Proprietors’ Income A Quantitative Assessment of the Costs of Under-Diversification Alan Moreira∗ September 5, 2008

Abstract In this paper I discuss two different exercises: in the first I use a standard consumption-CAPM to evaluate how narrowly held is aggregate proprietors’ income risk. I find that this risk is substantially concentrated on small business owners and my most conservative estimates indicate a wealth loss equivalent to 6% of proprietors’ lifetime consumption. In the second exercise I take the price of aggregate risks as given and explore the consequences of concentrated positions on idiosyncratic risk. I find that even for fairly small values of relative risk aversion the losses are high: expected excess returns increase from 2.5% to 7% in the preferred parametrization. Using household level consumption data I find no evidence that stocks allow small business owners to transfer idiosyncratic risk, but preliminary findings suggest that stock market participation does help business owners to transfer exposure to aggregate proprietors’ income risk.

1

Introduction

How much of aggregate proprietors’ income risk is held by proprietors? How much wealth is lost by this aggregate risk being narrowly ∗ University

of Chicago (GSB).

1

held? In the first exercise I use a power utility consumption-CAPM and household level consumption data and I find that even for fairly low levels of risk aversion, the difference in implied valuations between proprietors and non-proprietors is equivalent to 6% of proprietors’ lifetime consumption. In a second exercise I use county level proprietors’ income data to quantify the effect on valuation of under-diversification of idiosyncratic risk. I show that equilibrium expected real returns on proprietors’ income are between 2.5% and 3% if agents are fully diversified. These returns increase sharply when investor portfolio holdings are calibrated to match the average wealth concentration in small business, estimated to be around 50%. More precisely for a relative risk aversion of 5, expected returns increase from 2.5%/3% to 5.3%/4.8%, and up to 7% in the preferred parametrization. These results are important because proprietary business income makes up roughly 9% of US GDP and it is 60% of US private equity income. Small business owners are on average wealthier and are a substantial fraction of the United States stock holding population (Heaton and Lucas, 2000a), on a per capita or value weighted participation basis. They also have a very concentrated portfolio with an average of 50% of their total wealth in a single business (Moskowitz and Vissing-Joregessen, 2002). My work builds on a growing body of literature that is concerned with the non-tradable component of household wealth. The particular interest in small business owners’ portfolio decisions dates back to Heaton and Lucas (2000a), and the idea that this non-diversified position in idiosyncratic risk can have aggregate consequences dates back to Mankiw (1986) and Constantinides and Duffie (1996). More recent work by Benzoni et al. (2007) explores the effects of long run relations between the non-tradable labor income and traded dividends to individual portfolio decisions, and Cocco (2005) studies the effect of housing on the agent portfolio choice. Cochrane (2008) proposes a mean-variance framework to approach this class of problems.

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Cochrane (1999) suggests the importance of narrow holding of risks for asset pricing. Most of this literature takes the non-tradable income flow exogenously. Moskowitz and Vissing-Joregessen (2002) ask why investors choose these wealth allocations in the first place and document that the average return on private equity seems to be of the same order of magnitude as the return on the total market portfolio. They name this finding the Private Equity Premium Puzzle. I apply two different approaches to investigate two distinct costs of market incompleteness. First I use a standard power utility consumptionCAPM to ask what would be the increase in aggregate proprietary business income value if it’s risk was broadly held by all the investors in the economy. The exercise is simple and consists of comparing valuation ratios for aggregate proprietors’ income when we use proprietors’ consumption versus non-proprietors consumption. In the second exercise, I take the pricing of aggregate risk as given and ask what is the cost of non-diversification of pure idiosyncratic risk. I do that by estimating an ad-hoc stochastic discount factor that correctly prices aggregate risks, but calibrating the risk price that makes the decision to hold a concentrated portfolio in the business optimal. The data sources and the data construction procedures are described in section 2. In section 3, I document the costs of narrow holding of aggregate proprietary business income risk. In section (4), I study the costs of under-diversification of idiosyncratic risk. In section 5 I discuss evidence whether stock market participation can reduce these costs.

