# Selection of regions for entrepreneurship: an application of the CAPM.

INTRODUCTION

In the classic Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965), the total return (current income plus price growth) of an asset is expressed as a linear function of non-diversifiable, systematic risk. Systematic risk is measured by the covariance of the asset's total return with the total return of a market index such as the S&P 500 Index. An important factor that contributes to total return is the growth rate of income. Focusing on the growth rate of income, rather than total return, reduces the influence from variation of discount rates. In addition it eliminates the need to know the original price, or cost, of an investment when calculating return. This opens up the possibility of testing for risk/return relations when income data is available, but not prices or initial investments. Per capita personal income by state and by county are two data sets in which only income data is available.

This paper describes an application of the CAPM to regional growth. In this application the growth rate of personal income of a particular region, such as a state, is an increasing linear function of its systematic risk. Further, systematic risk is measured as the covariance of the growth rate of the personal income of the region with the growth rate of personal income of the entire United States. The results enable ranking of the states (and counties) based on both systematic risk and on excess growth.

The contribution of the paper is two fold. First, this paper is an application of Campbell and Vuolteenaho's (2004) assertion that the CAPM holds when the data considered is relatively free from variation of discount rates (or dividend yields). The regional economic growth rate data used in this study meets this criterion. Second, the paper demonstrates, for the first time, the application of the classic CAPM notion of systematic and unsystematic risks to regional economic growth and related location decisions.

A relation between personal income growth and growth systematic risk has several implications. First, individuals and firms deciding whether to locate in a particular state can temper the attractiveness of a state's high-growth rate with that state's accompanying higher systematic risk. Particularly for firms operating in multiple states or regions, the marginal effect on the firm's overall risk exposure from locating a new facility in another state or region may be quantified by referring to the systematic part of total risk which can not be diversified away. Second, entrepreneurial individuals and firms may consciously choose to seek out states or regions with above average non-systematic risk. Because individuals can effectively choose to live and work in only one state, knowledge of region-specific characteristics that make higher than risk-adjusted average personal income growth rates possible may influence the selection of one region over another. Entrepreneurs may choose to start their business in a region with these unique characteristics to grow their revenues quickly. Third, state and regional economic policy makers can measure the effect of their actions over time on the risk/return profile of their region in comparison to all other regions. For example, policies that improve personal income growth without raising the state's systematic risk can be emphasized. Fourth, economic development professionals seeking to position their state competitively in the minds of potential new residents and investors have a new tool to consider in designing their message. They can target their message to segments that are more likely to be attracted to their area due to its growth/risk profile.

The remainder of the paper is organized as follows. In Section 2, a review of the CAPM and an explanation of why the CAPM should hold for growth rates are presented. Considerable space is devoted to this review to explain the applicability of the CAPM to regional growth data in spite of the fact that the model has generally been rejected in the asset pricing literature. In Section 3, the applicability of the CAPM to personal income growth rates is specifically discussed. Section 4 contains a description of the data and methodology. Results are described in Section 5. Conclusions, limitations and areas for future research are presented in Section 6.

REVIEWING THE CAPM WITH AN APPLICATION TO INCOME GROWTH

The CAPM expresses expected next period return of an investment j in terms of a "risk-free" rate, [R.sub.f] and the systematic risk [[beta].sub.j] relative to the return of the common index [R.sub.M].

[R.sub.j] = [R.sub.f] + [[beta].sub.j] ([R.sub.M] - [R.sub.f]) (1)

Where,

[[beta].sub.j] = cov([R.sub.j], [R.sub.M])/[[sigma].sup.2.sub.M] (2)

The CAPM assumes that utility functions are normal (or quadratic), and there is no labor income (e.g. Campbell and Cochrane 2000). Early test results supported the CAPM (e.g. Fama and MacBeth, 1973). However, later cross-sectional studies such as Banz (1981) and Reinganum (1981), Basu, (1983), Rosenberg, Reid and Lanstein, (1985) and Chan, Hamao and Lakonishok, (1991) found that the CAPM did not hold because "value stocks", identified by the ratio of price-to-book value or price-to-earnings enabled returns in excess of those predicted by the CAPM. Due to the value stock anomaly, Fama and French (1992) advocate use of multifactor models that include other variables. However, researchers have not all given up on the usefulness of the CAPM.

GENERALIZING THE CAPM TO MULTIPLE PERIODS

The CAPM is a two-period model. Merton (1973) proposed an intertemporal CAPM (ICAPM) because the information set changes over time. The ICAPM necessitates adding other factors in addition to the market index. Jagannathan and Wang (1996) assume the CAPM holds conditional on the information set available at a particular time, but that betas and the market price of risk vary over time. They show that when the conditional version of the CAPM holds, a two beta model obtains unconditionally. One beta is the traditional one, based on the covariance of the asset's return with the market index. The other beta is based on the covariance of the asset's return with the market price of risk, which varies over time due to the business cycle. Their proxy for the variable market price of risk is interest rates. Campbell and Vuolteenaho (2004) argue that a two beta model is required because one beta is needed to measure the risk associated with an index proxy that captures market dividend yields and another beta is needed to measure the risk associated with an index proxy that captures expectations about long run cash flows (or earnings growth) of the firm. They find that when the assets under study have more or less constant ratios of the two types of risk, then the single index CAPM performs adequately. This was the case in the early Fama and French (1973) study. Thus, when the risk associated with variation in dividend yields is controlled for, the unconditional CAPM is supported.

ALTERNATIVES TO THE STOCK MARKET INDEX

Cochrane (2001, p.152) notes that the CAPM does not take into account labor income and shows that it is a specialized case of the Consumption Capital Asset Pricing Model (CCAPM) of Lucas (1978), Breeden (1979) and Brock (1982). The CCAPM is often tested using the growth rate of consumption as the "market index". One basic premise of the CCAPM is that any asset whose return covaries positively with consumption makes consumption more volatile, justifying a higher return because the investor's consumption-enabling income stream is made more volatile. Mankiw and Shapiro (1986) point out that the CCAPM is preferred on theoretical grounds because it takes into account the presence of other assets besides stocks in the wealth portfolio serving as the market index. The value of a broader index is supported in the few tests of the CAPM using return data for real assets such as commodities and agricultural or timberland, instead of stock returns. When these tests use the traditional stock market index to measure risk, the results generally do not support the CAPM (e.g. Holthausen and Hughes, 1978 and Bjornson and Innes, 1992). However, when a specially constructed market index is adapted for real assets, the CAPM is generally supported (e.g. Barry, 1980, Redmond and Cubbage, 1988 and Slade and Thille, 1997).

Jagannathan and Wang (1996, p.13) point out that the return on stocks will not measure the return on aggregate wealth because dividends from stocks represent less than 3% of household personal income, and labor income is a much more significant source of personal income. Thus, many of the assets included in investor wealth may not have a published market price available (e.g. privately held assets and the value of education). Despite its attractive theoretical generality, empirical tests of the CCAPM for total returns of stocks have been supported even less than the CAPM. Mankiw and Shapiro (1986), for example, find that even the unconditional CAPM performs better than the CCAPM. Campbell and Cochrane (2000) argue the reason the CAPM performs better than the CCAPM is that the growth rate of consumption does not capture the variation in dividend yield, whereas the stock market index does. This variation in dividend yield is the same discount rate risk identified by Campbell and Vuolteenaho (2004). Campbell and Vuolteenaho (2004, p. 1271) conclude that, when attempting to validate the CAPM, the cash flow-related beta is the most relevant, with the discount rate beta of only secondary influence. This conclusion is also supported by Lee (1998) who found earlier that the influence on stock prices of non-cash flow related fundamentals, such as discount rates, declines as the time horizon increases. The income growth rates investigated in this paper represents just such a data series: one which is free of non-cash flow related information.

