# A Model For Geopolitical CAPM And Long-Term Stock Market Returns

## Summary

- In the last year, geopolitical risk has emerged as a significant short-term driver of volatility.
- Firms with high sensitivity to geopolitical risk do not have long-term returns statistically different than firms with low sensitivities.
- Within the CAPM, geopolitical risk is not a statistically significant factor addition to the model, failing to invalidate the Sharpe-Lintner CAPM.

This article examines the capital-asset pricing model (CAPM) which has been extended with a factor for geopolitical risk. I use monthly stock return data for all stocks listed on a major US exchange from January 1990 to December 2016 and utilize a Fama-MacBeth Regression with Newey-West standard errors to test the geopolitical augmented Sharpe-Lintner CAPM. First, I determine if increased sensitivity to geopolitical risk leads to lower average returns, and second, assess if geopolitical risk as an explanatory variable is a significant enough to expose a failure of the CAPM to capture expected returns fully through beta. The results of my regressions do not confirm the hypothesis that firms with high sensitivities to geopolitical risk have expressly different returns in the long run. Furthermore, our Fama-MacBeth regression does not find expressly significant average slopes for geopolitical risk as a variable.

*I. Introduction*

The purpose of this paper is to use a more accurate proxy for real geopolitical risk to determine if firms with greater sensitivities to real geopolitical risk display significantly different returns than firms with low sensitivities. Using asset pricing theory, the rationale of this hypothesis is that during a geopolitical shock, firms with low sensitivities to geopolitical risk would be viewed by investors as safe heaven assets. This would lead to higher than expected returns in times of high geopolitical risk or perceived uncertainty. This paper aims to investigate whether geopolitical risk is priced in financial markets in the long run.

Geopolitical risk is not a new risk factor in the world of asset pricing. As perceived geopolitical risk changes, public markets appear to adjust in order to price in this risk. It should come as no surprise that investors do not like international uncertainty in policy or interruptions of free trade. Declarations of war as well as sovereign conflicts make investors uneasy and this sentiment can be seen in almost all stock markets. Indeed, numerous financial instruments aim to capture this risk factor for investors; popular measures include the CBOE Volatility Index, the doomsday clock, the Economic Policy Uncertainty Index. These measures, however, either focus on a financial economic aspect of risk or on a narrow element of greater geopolitical risk. In other words, these instruments fail to act as accurate proxies for real geopolitical risk.

The effect of geopolitical events and risk on the stock market is well-known. Large geopolitical events which surprise the market create uncertainty or stress international relations negatively impact stock market performance in the short term. That is to say, any international turmoil which poses to jeopardize expect returns to shareholders or erects barriers to free trade is viewed quite negatively by the stock market. This often leads to geopolitical events playing integral roles in daily stock market "moods". Even in recent years and months, during a bull market, we have witnessed short and periodic spouts of stocks moving into the red as tweets on social media or breaking news of high-coverage political policy change or scandal reach the public. It would, therefore, be rational to think that as investors digest geopolitical shocks, markets tend to experience some degree of emotional friction. As fear or unrest push investors into assets and securities considered to be portfolio insurance or safe heaven assets. For this reason, popular proxies like the CBEO Volatility Index have stood in for geopolitical risk as a geopolitical shock should be endogenously absorbed by market volatility. However, this may not be economically justifiable; the reason for which will be discussed later on in this paper.

Table 1, shown above, highlights the relatively short-term impact of key shocks to the stock market since World War II. Though several of the larger shocks were financial or economic in nature. The number which are entirely geopolitical and their respective impacts can be seen to be quite significant. It is important to note, however, that not all securities and firms are impacted the same way to each shock. Perhaps due to various firm-specific levels of exposures, geopolitical risk impacts the price of individual securities differently. If one believes that markets respond rationally to information, then a firm with more exposure to foreign markets would experience a larger drop in returns with increased geopolitical tensions than an entirely domestically operating firm. As a result, all firms can be said to have individual sensitives to risk factors such as geopolitical risk. But the impact of geopolitical risk and the impact of daily stock movements are not of much concern to long-term investors.

What is concerning and of interest is the effect that geopolitical risk has to individual stocks and the stock market in the long run. The marked divergence of popular financial proxies of geopolitical risk with real geopolitical risk in recent years makes the use of these proxies inaccurate as shown in Caldara and Iacoviello (2017). Though the divergence is not a consistent one, a prolonged deviation shows that these proxy measures fail to fully encapsulate real geopolitical risk and instead are measures of financial sensitivities to geopolitical risk. In the present world, financial markets are indeed experiencing a divergence between geopolitical risk and financial uncertainty. Traditional proxy measures such as the VIX index are showing very low levels of market volatility and risk, while nuclear and geopolitical risk proxies exogenous to financial markets reflect highly elevated levels of global uncertainty.