2 2.1

Description of the data Consumption data

The source of household level consumption data is the Consumption Expenditure Survey (CEX), produced by the Bureau of Labor Statistics. It consists of a series of cross-sections with around 5000 house-

3

holds interviewed each year. For each household we observe four quarters of consumption data, the household sources of income at the beginning and at the end of the interview period and the allocation of liquid wealth at the end of period. A variety of other information is provided for each household. I choose to focus on the surveys from 1987 to 2006. The sample can be potentially extended to 1981, but there are some issues regarding the data on food consumption that can significantly impact our measure of non durable consumption1 . From this sequence of cross-sections I construct a pseudo-panel, that consists of average growth rates for different groups of agents in each period. Specifically to construct this panel: • I drop households with less than $35,000 in annual before-tax income in 2006 dollars (inflation adjusted for the other years). • I drop households with fewer than 2 consecutive interviews or no data on income. (If I only keep households with all 4 interviews the sample does not change much). • I construct a measure of non durable consumption by excluding from total expenditure expenses for shelter, household equipment, health services, education services and purchases of motor vehicles. The interviews are distributed within each quarter and there is no obvious mapping between consumption reported in a specific interview with a specific calendar quarter. For each interview the CEX specifies what portion of the previous 3 months consumption was made this calendar quarter versus the past calendar quarter. For example if the second interview is in May and the household declares a consumption of $4,000 during the current quarter and $10,000 during the previous quarter, this means that $4,000 were consumed in April and the $ 10,000 in February and March. To form the first 1 Vissing-Jorgensen

(2002) discuss some of the issues.

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calendar quarter consumption I sum the consumption of the current quarter of the first interview (that will be consumption relative to January) with the previous quarter consumption of the second interview (February and March). I then adjust the consumption expenditures by household size using the mean of different equivalence scales as in Fernandez-Villaverde and Krueger (2007). This equivalence scale attempts to adjust for returns to scale in consumption. This adjustment is somewhat arbitrary but it is not critical for any of the results. Repeating this procedure, I can compute for each household three calendar quarters of consumption growth rates. As standard in studies that use this data and in an attempt to reduce measurement error I delete every household quarter consumption growth observation in which gross consumption growth is higher than 2 or less than .52 . To form the pseudo-panel I pick a given characteristic that defines a group, for example households for which proprietors’ income is more than $10,000 (2006). I delete all observations that do not comply with this criteria. I compute quarterly mean growth rates. I then run a regression of the quarterly growth rates on 4 seasonal variables. The final consumption growth variable is the residual of this regression plus the mean of the coefficients of the first regression.

2.2

County level income data

The income data is from the National Income and Product Accounts, produced by the Bureau of Economic Analysis. It consists of nonfarm proprietors’ income per county and it is constructed from the income tax return forms 1040 schedule C for sole proprietorships and form 1065 for partnerships. This data does not adjust for consumption that business owners make through their own business, the investments made in the business or the labor income of the business owner. These are potentially important adjustments that I ignore in this study. The CEX data however provide this information and can be potentially used in a future study. 2 See

for example Constantinides and Duffie (1996) or Vissing-Jorgensen (2002)

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The data is annual and ranges form 1969 to 2005 and includes 3100 counties. I delete any county that does not have all 37 years of observations. I pool counties in 733 commuting zones. Commuting zones are used in the labor literature as a proxy for local labor markets3 . This pooling greatly reduces spacial correlation between shocks to commuting zone proprietors’ income that are orthogonal to the aggregate shocks4 . This result is expected since my view of the uncertainty that small business owners face is closely related to the local labor market conditions.

2.3

Aggregate data

The aggregate data on consumption,corporate earnings and proprietors’ income is also from NIPA, has quarterly frequency and ranges from 1947 to 2007. From NIPA I also obtain the PCE inflation index on non durables, which I use to adjust all the series in this study.

2.4

Stock return data

The stock return data is obtained through Kenneth French’s website. I use the 25 portfolios sorted on size and book-to-market, the value-weighted CRSP total market portfolio and the long-short portfolios Small-minus-Big and High-minus-Low. For some exercises I need quarterly returns on the 25 portfolios, which I construct by compounding monthly returns to a quarterly frequency.

3

Narrow holding of aggregate risk

In this exercise I explore heterogeneity between proprietors’ and nonproprietors’ consumption decisions. I measure the consumption of 3 See 4I

for example Autor and Dorn (2007) follow Conley and Dupor (2003) and estimate non-parametrically a covariance

function between shocks as function of the distance between commuting zones.

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both groups by using CEX5 data on non durable consumption and income. I address the numerous issues with the construction of these measures in the data section. Since I assume agents to have power utility and be infinitely lived, the pricing of any cash flow (X)is straightforward:

Pt

=

Et [

∞ X

e

−jρ

(

Ct+j+1

j=0

Ct+j

)−γ Xt+j ]

Where γ is the coefficient of relative risk aversion,  denotes the agent type and C type i agent consumption.

