In their seminal paper Jegadeesh and Titman (1993) develop a statistical model to infer where moment comes from.
In practice they setup the following:
$r_{it}=\mu_i + b_i f_t +e_{it}$
$E(f_t)=E(e_{it})=cov(e_{it},f_t)=cov(e_{it},e_{j,t-1})=0$.
The implication of momentum is:
$E[(r_{it}-\bar{r_t})(r_{it-1}-\bar{r_{t-1}})]>0$
From the equation above they get:
$E[(r_{it}-\bar{r_t})(r_{it-1}-\bar{r_{t-1}})]=\sigma^2_\mu+\sigma^2_bCov(f_t,f_{t-1})+\bar{\text{Cov}}(e_{it},e_{it-1})$
This is equation (3) of the above paper. Does anyone know how do they get this?
Edit: I think I am close to the answer:
$E[(r_{it}-\bar{r_t})(r_{it-1}-\bar{r_{t-1}})]=E[r_{it} r_{it-1}]-E[r_{it}]E[ r_{it-1}] = E[(\mu_i + b_i f_t +e_{it})(\mu_i + b_i f_{t-1} +e_{it-1})]-\mu_i^2=E[\mu_i^2+\mu_ibf_{t-1}+\mu_ie_{it}+bf_t\mu_i+b^2f_tf_{t-1}+bf_te_{it-1}+\mu_ie_{it-1}+bf_{t-1}e_{it-1}+e_{it}e_{it-1}]-\mu_i^2=E[\mu_i^2+b^2f_tf_{t-1}+e_{it}e_{it-1}]-\mu_i^2$
I am just now not sure:
1) Why can I factor out from the expectation $b^2f_tf_{t-1}=\sigma^2_bCov(f_t,f_{t-1})$
2) Why is there a bar above the last covariance
3) What happened to the $\mu_i^2$ term?
Answer
Preliminary
This wonderful question is directly connected to the necessity of precise definitions and carefully writing in academic research: The decomposition you are asking for has two different solutions, regarding to the framework you are using. So a superficial look to the literature and quickly comparing the formulas yields to confusion, especially in this case.
Economic implications
The implication of momentum in Jegadeesh and Titman (1993) is $$E[(r_{it}-\bar{r_t})(r_{it-1}-\bar{r_{t-1}})]>0$$
i.e. that stocks that generate higher returns than average returns in one period, also generate higher than average returns in the subsequent period. It is important to note, that a bar above a variable denotes its cross-sectional average and not its time-series mean (p. 71).
The formula above describes a trading strategie, which weights stocks by their past returns less the past equally weighted index returns, i.e. $r_{it-1}-\bar{r_{t-1}}$ is the weighting $\omega_{i,t}$ for the excess return of stock $i$ in the current period. Jegadeesh and Titman (1990) p. 71 state:
The above cross-sectional covariance equals the expected profits from the zero-cost contrarian trading strategy examined by Lehman (1990) and Lo and MacKinlay (1990) that weights stocks by their past returns less the past equally weighted index returns.
So in fact, you have to follow this strategy and analyze a diversified portfolio of stocks, investing long in past "winners" and short in past "losers" and weighting stocks with their past return. Further:
This weighted relative strength strategy (WRSS) is closely related to our strategy.
Keep this carefully in mind. I will discuss the difference of their strategy and the WRSS later. Now let's formalize their investment strategy.
How NOT to derive equation (3) from Jegadeesh and Titman (1993)
Consider you invest equal-weighted in a broad range of $N$ stocks. The cross-sectional mean of stock returns equals the market return, so this results in $$\bar{r_{t-1}} = \sum_{i=1}^{N}{\frac{r_{it-1}}{N}} = r_{mt-1}$$ where $r_{mt-1}$ is the equal-weighted market return in $t_{-1}$.
Their equation (3)
$$E[(r_{it}-\bar{r_t})(r_{it-1}-\bar{r_{t-1}})]>0$$
is applicable for any return $r_{i}$ (and true for stock $i$, if $i$ provides momentum). What you have to look at is the cross-sectional application of this formula within market equilibrium, based on the return generating process
$$r_{it}=\mu_i + b_i f_t +e_{it}$$ $$E(f_t)=E(e_{it})=Cov(e_{it},f_t)=Cov(e_{it},e_{j,t-1})=0$$
where $\mu_i$ is the unconditional expected return on security $i$, $f_t$ is the unconditional unexpected return on a factor-mimicking portfolio, $e_{it}$ is the firm-specific component of return at time $t$, and $b_i$ is the factor sensitivity of security $i$.