Roll (1977) argued that a proxy for the "true" market index, consisting of all wealth could not be identified, making it impossible to test the CAPM. Fama (1990) finds that the growth rate of GDP explains about 43% of the average real return on stocks while proxies for discount rate changes explain about 30%. Thus, enhancing the stock market index with macroeconomic variables may improve models. Jagannathan and Wang (1996) use a market index that combines a traditional stock market index with the growth rate of labor income in a conditional CAPM and find that value stocks can no longer earn excess returns. Knez and Ready (1997) suggest that a risk premium on small stocks exists because small stock returns are correlated with investor's future labor income, but that this correlation is not captured with a stock index return. Jagannathan, Kubota and Takehara (1998) find that although a traditional stock market index beta alone could explain only 2% of the variation of Japanese security returns, including a labor income beta in the model explained 75%. Korniotis (2006) finds that a measure of regional risk that includes variance of consumption growth by state is priced in stock prices.

The collective results of these studies suggest the following: First, even though a stock market index does not perform as well as a conditional two beta model, the stock market index captures some influence of dividend yield changes that a macroeconomic index such as consumption growth does not capture. Second, creating a combined index containing influence from both the stock market and from macroeconomic growth variables improves the performance of the CAPM. Third, and most important to this paper, when the asset returns under study are controlled for the influence of dividend yield, the focus is on measuring cash flow beta and a single macroeconomic index, like consumption or income growth, is appropriate.

RESTRICTING ALL RETURN DATA TO GROWTH RATES

A simple way to control for effects of changing dividend yield is by restricting the measure of return on all assets under study to income growth rates rather than total return. The Gordon (1962) perpetual dividend growth model can be used to present the CAPM in terms of growth of cash flow. Gordon (1962) modeled the investor's expected return [R.sub.j] on a security j assuming an expected constant growth rate, [g.sub.j] of current dividends [D.sub.j0] and an initial investment price [P.sub.j0]. With the expected growth [g.sub.j] being conditional on the information set available at time 0, the return is expressed as:

[R.sub.j] = [D.sub.j0] (1 + [g.sub.j])/[P.sub.j0] + [g.sub.j] (3)

A similar expression for the return on the common market index, M is

[R.sub.M] = [D.sub.M0] (1 + [g.sub.M])/[P.sub.M0] + [g.sub.M] (4)

Assume, as did Jagannathan and Wang (1996, p.5), that the CAPM in (2) holds for total returns conditionally, i.e. based on the available information set at a particular point in time. Then, substituting (3) and (4) into (2) and solving for [g.sub.j]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

To simplify (5), note that [R.sub.f], [D.sub.j0]/[P.sub.j0] and [D.sub.M0]/[P.sub.M0] are all known at time 0, and note that [[beta].sub.j], is constant, given the information set and then let

[kappa] = ([R.sub.f] - [[beta].sub.j][R.sub.f] - [D.sub.j0]/[P.sub.j0] + [[beta].sub.j][D.sub.M0]/[P.sub.M0])/(1 + [D.sub.j0]/[P.sub.j0]) (6)

[[gamma].sub.j] = [[beta].sub.j](1 + [D.sub.M0]/[P.sub.M0])/(1 + [D.sub.j0]/[P.sub.j0]) (7)

With these substitutions, the expected growth rate is a linear function of the expected growth rate of the market index.

[g.sub.j] = [kappa] + [[gamma].sub.j][g.sub.M] (8)

Since the measure of systematic growth risk, gamma or [[gamma].sub.j] is a slope, the regression estimate for it [y.sub.j]' can be written as

[[gamma]'.sub.j] = cov([g.sub.j], [g.sub.M])/[[sigma].sup.2.sub.gM] (9)

Thus, if the CAPM is valid conditionally based on total returns, then it is also valid conditionally when the focus is restricted to growth in dividends. While the rate on treasury securities is often used as a proxy for the risk free rate, [R.sub.f], there is no security whose rate of return can proxy for the intercept [kappa] in (6). In this paper, [kappa] will be referred to as the "zero risk" growth rate. It can be interpreted as the rate of growth available when no systematic growth risk is assumed. The implication of (8) is that investors can form their expectation about the individual investment's growth rate [g.sub.j] knowing the estimate for the investment's systematic income growth risk provided in (9). In addition, recall from (3) that the investor's rate of return is also a function of the expected growth rate [g.sub.j], so that forming an expectation about future growth rate allows forming an expectation about future return, given the current status of dividend yields. For a firm, dividends cannot grow unless earnings grow. In addition, except for short run cost-cutting measures, earnings cannot grow unless revenues grow. Thus, the essential driver of returns to investors (entrepreneurs) is sales growth in the regions in which they operate.

APPLYING THE CAPM TO PERSONAL INCOME GROWTH BY REGION

A region such as a county or a state may be viewed as a portfolio of entrepreneurs or firms producing revenue from production. This revenue is divided among suppliers of labor, equity and debt capital, and intermediate materials and appears in measures of their personal income. These suppliers of resources may be viewed (loosely) as "investors" in the firm. For example, even though a worker may not be an owner of the firm, he commits (invests) his time (human capital) to the firm and receives a return. The return to these suppliers/investors can be measured as the growth rate in their incomes which results from their association with the firm. The form of the CAPM presented in equation (8) suggests a linear relation between growth risk and return, measured as income growth, even when there is no investment price data available. Measures of macroeconomic income such as the Personal Income series tabulated by the Bureau of Economic Analysis of the U.S. Department of Commerce can provide the income growth data required. A novel exploitation of (8) is, then, to test for the linear relation posited by the CAPM using the growth rate of personal income for each of the 48 contiguous states and also for county data. Thus, when a state's personal income growth systematic risk is high, as measured by its covariance with the United States, higher growth rates, higher rates of return and lower prices should be expected on the assets in that state. The wealth (assets) in the state will include privately held small businesses and the value of human capital, such as education and training.

There is some previous evidence to suggest that a tradeoff between income growth and volatility does exist in regions. For example, Barlevi (2004) documents the reduction in consumption associated with higher volatility of national income. For Italian households, Guiso, Japelli and Terlizzese (1996) construct a measure of income risk based on expectations of the variance of inflation and income growth and find that households reduce their ownership of risky assets when confronted with higher income risk. Similarly, Heaton and Lucas (2000) find that households with higher variance of business income hold less wealth in equity shares. When holdings are reduced, prices fall, indicating rates of return rise. Chandra (2002) documents a relation between a state's economic growth rate and the total risk or instability of growth as measured by the standard deviation. The innovation of this paper, compared to these studies, is that the measure of risk is systematic risk, measured in a CAPM context, rather than total risk.

Applying the CAPM to regional growth rate data, gives entrepreneurial individuals and firms a new analytical tool to help them form expectations about the growth rate of personal income in each state and county. Risk-adjusted expectations about growth should assist in selecting the preferred regional locations for investing both financial capital and human capital. A state, for example, may be selected as a location because of its tendency to grow at above systematic risk-adjusted rates due to non-systematic risk characteristics. Chatterjee, Lubatkin and Schulze (1999) posit that firm managers care about non-systematic risk and seek to control it at the individual firm level because investors cannot completely diversify their investments as well as the CAPM assumes. In addition, workers must choose to work in one state and many entrepreneurs, at least when starting out, can only operate in a few states. To the extent that they cannot achieve good diversification, these workers and entrepreneurs are forced to bear nonsystematic growth risk. Similarly, state economic policy makers and economic development professionals can seek to control, even design, their state's non-systematic risk characteristics. For example, Kalemli-Ozcan, Sorensen and Yosha, (2003) find that a high level of regional production specialization leads to not only a higher growth rate but also higher volatility and that regions can offset the increased volatility by risk-sharing through financial diversification between regions. Thus, diversifying the production activities in a region will tend to reduce nonsystematic risk. In addition, state economic policy makers and economic development professionals can seek to raise or reduce their state's systematic risks. This might be achieved by, for example, encouraging firms whose revenues have a high covariance with personal income to locate in their region.

