Quantitative law describing market dynamics before and after interest-rate change

We study the behavior of U.S. markets both before and after U.S. Federal Open Market Commission meetings and show that the announcement of a U.S. Federal Reserve rate change causes a financial shock, where the dynamics after the announcement is described by an analog of the Omori earthquake law. We quantify the rate $n\left(t\right)$ of aftershocks following an interest-rate change at time $T$ and find power-law decay which scales as $n\left(t-T\right)\sim {\left(t-T\right)}^{-\Omega }$, with $\Omega$ positive. Surprisingly, we find that the same law describes the rate $n^{\prime }\left(\left|t-T\right|\right)$ of “preshocks” before the interest-rate change at time $T$. This study quantitatively relates the size of the market response to the news which caused the shock and uncovers the presence of quantifiable preshocks. We demonstrate that the news associated with interest-rate change is responsible for causing both the anticipation before the announcement and the surprise after the announcement. We estimate the magnitude of financial news using the relative difference between the U.S. Treasury Bill and the Federal Funds effective rate. Our results are consistent with the “sign effect,” in which “bad news” has a larger impact than “good news.” Furthermore, we observe significant volatility aftershocks, confirming a “market under-reaction” that lasts at least one trading day.

Figure 1

(Color online) An illustration of the close relation between the Treasury Bill and the Federal Funds rate. (a) Time series of the Federal Reserve Target rate $R\left(t\right)$ and the Federal Reserve Effective rate $F\left(t\right)$ for Federal Funds dating from January 2000 to April 2008. The 6 month Treasury Bill $T\left(t\right)$ closely follows the effective rate, with speculation about future changes causing deviations in the relative values. United States Treasury Bills carry little risk and are considered to be one of the most secure investments. (b) A typical illustration of the Federal Funds effective rate and the Treasury Bill, where both gravitate around the Federal Funds target rate. The change in the relative spread $\delta \left(t\right)$, defined in Eq. (1), between the Treasury Bill and the Federal Funds effective rate, indicates changes in market speculation. (c) The relative spread $\delta \left(t\right)$, 15 days before and 15 days after the scheduled FOMC meeting on December 14, 2004, which corresponds to $\Delta t=0$. Note that the average value of the relative spread increases after the announcement, indicating a shift in market consensus and speculation.

Figure 2

(Color online) Average daily volatility trend defined in Eq. (4) exhibits increased market volatility on the day of FOMC meetings, corresponding to $\Delta t=0$. “Bank” refers to the portfolio of 18 stocks that belong to the S&P 100. There is a $15–20\mathrm{\%}$ increase in volatility on days corresponding to FOMC meetings. Standard deviation $\sigma \mathbf{\left(}v\left(\Delta t\right)\mathbf{\right)}\approx 0.4$ can be assigned to each data point in the time series and is calculated by randomizing the daily volatility time series of each company. (Inset) Probability density function (pdf) of normalized volatility $v\equiv r\left(t\right)/⟨r⟩$, where the quantity $r\left(t\right)\equiv \text{ln}\left[p_{hi}\left(t\right)/p_{low}\left(t\right)\right]$ is the range of the price time series of a given stock for a particular day. We plot the pdf of volatility values for the S&P 100 on the set of days with FOMC meetings and for the set of all “other” days. The distributions are approximately logarithmic normal, with a shift toward higher average volatility on FOMC days. The average values for the two data sets are ${⟨v⟩}_{FOMC}=1.12$ and ${⟨v⟩}_{other}=1.00$.