The profit $\pi_t$ from the diversified strategy investing in $N$ stocks is
$$\pi_t = \frac{1}{N} \sum_{i=1}^{N}{\omega_{i,t} \cdot r_{i,t}}$$
with the return weighting $\omega_{i,t} = r_{it-1}-\bar{r_{t-1}}$. The weighting $\omega_{i,t}$ is positiv for stocks, which were "winners" (with respect to $r_{mt-1}$) in $t_{-1}$ and negativ for "losers". As the weights sum up to zero, you have a zero cost arbitrage portfolio. If there is a "momentum anomaly", than the excess return $\pi_t^e = \pi_t - r_{mt}$ should be (positiv) statistical significant differently from zero.
Let's go on with analyzing the profit $\pi_t$ and decompose it:
$$\pi_t = \frac{1}{N} \sum_{i=1}^{N}{\left(r_{it-1}-\bar{r_{t-1}} \right) \cdot r_{i,t}}$$
$$=\frac{1}{N} \sum_{i=1}^{N}{r_{it-1}r_{it}} - \frac{1}{N} \sum_{i=1}^{N}{r_{mt-1} r_{i,t}}$$
$$= \left( \frac{1}{N} \sum_{i=1}^{N}{r_{it-1}r_{it}} \right)- {r_{mt-1}r_{m,t}}$$
Re-arranging and taking unconditional expectations yields
$$E[\pi_t] = \left( \frac{1}{N} \sum_{i=1}^{N}{E[r_{it-1}r_{it}}] \right)- E[{r_{mt-1}r_{m,t}}]$$
$$= \left( \frac{1}{N} \sum_{i=1}^{N}{Cov(r_{it-1},r_{it}}) + \mu_i^2 \right)- Cov({r_{mt-1},r_{m,t}}) - \mu_m^2$$
$$= \frac{1}{N} \sum_{i=1}^{N}{Cov(r_{it-1},r_{it}}) + \frac{1}{N} \sum_{i=1}^{N}{\mu_i^2} - Cov({r_{mt-1},r_{m,t}}) - \mu_m^2$$
$$= \frac{1}{N} \sum_{i=1}^{N}{Cov(r_{it-1},r_{it}}) + \frac{1}{N} \sum_{i=1}^{N}{(\mu_i-\mu_m)^2} - Cov({r_{mt-1},r_{m,t}})$$
The first term describes the cross-sectional average of first-order autocovariances of the individual securities, which can be re-written as $\bar{\text{Cov}}(e_{it},e_{it-1})$. The second term describes the cross-sectional variance of mean returns which can be denoted as $\sigma^2_\mu$. So we have
$$ E[\pi_t] = \bar{Cov(e_{it},e_{it-1})} + \sigma^2_\mu - Cov({r_{mt-1},r_{m,t}})$$
The third term is the first-order autocovariance of the equally-weighted market index. Further in eq. (4), Jegadeesh and Titman (1993) states*
$$Cov(\bar{r_t},\bar{r_{t-1}}) = \bar{b_i}^2 Cov(f_t,f_{t-1})$$
which is an approximation assuming that the contribution of the serial covariances of $e_{it}$ to the serial covariance of the equally weighted index becomes arbitrarily small as the number of stocks in the index becomes arbitrarily large. As $\bar{r_t}$ is the return of the equal-weighted market portfolio, this results in $$Cov(\bar{r_{mt}},\bar{r_{mt-1}}) = \bar{b_i}^2 Cov(f_t,f_{t-1}).$$
The term $\bar{b_i}^2$ describes the cross-sectional variance of factor-loadings, so finally you get
$$ E[\pi_t] = \bar{Cov(e_{it},e_{it-1})} + \sigma^2_\mu - \sigma_b^2 Cov(f_t,f_{t-1})$$
Why is the third term negative and how do i finally derive equation (3) from Jegadeesh and Titman (1993)?
First, don't worry about the negative sign: The above equation for $E[\pi_t]$ is exactly the same as Jegadeesh and Titman (2002) themselves use in eq. (3) (of the 2002-paper!) - using a different point of view. On p. 145 of this paper they claim
The trading strategy in Jegadeesh and Titman buys the decile of stocks with the highest past returns and sells the decile of stocks with the lowest past returns. The stocks in the buy and sell portfolios are equally weighted in the Jegadeesh and Titman strategy. To understand the sources of momentum profits, however, it is analytically more convenient to consider the weighting scheme originally proposed by Lo and MacKinlay (1990) [...].
So the above decomposition refers to the one of Lo/MacKinlay (1990). The methods used in empirical research (e.g. portfolio sorts) are generally based on calculating their relative excess return, but
the momentum literature can be divided into two streams:
- Relative strength strategies (RSS), proposed by DeBondt and Thaler (1985) and Jegadeesh and Titman (1993).
- Weighted relative strength strategies (WRSS), proposed by Lo and MacKinlay (1990) and Conrad and Kaul (1998).