When markets are

complete all agents in the economy have state-by-state equal consumption growth rates and consequently value cash flows identically. When markets are incomplete this is no longer true, but we can still use the same pricing equation to compute marginal valuations from the perspectives of both agents. If there are differences in these marginal valuations, there are further gains from trade not explored by the agents. I model log-consumption growth as a random walk. I choose to do so since all evidence for predictability is for very low frequencies and because my time series is relatively short.

ct+1 − ct = Δog(Ct+1 )

=

μc + σc εt+1

I model the log proprietors’ income growth as an affine function of the state variables:

t+1 − t = Δog(Xt+1 )

=

μ + GZt + σ εt+1 ,

where the state variable is a first order vector auto-regressive process: 5 CEX:Consumption

Expenditure Survey

7

Zt+1 = AZt + Cεt+1

Shocks {εt } are assumed to be normally distributed. The marginal valuation of this cash-flow given the consumption process above can be written as (I abstract from the superscript  but these equations are valid for both agents):

Pt

=

Et [

∞ X

exp(−jρ − γ(μc + σc εt+j ))Xt+j ]

(1)

j=0

(2) In Appendix 7.1 I discuss in detail how to solve this equation and write it as:

Pt

=

ƒ (Zt |Parameters)Xt

(3) (4)

Where ƒ is a known function,Zt are the state variables and Xt is the current income level.

3.1

Estimation

I choose state variables that do a good job of predicting proprietors’ income growth. I include 2 lags of log-proprietors’ income growth and 2 lags of log deviations of corporate earnings from proprietors’ income. This level term is motivated by the fact that the ratio between these two variables ought to be stationary and deviations from the trend should be followed by a subsequent mean-reversion. If this mechanism helps predict future proprietors’ income growth is ultimately an empirical question. The adjustment can come from adjustments in proprietors’ income growth or in corporate earnings growth. Evidence in, for example, Hansen et al. (2008), suggests that corporate earnings is a leading variable in the business cycle and the 8

estimates in table 2 suggest that such a specification does a good job in capturing the dynamics of proprietors’ income growth.

3.2

Identifying exposure to consumption risk

The proprietors consumption series is substantially shorter, ranging from 1987 to 2007. Therefore I first estimate the proprietors’ income dynamics, and from the identified shocks I estimate the consumption risk exposure. In table 3 we can see why there are further gains of diversification. Proprietors’ consumption is more correlated with proprietors’ income, and this correlation is especially higher for the permanent shocks to proprietors’ income growth. As intuition suggests, the transitory shocks to proprietors’ income seem to be smoothed out, but not the permanent shocks. In the same table we see that proprietors consumption growth is more volatile, which will also impact the valuation exercise. In table 4 we see the matrix C of exposures of the state variable decomposed into a consumption risk piece and a residual.

3.3

Results

In figure 5 I plot demeaned returns on aggregate proprietors’ income and proprietors’ income growth. We see that the return tracks the income growth, but not perfectly. The construction of this return series is described in the appendix 7.3. The correlation between both is 0.76 and the returns are 40% more volatile with an annualized quarterly standard deviation of 5%. Figure 5 presents the evolution of the calculated price-dividend ratio for aggregate proprietors’ income when proprietors’ consumption is used for valuation. It makes clear the substantial variation in valuation ratios. In figure 1 we can see the valuation ratios of proprietors’ income for proprietors and non-proprietors. These are the counterparts of price-dividend ratios for proprietors’ income. The very high level of

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the price-dividend ratio is not very surprising given the equity premium puzzle of Mehra and Prescott (1985). There is substantial heterogeneity in the level of valuation ratios but especially in the steepness of valuation ratios with respect to the relative risk aversion parameter. While valuations are insensitive to the risk aversion parameter for non-proprietors’ consumption, they are very sensitive when I use proprietors’ consumption. For a relative risk aversion of 30, the difference in marginal valuations is more than 30% of the value perceived by proprietors. This however does not mean that all this difference in valuations would disappear under market completeness by increases in proprietors’ valuations. As non-proprietors hold a greater share of proprietors income risk , their consumption ought to co-move more with it, and so valuations become more sensitive to the risk aversion. We know that complete market valuations will lie somewhere in between. To assess this quantitatively, we could use the relative wealth between groups, or more directly we can repeat the same exercise using aggregate consumption. This is arguably the right benchmark to think about valuation under market completeness. In figure 2 we see that, as expected, aggregate consumption valuations are more sensitive than non-proprietors’ consumption. However, they fall above from the proprietors’ valuations and the gap is still more than 20% of the proprietors valuation for a relative risk aversion of 30. Transforming these differences in valuation ratios into a dollar value of wealth losses is tricky, and is critically dependent on the risk aversion parameter. In figure 3 we see this wealth loss in 2005 as a proportion of total US consumption. Even for plausible risk aversion parameters the loss is substantial. For example, for a relative risk aversion of 5, the loss amounts to six months of total US consumption. Since these losses are concentrated in a subset of the population, on a per capita basis this loss is even more important. This comparison is still misleading since it compares a stock (wealth