Figure 3

(Color online) Demonstration of the relation between speculation of interest-rate change and market volatility in the S&P 100 and for the subset of banking stocks. We relate ${\Theta }_i$, the speculation in the market before a FOMC meeting defined in Eq. (2), to the market volatility $V_i$ defined in Eq. (5). A large absolute value of ${\Theta }_i$ reflects the high probability that an interest-rate change will happen. Interestingly, there are many instances where ${\Theta }_i\approx 0$, followed by large volatility. These values correspond to FOMC decisions to maintain interest-rate levels $\left(\Delta R=0\right)$ and suggest a fundamental difference in the dynamics following decisions to change versus decisions not change the Federal Funds target rate. Also, there is an underlying symmetry in $\Delta R$ since in the case of either a rate increase or a rate decrease, the FOMC also has the option of no increase. Hence, $\Delta R=0$ can occur as either good or bad news, whereas typically decisions of $\Delta R>0$ reflect situations with positive market sentiment whereas decisions of $\Delta R<0$ reflect situations with negative market sentiment. Hence, the asymmetry in market volatility is consistent with the sign effect [12]. Although the correlation between ${\Theta }_i$ and $V_i$ is dominated by residual error, it is nevertheless supporting that the regression captures the crossover at $\left(\theta ,V\right)=\left(0,1\right)$. Including all data points, the regression correlation coefficient is $r^2=0.34$, and the slope of the regression is $m=0.36\pm 0.13$ for (a), and $r^2=0.30$ and $m=0.54\pm 0.22$ for (b). Restricting data points corresponding only to interest-rate changes (red and green triangles): $r^2=0.48$ and $m=0.37\pm 0.12$ for (a), and $r^2=0.40$ and $m=0.53\pm 0.21$ for (b) (this second regression is not shown and is indistinguishable from the regression including all data points). All linear regressions pass the ANOVA (Analysis of variance) $F$ test, rejecting the null hypothesis that $m=0$ at the $\alpha =0.05$ significance level.

Figure 4

(Color online) The cumulative volatility time series $N\left(t\right)$ demonstrates Omori-law response dynamics; here in response to FOMC announcements occurring at the time $T$ indicated by a vertical solid red line. The abrupt change in the curvature of $N\left(t\right)$ around time $t\approx T$ illustrates the increased volatility caused by the announcement. The significant aftershocks which occur until the end of the trading day are consistent with market under-reaction [11, 12]. Market under-reaction and other market inefficiencies can result from increased levels of uncertainty among traders following market news [13]. Each time series $N\left(t\right)$ is calculated for a given volatility threshold $q$, where larger $q$ values correspond to $N\left(t\right)$ curves with smaller amplitude (smaller rate of large volatility events). (a)–(c) illustrate the dynamics around a scheduled announcement made at $T=285\text{min}$ (2:15 p.m. ET). For the S&P 100, we calculate $N\left(t\right)$ on 05/15/01 for (a) 1 min volatility and (b) 1 min total volume using Eq. (8). (c) We calculate $N^j\left(t\right)$ for MER on 08/21/01. (d) The Omori law also occurs for unscheduled FOMC announcements, as illustrated for the bank sector $N\left(t\right)$ on 04/18/01, when the surprise rate change was announced at $T=90\text{min}$ (11:00 a.m. ET), resulting in raised levels of volatility throughout the entire trading day. For (a)–(d), the dashed red lines are power-law fits beginning immediately after the announcement, with the corresponding exponents ${\Omega }_a\left(q\right)$ appearing in parentheses within the legends.

Figure 5

(Color online) The Omori-law relaxation can extend for several days. We compare the Omori exponents ${\Omega }_a\left(q\right)$ (indicated in legends) calculated for (a) the time series $N_a\left(\tau \right)$ of the bank sector and (b) the time series $N_a^j\left(\tau \right)$ of Merrill Lynch for 3 days (1275 min) after the announcement on Tuesday 08/21/01 at 2:15 p.m. ($\tau =0\text{min}$ corresponding to $T=285\text{min}$). For the remaining 3 days of the trading week, the Omori-law relaxation corresponding to an individual stock (MER) is quantitatively similar to the Omori-law relaxation of the bank sector over the final 105 min of the initial trading day. We do not use the bank sector $N_a\left(\tau \right)$ over the same 1275 min time period for comparison because “opening effects” occurring during the first 60 min of each trading day make power-law regression of conjoined $N_a\left(\tau \right)$ problematic.

Figure 6

(Color online) An illustration of the method used to calculate (a) $N_{b,i}\left(\tau \right)=N_i\left(T_i\right)-N_i\left(\left|t-T_i\right|\right)$ and (b) $N_{a,i}\left(\tau \right)=N_i\left(t-T_i\right)-N_i\left(T_i\right)$ for each intraday time series $N_i\left(t\right)$. The displaced time $\tau =\left|t-T_i\right|$ is defined symmetrically around the time of the announcement $T_i$. We plot the same data as in Fig. 4a, corresponding to the announcement on 05/15/01 which occurred at $T=285\text{min}$. (c) and (d) show that $N_{b,i}\left(\tau \right)$ and $N_{a,i}\left(\tau \right)$ are approximately linear on logarithmic scale.

