Removing biases in computed returns

Abstract

This paper presents a straightforward method for asymptotically removing the well-known upward bias in observed returns of equally-weighted portfolios. Our method removes all of the bias due to any random transient errors such as bid-ask bounce and allows for the estimation of short horizon returns. We apply our method to the CRSP equally-weighted monthly return indexes for the NYSE, Amex, and NASDAQ and show that the bias is cumulative. In particular, a NASDAQ index (with a base of 100 in 1973) grows to the level of 17,975 by 2006, but nearly half of the increase is due to cumulative bias. We also conduct a simulation in which we simulate true prices and set spreads according to a discrete pricing grid. True prices are then not necessarily at the midpoint of the spread. In the simulation we compare our method to calculating returns based on observed closing quote midpoints and find that the returns from our method are statistically indistinguishable from the (simulated) true returns. While the mid-quote method results in an improvement over using closing transaction prices, it still results in a statistically significant amount of upward bias. We demonstrate that applying our methodology results in a reversal of the relative performance of NASDAQ stocks versus NYSE stocks over a 25 year window.

Appendices

Appendix 1

1.1 Proof that a log normally distributed error term is approximately equal to a binomially distributed error term

Let $Y_{it} = { \log }_{e} (1 + e_{it} )$. If $( 1+ e_{it} )$ is log normally distributed, Y _it is normally distributed with mean $\mu_{it}$ and variance $\sigma^{2} (Y_{it} )$. From Aitchison and Brown (1957, pp. 8–10),

$$ {\text{E}}(1 + e_{i,t - 1} ) = \exp \left[ {\mu_{it} + 0.5\sigma^{2} (Y_{it} )} \right] $$

(A1)

and

$$ {\text{E}}\left[ {{\frac{1}{{1 + e_{i,t - 1} }}}} \right] = \exp \left[ { - \mu_{it} + 0.5\sigma^{2} (Y_{it} )} \right] $$

(A2)

By assumption${\text{ E(1}} + e_{it} ) = 1$, then since exp(0) = 1 it must be that

$$ \mu_{it} + 0.5\sigma^{2} (Y_{it} ) = 0 $$

(A3)

Solving for $\mu_{it}$ and substituting the result into Eq. (A2) yields

$$ {\text{E}}\left[ {{\frac{1}{{1 + e_{i,t - 1} }}}} \right] = \exp \left[ {\sigma^{2} (Y_{it} )} \right] $$

(A4)

Aitchison and Brown (1957, Eq. 2.9) state that the exponential function of the variance of the transformed distribution (i.e., the right-hand side of Eq. (A4)) is equal to one plus the squared coefficient of variation of the parent, $(1 + e_{it} )$, distribution or $e^{{\sigma^{2} }} = 1 + \eta^{2}$. Therefore, Eq. (A4) can be rewritten as

$$ {\text{E}}\left[ {{\frac{1}{{1 + e_{i,t - 1} }}}} \right] = 1 + {\frac{{\sigma^{2} \left( {1 + e_{i,t - 1} } \right)}}{{\left[ {{\text{E}}(1 + e_{i,t - 1} )} \right]^{2} }}} $$

(A5)

Since ${\text{E(1}} + e_{i,t - 1} ) = 1$ and $\sigma^{ 2} ( 1+ e_{it} ) = \sigma^{2} (e_{i,t - 1} )$, Eq. (A5) becomes

$$ {\text{E}}\left[ {{\frac{1}{{1 + e_{i,t - 1} }}}} \right] = 1 + \sigma^{2} \left( {e_{i,t - 1} } \right) $$

(A6)

.

Appendix 2

2.1 Conditional one-period expected index relatives assuming lagged adjustments to new information

Recall Eq. (5) from the text:

$$ \hat{P}_{{_{i,t} }} = \left[ {\left( {1 - a_{it} } \right)P_{i,t - 1} + a_{it} P_{it} } \right]\left( {1 + e_{it} } \right) $$

(5)

where: a _i,t  = adjustment coefficient which shows the extent to which prices have adjusted to information released since t − 1. If there is no lag process, a _i,t  = 1. For tractability we assume either that all prices fully adjust or that they do not adjust at all.