The two period CAPM assumes that betas and the market price of risk are stable over time. For a state or region, this implies stability of the circumstances that make it more or less risky than the average. The large body of literature attempting to explain regional growth rates can be categorized based on what is said about this assumption of stability. Martin and Sunley (1998), for example contrast the traditional theory that regional economies will "converge" and become more alike, with the newer idea that regional economies will "diverge", resulting in regions with unique characteristics. The rationale for convergence is that, with mobility, self-correcting free market processes will adjust prices and wages and the supply of labor and capital will equalize across regions. The argument for divergence is that economies of scale, specialization and the achievement of critical masses of resources will lead to further accumulations of resources and capabilities within regions that tend to be self-perpetuating. Mankiw, Romer and Weil (1992) have argued for blending of these two approaches so that regions reach "conditional convergence" to the extent they have similar government policies, societal preferences, access to technologies and other structural circumstances, but each region will ultimately reach a unique steady state of relative volatility and growth. Evans and Karras (1996) find that differences in the levels of technology, the share of total income paid to capital and the per cent return to capital in the 48 contiguous states are evidence that the states converge rapidly to stable growth levels that are significantly different. Carlino and Sill (2001) find that the regions of the United States differ significantly in terms of long term volatility of economic growth. In this paper, it is assumed that, while the price of growth risk is stable over the long run, the practical possibility exists for economic policy makers to target policy changes in a particular state that may, over time, result in changes to the state's systematic growth risk.

DATA AND METHODOLOGY

The empirical tests to be performed in this paper use per capita personal income by state and county data available from the Bureau of Economic Analysis (BEA) website (http://www.bea.gov). Per capita data is used, rather than aggregate personal income data to follow the convention established in the CCAPM literature (e.g. Mankiw and Shapiro, 1986). The tests are restricted to annual data for two reasons. First, Fama (1990, p.1106) finds that income growth and asset returns are related over more than one quarter and this leads to autocorrelation and measurement problems when using quarterly data. Second, some information related to proprietor income and dividend and interest income are not included in the quarterly personal income estimates - only the annual figures. Annual personal income data for the 48 contiguous states was available from 1929 through 2006, on a per capita basis. In addition to these data, personal incomes per capita by county are available from 1969 to 2005. For each of the data sets, "gross" growth rates of personal income per capita are calculated as the simple ratio of current year per capita personal income to that for the previous year. Thus, positive "net" growth rates are reflected in the data as "gross" rates which are greater than one and negative net growths rates are reflected as gross rates which are less than one (but greater than zero).

Using the annual growth rates over the years for which data is available, an estimate for systematic risk for each state and region is calculated using (9). In (9), the growth rate of per capita personal income of the United States serves as the proxy for [g.sub.M]. For the same time period over which the regional estimates of systematic risk, [[gamma].sub.j] are calculated, the simple average of the regions' growth rates are also calculated. Then, having pairs of estimates of systematic risk and average growth rates for each region, the linear relation between risk and growth posited by the CAPM can be evaluated. Cross-sectional regressions of these average returns on their respective estimated systematic risks are performed using Generalized Least Squares (GLS) to determine whether the prices of risk (slopes) are different from zero, at the 99% confidence level. Because a state is a "portfolio" of multiple counties, the availability of these data sets makes it possible to look for the effect of diversification. For example, a higher [R.sup.2] should be expected for the regressions of state data versus the county data sets because more unsystematic risk has been diversified away in the portfolios. In addition, because counties are smaller, in terms of population and total personal income, the possibility of a CAPM-compromising "size effect" can be examined. Two sets of 48 portfolios of counties (about 65 counties in each portfolio) are created: one set created at random, based on an alphabetical listing and the other set created after sorting the counties based on total personal income. If a size effect exists, then the price of risk (slope) of the regressions of size-sorted portfolios of counties will be statistically different from that for the states and for the random (alphabetically sorted) portfolios. To make these comparisons, the two regressions are tested for equivalence using a version of the Chow (1960) test. When the regressions are found to not be equal, tests of the equality of the two slopes and intercepts are carried out using the dummy variable method, as described by Gujarati (1970). The test methodology is more fully described in the Appendix.

RESULTS

Table 1 shows the results of the cross-sectional regressions for the various data sets. The regressions for the 48 states over the entire period, 1930-2006 and for each of the two subperiods, 1930-1969 and 1970-2006 were statistically significant at the 99% confidence level since the confidence intervals for the slopes do not contain zero. The [R.sup.2] for the entire period, 1930-2006 was .175 and that for the early and later sub-period regressions were .264 and .158, respectively. The regressions for individual counties over the period 1970-2005 were also statistically significant, with an [R.sup.2] of .300. When the counties were sorted into 48 portfolios of counties created both randomly and sorted based on size, these regressions were also all statistically significant. The [R.sup.2] for the portfolios of counties increased to .379 and .843, for the random and size-sorted, respectively.

Table 2 records the results of tests for equality of slopes and intercepts for various pairs of regressions. The comparison of the regressions for the two time periods (1930-1970 and 1970-2005) for the 48 states indicates that while the slopes (prices of risk) are equal, the intercepts are not. This indicates the "zero risk" rate has declined in recent times for the states. The comparison of the regression of 48 states to the regression of 48 portfolios of counties sorted by size also showed that the slopes are equal but the intercepts are statistically different. The comparison of the regression of randomly created portfolios of counties to that for the 48 portfolios of counties created by sorting them based on size were not statistically different.

Figure 1 shows the fit of per capita personal income growth versus systematic risk for the 50 states, including Alaska and Hawaii. Note that the regression results given in Table 1 include only the 48 contiguous states to compare to the early time period in which data for the last two states were not available.

Figure 2 shows the fit of per capita personal income growth versus systematic risk for 3089 individual U.S. counties. It can be seen that the data is clustered with a few outliers.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Figure 3 shows the fit of per capita personal income growth versus systematic risk for 48 randomly created (based on alphabetical order) portfolios of U.S. counties. The visual fit appears to improve with portfolios compared to individual counties and this is confirmed by the [R.sup.2] 's given in Table 1.

[FIGURE 3 OMITTED]

Figure 4 shows a scatter plot of a pairing of the size rank of each portfolio of counties that constructed based on size (total personal income) with the percent residual

[FIGURE 4 OMITTED]

Table 3 shows the ranking of the 50 states based on systematic risk. The five states with the lowest systematic risk are Hawaii, West Virginia, Montana, Utah, and Idaho. The five states with the highest systematic risk levels are Connecticut, Louisiana, Texas, Oklahoma, and North Dakota.

Table 4 shows the ranking of the 50 states based on historic ability to grow at rates in excess of the expected growth rate on a risk-adjusted basis. The five states with the highest excess growth are Alabama, South Dakota, Wyoming, Mississippi and Tennessee. The five states with the lowest excess return averages are Ohio, Nevada, Michigan, California, and Alaska.

CONCLUSIONS, LIMITATIONS AND FUTURE RESEARCH

We did not conduct a strict Fama and McBeth (1973) three-step test to confirm the validity of the CAPM using our growth data. Rather, the CAPM's validity is simply assumed so that it can be applied to our growth data. Overall, the results support this application of the CAPM to personal income growth rates. First, the slopes of the relations between growth risk and growth are all statistically significant. Second, regressions of aggregated data result in higher [R.sup.2], as would be expected in data for which the CAPM holds. However, the [R.sup.2] for the states is lower than that for the regression of individual counties. This would not be expected and is a divergence from the CAPM. Outliers may be a possible reason for this failure since it is appears in Figure 2 that a few data points may have significant influence on the results and bias the [R.sup.2] of that regression upward. Third, no preliminary evidence of a size effect was identified in the portfolios of counties. These results support previous studies (e.g. Campbell and Vuolteenaho , 2004) which find that datasets which have low influence from discount rate variation can be modeled using the CAPM.

