Proximity matters!

Figure based on Federle et al. (2022).

Fama-French-Carhart four-factor model

To quantify the effect of the Russia-Ukraine war on the index returns, we control for four relevant factors by using an event study approach

Fama-French-Carhart four-factor model

To quantify the effect of the Russia-Ukraine war on the index returns, we control for four relevant factors by using an event study approach

Pre-event estimation window is defined as (traded days -350, -101), the event window is (-100,0), as the common event date (day 0) is 2022 February 24

Fama-French-Carhart four-factor model

We estimated the country-specific \(\beta\) parameters with the observations of the pre-event window, and apply them to calculate the Abnormal returns in the event window.

\[ AR_{i,t} = r_{i,t} - \hat{\beta}_{0,i} - \hat{\beta}_{1,i} MKT_t - \hat{\beta}_{2,i}SMB_t - \hat{\beta}_{3,i}HML_t-\hat{\beta}_{4,i}UMD_t \]

• AR shows the daily change of an index after controlling for the described factors

• We derive adjusted price indexes as follows:

\[ p_{i, t} = (1 + AR_{i,t}) p_{i,t-1} \:\:\text{and}\:\: p_{i,0}=1 \]

Rolling OLS

• Traders anticipating the outbreak of war will avoid certain regions whose markets will be disrupted by the conflict.

• We assume that the traders may expect which markets are the ones that will fall

• Liquidation value (\(v\)): the price of the given index on the event day

• Informed traders will buy stocks at markets where the expected liquidation value is higher than the actual

These result the following price movement, if the traders anticipate the war:

\[\Delta p_{i,t}= \beta_0 + \beta_{1}\left(v-p_{i,t-1}\right)+\epsilon_{i,t}\]

Rolling OLS

\[\Delta p_{i,t}= \beta_0 + \beta_{1}\left(v-p_{i,t-1}\right)+\epsilon_{i,t}\]

Rolling OLS

\[\Delta p_{i,t}= \beta_0 + \beta_{1}\left(v-p_{i,t-1}\right)+\epsilon_{i,t}\]

• We estimate the coefficient related to the \(v-p_{i,t-1}\) term, in 10-days rolling windows

• PRIOR EXPECTATION: price changes should depend positively on \((v − p_{t−1})\), and that this dependence should grow as we are getting closer in time (Ellison and Mullin, 1997) to the outbreak of war

\[\Delta p_{i,t}= \beta_0 + \beta_{j}\left(v-p_{i,t-1}\right)D_j+\epsilon_{i,t}\]

, where \(D_{j}\) is a categorical variable referring to a 10-day (trading day) rolling window

Rolling OLS

\[\Delta p_{i,t}= \beta_0 + \beta_{j}\left(v-p_{i,t-1}\right)D_j+\epsilon_{i,t}\]

Rolling OLS

\[\Delta p_{i,t}= \beta_0 + \beta_{j}\left(v-p_{i,t-1}\right)D_j+\epsilon_{i,t}\]

This estimation follows the framework applied by Ellison and Mullin (1997)

Rolling OLS

\[\Delta p_{i,t}= \beta_0 + \beta_{j}\left(v-p_{i,t-1}\right)D_j+\epsilon_{i,t}\]

This estimation follows the framework applied by Ellison and Mullin (1997)

Rolling OLS

\[\Delta p_{i,t}= \beta_0 + \beta_{j}\left(v-p_{i,t-1}\right)D_j+\epsilon_{i,t}\]

This estimation follows the framework applied by Ellison and Mullin (1997)

• Constantly hold intercept could lead to error clusters

• We re-estimate the model with changing intercept to ensure that this does not distort our results (like separate regressions)

\[\Delta p_{i,t}= \beta_{0,j}D_j + \beta_{1,j}\left(v-p_{i,t-1}\right)D_j+\epsilon_{i,t}\]

Rolling OLS

\[\Delta p_{i,t}= \beta_{0,j}D_j + \beta_{1,j}\left(v-p_{i,t-1}\right)D_j+\epsilon_{i,t}\]

Nonlinear regression

Motivation

Nonlinear regression

Motivation

• The previous methodologies could only detect the time window, when markets started to price the risk of the war