Asset pricing is a topic which is of great interest to investors. As such, numerous models exist for various asset classes to model out portfolios or asset pricings. These models range from focuses on individual securities to macro-market models. Yet as more accurate and valid models of asset pricing are developed, the Sharpe-Linter Capital Asset Pricing Model remains arguably the most well-known.

Existing instruments of geopolitical risk rely on financial indicators and economic measures rather than intrinsically capturing geopolitical risk. In addition to this, many of these financial or economically tied instruments are correlated to many nationalistic financial risk levels that may not necessarily be correlated to geopolitical risk but rather to economic policy. Furthermore, this paper examines geopolitical risk as a factor in CAPM to determine if a security's beta to the market risk premium can fully capture the relationship as postulated in the Sharpe-Lintner CAPM. One hypothesis for why geopolitical risk may be a significant augmentation to CAPM is that an accurate proxy for geopolitical risk would theoretically be more exogenous than economic and financial uncertainty and risk instruments.

*II. Literature Review*

The capital asset pricing model is a widely used and well-known model for observing the relationship between expected returns and risk. The model of William Sharpe (1964) and Lintner (1965) adapted the model of portfolio optimization of Markowitz (1959), also known as the mean-variance model. The Markowitz mean-variance model suggested that investors are risk-averse and choose a portfolio that aims to minimize the variance of a portfolio while maximizing expected returns. Algebraically, this model is given by:

Sharpe (1964) and Lintner (1965) extended the Markowitz model by adding two key assumptions. The first assumption states that given market-clearing asset prices in the previous period, all investors agree on the joint distribution of asset returns from to. The second assumption postulates that all investors have the ability to borrow and lend at a risk-free rate that is identical for all investors and does not vary on the amount borrowed or lent. Given these assumptions, Sharpe (1964) and Lintner (1965) showed that the expected return of any asset can be quantified by the sum of the risk-free rate and the market risk premium times the asset's market beta. This allowed for the pricing of individual assets as well as entire portfolios.

The Sharpe-Linter CAPM is shown above in equation (2). The model can be summarized in a simple algebraic equation which shows that any equity or portfolio of equities is the sum of the right-hand side (RHS) variables. Even though this model has been proven too simple to fully explain asset pricing, it is still the cornerstone equation taught in financial economics for asset pricing.

The Sharpe-Linter CAPM has received much criticism both on its empirically applications as well as its theoretical validity since its original inception. The largest and most concise critique was published by Fama and French (2004), in which the empirical track record of the model in early years is shown to reject its validity. The pair conclude their findings by saying "we also warn students that despite its seductive simplicity, the CAPM's empirical problems probably invalidate its use in applications".

One problem with the model is the assumption of unrestricted access to borrow at the risk-free rate and was addressed by Black (1972); the model is commonly referred to as the Fischer Black CAPM. The model assumes that investors can hold unrestricted long and short positions but assumes that there is no risk-less asset. However, this assumption of unlimited shorting was shown to be as unrealistic as unlimited borrowing and lending by Fama and French (2004).

Another solution to the assumption of a risk-less asset is addressed with the extension to a zero-beta version of CAPM. This model removes the assumption of a risk-less asset and instead replaces it with the lowest variance portfolio of zero-beta assets. This version was tested by Black, Jensen, and Scholes (1972) and found an almost linear relationship between expected returns and beta. In other words, firms with lower betas experienced lower average returns while firms with higher betas experienced higher average returns.

Fama and MacBeth (1973) developed a method to test the CAPM more thoroughly and solve the implicit inference problem which results from the correlation of the residuals of a cross-section regression of average monthly returns. Fama and MacBeth (1973) presented a working model of the CAPM, similar to eqs. (2) and suggested that it had three testable implications: first, in any efficient portfolio *p,* the relationship between the expected return on a security and its risk must be linear; second, wholly captures any associated risk of security *i* in the efficient portfolio *p*; and finally, if investors are risk-averse, higher risk should clearly be positively associated with higher expected returns. Given these conditions, Fama and MacBeth (1973) proposed the following stochastic generalization of eqs. (2):

The model specified above was used to test several the given implications about CAPM. First, if the variation in expected return of portfolio *p* is linear to, then the linearity assumption would suggest that the coefficients of additional explanatory variables would be, statistically, zero. Secondly, that a positive tradeoff between risk and reward exists.