Figure 7

(Color online) Comparison of Omori-law exponents for both volatility dynamics and volume dynamics on the day of 19 FOMC meetings during the 2 year period of January 2001–December 2002. (a)–(d) correspond to the S&P 100 and (e)–(h) correspond to the bank sector. The average value of $\Omega$ for the 16 scheduled FOMC meetings (excluding the three unannounced meetings $i=\left\{1,4,8\right\}$) are (a) ${\overline{\Omega }}_b=0.10\pm 0.13$, (b) ${\overline{\Omega }}_a=0.24\pm 0.08$, (c) ${\overline{\Omega }}_b=0.04\pm 0.07$, (d) ${\overline{\Omega }}_a=0.24\pm 0.07$, (e) ${\overline{\Omega }}_b=0.11\pm 0.16$, (f) ${\overline{\Omega }}_a=0.23\pm 0.09$, (g) ${\overline{\Omega }}_b=0.01\pm 0.10$, and (h) ${\overline{\Omega }}_a=0.26\pm 0.08$. The similarity in exponents for 1 min volatility and 1 min cumulative volume suggests a universal underlying mechanism. Solid symbols (◼ and ●) refer to $\Omega$ computed from $N\left(t\right)$. Open symbols (◻ and ○) refer to $⟨\Omega ⟩$ computed from $S$ individual Omori exponents ${\Omega }^j$, with $S_{bank}=18$. Note the relatively low values of ${\Omega }_a$ and $⟨{\Omega }_a⟩$ for unscheduled FOMC announcements $i=1$ and 8, which indicates that volatility rate following the announcement increased throughout the day.

Figure 8

(Color online) (a) A comparison of ${\Omega }_a$ for five sectors with volatility threshold $q=3$ suggests a broad universal market response to FOMC news. The technology sector tends to have the largest average ${\Omega }_a$, where large $\Omega$ values correspond to faster relaxation. The horizontal straight line represents the mean ${\overline{\Omega }}_a=0.24\pm 0.08$, averaged over all stocks in the S&P 100 and all scheduled meetings (excluding the unscheduled meetings $i=\left\{1,4,8\right\}$). (b) Probability density function $P\left(x\right)$ of the variable $x\equiv x_{a,i}^j={\Omega }_{a,i}^j-⟨{\Omega }_{a,i}⟩$, which corresponds to individual ${\Omega }_{a,i}^j$ values centered around the average exponent $⟨{\Omega }_{a,i}⟩$ of a given meeting $i$. We conclude from a $Z$ test at the $\alpha =0.0005$ significance level that technology sector Omori exponents are larger on average, ${⟨x⟩}_{Tech}>{⟨x⟩}_{SP100}$. Hence, since larger $\Omega$ values correspond to shorter relaxation time, we find that the technology sector stocks respond more quickly to FOMC news, possibly as a result of relatively intense trading activity among these stocks.

Figure 9

(Color online) The relation between the size of the financial shock, quantified by the S&P 100 volatility Omori-law parameters $⟨{\beta }_b⟩$, $⟨{\beta }_a⟩$, $⟨{\Omega }_b⟩$, $⟨{\Omega }_a⟩$, and $\Delta \left(⟨\Omega ⟩\right)=⟨{\Omega }_a⟩-⟨{\Omega }_b⟩$, and the size of the FOMC news, quantified through the metrics $\Theta$ representing market anticipation and $\Delta$ representing market surprise. All trends are consistent with the hypothesis that a strong anticipation of an interest-rate change, and the element of surprise inherent in the FOMC decision, results in a market perturbation that is significant in scale, and broad across the market. Linear regressions of (a) and (b) S&P 100 data, and (c) and (d) bank sector data are provided for visual aid. Linear regressions that pass the ANOVA $F$ test (rejecting null hypothesis that regression slope $m=0$) at the $\alpha =0.05$ significance level are solid green line; regressions that fail to pass the $F$ test at the $\alpha =0.05$ significance level are dashed red line.