Market Timing

\(\underline{Introduction}\)

• The topic of our paper is market timing and portfolio rotation. Our definition of market timing is trading based on the change of the \(CBOE\space Volatility\space Index\), and we used an SMA 75 of the VIX to serve the purpose of smoothing the noisiness of the fluctuations. Then, set up various testing sets, and examine the type of portfolios that could possibly generate the highest return over the period of high volatility.
• Based on the literature we found, Copeland claims the following: “On days that follow increases in the VIX, portfolios of large-capitalization stocks outperform portfolios of small-capitalization stocks and value-based portfolios outperform growth-based portfolios”. However, we shared a slightly different conclusion based on our analysis: On days that follow increases in the VIX, value-based portfolios do not outperform growth-based portfolios, but portfolios of large-capitalization stocks do outperform portfolios of small-capitalization stocks.

\(\underline {Data}\)

• We grant out sets of Wilshire indexes data and Vix from “FRED”, which contains the following five portfolios :

• Wilshire US Small-Cap Total Market Index (WILLSMLCAP)

• Wilshire US Large-Cap Total Market Index (WILLLRGCAP)

• Wilshire 2500 Value Total Market Index (WILL2500INDVAL)

• Wilshire 2500 Growth Total Market Index (WILL2500INDGR) CBOE Volatility Index: VIX (VIXCLS)

Our dataset has 5549 rows of data from 2000-4-18 to 2022-05-09, in which we used na.omitted(.) to eliminate the first 3 months of the year 2000 because of the nature of SMA 75, and a few NA value down the road that appears in another column of data.

\(\underline{Survey\space of\space literature}\)

• In the article “Market Timing: Style and Size Rotation Using the VIX”, they first tested the relationship between different styles of indexes and volatility. The result is that the slopes are all negative, however, the slopes for the Value indexes are consistently less negative than the slopes for the Growth indexes. And that the slopes for Large-Cap indexes are consistently less negative than the slopes for the Small-Cap indexes \((see\space fig. 1)\). They then went on to run regressions of return of the portfolio, buy Value Indexes and sell Growth indexes, on three variations of the volatility: level of volatility, the change in volatility, and the percentage change in volatility. The slopes are all positive \((see\space fig. 2)\), meaning as the volatility, change in volatility, or percentage change in volatility goes up, the return of the portfolio will increase. One issue with the three variations of volatility is that they are not scaled, for example, let’s say the average value of volatility is \(20\), and the volatility for yesterday is \(2\), and the volatility for today is \(3\) which is still low volatility, however, the percentage change in volatility is 50% which is really high. To mitigate this issue, they calculated a new percentage change with relative to the past \(75-day\) \(moving average\). And then they regressed the return of the same portfolio, buy Value indexes and sell Growth indexes, held for \(1\) to \(20\) days on the new percentage change. The slopes are all positive \((see\space fig. 3)\), which affirms that as the percentage change in volatility increases, the return of our portfolio increases. Lastly, they wanted to test a trading rule in which we buy the portfolio when the percentage change is greater than a specified threshold, and sell when the percentage change is less than the the threshold for a specified number of days, called Holding Period. The cumulative returns are mostly positive, especially when the threshold is more than 10% \((see\space fig. 4)\). They went on to test the portfolio buy Large-Cap indexes and sell Small-Cap indexes, the results turned out to be very similar to the results using the Value and Growth portfolio.

Table1. Contemporaneous Relationship between Change in Historical Volatility and Index Return

Index	Intercept	Intercept_T_Stat	Slope	Slope_T_Stat	R2
Small	0.0003526	2.639002	-0.0060566	-82.75777	0.5529576
Large	0.0002832	2.910240	-0.0054828	-102.83937	0.6563631
Value	0.0003150	3.184941	-0.0052412	-96.75023	0.6283296
Growth	0.0002627	2.321734	-0.0058301	-94.07354	0.6151341

Table 2. Contemporaneous Relationship between Timing and Style

Variables	Intercept	Intercept_T_Stat	Slope	Slope_T_Stat	R2
Level of historical variance	-0.0002471	-1.2363489	0.0000150	1.635554	0.0004829
Change in level of historical variance	0.0000523	0.6600147	0.0005889	13.568094	0.0321780
Percentage change in variance	0.0000523	0.6648245	0.0179164	16.241821	0.0454760

Table 3. Value and Growth Index Returns versus Percentage Change in the VIX Today