The intercept was found to be lower for the early 1930-1969 state data than for the later 1970-2006 period. Thus, the "zero risk rate" of growth appears to have risen in recent times. One possible explanation for this is the idea that government transfer payments and other public and private insurance programs which have been instituted in more recent times have made it possible to achieve higher growth rates without assuming systematic risk.

The fact that the intercept from the regression of the 48 states was different from that for the 48 portfolios of counties sorted by size is not a violation of the CAPM, because the prices of systematic risk are the same as posited by the CAPM. Thus, this result indicates there is no size effect. However, the difference in intercepts is a curiosity, since it indicates that counties have a lower "zero risk" growth rate than states, on average. It may be speculated that, as in the case of 1930-1969 dataset compared to the 1970-2006 dataset for states, this may be due to risk mitigation programs at the federal or state level that have not reached or been targeted at small counties.

Another way to test for a size effect in the size-sorted portfolios of counties is to examine the residuals (predicted growth versus actual growth) to see if there is a tendency for large or small county portfolios to have positive excess returns. In Figure 4, the smallest portfolio (rank = 1) appears to be an outlier. Ignoring that point, there appears to be slight upward slope, indicating that portfolios of larger counties tend to earn positive excess returns. However, even after removing this "outlier", the slope of the regression fit of percent error versus size rank is not statistically significant at the .01 level (F statistic = 5.2, confidence level = .027). Thus, the portfolios of counties sorted based on total personal income do not exhibit a size effect.

The results support the application of the CAPM to regional economic growth rates. As in other applications of the CAPM, the individual analyst must decide whether a high, medium or low level of risk is preferred. The ranking in Table 3 and in Table 4 can assist individuals and firms in selecting regions based on their preferences. From the point of view of fully diversified firms with investments in many states, the measure of systematic risk (Table 3) is relevant but the ranking of the states by excess growth (Table 4) is not. Table 5 shows a grouping of the states that takes into account both their systematic risk and their excess growth (i.e. unsystematic risk).

For these multi-state, well-diversified firms, Table 5, in its entirety, is relevant. If, for example, a firm's management is willing to tolerate medium risk levels, then all 18 states contained in the middle column of Table 5 would be viable choices for location of a new establishment or plant. However, for small firms or entrepreneurs considering where to locate their first (or perhaps second or third) establishment, both systematic risk and excess growth are relevant. This is also the case for workers who wish to choose from among a few possible job offerings based on the locations. For these investors, the states listed in the second row of Table 5, which had negative excess growth rates, would probably not be good choices. Under these circumstances, a firm, for example that is comfortable with high levels of risk, should select a location from the states listed in the upper right-hand grouping in Table 5 (i.e. AR, NC, GA, MN, CO, NH, CT, ND, or LA).

Consider also an individual who works (say) for a national restaurant chain and is given the choice of transferring as an assistant manager to one of two states. It is reasonable to assume that the relative success of the restaurant in which he works is dependent on regional personal income growth. If his forecast of the national rate of personal income growth is up (down), such an individual may choose to locate in the state with the higher (lower) systematic risk.

State economic development officials can use Table 5 to assist them in defining the target market for their promotional message. For example, the states listed in the second row of Table 5, which had negative excess growth rates, might want to focus their promotional efforts on larger well-diversified firms to whom only systematic risk is relevant. Other states with positive excess growth, may wish to advertise the special circumstances that they feel result in their state's unique risk character.

The focus of this paper is on the growth/risk tradeoff and not on identifying specific factors which "explain" different risk levels. There is a need for more research to identify what economic variables may be adjusted by policy makers to, in turn, adjust a region's systematic risk. However, economic developers may wish to encourage firms having certain systematic risk levels which will raise or lower the average of the state (or region) to locate in their region.

This paper was limited in scope to regional personal income growth rates. The results suggest that a similar study using revenue growth rates of stocks with U.S. personal income growth, or perhaps consumption growth, as the market index may yield interesting results. Such a study will help identify firms and industries that can be added to the existing regional portfolio to adjust its risk. In addition, the model can be tested using other income datasets such as GDP by state or income data from other countries for interesting comparisons.

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Jon D. Pratt, Louisiana Tech University

In the classic Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965), the total return (current income plus price growth) of an asset is expressed as a linear function of non-diversifiable, systematic risk. Systematic risk is measured by the covariance of the asset's total return with the total return of a market index such as the S&P 500 Index. An important factor that contributes to total return is the growth rate of income. Focusing on the growth rate of income, rather than total return, reduces the influence from variation of discount rates. In addition it eliminates the need to know the original price, or cost, of an investment when calculating return. This opens up the possibility of testing for risk/return relations when income data is available, but not prices or initial investments. Per capita personal income by state and by county are two data sets in which only income data is available.

This paper describes an application of the CAPM to regional growth. In this application the growth rate of personal income of a particular region, such as a state, is an increasing linear function of its systematic risk. Further, systematic risk is measured as the covariance of the growth rate of the personal income of the region with the growth rate of personal income of the entire United States. The results enable ranking of the states (and counties) based on both systematic risk and on excess growth.

The contribution of the paper is two fold. First, this paper is an application of Campbell and Vuolteenaho's (2004) assertion that the CAPM holds when the data considered is relatively free from variation of discount rates (or dividend yields). The regional economic growth rate data used in this study meets this criterion. Second, the paper demonstrates, for the first time, the application of the classic CAPM notion of systematic and unsystematic risks to regional economic growth and related location decisions.

A relation between personal income growth and growth systematic risk has several implications. First, individuals and firms deciding whether to locate in a particular state can temper the attractiveness of a state's high-growth rate with that state's accompanying higher systematic risk. Particularly for firms operating in multiple states or regions, the marginal effect on the firm's overall risk exposure from locating a new facility in another state or region may be quantified by referring to the systematic part of total risk which can not be diversified away. Second, entrepreneurial individuals and firms may consciously choose to seek out states or regions with above average non-systematic risk. Because individuals can effectively choose to live and work in only one state, knowledge of region-specific characteristics that make higher than risk-adjusted average personal income growth rates possible may influence the selection of one region over another. Entrepreneurs may choose to start their business in a region with these unique characteristics to grow their revenues quickly. Third, state and regional economic policy makers can measure the effect of their actions over time on the risk/return profile of their region in comparison to all other regions. For example, policies that improve personal income growth without raising the state's systematic risk can be emphasized. Fourth, economic development professionals seeking to position their state competitively in the minds of potential new residents and investors have a new tool to consider in designing their message. They can target their message to segments that are more likely to be attracted to their area due to its growth/risk profile.

The remainder of the paper is organized as follows. In Section 2, a review of the CAPM and an explanation of why the CAPM should hold for growth rates are presented. Considerable space is devoted to this review to explain the applicability of the CAPM to regional growth data in spite of the fact that the model has generally been rejected in the asset pricing literature. In Section 3, the applicability of the CAPM to personal income growth rates is specifically discussed. Section 4 contains a description of the data and methodology. Results are described in Section 5. Conclusions, limitations and areas for future research are presented in Section 6.

REVIEWING THE CAPM WITH AN APPLICATION TO INCOME GROWTH

The CAPM expresses expected next period return of an investment j in terms of a "risk-free" rate, [R.sub.f] and the systematic risk [[beta].sub.j] relative to the return of the common index [R.sub.M].