To test the stochastic generalized model in eqs (3), a time-series regression is estimated for each security to determine each security's beta coefficient. The securities are then sorted into twenty portfolios based on their betas and a cross-sectional regression for each month is estimated to determine the average of the coefficients over time. Using monthly NYSE equities returns from January 1926 to June 1968, the paper found a positive relationship between risk and return.

Works highlighting the contradictions of the Sharpe-Lintner CAPM expose failings of the model. Fama and French (1992) used Fama-MacBeth regressions of the cross-section of stock returns on size, beta, leverage, earnings-to-price, and book-to-market equity to outline empirical failures of the CAPM. Using a time frame of July 1963 to December 1990, Fama and French (1992) found a reliably negative relationship between firm size and the cross-section average stock returns. In addition, they found a strong relationship between average returns and book-to-market which was more dominant than the size effect. These two factors also encapsulated the impact of leverage and earnings-to-price on average stock returns. They found that book-to-market, debt-equity, earnings-price, and size additionally explained variations in expected stock returns; something not possible under the Sharpe-Lintner CAPM. In a nutshell, when allowing for variations in beta that is not correlated to the size effect, there was no significant relationship between average returns and betas for firms listed on the NYSE, AMEX, and NASDAQ for the 1963-1990 period.

Though few models exist to quantify geopolitical risk, the effect it has on nations, policymakers, and international relations is instrumental. Though the relation to geopolitical risk and macroeconomic and financial measures is known, formal research into the correlation is limited. However, research has been done on uncertainty as well as key geopolitical events and their economic consequences. In addition, papers discussing disaster events and risk and stock returns also show this correlation. In Baker, Bloom, and Davis (2016), an economic policy uncertainty index (EPU) was developed based on newspaper coverage frequency. The paper found that increased policy uncertainty resulted in increased stock price volatility and lower investment.

Real geopolitical risk differs from other more traditional proxies for uncertainty which are largely based on financial or economic indicators such as the CBOE Volatility Index (VIX) which measures the expected stock market volatility implied from S&P 500 index options. Though the VIX is colloquially known as the "fear gauge", its measure of risk is shown in Caldara and Iacoviello (2017) to likely be endogenously explained by the GPR Index. In other words, though a higher GPR value is correlated with increases in the VIX, the relationship seems to lead from the GPR index to the VIX. Though several reasons for this relationship are possible, the most likely and most common sense can be concluded based on the VIX itself: as a financial instrument, the CBOE Volatility Index does not aim to accurately measure geopolitical risk but rather stock market volatility. However, it has incorrectly been assumed that the stock market volatility can be used as an accurate proxy for geopolitical risk. As the instrument is being over-extended and perhaps misused in this regard, the VIX was shown to react to isolated financial and economic events which have very minor impact on global geopolitical risk. As a result, the VIX's movements do not track geopolitical risk; though it does appear to be impacted by changes in real geopolitical risk.

*III. Data*

The data used in this paper and in our regressions come from three sources and all data were monthly observations. Data on the market risk premium was taken from the data library of Kenneth R. French of Dartmouth University. We used stock market returns for all stocks listed on the NYSE (January 1990 - December 2016), AMEX (January 1990 - December 2016), and NASDAQ (January 1990 - December 2016) as well as the returns of the one-month treasury bill (January 1990 - December 2016) from the Center for Research in Security Prices of the University of Chicago (CRSP). Firms in the context of our regressions were identified by the CRSP PERMNO associated with them. The reason for this is that a number of securities changed their stock symbols or exchanges over time. The PERMNO is an integer which is assigned to a firm and permanently tracks the firm, irrespective of these superficial changes.

For geopolitical risk, we used the monthly GPR Index as described by Caldara and Iacoviello (2017). The index is constructed by compiling text-search results from archives of eleven national and international newspapers. The index is built on the number of articles which are related to geopolitical risk for each month. The searches identify and sort articles into six groups based on word associations. The groups rank articles using text criteria terms involving explicit mentions of geopolitical risk, military tensions, nuclear tensions, war and terrorist threats and media coverage of "actual adverse geopolitical events". According to Caldara and Iacoviello (2017), the rationale behind this is to control for the impact of the event itself as well as to obtain accurate identification of "risk-inducing shocks".

