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Slide 1 - Will the Stock Market crash with the upcoming rise in interest rates? A study of the relationship between interest rates and the stock market Gaetan Lion, February 19, 2022 1
Slide 2 - Both the Federal Reserve and the Futures markets anticipate that the Fed Funds rate may reach 1.50% or higher during the first half of 2023. 2
Slide 3 - 3 Source: H.15 for FF and MacroTrends for S&P 500. Data ranges from 1954 Q3 to 2021 Q3 The relationship between Rates and the Stock Market appears weak We looked at 4 quarter increase in Federal Funds (FF) vs. the change in the S&P 500 level over same period. As shown above, the S&P 500 performance is not clearly differentiated vs. rise in FF. * The S&P 500 change were calculated by adding 4 quarterly % changes. If I had used compounding, the mentioned changes would have been immaterially higher.
Slide 4 - 4 Different data segmentations and quantitative methods all confirmed the relationship between interest rates and the Market is weak Models such as OLS and VAR did not work well (adjusted R Squares were very low. They only confirmed the weakness of the relationship between rates and stock market. This was true even when including other macroeconomic variables; Disaggregating FF rates into “Up” and “Down” directions. The relevant correlations with the S&P 500 were even weaker than otherwise; Using lagged variables and forward variables. Again, the relevant correlations weakened.
Slide 5 - Models 5
Slide 6 - 6 Macroeconomic variables relationship (correlation) with S&P 500 Source: H15, Z1 Federal Reserve, BEA, MacroTrends Data from 1954 Q4 to 2021 Q3 Each variable is fully detrended either as quarterly % change for nominal variables or as First Difference in % for rate like variables. Glossary: Federal Funds rate t10. 10 Year Treasury rgdp. Real GDP growth CPI. Consumer Price Index qe. Quantitative Easing. It represents the quarterly first difference in the ratio of the Federal Reserve securities holdings divided by GDP. fiscal. This stands for fiscal policy. It is captured by the quarterly first difference in the ratio of total Treasury securities outstanding divided by GDP. None of the interest rates, macroeconomics, monetary policy or fiscal policy variables have high enough correlations to explain much about the change in S&P 500 level. The correlations are low. The resulting R Squares are close to Zero.
Slide 7 - An OLS Regression model 7 All the variables are stat. significant at the alpha p-value < = 0.10 level. A 1 percentage point change in 10 Year Treasury has more than twice the impact on the S&P 500 vs. the same change in FF. Notice that overall this is a weak model with an adjusted R Square of only 0.19. Yet, it includes 5 of the 6 main independent variables we considered. When factoring FF much higher volatility than 10 Year Treasury, their respective impact on the S&P 500 converges significantly (similar standardized coefficient or “Std. Beta”).
Slide 8 - 8 All models fit the historical data rather poorly confirming the weak relationships between rates (and other macroeconomics variables) and the S&P 500 Leveraging the variable selection of the OLS model, we built Vector Autoregression (VAR) models with 1 lag and 3 lags (selected as best lags using information criteria). As shown all three models do not fit the historical S&P 500 data. Just using a Naïve model taking the average S&P 500 quarterly % change as the one single estimate and accepting this variable standard deviation as the standard error of such a model residuals… gives you nearly as much info as using the three mentioned models (OLS, VAR 1, VAR 3).
Slide 9 - 9 The OLS Regression (Step 5) shows a lot more sensitivity to macroeconomic variables. See how the Estimates in green are a lot more sensitive (volatile) within the top graph vs. the bottom graph disclosing the VAR 1 model. In both cases, the majority of the S&P 500 quarterly % change remains unexplained as the Residuals (red) are far larger and more sensitive to the historical data than the model estimates (green). We don’t show the graph for the VAR 3 model because it is visually nearly identical to the VAR 1 model.
Slide 10 - Impulse Response Function Conundrum 10 Impulse Response Functions (IRFs) represent the response of a dependent variable to an unanticipated shock in a specific independent variable (the impulse variable) over several periods. But, as we will see the IRFs generated by the R software appear flawed. The ones generated by Python are consistent.
Slide 11 - Using R, it is difficult to interpret the IRFs 11 It is impossible to make any connection between the IRF graph and the coefficients of the FF variable with the 3 different lags. The IRF shows a S&P 500 response to a 1 percentage point unexpected shock in FF that is a lot closer to Zero over the first 3 quarterly periods of the IRF graph than what the actual VAR regression coefficients of the three FF lagged variables suggest.