[R.sub.j] = [R.sub.f] + [[beta].sub.j] ([R.sub.M] - [R.sub.f]) (1)

Where,

[[beta].sub.j] = cov([R.sub.j], [R.sub.M])/[[sigma].sup.2.sub.M] (2)

The CAPM assumes that utility functions are normal (or quadratic), and there is no labor income (e.g. Campbell and Cochrane 2000). Early test results supported the CAPM (e.g. Fama and MacBeth, 1973). However, later cross-sectional studies such as Banz (1981) and Reinganum (1981), Basu, (1983), Rosenberg, Reid and Lanstein, (1985) and Chan, Hamao and Lakonishok, (1991) found that the CAPM did not hold because "value stocks", identified by the ratio of price-to-book value or price-to-earnings enabled returns in excess of those predicted by the CAPM. Due to the value stock anomaly, Fama and French (1992) advocate use of multifactor models that include other variables. However, researchers have not all given up on the usefulness of the CAPM.

GENERALIZING THE CAPM TO MULTIPLE PERIODS

The CAPM is a two-period model. Merton (1973) proposed an intertemporal CAPM (ICAPM) because the information set changes over time. The ICAPM necessitates adding other factors in addition to the market index. Jagannathan and Wang (1996) assume the CAPM holds conditional on the information set available at a particular time, but that betas and the market price of risk vary over time. They show that when the conditional version of the CAPM holds, a two beta model obtains unconditionally. One beta is the traditional one, based on the covariance of the asset's return with the market index. The other beta is based on the covariance of the asset's return with the market price of risk, which varies over time due to the business cycle. Their proxy for the variable market price of risk is interest rates. Campbell and Vuolteenaho (2004) argue that a two beta model is required because one beta is needed to measure the risk associated with an index proxy that captures market dividend yields and another beta is needed to measure the risk associated with an index proxy that captures expectations about long run cash flows (or earnings growth) of the firm. They find that when the assets under study have more or less constant ratios of the two types of risk, then the single index CAPM performs adequately. This was the case in the early Fama and French (1973) study. Thus, when the risk associated with variation in dividend yields is controlled for, the unconditional CAPM is supported.

ALTERNATIVES TO THE STOCK MARKET INDEX

Cochrane (2001, p.152) notes that the CAPM does not take into account labor income and shows that it is a specialized case of the Consumption Capital Asset Pricing Model (CCAPM) of Lucas (1978), Breeden (1979) and Brock (1982). The CCAPM is often tested using the growth rate of consumption as the "market index". One basic premise of the CCAPM is that any asset whose return covaries positively with consumption makes consumption more volatile, justifying a higher return because the investor's consumption-enabling income stream is made more volatile. Mankiw and Shapiro (1986) point out that the CCAPM is preferred on theoretical grounds because it takes into account the presence of other assets besides stocks in the wealth portfolio serving as the market index. The value of a broader index is supported in the few tests of the CAPM using return data for real assets such as commodities and agricultural or timberland, instead of stock returns. When these tests use the traditional stock market index to measure risk, the results generally do not support the CAPM (e.g. Holthausen and Hughes, 1978 and Bjornson and Innes, 1992). However, when a specially constructed market index is adapted for real assets, the CAPM is generally supported (e.g. Barry, 1980, Redmond and Cubbage, 1988 and Slade and Thille, 1997).

Jagannathan and Wang (1996, p.13) point out that the return on stocks will not measure the return on aggregate wealth because dividends from stocks represent less than 3% of household personal income, and labor income is a much more significant source of personal income. Thus, many of the assets included in investor wealth may not have a published market price available (e.g. privately held assets and the value of education). Despite its attractive theoretical generality, empirical tests of the CCAPM for total returns of stocks have been supported even less than the CAPM. Mankiw and Shapiro (1986), for example, find that even the unconditional CAPM performs better than the CCAPM. Campbell and Cochrane (2000) argue the reason the CAPM performs better than the CCAPM is that the growth rate of consumption does not capture the variation in dividend yield, whereas the stock market index does. This variation in dividend yield is the same discount rate risk identified by Campbell and Vuolteenaho (2004). Campbell and Vuolteenaho (2004, p. 1271) conclude that, when attempting to validate the CAPM, the cash flow-related beta is the most relevant, with the discount rate beta of only secondary influence. This conclusion is also supported by Lee (1998) who found earlier that the influence on stock prices of non-cash flow related fundamentals, such as discount rates, declines as the time horizon increases. The income growth rates investigated in this paper represents just such a data series: one which is free of non-cash flow related information.

Roll (1977) argued that a proxy for the "true" market index, consisting of all wealth could not be identified, making it impossible to test the CAPM. Fama (1990) finds that the growth rate of GDP explains about 43% of the average real return on stocks while proxies for discount rate changes explain about 30%. Thus, enhancing the stock market index with macroeconomic variables may improve models. Jagannathan and Wang (1996) use a market index that combines a traditional stock market index with the growth rate of labor income in a conditional CAPM and find that value stocks can no longer earn excess returns. Knez and Ready (1997) suggest that a risk premium on small stocks exists because small stock returns are correlated with investor's future labor income, but that this correlation is not captured with a stock index return. Jagannathan, Kubota and Takehara (1998) find that although a traditional stock market index beta alone could explain only 2% of the variation of Japanese security returns, including a labor income beta in the model explained 75%. Korniotis (2006) finds that a measure of regional risk that includes variance of consumption growth by state is priced in stock prices.

The collective results of these studies suggest the following: First, even though a stock market index does not perform as well as a conditional two beta model, the stock market index captures some influence of dividend yield changes that a macroeconomic index such as consumption growth does not capture. Second, creating a combined index containing influence from both the stock market and from macroeconomic growth variables improves the performance of the CAPM. Third, and most important to this paper, when the asset returns under study are controlled for the influence of dividend yield, the focus is on measuring cash flow beta and a single macroeconomic index, like consumption or income growth, is appropriate.

RESTRICTING ALL RETURN DATA TO GROWTH RATES

A simple way to control for effects of changing dividend yield is by restricting the measure of return on all assets under study to income growth rates rather than total return. The Gordon (1962) perpetual dividend growth model can be used to present the CAPM in terms of growth of cash flow. Gordon (1962) modeled the investor's expected return [R.sub.j] on a security j assuming an expected constant growth rate, [g.sub.j] of current dividends [D.sub.j0] and an initial investment price [P.sub.j0]. With the expected growth [g.sub.j] being conditional on the information set available at time 0, the return is expressed as:

[R.sub.j] = [D.sub.j0] (1 + [g.sub.j])/[P.sub.j0] + [g.sub.j] (3)

A similar expression for the return on the common market index, M is

[R.sub.M] = [D.sub.M0] (1 + [g.sub.M])/[P.sub.M0] + [g.sub.M] (4)

Assume, as did Jagannathan and Wang (1996, p.5), that the CAPM in (2) holds for total returns conditionally, i.e. based on the available information set at a particular point in time. Then, substituting (3) and (4) into (2) and solving for [g.sub.j]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

To simplify (5), note that [R.sub.f], [D.sub.j0]/[P.sub.j0] and [D.sub.M0]/[P.sub.M0] are all known at time 0, and note that [[beta].sub.j], is constant, given the information set and then let

[kappa] = ([R.sub.f] - [[beta].sub.j][R.sub.f] - [D.sub.j0]/[P.sub.j0] + [[beta].sub.j][D.sub.M0]/[P.sub.M0])/(1 + [D.sub.j0]/[P.sub.j0]) (6)

[[gamma].sub.j] = [[beta].sub.j](1 + [D.sub.M0]/[P.sub.M0])/(1 + [D.sub.j0]/[P.sub.j0]) (7)

With these substitutions, the expected growth rate is a linear function of the expected growth rate of the market index.