10

loss due to market incompleteness) with a flow (consumption). So to have a better sense of magnitudes I add up all proprietors’ consumption in the year 2006 and all their proprietors’ income in this same year. I then use the difference in valuations ratios between the representative agent and proprietors’ to compute the total wealth loss for proprietors in the 2006 CEX sample. I then transform this wealth loss into a perpetuity by using the average real risk-free rate (1%) and divide this number by 2006 total consumption. This computation tells us how much higher would be the entire consumption path of proprietors if aggregate proprietors’ income risk was broadly held. We see this computations results in figure 4. Note that even for a relative risk aversion of 5, the lifetime consumption increases by 6%.

4

Non-diversification of idiosyncratic risk

In this exercise my goal is to quantify the wealth cost of not being able to trade the idiosyncratic piece of shocks to proprietors’ income. It is a partial equilibrium exercise by construction as I take the price of aggregate risk as given and consider optimal investors decisions. I do so by calibrating a stochastic discount factor that prices the three Fama-French factors (Fama and French, 1996) and the return on aggregate proprietors’ income. In this exercise, my unit of analysis will be a commuting zone6 and I assume that risks within the commuting zone are pooled. This assumption arguably underestimates the amount of idiosyncratic risk that an individual business owner bears. Nevertheless, I believe it captures an important piece of the actual risk burden on proprietors: Heaton and Lucas (2000b) estimates a median household level proprietors’ income growth standard deviation of 29%. So my exercise is capturing roughly one third of this risk as the median standard deviation of commuting zone level pro6 As

explained in the data section, commuting zones are a pool of counties (average

of 4) that is uses in the labor literature as a proxy for local labor markets.

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prietors’ income growth is slightly higher than 9%. If we believe7 that commuting-zone wide shocks make a larger portion of permanent shocks, then this exercise is capturing even a larger fraction of the shocks that agents care about most8 .

4.1

The stochastic discount factor

The motivation for including returns on proprietors’ income is that business wealth is part of the total wealth portfolio. How this risk will be priced in equilibrium is an empirical question, and work by Heaton and Lucas (2000a) suggests that this is an important source of risk. I assign risk prices by estimating the following linear discount factor using as test assets the 25 size and book-to-market sorted portfolios. I use the no-arbitrage restriction E[Δs∗ Re ] = 0 to estimate risk t ,t prices bj , where

Δs∗ t

mkt−rƒ

− E[r mkt−rƒ ]) − bsmb (rtsmb − E[r smb ]) −

=

1 − bmkt (rt

−

bhm (rthm − E[r hm ]) − bp (rt − E[r p ])

p

(5)

With these risk prices (bj ) I specify the stochastic discount factor st in the log form. Where b is the unobservable risk price of the pure idiosyncratic piece of proprietors’ income risk for some individual i (zero in the full diversification benchmark). To complete the specification of the stochastic discount factor, I add μs to match the level of the risk-free rate9 . I can then write the stochastic discount factor as: 7 MaCurdy

(1982) presents estimates of -0.4 for the autocorrelation coefficient of

shocks to total household earnings and -0.5 for shocks to total wage income.So individual level shocks are mostly transitory. 8 Empirical and theoretical research shows that credit markets are efficient in smoothing transitory shocks. 9μ

s

= rƒ +

1 |[ 2

bmkt

bsmb

bhm

bp

b ]ƒ |2 where ƒ is the Cholesky decom-

position of the shocks covariance matrix, that is from the three Fama-French factors, the aggregate proprietors’ income and own proprietors’ income (when idiosyncratic risk price is not zero)

12



st+1 − st

=

−μs − [ bmkt

st+1 − st

=

−μs − bƒt+1

p

bsmb

bhm

bp

mkt−rƒ

− E[r mkt−rƒ ]  rt+1  smb − E[r smb ]  rt+1   hm − E[r hm ] b ]  rt+1  p  rt+1 − E[r p ]   p p r,t+1 − E[r ]

p

Where r,t − E[r,t ] is the return on commuting zone level proprietors’ income. Define ƒ such that: 0 0ƒ ƒ = E[ƒt+1 ƒt+1 ],

and restrict ƒ to be a lower diagonal matrix. So ƒ is the Cholesky decomposition of the factors’ covariance matrix. Formally the vector of factors should have a superscript  since the vector has as one of its components the return on individual level proprietors’ income. However, for exposition reasons I omit it. The discount factor can be rewritten as:

st+1 − st

=

−μs − bƒ ξt+1 ,

where ξ is a 5 by 1 vector of normal iid shocks. In the Appendix (7.2) I show why a SDF so constructed prices correctly the factors if they are traded.

