


W inner Loser W−L


dc(%) 1.507 0.989 0.518 (2.31)

rb(%) 1.579 1.315 0.264 (1.12)


rb(%) 1.61 1.34 0.27 (1.10)




Small B ig S−B


dc(%) 1.396 0.990 0.407 (1.38)

rb(%) 1.638 1.012 0.626 (2.04)


rb(%) 1.68 1.03 0.65 (2.02)


W inner Loser W−L


dc(%) 1.507 0.989 0.518 (2.31)

rb(%) 1.579 1.315 0.264 (1.12)


rb(%) 1.61 1.34 0.27 (1.10)




Small B ig S−B


dc(%) 1.396 0.990 0.407 (1.38)

rb(%) 1.638 1.012 0.626 (2.04)


rb(%) 1.68 1.03 0.65 (2.02)


W inner Loser W−L


dc(%) 1.507 0.989 0.518 (2.31)

rb(%) 1.579 1.315 0.264 (1.12)


rb(%) 1.61 1.34 0.27 (1.10)




Small B ig S−B


dc(%) 1.396 0.990 0.407 (1.38)

rb(%) 1.638 1.012 0.626 (2.04)


rb(%) 1.68 1.03 0.65 (2.02)


W inner Loser W−L


dc(%) 1.507 0.989 0.518 (2.31)

rb(%) 1.579 1.315 0.264 (1.12)


rb(%) 1.61 1.34 0.27 (1.10)


Degree of rebalance

Traded value pm of longing $100 L and shorting $100 of S

Decomposed Re-balanced Difference

Buy Sell Buy Sell DifBuy DifSell

MV -based decile portfolios (1951–2003)

L 2.71 4.10 6.40 7.77 3.693 3.673

S 2.48 1.50 4.53 3.61 2.052 2.108


L 13.9 15.6 17.6 19.2 3.721 3.543

S 15.4 14.3 19.4 18.3 4.026 3.950

• For MV portfolios, the traded value is more than doubled with rb

� For longing $100 of L, the traded value (buy+sell) is $6.81 pm

($81.72 pa) with dc, but it is $14.17 pm ($170 pa) with rb

• For momentum portfolios, the traded value is large since winning and

losing stocks rarely retain their status in successive holding periodsLiu Weimin Decompose Portfolio Return December 12, 2014 18 / 21

Degree of rebalance





L 2.71 4.10 6.40 7.77 3.693 3.673

S 2.48 1.50 4.53 3.61 2.052 2.108


L 13.9 15.6 17.6 19.2 3.721 3.543

S 15.4 14.3 19.4 18.3 4.026 3.950






Degree of rebalance





L 2.71 4.10 6.40 7.77 3.693 3.673

S 2.48 1.50 4.53 3.61 2.052 2.108


L 13.9 15.6 17.6 19.2 3.721 3.543

S 15.4 14.3 19.4 18.3 4.026 3.950






Degree of rebalance





L 2.71 4.10 6.40 7.77 3.693 3.673

S 2.48 1.50 4.53 3.61 2.052 2.108


L 13.9 15.6 17.6 19.2 3.721 3.543

S 15.4 14.3 19.4 18.3 4.026 3.950






Degree of rebalance





L 2.71 4.10 6.40 7.77 3.693 3.673

S 2.48 1.50 4.53 3.61 2.052 2.108


L 13.9 15.6 17.6 19.2 3.721 3.543

S 15.4 14.3 19.4 18.3 4.026 3.950






Transaction-cost-adjusted performance

• Transaction costs are estimated using results of KM (1997)


dc(%) 1.154 0.923 0.231 (1.22)

rb(%) 1.143 0.923 0.219 (1.14)


dc(%) 1.442 0.918 0.525 (3.36)

rb(%) 1.505 1.020 0.485 (3.09)

• Transaction costs reduce the size premium by 26% (from 0.313 to

0.231) with dc, but by 51% (from 0.448 to 0.219) with rb. Similarly

for the value premium





dc(%) 1.154 0.923 0.231 (1.22)

rb(%) 1.143 0.923 0.219 (1.14)


dc(%) 1.442 0.918 0.525 (3.36)

rb(%) 1.505 1.020 0.485 (3.09)








dc(%) 1.154 0.923 0.231 (1.22)

rb(%) 1.143 0.923 0.219 (1.14)


dc(%) 1.442 0.918 0.525 (3.36)

rb(%) 1.505 1.020 0.485 (3.09)








dc(%) 1.154 0.923 0.231 (1.22)

rb(%) 1.143 0.923 0.219 (1.14)


dc(%) 1.442 0.918 0.525 (3.36)

rb(%) 1.505 1.020 0.485 (3.09)





Transaction-costs-adjusted performance (cont’d)

L S L−SR−6-based decile portfolios (1951–2003)

dc(%) 1.422 1.163 0.259 (1.44)

rb(%) 1.336 1.613 −0.277 (−1.39)

• Transaction costs have the greatest effect on momentum profits

because winning and losing stocks rarely retain their positions in

consecutive holding periods

• After adjusting for transaction costs, while the momentum premium

is reduced by 70% (from 0.875 to 0.259) with dc, the reduction is

148% (from 0.583 to −0.277) with rb

• Thus, rebalanced method is a poor, impractical investment strategy




dc(%) 1.422 1.163 0.259 (1.44)

rb(%) 1.336 1.613 −0.277 (−1.39)






148% (from 0.583 to −0.277) with rb





dc(%) 1.422 1.163 0.259 (1.44)

rb(%) 1.336 1.613 −0.277 (−1.39)






148% (from 0.583 to −0.277) with rb





dc(%) 1.422 1.163 0.259 (1.44)

rb(%) 1.336 1.613 −0.277 (−1.39)






148% (from 0.583 to −0.277) with rb





dc(%) 1.422 1.163 0.259 (1.44)

rb(%) 1.336 1.613 −0.277 (−1.39)






148% (from 0.583 to −0.277) with rb



Summary

• A growing number of studies base statistical tests on single-period

portfolio returns from a multiperiod holding period

• In this paper, we provide a formal analysis to decompose the

multiperiod buy-and-hold return

• The commonly used re-balanced method is unrealistic, and can lead

to spurious inferences

• Dangers of using the rebalanced method are relevant for tests of

asset pricing model, of market efficiency, and for assessing investment

strategies


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Documents

@let@token Biases in Decomposing Holding Period Portfolio ...rcaf.dufe.edu.cn/upload/2014/1204/20141204090229774.pdf · A question Suppose that we have an equally-weighted (ew) portfolio