[g.sub.j] = [kappa] + [[gamma].sub.j][g.sub.M] (8)

Since the measure of systematic growth risk, gamma or [[gamma].sub.j] is a slope, the regression estimate for it [y.sub.j]' can be written as

[[gamma]'.sub.j] = cov([g.sub.j], [g.sub.M])/[[sigma].sup.2.sub.gM] (9)

Thus, if the CAPM is valid conditionally based on total returns, then it is also valid conditionally when the focus is restricted to growth in dividends. While the rate on treasury securities is often used as a proxy for the risk free rate, [R.sub.f], there is no security whose rate of return can proxy for the intercept [kappa] in (6). In this paper, [kappa] will be referred to as the "zero risk" growth rate. It can be interpreted as the rate of growth available when no systematic growth risk is assumed. The implication of (8) is that investors can form their expectation about the individual investment's growth rate [g.sub.j] knowing the estimate for the investment's systematic income growth risk provided in (9). In addition, recall from (3) that the investor's rate of return is also a function of the expected growth rate [g.sub.j], so that forming an expectation about future growth rate allows forming an expectation about future return, given the current status of dividend yields. For a firm, dividends cannot grow unless earnings grow. In addition, except for short run cost-cutting measures, earnings cannot grow unless revenues grow. Thus, the essential driver of returns to investors (entrepreneurs) is sales growth in the regions in which they operate.

APPLYING THE CAPM TO PERSONAL INCOME GROWTH BY REGION

A region such as a county or a state may be viewed as a portfolio of entrepreneurs or firms producing revenue from production. This revenue is divided among suppliers of labor, equity and debt capital, and intermediate materials and appears in measures of their personal income. These suppliers of resources may be viewed (loosely) as "investors" in the firm. For example, even though a worker may not be an owner of the firm, he commits (invests) his time (human capital) to the firm and receives a return. The return to these suppliers/investors can be measured as the growth rate in their incomes which results from their association with the firm. The form of the CAPM presented in equation (8) suggests a linear relation between growth risk and return, measured as income growth, even when there is no investment price data available. Measures of macroeconomic income such as the Personal Income series tabulated by the Bureau of Economic Analysis of the U.S. Department of Commerce can provide the income growth data required. A novel exploitation of (8) is, then, to test for the linear relation posited by the CAPM using the growth rate of personal income for each of the 48 contiguous states and also for county data. Thus, when a state's personal income growth systematic risk is high, as measured by its covariance with the United States, higher growth rates, higher rates of return and lower prices should be expected on the assets in that state. The wealth (assets) in the state will include privately held small businesses and the value of human capital, such as education and training.

There is some previous evidence to suggest that a tradeoff between income growth and volatility does exist in regions. For example, Barlevi (2004) documents the reduction in consumption associated with higher volatility of national income. For Italian households, Guiso, Japelli and Terlizzese (1996) construct a measure of income risk based on expectations of the variance of inflation and income growth and find that households reduce their ownership of risky assets when confronted with higher income risk. Similarly, Heaton and Lucas (2000) find that households with higher variance of business income hold less wealth in equity shares. When holdings are reduced, prices fall, indicating rates of return rise. Chandra (2002) documents a relation between a state's economic growth rate and the total risk or instability of growth as measured by the standard deviation. The innovation of this paper, compared to these studies, is that the measure of risk is systematic risk, measured in a CAPM context, rather than total risk.

Applying the CAPM to regional growth rate data, gives entrepreneurial individuals and firms a new analytical tool to help them form expectations about the growth rate of personal income in each state and county. Risk-adjusted expectations about growth should assist in selecting the preferred regional locations for investing both financial capital and human capital. A state, for example, may be selected as a location because of its tendency to grow at above systematic risk-adjusted rates due to non-systematic risk characteristics. Chatterjee, Lubatkin and Schulze (1999) posit that firm managers care about non-systematic risk and seek to control it at the individual firm level because investors cannot completely diversify their investments as well as the CAPM assumes. In addition, workers must choose to work in one state and many entrepreneurs, at least when starting out, can only operate in a few states. To the extent that they cannot achieve good diversification, these workers and entrepreneurs are forced to bear nonsystematic growth risk. Similarly, state economic policy makers and economic development professionals can seek to control, even design, their state's non-systematic risk characteristics. For example, Kalemli-Ozcan, Sorensen and Yosha, (2003) find that a high level of regional production specialization leads to not only a higher growth rate but also higher volatility and that regions can offset the increased volatility by risk-sharing through financial diversification between regions. Thus, diversifying the production activities in a region will tend to reduce nonsystematic risk. In addition, state economic policy makers and economic development professionals can seek to raise or reduce their state's systematic risks. This might be achieved by, for example, encouraging firms whose revenues have a high covariance with personal income to locate in their region.

The two period CAPM assumes that betas and the market price of risk are stable over time. For a state or region, this implies stability of the circumstances that make it more or less risky than the average. The large body of literature attempting to explain regional growth rates can be categorized based on what is said about this assumption of stability. Martin and Sunley (1998), for example contrast the traditional theory that regional economies will "converge" and become more alike, with the newer idea that regional economies will "diverge", resulting in regions with unique characteristics. The rationale for convergence is that, with mobility, self-correcting free market processes will adjust prices and wages and the supply of labor and capital will equalize across regions. The argument for divergence is that economies of scale, specialization and the achievement of critical masses of resources will lead to further accumulations of resources and capabilities within regions that tend to be self-perpetuating. Mankiw, Romer and Weil (1992) have argued for blending of these two approaches so that regions reach "conditional convergence" to the extent they have similar government policies, societal preferences, access to technologies and other structural circumstances, but each region will ultimately reach a unique steady state of relative volatility and growth. Evans and Karras (1996) find that differences in the levels of technology, the share of total income paid to capital and the per cent return to capital in the 48 contiguous states are evidence that the states converge rapidly to stable growth levels that are significantly different. Carlino and Sill (2001) find that the regions of the United States differ significantly in terms of long term volatility of economic growth. In this paper, it is assumed that, while the price of growth risk is stable over the long run, the practical possibility exists for economic policy makers to target policy changes in a particular state that may, over time, result in changes to the state's systematic growth risk.

DATA AND METHODOLOGY

The empirical tests to be performed in this paper use per capita personal income by state and county data available from the Bureau of Economic Analysis (BEA) website (http://www.bea.gov). Per capita data is used, rather than aggregate personal income data to follow the convention established in the CCAPM literature (e.g. Mankiw and Shapiro, 1986). The tests are restricted to annual data for two reasons. First, Fama (1990, p.1106) finds that income growth and asset returns are related over more than one quarter and this leads to autocorrelation and measurement problems when using quarterly data. Second, some information related to proprietor income and dividend and interest income are not included in the quarterly personal income estimates - only the annual figures. Annual personal income data for the 48 contiguous states was available from 1929 through 2006, on a per capita basis. In addition to these data, personal incomes per capita by county are available from 1969 to 2005. For each of the data sets, "gross" growth rates of personal income per capita are calculated as the simple ratio of current year per capita personal income to that for the previous year. Thus, positive "net" growth rates are reflected in the data as "gross" rates which are greater than one and negative net growths rates are reflected as gross rates which are less than one (but greater than zero).