Slide 12 - Depending on how you order the variables within the VAR model and IRF function, the IRF graph changes directional sign 12 Here the IRF denotes a negative impact of a FF rate shock on the S & P 500. Here the IRF denotes a negative impact of a FF rate shock on the S & P 500. The actual VAR model and the related FF coefficients do not change regardless of the order of the variables; But, the IRFs do change as shown above.
Slide 13 - With Python you get consistent IRF graphs 13 Regardless of how you order your impulse variables within the VAR model or the IRF, the IRF graphs will be consistent and readily interpretable. The first, second, and third quarterly periods of the IRF show the exact same impact as the cumulative change derived from the VAR regression coefficients. Notice that the IRF shows the S&P 500 having a positive response to a FF shock. This contradicts investment theory and monetary policy. This is no fault of Python. It reflects the weak relationship between the variables leading to spurious VAR models. The OLS regression did not run into that issue. 0.04 0.31 0.74
Slide 14 - Python IRFs with 10 Year Treasury as impulse variable 14 - 0.87 - 1.20 - 1.40 Here the third period of the IRF shows a value around – 0.80 that does not perfectly match the - 1.20 cumulative change derived from the VAR regression coefficients. Mathematically, the two are not expected to be identical. But, it is reassuring to observe a convergence between the two. When using R, you don’t get that.
Slide 15 - Data Visualization: R (ggplot2) vs. Python (Matplotlib & Seaborn) 15
Slide 16 - 16 Scatter Plot Matrix Python R The R matrix adds correlation coefficients. The Python one includes the Confidence Interval of the location of the regression line data points.
Slide 17 - Scatter Plot 17 R Python The coding challenge is relatively similar. Both software are relatively competitive when generating such scatter plots. The scatter plots confirm the weak relationship between FF and the S&P 500. The slope of the regression line is pretty flat. The width of the 95% Confidence Interval associated with the location of the mean data point on the regression line is unusually wide.
Slide 18 - 18 The scatter plots confirm the very weak relationship between the S&P 500 and FF or 10 Year Treasury Both scatter plots show you that rates don’t impart much info to any model fitting the S&P 500. The two scatter plots are not far from randomness. Earlier, we had observed weak correlations between the rate variables and the S&P 500. The scatter plots give you a visual representations of such weak correlations.
Slide 19 - Scatter Plot Themes with R ggplotthemes() 19 The Economist Wall Street Journal Business Week R offers numerous excellent graph theme templates within its ggplotthemes() library. The template to replicate The Economist format is, in my mind, the most successful. With a single line of code, you generate a very polished plot-theme. Python does not have such capabilities. I have seen one set of codes to attempt to replicate The Economist format, it appears to have more than 15 lines of codes and the result was not nearly as good as in R.
Slide 20 - Scatter Plot Themes with R ggthemes() continued 20 Five Thirty Eight Google Docs Solarized2 ~ Financial Times The trade off when using these templates is that you can’t readily customize them. What you see is what you get.
Slide 21 - Scatter Plot Themes with R ggplotthemes() continued 21 Stata Base R Clean
Slide 22 - Scatter Plot Themes with R ggplotthemes() continued 22 Get Foundation Excel New I believe the “Foundation” theme template in the middle is the only one that is catered to facilitate further customization.
Slide 23 - Customized vs. template themes with R 23 Coding for the theme component: Coding for the theme component: The theme template saves a ton of coding
Slide 24 - Facet grid plot. Customized vs. The Economist template 24 Again, The Economist theme template generates a pretty cool graph theme in just one word of coding.
Slide 25 - 25 Histogram. R vs. Python R Python It is a lot easier to add a Kernel Density Function curve and a Normal distribution fit with Python than it is with R. I am sure you can do it with R, but it takes a lot more efforts.
Slide 26 - Histogram in R. Customized vs. The Economist template 26 Again, The Economist theme template generates a pretty cool graph-theme in just one word of coding. But, you loose ability to customize any graph attributes. The customized approach is burdensome. But, you have control on the size of the fonts for the titles. And, whether you want to bold such titles. You also have control on the color of the bars and background. Note that the size of the OLS regression residuals (step 5) are rather huge. If we did a histogram of the S&P 500 quarterly % change, it would look fairly similar, indicating that the model offers little information.
 

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