• We would like to apply an empirical model to identify the exact day

• Dependence should grow by time

• We also saw this pattern empirically

• Traders gradually incorporate their information into their behaviour

Nonlinear regression

Motivation

• Dependence should grow by time

The price movement should follow the following formula, when the markets price the war:

\[\Delta p_{t}=\gamma_{1}\left(v-p_{t-1}\right)+\gamma_{2}\left(v-p_{t-1}\right) t+\epsilon_{t}\]

Nonlinear regression

SOLUTION: Let us multiply the investigated effects with indicator functions (Ellison and Mullin, 1997)

\[\Delta p_{t}=\xi_{0}+\gamma_{1}\left(v-p_{t-1}\right) \frac{e^{\left(t-t_{0}\right)}}{1+e^{\left(t-t_{0}\right)}}+ \\ \gamma_{2}\left(v-p_{t-1}\right)\left(\frac{t-t_{0}}{T-t_{0}}\right) \frac{e^{\left(t-t_{0}\right)}}{1+e^{\left(t-t_{0}\right)}} + \epsilon_t,\]

where \(t_0\) denotes the estimated starting point of the trading process, \(t\) refers to calendar days from the first day of the investigated time window and \(T\) equals to the length of the window (100 in our case).

Interpretation: \(\frac{e^{\left(t-t_{0}\right)}}{1+e^{\left(t-t_{0}\right)}}\)

Interpretation: \(\frac{t-t_{0}}{T-t_{0}}\)

Interpretation: \(\left(\frac{t-t_{0}}{T-t_{0}}\right) \frac{e^{\left(t-t_{0}\right)}}{1+e^{\left(t-t_{0}\right)}}\)

Nonlinear regression

\[\Delta p_{t}=\xi_{0}+\gamma_{1}\left(v-p_{t-1}\right) \frac{e^{\left(t-t_{0}\right)}}{1+e^{\left(t-t_{0}\right)}}+ \\ \gamma_{2}\left(v-p_{t-1}\right)\left(\frac{t-t_{0}}{T-t_{0}}\right) \frac{e^{\left(t-t_{0}\right)}}{1+e^{\left(t-t_{0}\right)}} + \epsilon_t\]

Nonlinear regression

Grid search

• We generate grid of the four estimated parameters (\(0 \leq \xi_0 \leq 1; 0 \leq \gamma_1 \leq 1; 0 \leq \gamma_2 \leq 1; 1 \leq t_0 \leq 100\)), and pick 1000 random combinations, to ensure that we find the global optimum

• We choose the model with the highest log-likelihood

#> # A tibble: 845 × 5
#>       xi gamma1    t0 gamma2 fit   
#>    <dbl>  <dbl> <dbl>  <dbl> <list>
#>  1 0.632  0.632  94.8 0.737  <nls> 
#>  2 0.421  0.526  73.9 0.316  <nls> 
#>  3 0.737  0.368  94.8 0.158  <nls> 
#>  4 0.895  0.684  89.6 0.211  <nls> 
#>  5 0.632  0.421  94.8 0.789  <nls> 
#>  6 0.895  0.368  79.2 0      <nls> 
#>  7 0.211  0.684  42.7 0.842  <nls> 
#>  8 0.895  0.526  16.6 0.158  <nls> 
#>  9 0.368  0.684  21.8 0.0526 <nls> 
#> 10 0.842  0.368  27.1 0.947  <nls> 
#> # … with 835 more rows

Nonlinear regression

The best fitted model

The table shows that:

1. Information incoroporated gradually (\(\gamma_2\) is significant)

2. The market started to price the risk of the war 49 days before its outbreak (\(100-t_0\))

References

• Ellison, S. F. and W. Mullin (1997). Gradual incorporation of information into stock prices: empirical strategies. URL: http://link.springer.com/10.1007/978-0-387-09616-2.

• Ritz, C. and J. C. Streibig, ed. (2009). Nonlinear Regression with R. New York, NY: Springer New York. ISBN: 978-0-387-09615-5 978-0-387-09616-2. DOI: 10.1007/978-0-387-09616-2. URL: http://link.springer.com/10.1007/978-0-387-09616-2 (visited on Jul. 08, 2022).