4.2

Cash flows & pricing

As in the previous exercise I model log proprietors’ income growth (Δ) as an affine function of a first order Markov state variable (Z):

13

          

t+1 − t

=

μ + GZt + σ ξt+1

 Zt+1

=

where, ξt+1

=

AZt + Cξt+1   mkt ϵ  t   smb   ϵ   t   hm   ϵ   t  p   ϵ   t    ϵt

(6)

Pricing is given by:

Pt

=

Et [

∞ X

exp(st+1+j ) exp(t+1+j )]

(7)

exp(st+1+j − st+j ) exp(t+1+j − t+j )]

(8)

j=0

Pt Xt

=

Et [

∞ X j=0

(9) Given the assumed dynamics for proprietors’ income and the stochastic discount factor, evaluation of this expectation is straight forward as explained in appendix 7.1.

4.3

The portfolio problem and the shadow price of risk

With a pricing function that links observable state variables to prices, I can recover the time-series of one period returns10 on commuting zone proprietors’ income for different values of the shadow price of risk (b ) of the idiosyncratic piece of proprietors’ income risk. With these returns I can estimate expected returns and the covariance matrix of one period returns on business wealth and the remaining traded assets. I need to pin down the portfolio behavior of the investor in order to fully characterize how these measures of valuation and risk change 10 see

appendix 7.3

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as the investor is forced to hold a higher fraction of his total wealth in a specific asset. This behavior will of course depend on the investor’s preferences for risk. As before, I assume power utility and an infinitely lived investor. In an environment of complete markets and constant expected returns, optimality implies a very simple portfolio allocation rule due to Merton (1971): θ=

1 γ

−1 (μr − rƒ ), r

(10)

where r is the covariance matrix of one-period returns, μr are the expected returns and γ is the relative risk aversion. To perform computations when markets are no longer complete I solve for the risk price (of the commuting zone idiosyncratic risk) that makes the investor willing to hold a concentrated position in the commuting zone proprietors’ income. This artificial risk price can be loosely interpreted as a multiplier on the constraint that the investor must hold a concentrated position in his own business, or the price of risk that the agent would demand if markets were complete. This is why the literature calls it the shadow price of risk. To calibrate for the relevant shadow price of risk I match the portfolio allocation of the business to what is the median allocation between business and other assets of a small business owner. Moskowitz and Vissing-Joregessen (2002) estimate this number to be 50% using the Survey of Consumer Finances data, but it can be substantially higher for wealthier households.

4.4

Return on aggregate proprietors’ income

Actual returns on aggregate proprietors’ income are not observed. I follow four different strategies to construct the unobservable return on business wealth: in three of them I assume a constant pricedividend ratio, and use the fact that in this particular case returns are proportional to dividend growth:

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Rt+1 =

Pt+1 + Dt+1 Pt

=

Pt+1 Dt+1 Dt+1 Dt Pt Dt

+

Dt+1 Dt

=

1+

P D

P D

Dt+1 Dt

∝

Dt+1 Dt

I them use three different measure of dividend growth: (i) a smoothed growth rate as in Heaton and Lucas (2000a), (ii) a lagged log growth rate and (iii) a contemporaneous log growth rate.

p

=

og(Xt−1 + Xt−2 ) − og(Xt−3 + Xt−4 )

p

=

og(Xt−1 ) − og(Xt−2 )

p

=

og(Xt ) − og(Xt−1 )

(i) rt (ii) rt (iii) rt

In measure (iv) I relax the constant price-dividend ratio assumption and use the returns obtained from the standard consumptionCAPM exercise. As a matter of fact the variation in the computed returns is consistent with any sdf of the log-linear form, being independent of either the assumed risk aversion or any assumed constant discount rate. The construction of this return series is described in the appendix 7.3. The idea is that there are permanent and transitory shocks to income growth. The first two specifications assume that all the shocks are permanent. With the VAR specification of the standard consumption-CAPM I can disentangle permanent from transitory shocks, and since both have different implications for future growth rates, they affect one period returns differently. Since I will be estimating risk prices using a set of traded portfolios, getting the level of returns on aggregate proprietors’ income is irrelevant at this stage. All that matters is to get the variation right.