a i,t−1	a i,t	$\hat{P}_{i,t - 1}$	$\hat{P}_{i,t}$	Conditional $E\left( {\hat{W}_{t} } \right)$
0	0	$P_{i,t - 2} \left( {1 + e_{i,t - 1} } \right)$	$P_{i,t - 1} \left( {1 + e_{it} } \right)$	$W_{t - 1} \overline{{(1 + \sigma^{2} (e_{t - 1} )}} )$
0	1	$P_{i,t - 2} \left( {1 + e_{i,t - 1} } \right)$	$P_{it} \left( {1 + e_{it} } \right)$	$W_{t - 1} W_{t} \overline{{(1 + \sigma^{2} (e_{t - 1} )}} )$
1	0	$P_{i,t - 1} \left( {1 + e_{i,t - 1} } \right)$	$P_{i,t - 1} \left( {1 + e_{it} } \right)$	$\overline{{(1 + \sigma^{2} (e_{t - 1} )}} )$
1	1	$P_{i,t - 1} \left( {1 + e_{i,t - 1} } \right)$	$P_{it} \left( {1 + e_{it} } \right)$	$W_{t} \overline{{(1 + \sigma^{2} (e_{t - 1} )}} )$

Define θ as the probability that a = 0 and α as (1 − θ) then

$$ {\text{E}}\left( {\hat{W}_{t} } \right) \approx \overline{{(1 + \sigma^{2} (e_{t - 1} )}} )\left[ {\theta^{2} W_{t - 1} + \alpha \theta W_{t - 1} W_{t} + \alpha \theta + \alpha^{2} W_{t} } \right] $$

$$ \approx \overline{{(1 + \sigma^{2} (e_{t - 1} )}} \left( {\alpha + W_{t - 1} } \right)\left( {\theta + \alpha W_{t} } \right) $$

(A7)

and since W = (1 + R), where R is the return on the true aggregate portfolio, and α = 1 − θ, then Eq. (A7) can be rewritten as.

$$ {\text{E}}(\hat{W}_{t} ) \approx \overline{{(1 + \sigma^{2} (e_{t - 1} ))}} \left[ {1 - \theta + \theta \left( {1 + R_{t - 1} } \right)} \right]\left\langle {\theta + \left( {1 - \theta } \right)\left( {1 + R_{t} } \right)} \right\rangle $$

(A8)

or

$$ {\text{E}}(\hat{W}_{t} ) \approx \overline{{(1 + \sigma^{2} (e_{t - 1} ))}} \left( {1 + \theta R_{t - 1} } \right)\left( {1 + \alpha R_{t} } \right) $$

(18)

Appendix 3

3.1 The ratio of a two-period expected index relative to a one-period expected index relative assuming lagged adjustment to new information

Following the methodology of “Appendix 2”, finding the expected one-period index relative ending at time t − 1 yields the following conditional table.

a i,t−2	a i,2	$\hat{P}_{i,t - 2}$	$\hat{P}_{i,t - 1}$	Conditional ${\text{E}}\left( {\hat{W}_{t - 1} } \right)$
0	0	$P_{i,t - 3} \left( {1 + e_{i,t - 2} } \right)$	$P_{i,t - 2} \left( {1 + e_{i,t - 1} } \right)$	$W_{t - 2} \overline{{(1 + \sigma^{2} (e_{t - 2} )}} )$
0	1	$P_{i,t - 3} \left( {1 + e_{i,t - 2} } \right)$	$P_{i,t - 1} \left( {1 + e_{i,t - 1} } \right)$	$W_{t - 2} W_{t - 1} \overline{{(1 + \sigma^{2} (e_{t - 2} )}} )$
1	0	$P_{i,t - 2} \left( {1 + e_{i,t - 2} } \right)$	$P_{i,t - 2} \left( {1 + e_{i,t - 1} } \right)$	$\overline{{(1 + \sigma^{2} (e_{t - 2} )}} )$
1	1	$P_{i,t - 2} \left( {1 + e_{i,t - 2} } \right)$	$P_{i,t - 1} \left( {1 + e_{i,t - 1} } \right)$	$W_{t - 1} \overline{{(1 + \sigma^{2} (e_{t - 2} )}} )$

Also as in “Appendix 2”, define θ as the probability that a = 0 and α as 1 − θ. Then

$$ {\text{E}}\left( {\hat{W}_{t - 1} } \right) \approx \overline{{(1 + \sigma^{2} (e_{t - 2} )}} )\left[ {\theta^{2} W_{t - 2} + \alpha \theta W_{t - 2} W_{t - 1} + \alpha \theta + \alpha^{2} W_{t - 1} } \right] $$

(A9)

$$ \approx \overline{{(1 + \sigma^{2} (e_{t - 2} )}} )\left( {\alpha + W_{t - 2} } \right)\left( {\theta + \alpha W_{t - 1} } \right) $$