Holding_Period	Intercept	Intercept_T_Stat	Slope	Slope_T_Stat
1	-0.0000974	-1.202428	-0.0011055	-3.415996
2	-0.0001706	-1.492402	-0.0016127	-3.532563
3	-0.0002424	-1.752161	-0.0022568	-4.085002
4	-0.0003233	-2.051256	-0.0029203	-4.640104
5	-0.0003946	-2.258433	-0.0036071	-5.171035
10	-0.0006914	-2.776967	-0.0057260	-5.762423
20	-0.0015047	-4.163514	-0.0104528	-7.250302

Table 4. Trading Rule Results: Using Changes in the VIX To Switch between Value and Growth Strategies

Holding_Period_Days	Change_in_the_VIX	Number_of_Days	Number_of_Round_Trip	Culmulative_Return
1	10 %	1466	199	-8.2 %
1	20 %	949	164	-26.4 %
1	30 %	609	110	-17.7 %
1	40 %	383	74	-15.4 %
1	50 %	268	55	-16.6 %
1	60 %	183	39	-11.7 %
1	70 %	130	26	-18.8 %
1	80 %	98	20	-15.6 %
1	-10 %	2298	296	0.9 %
1	-20 %	697	141	16 %
2	10 %	1618	152	-9.3 %
2	20 %	1063	114	-12.7 %
2	30 %	697	88	-25.5 %
2	40 %	443	60	-25.5 %
2	50 %	314	46	-18.4 %
2	60 %	217	34	-10 %
2	70 %	152	22	-12.4 %
2	80 %	117	19	-11 %
2	-10 %	2507	209	-12.9 %
2	-20 %	802	105	2.5 %
3	10 %	1738	120	0.2 %
3	20 %	1165	102	-19 %
3	30 %	774	77	-26.6 %
3	40 %	497	54	-22.3 %
3	50 %	355	41	-18.9 %
3	60 %	248	31	-18.1 %
3	70 %	171	19	-13.1 %
3	80 %	133	16	-8.6 %
3	-10 %	2671	164	-26.7 %
3	-20 %	886	84	7.4 %
10	10 %	2396	80	5.4 %
10	20 %	1714	66	-7.2 %
10	30 %	1174	53	-24.1 %
10	40 %	769	36	-24.9 %
10	50 %	577	29	-24.1 %
10	60 %	415	22	-20.3 %
10	70 %	291	15	-17.6 %
10	80 %	231	13	-20.9 %
10	-10 %	3339	72	-28.8 %
10	-20 %	1288	51	-0.3 %

Table 5. Contemporaneous Relationship between Timing and Style

Variables	Intercept	Intercept_T_Stat	Slope	Slope_T_Stat	R2
Level of historical variance	-0.0007007	-3.5258156	0.0000316	3.470007	0.0021699
Change in level of historical variance	-0.0000693	-0.8786305	0.0005738	13.274981	0.0308451
Percentage change in variance	-0.0000693	-0.8805118	0.0156021	14.134787	0.0348265

Table 6. Large and Small Index Returns versus Percentage Change in the VIX Today

Holding_Period	Intercept	Intercept_T_Stat	Slope	Slope_T_Stat
1	-0.0002721	-3.405647	0.0004965	1.555840
2	-0.0005191	-4.611332	0.0013652	3.036899
3	-0.0007691	-5.639854	0.0011634	2.136528
4	-0.0010244	-6.527865	0.0013656	2.179263
5	-0.0012666	-7.321569	0.0014297	2.069708
10	-0.0023907	-10.188201	0.0037842	4.040583
20	-0.0047474	-13.881327	0.0042688	3.128807

Table 7. Trading Rule Results: Percentage Changes in the VIX To Switch between Large- and Small-Cap Strategies

Holding_Period_Days	Change_in_the_VIX	Number_of_Days	Number_of_Round_Trip	Culmulative_Return
1	10 %	1466	199	26.2 %
1	20 %	949	164	11.8 %
1	30 %	609	110	12.9 %
1	40 %	383	74	10.7 %
1	50 %	268	55	12.9 %
1	60 %	183	39	10 %
1	70 %	130	26	8.7 %
1	80 %	98	20	10 %
1	-10 %	2298	296	29.9 %
1	-20 %	697	141	-15.6 %
2	10 %	1618	152	19.9 %
2	20 %	1063	114	25.5 %
2	30 %	697	88	6.4 %
2	40 %	443	60	3.6 %
2	50 %	314	46	14.8 %
2	60 %	217	34	18.1 %
2	70 %	152	22	3.9 %
2	80 %	117	19	-0.2 %
2	-10 %	2507	209	23 %
2	-20 %	802	105	-10.5 %
3	10 %	1738	120	4.8 %
3	20 %	1165	102	21.8 %
3	30 %	774	77	18.3 %
3	40 %	497	54	9.8 %
3	50 %	355	41	15.1 %
3	60 %	248	31	9.3 %
3	70 %	171	19	3.4 %
3	80 %	133	16	0 %
3	-10 %	2671	164	8.7 %
3	-20 %	886	84	-8.1 %
10	10 %	2396	80	-16.8 %
10	20 %	1714	66	-12.5 %
10	30 %	1174	53	-17.3 %
10	40 %	769	36	-3.1 %
10	50 %	577	29	3.8 %
10	60 %	415	22	11.7 %
10	70 %	291	15	16.1 %
10	80 %	231	13	4 %
10	-10 %	3339	72	0.5 %
10	-20 %	1288	51	6.7 %