Using the annual growth rates over the years for which data is available, an estimate for systematic risk for each state and region is calculated using (9). In (9), the growth rate of per capita personal income of the United States serves as the proxy for [g.sub.M]. For the same time period over which the regional estimates of systematic risk, [[gamma].sub.j] are calculated, the simple average of the regions' growth rates are also calculated. Then, having pairs of estimates of systematic risk and average growth rates for each region, the linear relation between risk and growth posited by the CAPM can be evaluated. Cross-sectional regressions of these average returns on their respective estimated systematic risks are performed using Generalized Least Squares (GLS) to determine whether the prices of risk (slopes) are different from zero, at the 99% confidence level. Because a state is a "portfolio" of multiple counties, the availability of these data sets makes it possible to look for the effect of diversification. For example, a higher [R.sup.2] should be expected for the regressions of state data versus the county data sets because more unsystematic risk has been diversified away in the portfolios. In addition, because counties are smaller, in terms of population and total personal income, the possibility of a CAPM-compromising "size effect" can be examined. Two sets of 48 portfolios of counties (about 65 counties in each portfolio) are created: one set created at random, based on an alphabetical listing and the other set created after sorting the counties based on total personal income. If a size effect exists, then the price of risk (slope) of the regressions of size-sorted portfolios of counties will be statistically different from that for the states and for the random (alphabetically sorted) portfolios. To make these comparisons, the two regressions are tested for equivalence using a version of the Chow (1960) test. When the regressions are found to not be equal, tests of the equality of the two slopes and intercepts are carried out using the dummy variable method, as described by Gujarati (1970). The test methodology is more fully described in the Appendix.

RESULTS

Table 1 shows the results of the cross-sectional regressions for the various data sets. The regressions for the 48 states over the entire period, 1930-2006 and for each of the two subperiods, 1930-1969 and 1970-2006 were statistically significant at the 99% confidence level since the confidence intervals for the slopes do not contain zero. The [R.sup.2] for the entire period, 1930-2006 was .175 and that for the early and later sub-period regressions were .264 and .158, respectively. The regressions for individual counties over the period 1970-2005 were also statistically significant, with an [R.sup.2] of .300. When the counties were sorted into 48 portfolios of counties created both randomly and sorted based on size, these regressions were also all statistically significant. The [R.sup.2] for the portfolios of counties increased to .379 and .843, for the random and size-sorted, respectively.

Table 2 records the results of tests for equality of slopes and intercepts for various pairs of regressions. The comparison of the regressions for the two time periods (1930-1970 and 1970-2005) for the 48 states indicates that while the slopes (prices of risk) are equal, the intercepts are not. This indicates the "zero risk" rate has declined in recent times for the states. The comparison of the regression of 48 states to the regression of 48 portfolios of counties sorted by size also showed that the slopes are equal but the intercepts are statistically different. The comparison of the regression of randomly created portfolios of counties to that for the 48 portfolios of counties created by sorting them based on size were not statistically different.

Figure 1 shows the fit of per capita personal income growth versus systematic risk for the 50 states, including Alaska and Hawaii. Note that the regression results given in Table 1 include only the 48 contiguous states to compare to the early time period in which data for the last two states were not available.

Figure 2 shows the fit of per capita personal income growth versus systematic risk for 3089 individual U.S. counties. It can be seen that the data is clustered with a few outliers.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Figure 3 shows the fit of per capita personal income growth versus systematic risk for 48 randomly created (based on alphabetical order) portfolios of U.S. counties. The visual fit appears to improve with portfolios compared to individual counties and this is confirmed by the [R.sup.2] 's given in Table 1.

[FIGURE 3 OMITTED]

Figure 4 shows a scatter plot of a pairing of the size rank of each portfolio of counties that constructed based on size (total personal income) with the percent residual

[FIGURE 4 OMITTED]

Table 3 shows the ranking of the 50 states based on systematic risk. The five states with the lowest systematic risk are Hawaii, West Virginia, Montana, Utah, and Idaho. The five states with the highest systematic risk levels are Connecticut, Louisiana, Texas, Oklahoma, and North Dakota.

Table 4 shows the ranking of the 50 states based on historic ability to grow at rates in excess of the expected growth rate on a risk-adjusted basis. The five states with the highest excess growth are Alabama, South Dakota, Wyoming, Mississippi and Tennessee. The five states with the lowest excess return averages are Ohio, Nevada, Michigan, California, and Alaska.

CONCLUSIONS, LIMITATIONS AND FUTURE RESEARCH

We did not conduct a strict Fama and McBeth (1973) three-step test to confirm the validity of the CAPM using our growth data. Rather, the CAPM's validity is simply assumed so that it can be applied to our growth data. Overall, the results support this application of the CAPM to personal income growth rates. First, the slopes of the relations between growth risk and growth are all statistically significant. Second, regressions of aggregated data result in higher [R.sup.2], as would be expected in data for which the CAPM holds. However, the [R.sup.2] for the states is lower than that for the regression of individual counties. This would not be expected and is a divergence from the CAPM. Outliers may be a possible reason for this failure since it is appears in Figure 2 that a few data points may have significant influence on the results and bias the [R.sup.2] of that regression upward. Third, no preliminary evidence of a size effect was identified in the portfolios of counties. These results support previous studies (e.g. Campbell and Vuolteenaho , 2004) which find that datasets which have low influence from discount rate variation can be modeled using the CAPM.

The intercept was found to be lower for the early 1930-1969 state data than for the later 1970-2006 period. Thus, the "zero risk rate" of growth appears to have risen in recent times. One possible explanation for this is the idea that government transfer payments and other public and private insurance programs which have been instituted in more recent times have made it possible to achieve higher growth rates without assuming systematic risk.

The fact that the intercept from the regression of the 48 states was different from that for the 48 portfolios of counties sorted by size is not a violation of the CAPM, because the prices of systematic risk are the same as posited by the CAPM. Thus, this result indicates there is no size effect. However, the difference in intercepts is a curiosity, since it indicates that counties have a lower "zero risk" growth rate than states, on average. It may be speculated that, as in the case of 1930-1969 dataset compared to the 1970-2006 dataset for states, this may be due to risk mitigation programs at the federal or state level that have not reached or been targeted at small counties.

Another way to test for a size effect in the size-sorted portfolios of counties is to examine the residuals (predicted growth versus actual growth) to see if there is a tendency for large or small county portfolios to have positive excess returns. In Figure 4, the smallest portfolio (rank = 1) appears to be an outlier. Ignoring that point, there appears to be slight upward slope, indicating that portfolios of larger counties tend to earn positive excess returns. However, even after removing this "outlier", the slope of the regression fit of percent error versus size rank is not statistically significant at the .01 level (F statistic = 5.2, confidence level = .027). Thus, the portfolios of counties sorted based on total personal income do not exhibit a size effect.

The results support the application of the CAPM to regional economic growth rates. As in other applications of the CAPM, the individual analyst must decide whether a high, medium or low level of risk is preferred. The ranking in Table 3 and in Table 4 can assist individuals and firms in selecting regions based on their preferences. From the point of view of fully diversified firms with investments in many states, the measure of systematic risk (Table 3) is relevant but the ranking of the states by excess growth (Table 4) is not. Table 5 shows a grouping of the states that takes into account both their systematic risk and their excess growth (i.e. unsystematic risk).

For these multi-state, well-diversified firms, Table 5, in its entirety, is relevant. If, for example, a firm's management is willing to tolerate medium risk levels, then all 18 states contained in the middle column of Table 5 would be viable choices for location of a new establishment or plant. However, for small firms or entrepreneurs considering where to locate their first (or perhaps second or third) establishment, both systematic risk and excess growth are relevant. This is also the case for workers who wish to choose from among a few possible job offerings based on the locations. For these investors, the states listed in the second row of Table 5, which had negative excess growth rates, would probably not be good choices. Under these circumstances, a firm, for example that is comfortable with high levels of risk, should select a location from the states listed in the upper right-hand grouping in Table 5 (i.e. AR, NC, GA, MN, CO, NH, CT, ND, or LA).

Consider also an individual who works (say) for a national restaurant chain and is given the choice of transferring as an assistant manager to one of two states. It is reasonable to assume that the relative success of the restaurant in which he works is dependent on regional personal income growth. If his forecast of the national rate of personal income growth is up (down), such an individual may choose to locate in the state with the higher (lower) systematic risk.