(A10)

and once again, since W = (! + R), where R is the return on the true aggregate portfolio and α = 1 − θ, then Eq. (A7) can be rewritten as

$$ {\text{E}}(\hat{W}_{t - 1} ) \approx \overline{{(1 + \sigma^{2} (e_{t - 2} ))}} \left[ {1 - \theta + \theta \left( {1 + R_{t - 2} } \right)} \right]\left\langle {\theta + \left( {1 + \theta } \right)\left( {1 + R_{t - 1} } \right)} \right\rangle $$

(A11)

or

$$ {\text{E}}(\hat{W}_{t - 1} ) \approx \overline{{(1 + \sigma^{2} (e_{t - 2} )}} )\left( {1 + \theta R_{t - 2} } \right)\left( {1 + \alpha R_{t - 1} } \right) $$

(A12)

Similarly, the following table gives the two-period aggregate wealth relative.

a i,t−2	a it	$\hat{P}_{i,t - 2}$	$\hat{P}_{it}$	Conditional ${\text{E}}\left( {{}_{2}\hat{W}_{t} } \right)$
0	0	$P_{i,t - 3} \left( {1 + e_{i,t - 2} } \right)$	$P_{i,t - 1} \left( {1 + e_{it} } \right)$	$W_{t - 2} W_{t - 1} \overline{{(1 + \sigma^{2} (e_{t - 2} )}} )$
0	1	$P_{i,t - 3} \left( {1 + e_{i,t - 2} } \right)$	$P_{it} \left( {1 + e_{it} } \right)$	$W_{t - 2} W_{t - 1} W_{t} \overline{{(1 + \sigma^{2} (e_{t - 2} )}} )$
1	0	$P_{i,t - 2} \left( {1 + e_{i,t - 2} } \right)$	$P_{i,t - 1} \left( {1 + e_{it} } \right)$	$W_{t - 1} \overline{{(1 + \sigma^{2} (e_{t - 2} )}} )$
1	1	$P_{i,t - 2} \left( {1 + e_{i,t - 2} } \right)$	$P_{it} \left( {1 + e_{it} } \right)$	$W_{t - 1} W_{t} \overline{{(1 + \sigma^{2} (e_{t - 2} )}} )$

Defining θ as the probability that a = 0 and α as (1 − θ) then

$$ {\text{E}}\left( {{}_{2}\hat{W}_{t} } \right) \approx \left( {\overline{{1 + \sigma^{2} (e_{t - 2} )}} } \right)\left[ {\theta^{2} W_{t - 2} W_{t - 1} + \alpha \theta W_{t - 2} + \alpha \theta W_{t - 1} + \alpha^{2} W_{t - 1} W_{t} } \right] $$

(A13)

and in return form

$$ {\text{E}}({}_{2}\hat{W}_{t} ) \approx \left( {\overline{{1 + \sigma^{2} (e_{t - 2} )}} } \right)\left[ {\left( {1 + R_{t - 1} } \right)\left( {1 + \theta R_{t - 2} } \right)\left( {1 + \alpha R_{t} } \right)} \right] $$

(A14)

then the ratio of (A14) to (A12) is

$$ {\frac{{{\text{E}}\left( {_{2} \hat{W}_{t} } \right)}}{{{\text{E}}\left( {\hat{W}_{t - 1} } \right)}}} = {\frac{{\left( {\overline{{1 + \sigma^{2} (e_{t - 2} )}} } \right)\left[ {\left( {1 + R_{t - 1} } \right)\left( {1 + \theta R_{t - 2} } \right)\left( {1 + \alpha R_{t} } \right)} \right]}}{{\left( {\overline{{1 + \sigma^{2} (e_{t - 2} )}} } \right)\left( {1 + \theta R_{t - 2} } \right)\left( {1 + \alpha R_{t - 1} } \right)}}} $$

(A15)

Since $\left\{ {\overline{{ 1+ \sigma^{ 2} \left( {e_{i,t - 2} } \right)}} } \right\}$ is in the numerator and denominator, Jensen’s inequality does not apply, thus

$$ {\text{E}}\left( {{\frac{{{}_{2}\hat{W}_{t} }}{{\hat{W}_{t - 1} }}}} \right) \approx {\frac{{\left( {1 + R_{t - 1} } \right)\left( {1 + \alpha R_{t} } \right)}}{{\left( {1 + \alpha R_{t - 1} } \right)}}} $$

(A16)

Therefore, in the presence of a lagged adjustment process longer than the holding period under consideration, our method will remove all of the random transient error bias, but not all of the lagged adjustment process bias.