4.5 4.5.1

Estimation The stochastic discount factor

In table 5 we see estimated risk prices for four different measures of the return on aggregate proprietors’ income. The lagged log growth 16

rate, in column (ii), is the only one that fails to have the exclusion restriction rejected. The other two growth rates have their exclusions of the SDF rejected with 1.5% and 3% p-values. The returns constructed using constant risk prices, but time-varying expected growth rates has it exclusion rejected with a p-value under 0.1%. It seems to be a solid case for the inclusion of returns on aggregate proprietors’ income in the SDF. However, we do see that depending on the assumed proxy for returns the risk price changes widely, from -25 in the contemporaneous growth rate specification, to +11 in the smoothed growth rate specification. 4.5.2

Cash flow dynamics

To complete the valuation exercise I need to estimate the dynamics of proprietors’ income. In the baseline specification I treat the crosssectional differences among commuting zones(CZs) as different realizations of the same process. This means that I fix all the parameters (but the mean growth rate) across commuting zones and estimate them using one pooled regression. In the preferred specification, the state variables are two lags of the commuting zone proprietors’ income growth. Unfortunately there is no corporate earnings data disaggregated by either commuting zone, county or state level and all the other variables considered do not add any meaningful prediction power, so I opt for a more parsimonious specification without any level term. In light of the different risk prices obtained from different ways of constructing a return on aggregate proprietors’ income,I perform the valuation exercise using both the specification that assumes constant price-dividend ratio and construct returns by smoothing growth rates (specification A from here on), and the specification that allows for time-variation in price-dividend ratios, but holds the discount rate fixed (specification B form here on). Table 6 presents VAR estimates for the two different ways of computing returns on proprietors’ income. The estimates suggest that 17

commuting zone proprietors’ income growth depends strongly on aggregate proprietors’ income returns and the three Fama-French factors. Figure 6 displays histograms for the estimates and it is clear that there is substantial heterogeneity in the dynamics and exposures to shocks. The specification(B) has this heterogeneity mostly concentrated in the exposure to the three Fama-French factors (SmB in particular), with the dynamics being reasonably similar across commuting zones. This goes in sharp contrast with specification (A) in which not only is there more heterogeneity in exposures to Mkt and HmL (SmB is about the same as in (B)), but the autoregressive parameters vary widely from commuting zone to commuting zone. These estimates suggest that specification (B) is closer to my assumption that commuting zones are ex-ante equal. From the estimates in table 6, it is trivial to form the parameters of equation (6). The construction of matrix C requires explanation: I construct it by multiplying the exposure estimates [ cmkt

csmb

chm

by a Cholesky decomposition of the covariance matrix of the components of the stochastic discount factor. In this way I transform the risk exposures to the four factors to exposures to four independent white noise shocks. In appendix (7.3) I present the mapping from the VAR to the state space representation.

4.6

Results

In table 8, column (FD), we see that in the full-diversification11 benchmark the price-dividend ratios are of the same order of magnitude as stocks, the time-series volatility and the dispersion across business owners is by all means tiny, staying 95% of the time in the range between 24 and 27 (specification A) or between 28 and 32 (specification B). Given this small variation, it is not surprising that the one period return volatility tracks the proprietary income growth volatility pretty closely. In the same table we also see that the maximum 11 By

full-diversification I mean the scenario where the price of idiosyncratic risk is

zero.

18

cp ]

Sharpe ratio portfolio is concentrated in the Market , HmL and aggregate proprietors’ income. While the maximum Sharpe ratio weights are almost the same for the three Fama-French factors across specifications, the weight on aggregate proprietors’ income changes from 0.22(A) to 0.70(B). This is due to the fact that the factor mimicking portfolio of specification B has a larger Sharpe ratio, and the resulting maximum Sharpe ratio portfolio has a large Sharpe ratio in specification B. This difference is mostly reflected in holdings of risk free bond that goes from -0.25(A) to -0.54(B). Equilibrium expected real returns on commuting zone proprietors’ income are 3%(A) and 2.5%(B) , slightly higher than the real return (2.63%(A),2.65%(B)) on the aggregate proprietors’ income factor mimicking portfolio for specification A, and slightly lower for specification B. In figure 7 we see that mean-reversion in the proprietors’ income process generates an interesting term structure of discount rates. Specification A implies a downward sloped term structure, while B implies an upward sloped one in the short run. Either way the slopes are not very steep. Although both measures of aggregate proprietors’ income risk imply fairly similar expected returns for the commuting zone proprietors’ income, the difference in term structure tell us that they have very different implications for times when the state variables are far from their means. In particular specification B discounts more heavily future expected growth. These different shapes are closely connected with the estimates for the mean reversion matrix A in equation (6) presented in table 6: It is clear that while both specifications have a medium run(two years) reversal component, only specification B has a strong short term(one year) persistence component. When investor portfolio holdings are constrained these valuation measures for commuting zone proprietors’ income become dependent on the investor’s preferences for risk. In table 8 we see the measures of risk and valuation for different values of relative risk aver-