*IV. Methodology*

I postulate that geopolitical risk is a priced factor addition to the Sharpe-Lintner CAPM shown in eqs. (2). That is to say, stocks with higher sensitives to geopolitical risk will have lower returns as in times of high geopolitical uncertainty, investors move to safe heaven assets. Risk factor additions to the Sharpe-Lintner CAPM are not constrained to be return factors. Factor additions such as risk factors are variables which instead aim to capture an assets exposure rather than directly explain returns. In this case, geopolitical risk impacting stocks is assumed to be surprises or shocks to the market. To examine the effect of geopolitical risk on the CAPM model as well as the effect sensitivity to geopolitical risk played in stock returns, we used a similar methodology as outlined in Fama and MacBeth (1973) to observe individual firm sensitivities to the geopolitical risk factor. We extended the Sharpe-Lintner CAPM model of eqs. (2) with a factor for geopolitical risk. Each security's sensitivity to geopolitical risk would be captured by the Δ_{iGPR} coefficient. Thus, the extension and the linear transformation applied to the risk-free rate on equation (2) are outlined below.

This extended model shown above in Eqs. (4) is then used to run our first stage Fama-MacBeth regression. In the first stage regression, for *n* securities, *n* time-series regressions are calculated regressing excess returns over the risk-free rate on the RHS factors of the extended CAPM shown in Eqs. (4):

The results of this first stage regression are then used to run the second stage cross-sectional regression as well as to be sorted into deciles. This branching and dual use of the first stage regression results allowed the testing of both parts of this paper's hypothesis. First, I will describe the second stage cross-sectional regression. The estimates from the first stage regression are used in the second stage regression as independent variables using average excess returns as the dependent variable for each firm.

Thus, we can say that third term on the RHS of the above equations represents the factor load of the *n*th firm by virtue of our second stage regression using the coefficients estimated from the first stage regressions. In other words, the coefficients estimated by the second stage regression calculate the exposure of the *n*th firm to its respective factor; in the case of Eqs. (6), beta and geopolitical risk; one a return factor and other a risk factor, respectively. This factor load exposure can be said to be the *n*th firm's sensitivity to geopolitical risk.

It is also important to note that unlikely the procedure outlined in Fama and MacBeth (1973), this paper uses Newey-West standard errors instead of Fama-MacBeth standard errors as the Newey-West procedure corrects for autocorrelation and heteroscedasticity in the error terms. The Newey-West procedure outlined in Newey and West (1987) requires a lag length which as outlined in the original paper as having two conditions:

Given these conditions, the lag length used in this paper to calculate the Newey-West standard errors simply sets the lag length *m* = 4 as it satisfied both conditions The modification to the Fama-MacBeth method outlined above involves the beta term and GPR term coefficients of eqs. (5) before the second stage cross-sectional regression was estimated. Under a standard Fama-MacBeth method, these coefficients are only used for the second stage cross-sectional regression. For our method, these coefficients were captured and stored along each coefficient's relative security. The results from the first stage regressions are then used to compute the second stage cross-sectional regression to identify the significance of the geopolitical risk factor in the context of the modified Sharpe-Lintner CAPM outlined in Equation (2).

After the compete two-stage Fama MacBeth regression was calculated, securities were then sorted into deciles by their sensitivities to geopolitical risk. These coefficients were the stored outputs of our first stage time-series regressions. Once sorted, the long-term average returns for each security were then computed. In order to account for the possibility of statistically different decile betas and to therefore allow comparative analysis, the long-term average returns are then transformed to abnormal excess returns and the average of those abnormal excess returns taken using the transformation described below:

These average abnormal excess decile returns are then used to perform a T-test for significance. The hypothesis being that firms with high sensitivities to geopolitical risk would have lower average abnormal excess returns than firms with low sensitivities due to the asset insurance rationale discussed earlier.

*V. Results*

After running our regressions, I found insufficient statistical evidence to reject the null hypothesis regarding firm's sensitives to geopolitical risk. The results are presented below. Summary statistics of the coefficients from the first stage time series regressions which were sorted into deciles are listed in Table 2.

The decile betas are indeed similar to 1, which would be considered a long-run market beta. The variation from this might be explained by a number of factors. Firstly, it could simply be sampling error or explained by variations in sampling. Secondly, as one of the criteria for firms to be in the dataset for this paper was active public trading during the entire period, there may be some survivor bias; meaning: firms that tend to survive may as a population have betas skewed below the market returns suggesting that less volatile firms last longer.