State economic development officials can use Table 5 to assist them in defining the target market for their promotional message. For example, the states listed in the second row of Table 5, which had negative excess growth rates, might want to focus their promotional efforts on larger well-diversified firms to whom only systematic risk is relevant. Other states with positive excess growth, may wish to advertise the special circumstances that they feel result in their state's unique risk character.

The focus of this paper is on the growth/risk tradeoff and not on identifying specific factors which "explain" different risk levels. There is a need for more research to identify what economic variables may be adjusted by policy makers to, in turn, adjust a region's systematic risk. However, economic developers may wish to encourage firms having certain systematic risk levels which will raise or lower the average of the state (or region) to locate in their region.

This paper was limited in scope to regional personal income growth rates. The results suggest that a similar study using revenue growth rates of stocks with U.S. personal income growth, or perhaps consumption growth, as the market index may yield interesting results. Such a study will help identify firms and industries that can be added to the existing regional portfolio to adjust its risk. In addition, the model can be tested using other income datasets such as GDP by state or income data from other countries for interesting comparisons.

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Jon D. Pratt, Louisiana Tech University

Table 1. Regression results for per capita personal income growth versus risk Zero Risk Conf. Int. Price of Intercept Intercept * Risk (slope) 48 States 1.0407 1.0161-1.0652 .0280 1930-2006 48 States 1.0276 1.0034-1.0519 .0350 1930-1969 48 States 1.0508 1.0285-1.0730 .0247 1970-2006 3089 Counties 1.0510 1.0498-1.0521 .0141 1970-2005 48 County 1.0494 1.0411-1.0577 .0155 Portfolios Random 48 County 1.0386 1.0339-1.0433 .0255 Portfolios Size-sorted Conf. Int. slope * [R.sup.2] 48 States .0053-.0507 .175 1930-2006 48 States .0127-.0573 .264 1930-1969 48 States .0035-0460 .158 1970-2006 3089 Counties .0131-.0150 .300 1970-2005 48 County .0078-.0231 .379 Portfolios Random 48 County .0212-.0298 .843 Portfolios Size-sorted * 99% Confidence level Table 2. Comparisons of two regressions Equal Equal First Regression Second Regression Slopes? Intercepts? 48 States 48 States YES NO 1970-2006 1930-1969 48 States 48 County YES NO 1970-2005 Portfolios Size-sorted 48 Country 48 County YES NO Portfolios Portfolios Size-sorted Random Table 3. Ranking of 50 states by Systematic Risk (lowest to highest) Average Systematic Excess State Growth Risk Growth HI 1.0583 0.8220 -0.00383 WV 1.0646 0.8444 0.00223 MT 1.0629 0.8790 0.00009 UT 1.0628 0.8815 -0.00003 ID 1.0625 0.8875 -0.00049 OR 1.0622 0.8935 -0.00087 NV 1.0590 0.8947 -0.00403 RI 1.0637 0.8982 0.00066 DE 1.0611 0.8989 -0.00201 MI 1.0588 0.9008 -0.00434 NM 1.0652 0.9024 0.00209 MD 1.0660 0.9038 0.00282 AL 1.0685 0.9070 0.00534 WI 1.0624 0.9188 -0.00097 PA 1.0635 0.9253 0.00005 MO 1.0621 0.9304 -0.00139 MS 1.0677 0.9349 0.00414 ME 1.0656 0.9416 0.00197 VT 1.0651 0.9419 0.00147 NY 1.0623 0.9420 -0.00128 TN 1.0672 0.9427 0.00360 NE 1.0638 0.9503 0.00006 IN 1.0611 0.9530 -0.00271 MA 1.0672 0.9537 0.00343 IL 1.0609 0.9580 -0.00290 OH 1.0599 0.9625 -0.00392 WA 1.0622 0.9895 -0.00203 AK 1.0589 0.9973 -0.00542 SC 1.0660 1.0052 0.00161 VA 1.0674 1.0083 0.00299 NJ 1.0654 1.0116 0.00093 SD 1.0690 1.0183 0.00447 KS 1.0641 1.0206 -0.00052 WY 1.0689 1.0256 0.00421 AR 1.0668 1.0273 0.00211 NC 1.0664 1.0279 0.00170 GA 1.0651 1.0308 0.00034 FL 1.0641 1.0321 -0.00065 KY 1.0645 1.0339 -0.00026 AZ 1.0618 1.0484 -0.00310 MN 1.0655 1.0607 0.00044 IA 1.0623 1.0842 -0.00309 CO 1.0666 1.0898 0.00112 CA 1.0604 1.0906 -0.00509 NH 1.0663 1.0951 0.00075 CT 1.0656 1.0963 0.00011 LA 1.0676 1.1043 0.00196 TX 1.0654 1.1963 -0.00137 OK 1.0651 1.1967 -0.00164 ND 1.0696 1.3280 0.00124 Table 4. Ranking of the 50 states by Excess Return (highest to lowest) Average Systematic Excess State Growth Risk Growth AL 1.0685 0.9070 0.00534 SD 1.0690 1.0183 0.00447 WY 1.0689 1.0256 0.00421 MS 1.0677 0.9349 0.00414 TN 1.0672 0.9427 0.00360 MA 1.0672 0.9537 0.00343 VA 1.0674 1.0083 0.00299 MD 1.0660 0.9038 0.00282 WV 1.0646 0.8444 0.00223 AR 1.0668 1.0273 0.00211 NM 1.0652 0.9024 0.00209 ME 1.0656 0.9416 0.00197 LA 1.0676 1.1043 0.00196 NC 1.0664 1.0279 0.00170 SC 1.0660 1.0052 0.00161 VT 1.0651 0.9419 0.00147 ND 1.0696 1.3280 0.00124 CO 1.0666 1.0898 0.00112 NJ 1.0654 1.0116 0.00093 NH 1.0663 1.0951 0.00075 RI 1.0637 0.8982 0.00066 MN 1.0655 1.0607 0.00044 GA 1.0651 1.0308 0.00034 CT 1.0656 1.0963 0.00011 MT 1.0629 0.8790 0.00009 NE 1.0638 0.9503 0.00006 PA 1.0635 0.9253 0.00005 UT 1.0628 0.8815 -0.00003 KY 1.0645 1.0339 -0.00026 ID 1.0625 0.8875 -0.00049 KS 1.0641 1.0206 -0.00052 FL 1.0641 1.0321 -0.00065 OR 1.0622 0.8935 -0.00087 WI 1.0624 0.9188 -0.00097 NY 1.0623 0.9420 -0.00128 TX 1.0654 1.1963 -0.00137 MO 1.0621 0.9304 -0.00139 OK 1.0651 1.1967 -0.00164 DE 1.0611 0.8989 -0.00201 WA 1.0622 0.9895 -0.00203 IN 1.0611 0.9530 -0.00271 IL 1.0609 0.9580 -0.00290 IA 1.0623 1.0842 -0.00309 AZ 1.0618 1.0484 -0.00310 HI 1.0583 0.8220 -0.00383 OH 1.0599 0.9625 -0.00392 NV 1.0590 0.8947 -0.00403 MI 1.0588 0.9008 -0.00434 CA 1.0604 1.0906 -0.00509 AK 1.0589 0.9973 -0.00542 Table 5. Categorization of the 50 states based on both risks Low Medium High Systematic Systematic Systematic Risk Risk Risk Positive WV MT RI MS ME VT WY TN NE AR NC GA MN CO Excess NM MD MA SC VA NH CT ND LA Growth AL PA NJ SD Negative HI UT ID NY IN IL FL KY AZ Excess OR NV DE OH WA AK IA CA TX Growth MI WI MO KS OK

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Title Annotation: | Capital Asset Pricing Model |
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Author: | Pratt, Jon D. |

Publication: | Academy of Entrepreneurship Journal |

Geographic Code: | 1USA |

Date: | Jul 1, 2010 |

Words: | 7655 |

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