19

sion. It is surprising that market incompleteness dampens variation in the price dividend ratio while increasing the variation in perceived one-period returns(see figure 8,panel A). The intuition is that as a big piece of the predictability in proprietors’ income growth is due to idiosyncratic shocks. The higher shadow price of risk induces the small business owner to discount more heavily this exposure, making valuations less sensitive to the predictability in cash flows. Figure (9) presents hedging demands due to the constrained position in commuting zone proprietors’ income. Aggregate proprietors’ income, when traded (T), is by far the most important asset in the investor hedging portfolio (-0.2% to -0.32%, relative to the unconstrained portfolio). In the traded case the Fama-French factors have non-zero hedging demands, with the investor being relatively short in the market (-0.04%) and HmL(-0.025%(A) to -0.06%(B)) and long in SmB (+0.04%). These demand substantially increase if the agent is not allowed to trade aggregate proprietors’ income: -0.08% for the market portfolio and between -0.07%(A) and -0.09%(B) for HmL. Hedging demands for SmB are roughly insensitive to the trading environment. In figure 8, panel B, note that when the investor is not allowed to trade aggregate proprietors’ income(NT), the price-dividend ratio is higher (than T) for low risk aversion, but it is lower for higher risk aversion. This effect is intuitive: on one hand the inability to directly trade the aggregate proprietors’ income makes the own proprietors’ income relatively more attractive, since commuting zone proprietary business income is almost a mean preserving spread of the aggregate. On the other hand, the inability to trade the aggregate proprietors’ income makes hedging less effective, making higher expected returns necessary to induce the concentrated portfolio. Figure 8, panel C, shows that consumption volatility keeps decreasing with risk aversion, as the agent can more effectively hedge the flows by shorting aggregate proprietors’ income(see hedging demand in figure 9). In the NT case it achieves a lower bound of 5.7%

20

very rapidly as the investor withdraws completely from the stock market splitting his wealth solely between the business and risk free bonds. My results show that overall non-diversification of idiosyncratic risk plays a big role on valuation of proprietors’ income. Required expected returns increase sharply with the portfolio concentration. For example, in specification B expected returns increase from 2.5% to 5.3% when the entrepreneur is constrained to have 50% of his wealth in the business (for a relative risk aversion of 5). I showed that the under-diversified portfolio induces hedging demands for the Market and HmL portfolios. The magnitudes are economic important as liquid wealth of proprietors’ is in the 2006 CEX sample 20% of the entire population liquid wealth and 22% of total population stock-holdings. A back of envelope computation

4.7

Linking to the consumption data

How reasonable is the consumption volatility implied by the model? The consumption expenditure survey data tells us that it is pretty reasonable.

Small business owners who hold stocks have an es-

timated annual consumption growth volatility of 6.8%, that is the model implied volatility for a risk aversion of 10-15. The volatility of non-stockholder proprietors is however too high relative the observed in the data: while the model suggests that for risk aversion above 25 the investor does not participate in the stock market and attains a volatility lower bound of 5.7%, in the CEX their estimated consumption growth volatility is 4%, suggesting that either the VAR overestimates how permanent the shocks are or the extent to which other sources of income reduce small business owner exposure to proprietors’ income risk. A relative risk aversion between 10 and 15 matches the observed consumption volatility of small business owners that hold stocks, and predicts stock holding in the range of 50% to 85% of liquid wealth. In the CEX data, the mean proportion of stock holdings to liquid wealth 21

is 70%, in the same ballpark as the quantitative exercise. This number however is likely to overestimate household position in stocks since in this work I do not take into account real estate wealth.

5

Does stock market participation help?

5.1

Transferring aggregate risk

Table 10 suggests that stock market participation does help business owners to transfer exposure to aggregate proprietors’ income risk. We see a clear pattern indicating that consumption is increasingly more exposed to permanent shocks to proprietors’ income as we refine our classification of small business owners. This pattern completely disappears as we condition on stock holding participation. I interpret this finding as preliminary evidence that agents can and do use stock markets to hedge their exposure to aggregate proprietors’ income risk. It is true that the CEX data does not leave too many degrees of freedom to split the data into different sub-samples without increasing noise substantially as the number of individuals per quarter decreases, but I believe that the pattern is suggestive.

5.2

Diversifying idiosyncratic risk

A simple way to have a first take on understanding if stock market participation allows business owners to transfer exposure in idiosyncratic risk is by comparing participants with non-participants. In particular, I compare the cross-sectional dispersion in consumption growth between groups. If stock market participation helps, we should observe a decrease in this measure of consumption idiosyncratic risk. Unfortunately as is clear from table ?? there is no meaningful variation between groups. This suggests that I am probably measuring more noise than true idiosyncratic risk. Another way to attack this problem is to look for portfolios reliably correlated with 22

shocks to proprietors’ income. This however requires identification of meaningful heterogeneity that is not related with the aggregate exposures. This is hard and requires discipline since the time-series available for proprietors’ income is relatively short. In unreported results I begin to explore such a strategy but did not find any meaningful observable heterogeneity that correlates in interesting ways with tradable portfolios (for example industry portfolios).