\(\underline{Table\space4/7}\)

• We shared a similar backtesting strategy as what Copeland did in their paper. “To overcome the noisiness in the VIX, we examined “today’s” VIX estimate relative to a 75-day (roughly three-month) moving average.” According to this statement, we made up the below formula to get the “Change In \(Vix\):

\[ {\Delta} Vix = \frac{(VIX_{today} - SMA75_{today})}{SMA 75_{today}} \]

• After getting the “Change In Vix”, we were trying to find out a way to enter and exist our position. Obviously, the best way would be replicating the same trading method as Copeland has, but they did not exposed their trading rules on the table. Therefore, we decided to set up our own rules as following:

\(Poistion\) \(Get\) \(In\) \(=\) \({\Delta}Vix\) \(>\) \({\Delta}Vix\) \(Column\) \(Two\) \(of\) \(Table\) \(4/7\) \(|\) \(Positive\) \(Poistion\) \(Get\) \(In\) \(=\) \({\Delta}Vix\) \(>\) \({\Delta}Vix\) \(Column\) \(Two\) \(of\) \(Table\) \(4/7\) \(|\) \(Negative\)

\(Poistion\) \(Get\) \(Out\) \(=\) When the number of Holding Days Decayed to Zero

• Getting into the position is by doing a spread trade, which means long on a portfolio and short another at the same time. In our case is by longing “Value” , shorting “Growth”, longing “Large” and shorting “Small”. As an example, if the “Change in Vix” surpassed 10%, we would have traded into the position (by long value and short growth), hold the position for one day, and check if the next value in “Change in Vix”, if it is still above 10%, we keep holding our position. If not, we trade out our position and make a round trip. In contrast, if “Change In Vix Column Two of Table 4/7” is less than -10%, we do the exact opposite by shorting the value and long the growth portfolio at the same time.

\(\underline{Empirical\space application}\)

• Although we had little to no worries in regards to testing for omitted variable bias, residual spread (heteroscedasticity and homoscedasticity), and checking for serial dependency as our research did not cover the need for such analysis, our issue was in picking a prime volatility factor as our model’s regressor:
  • We preferred using the \(VIX\) over traditional volatility models like \(GARCH\), \(GJR-GARCH\), \(EMA\), etc which are better estimators of market volatility due to the VIX’s forward looking feature and its ability to rapidly incorporate new information. In addition, we considered conclusions from outside research referenced in the literature as “Fleming et al. found that the VIX of implied volatility exhibits a strong relationship to future realized stock market volatility” (Copeland et al. 74) to support our preference for the VIX.

\(\underline{Conclusion}\)

Furthermore, as conventional wisdom assumes that small-cap stocks perform better than large-cap stocks (Basu 1983 and Fama and French 1992), our results proved otherwise as large-cap stocks performed better with implied volatility jumps and small-cap stocks performed better than large-cap as implied volatility decreased.

Subsequently, from using the percentage change in the VIX (in relation to its 75 day moving average) as the regressor against the index returns (with high liquidity in the futures market) as our regressand, we were able to time the market with only the size strategy (large minus small) as its cumulative returns were mostly positive over holding periods with percentage increases in the VIX (Table 7).

However, the style strategy (value minus growth) fell short of our market timing goal indicated by the low cumulative returns over the holding periods with percentage increases in the VIX (Table 4) due to growth returns overshadowing value returns as shown by the low R squared and low significant Beta (Table 3). This contradicts the analysis done in the literature (both style and strategy timed the market successfuly) which used a different data set from ours.

Nonetheless, it is important to note that switching to value stocks alone in the advent of the VIX rising (signaling falling investor confidence in the market) does produce positive returns whereas switching to growth stocks as the VIX falls (signaling strong investor confidence in the market) produces positive returns as well. In effect, we recommend the size timing strategy for both day traders and money managers alike in terms of allocating assets in the wilshire index.