A two-tailed T-test for significance run on decile betas found that the difference in decile betas observed were statistically no different than zero, shown above in Table 3. This would validate comparative analysis of decile average returns. However, regardless of this finding, average abnormal excess return for deciles was computed and utilized, instead of average excess returns.

Decile 1 (Table 6, Panel A) against Decile 10 (Table 6, Panel J) resulted in P value of 0.93, under the stricter heteroscedasticity assumption. With such a high P value, we fail to reject the null hypothesis that firms with high sensitivities to geopolitical risk have expressly the same average abnormal excess returns in the long-run as firms with low sensitivities to geopolitical risk.

From the results seen in Table 4, the difference in average abnormal excess returns among firms sorted by their respective sensitivities to geopolitical risk was statistically not significant between any decile pairing.

Table 5 shows the results of the second stage cross sectional regression. With a t-stat of -0.11, geopolitical risk as a factor to explain long-term average stock returns is insignificant. This result is not surprising considering the results of Table 4 which showed that firms with high sensitivity to geopolitical risk do not have significantly lower long-term average abnormal excess returns.

The implication of this finding is that though geopolitical risk notably impacts short-term stock returns and stock market movement, as a risk factor, geopolitical risk has no statistically significant impact on long-term stock returns. This dichotomy between the short-term and long-term might be reconciled with asset pricing theory which would suggest that such exogenous factors as geopolitical risk would not impact long-term stock returns as the associated risk of each geopolitical shock would asymptotically approach zero as time approaches infinity.

*VI. Conclusion*

In this paper, we used a modified Fama-MacBeth procedure to observe the link between an accurate measure for geopolitical risk and long-term stock returns as well as the modification of the Sharpe-Lintner CAPM with a geopolitical risk factor. The results showed that firms which had high sensitives to geopolitical risk did not have statistically lower returns than firms with low sensitives to geopolitical risk. The implication of this suggests that sensitivity to geopolitical risk does not play a significant role in long-term returns for securities listed on major US exchanges. Furthermore, in the context of the Sharpe-Lintner CAPM shown in Eq. (2) and my extended version shown in Eq. (4), geopolitical risk was not a statistically significant factor in explaining excess returns over the market returns.

This is not to say that geopolitical risk as a factor is entirely insignificant in its effect on stock market return. This paper focused specifically on long-term average returns and long-term average abnormal excess returns. Further papers and analysis can be done to quantify the short-term effect of high sensitivities to geopolitical risk on stock market returns and security returns. In addition, to determine and model out the shock-specific risks as time *t* increases. That is to say, how geopolitical shock risk changes from the short-term to asymptotically zero in the long-run.

*VII. References*

Baker, S. R., N. Bloom, and S. J. Davis, 2016. "Measuring Economic Policy Uncertainty*." The Quarterly Journal of Economics, 131(4), 1593.

Black, Fischer. 1972. "Capital Market Equilibrium with Restricted Borrowing." Journal of Business. 45:3, pp. 444-454.

Black, Fischer, Michael C. Jensen and Myron Scholes. 1972. "The Capital Asset Pricing Model: Some Empirical Tests," in Studies in the Theory of Capital Markets. Michael C. Jensen, ed. New York: Praeger, pp. 79-121.

Caldara, Dario and Matteo Iacoviello, "Measuring Geopolitical Risk," working paper, Board of Governors of the Federal Reserve Board, August 2017

Fama, Eugene F. and James D. MacBeth. 1973. "Risk, Return, and Equilibrium: Empirical Tests." Journal of Political Economy. 81:3, pp. 607-636.

Fama, Eugene F. and Kenneth R. French. 1992. "The Cross-Section of Expected Stock Returns." Journal of Finance. 47:2, pp. 427-465.

Lintner, John. 1965. "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets." Review of Economics and Statistics. 47:1, 13-37.

Markowitz, Harry. 1959. "Portfolio Selection: Efficient Diversification of Investments." Cowles Foundation Monograph No. 16. New York: John Wiley & Sons, Inc.

Newey, Whitney K. and West, Kenneth D. 1987. "A Simple, Positive Semi-Definite, Heteroscedasticity and Autocorrelation Consistent Covariance Matrix." Econometrica, Vol. 55, No. 3, pp. 703-708

Sharpe, William F. 1964. "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk". Journal of Finance. 19:3, pp. 425-442.

Sharpe, William F. and Cooper, GM. 1972. "Risk-return classes of New York Stock Exchange common stocks". 1931-1967. Financial Analysts Journal 28(2), 46-81

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