When we did the weighting
$$\pi_t = \frac{1}{N} \sum_{i=1}^{N}{\omega_{i,t} \cdot r_{i,t}}$$
we follow the (originally) WRSS approach. The RSS however considers stock excess returns in the current period, i.e. instead of $r_{it}$ you are interested in $r_{it}-\bar{r_{t}}$. Remember Jegadeesh/Titman (1990) p. 71:
This weighted relative strength strategy (WRSS) is closely related to our strategy.
The WRSS approach (in terms of Lo/MacKinlay) equals the approach from Jegadeesh/Titman (1990) only if you ignore excess returns in the current period. But that is not what Jegadeesh/Titman refer to on p.72:
[...] the WRSS profits given in expression (2) [...]
Expression (2) however is
$$E[(r_{it}-\bar{r_t})(r_{it-1}-\bar{r_{t-1}})]>0$$
and explicitly considers excess returns in the current period, which makes their approach different from the originally WRSS approach. This is what leeds to confusion when reading other papers: Also Jegadeesh/Titman (1993) call it WRSS, their approach is (mathematical slightly but in fact substantial) different from the originally WRSS approach.
So considering excess returns in the current period leads to
$$E\left[\pi_t^{RSS}\right]=E[(r_{it}-\bar{r_t})(r_{it-1}-\bar{r_{t-1}})]= E\left[\frac{1}{N} \sum_{i=1}^{N}{\omega_{i,t} \cdot \left( r_{it}-\bar{r_{t}} \right)}\right]$$
This leads with minor algebraic rearranging to
$$E[\pi_t^{RSS}]=\frac{1}{N} \sum_{i=1}^{N}{\left( E[r_{it-1}r_{it}] - E[r_{mt-1}r_{mt}] \right)}$$
Taking unconditional expectations yields
$$E[\pi_t^{RSS}]= \left[ \frac{1}{N} \sum_{i=1}^{N}{\mu_i^2} + \frac{1}{N} \sum_{i=1}^{N}{\beta_i^2 Cov(f_t,f_{t-1})} + \frac{1}{N} \sum_{i=1}^{N}{Cov(\epsilon_t,\epsilon_{t-1})}\right]- \left[ \mu_m^2 + \beta_m^2 Cov(f_t,f_{t-1}) +\frac{1}{N^2} \sum_{i=1}^{N}{Cov(\epsilon_t,\epsilon_{t-1})} \right]$$
$$= \sigma_\mu^2 + \sigma_\beta^2 Cov(f_t,f_{t-1}) + \frac{N-1}{N^2} \sum_{i=1}^{N}{Cov(\epsilon_t,\epsilon_{t-1})}$$
The third term equals the cross-sectional average of first-order autocovariances of the individual securities, so after rearranging you get
$$ E[(r_{it}-\bar{r_t})(r_{it-1}-\bar{r_{t-1}})] = \bar{Cov(e_{it},e_{it-1})} + \sigma^2_\mu + \sigma_b^2 Cov(f_t,f_{t-1})$$
which is the equation you asked for.
Additional remarks
- If you read the literature to the WRSS approach through the eyes of a "momentum researcher", you have to switch signs again, as the strategy has it's root in the analysis of a contrarian investment strategy, i.e. Lo/MacKinlay (1990) p. 6:
[...], consider buying stocks at time $t$ that were "losers" at time $t-k$ and selling stocks at time $t$ that were "winners" at time $t-k$, [...].
That is the economic equivalent of a momentum strategy if you switch the signs within this framework.
- A slightly more comprehensive representation on how to decompose the momentum based portfolio return is provided by Dittmar/Kaul/Lei (2007) and in even more detail in this PhD thesis (pp. 21).
* Further calculations on the derivation of this equation (4) is provided by Campbell/Lo/MacKinlay (1997), pp. 74-78.
References
Campbell/Lo/MacKinlay (1997), The Econometrics of Financial Markets, ed. 2, Princeton University Press.
Conrad/Kaul (1998), An Anatomy of Trading Strategies, The Review of Financial Studies, Vol. 11(3).
DeBondt/Thaler (1985), Does the Stock Market Overreact?, The Journal of Finance, Vol. 40(3).
Dittmar/Kaul/Lei (2007), Momentum is Not an Anomaly, available at SSRN: https://ssrn.com/abstract=1027057
Jegadeesh/Titman (1993), Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency, The Journal of Finance, Vol. 48(1).
Jegadeesh/Titman (2002), Cross-Sectional and Time-Series Determinants of Momentum Returns, The Review of Financial Studies, Vol. 15(1).
Lehman (1990), Fads, Martingales, and Market Efficiency, Quarterly Journal of Economics, Vol. 105.
Lo/MacKinlay (1990), When Are Contrarian Profits Due to Stock Market Overreaction?, The Review of Financial Studies, Vol. 3(2).
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