• Boubaker, S., J. W. Goodell, D. K. Pandey, et al. (2022). “Heterogeneous impacts of wars on global equity markets: Evidence from the invasion of Ukraine”. In: Finance Research Letters 48, p. 102934. ISSN: 15446123. DOI: 10.1016/j.frl.2022.102934. URL: https://linkinghub.elsevier.com/retrieve/pii/S1544612322001969 (visited on Jun. 14, 2022).

References

• Federle, J., A. Meier, G. J. Müller, et al. (2022). Proximity to War: The Stock Market Response to the Russian Invasion of Ukraine. SSRN Scholarly Paper 4060222. Rochester, NY: Social Science Research Network. DOI: 10.2139/ssrn.4060222. URL: https://papers.ssrn.com/abstract=4060222 (visited on Jun. 15, 2022).

References

• Asness, C. S., J. M. Liew, and R. L. Stevens (1997). “Parallels between the cross-sectional predictability of stock and country returns”. In: Journal of Portfolio Management 23.3, p. 79.

• Becker, R. A., A. R. Wilks, R. Brownrigg, et al. (2021). maps: Draw Geographical Maps. R package version 3.4.0. URL: https://CRAN.R-project.org/package=maps.

• Asness, C. S., J. M. Liew, and R. L. Stevens (1997). “Parallels between the cross-sectional predictability of stock and country returns”. In: Journal of Portfolio Management 23.3, p. 79.

• Fama, E. F. and K. R. French (1993). “Common risk factors in the returns on stocks and bonds”. In: Journal of Financial Economics 33.1, pp. 3–56.

References

• Jegadeesh, N. and S. Titman (1993). “Returns to buying winners and selling losers: Implications for stock market efficiency”. In: The Journal of Finance 48.1, pp. 65–91.

• Carhart, M. M. (1997). “On persistence in mutual fund performance”. In: The Journal of Finance 52.1, pp. 57–82.

• MacKinlay, A. C. (1997). “Event studies in economics and finance”. In: Journal of economic literature 35.1, pp. 13–39.

• Kothari, S. P. and J. B. Warner (2007). “Econometrics of event studies”. In: Handbook of empirical corporate finance. Elsevier, pp. 3–36.

Rolling OLS params

#> # A tibble: 15 × 5
#>    Term                    Estimate `Std. error` Statistic `P-value`
#>    <chr>                   <chr>    <chr>        <chr>     <chr>    
#>  1 Intercept               0.0002   0.0001       1.2028    0.2291   
#>  2 2021-10-07 - 2021-10-15 0.0090   0.0044       2.0281    0.0426** 
#>  3 2021-10-18 - 2021-10-26 0.0066   0.0043       1.5504    0.1211   
#>  4 2021-10-27 - 2021-11-05 0.0046   0.0040       1.1367    0.2557   
#>  5 2021-11-08 - 2021-11-15 0.0145   0.0049       2.9653    0.0030***
#>  6 2021-11-16 - 2021-11-25 0.0105   0.0043       2.4280    0.0152** 
#>  7 2021-11-26 - 2021-12-03 0.0177   0.0052       3.3894    0.0007***
#>  8 2021-12-06 - 2021-12-15 0.0007   0.0047       0.1582    0.8743   
#>  9 2021-12-16 - 2021-12-24 0.0113   0.0051       2.2283    0.0259** 
#> 10 2021-12-27 - 2022-01-04 0.0051   0.0050       1.0302    0.3029   
#> 11 2022-01-05 - 2022-01-14 0.0279   0.0050       5.6082    0.0000***
#> 12 2022-01-17 - 2022-01-24 0.0511   0.0069       7.4059    0.0000***
#> 13 2022-01-25 - 2022-02-03 0.0322   0.0080       4.0479    0.0001***
#> 14 2022-02-04 - 2022-02-11 0.0422   0.0096       4.4158    0.0000***
#> 15 2022-02-14 - 2022-02-23 0.0910   0.0110       8.2644    0.0000***