5.3

Discussion

The first exercise has one clear message:aggregate proprietors’ income risk is disproportionably concentrated in the hands of proprietors. There is no reason to be so: holding aggregate risks does not help with incentives. In fact I show some weak evidence in section 5 that stock market participation might help with this risk transfer. The second exercise is partial equilibrium by nature. I model the SDF in an ad-hoc way, and by taking risk prices as given I can explore the consequences of under-diversification for valuation measures, portfolio and consumption decisions. My approach is simple, but has weaknesses. The shadow risk price is constant, meaning that as there is, for example, a high enough positive shocks to individual proprietors’ income, the value of business wealth increases and the investor will re-balance back his portfolio by selling a piece of the business. This is not quite right, as the main characteristic of small business is the inability to sell small shares of the claims to business cash flow. So shadow risk prices of the true incomplete market problem faced by the small business owners are state dependent. In a framework with iid income growth, Schwartz and Tebaldi (2006) show that non-tradable income to liquid wealth ratio fully describes the state. In the environment that I explore, this is not true as there is substantial predictability in proprietors’ income growth. My approach also ignores borrowing constraints, that are potentially important in face of substantial transitory shocks as Heaton and Lucas (2000b) show in a numerical exercise. 23

Of course holding of idiosyncratic risks imposes costs and benefits. For example, it gives the right incentives for the business owner to manage the business optimally. However, a large portion of the risk hold by business owners are not linked to their actions and these risks can be diversified without any incentive costs. In this study I did not find any observable regional/local pattern of proprietors’ income growth that could help business owners with this task. Further investigation is important as the costs of under-diversification are fairly high, as documented in this paper.

6

Conclusion

In this paper I showed that permanent shocks to proprietors’ income are correlated with consumption of proprietors, this correlation is higher than with consumption of non-proprietors or aggregate percapita consumption. In a quantitative exercise, I use the power utility consumption-CAPM to show that this difference in correlation imply important welfare costs for proprietors: for a relative risk aversion of 5, it is equivalent to a decrease of 6% in lifetime consumption . I also present evidence that stocks might be effective in transferring aggregate proprietors’ income risk. In a second exercise I explore the costs for small business owners of concentrated portfolio positions in pure idiosyncratic risk. I show that the effect in portfolio positions, dynamics of valuation ratios and expected returns is substantial. The undiversified position generates negative hedging demands for the Market and HmL portfolios, and positive hedging demand for SmB. It also increases volatility of one period returns. I contrast with the first exercise,I did not find any evidence that stock-market participation can mitigate exposure of small business owners to idiosyncratic risk.

24

References Autor, D., Dorn, D., 2007. Inequality and Specialization: The Growth of Low-Skilled Jobs in the United States. Tech. rep., mimeo MIT. Benzoni, L., Collin-Dufresne, P., Goldstein, R., 2007. Portfolio Choice over the Life-Cycle when the Stock and Labor Markets Are Cointegrated. The Journal of Finance 62 (5), 2123–2167. Cocco, J., 2005. Portfolio Choice in the Presence of Housing. Review of Financial Studies 18 (2), 535–567. Cochrane, J., 1999. Portfolio advice for a multifactor world. Economic Perspectives-Federal Reserve Bank Of Chicago 23, 59–76. Cochrane, J., 2008. A Mean Variance Benchmark for Intertemporal Portfolio Theory. Manuscript, University of Chicago. Conley, T., Dupor, B., 2003. A Spatial Analysis of Sectoral Complementarity. Journal of Political Economy 111 (2), 311–352. Constantinides, G., Duffie, D., 1996. Asset Pricing with Heterogeneous Consumers. Journal of Political Economy 104 (2), 219. Fama, E., French, K., 1996. Multifactor explanations of asset pricing anomalies. Journal of Finance 51 (1), 55–84. Fernandez-Villaverde, J., Krueger, D., 2007. Consumption over the Life Cycle: Facts from Consumer Expenditure Survey Data. The Review of Economics and Statistics 89 (3), 552–565. Hansen, L., Heaton, J., Li, N., 2008. Consumption Strikes Back? Measuring Long-Run Risk. Journal of Political Economy 116 (2), 260–302. Heaton, J., Lucas, D., 2000a. Portfolio Choice and Asset Prices: The Importance of Entrepreneurial Risk. The Journal of Finance 55 (3), 1163–1198. Heaton, J., Lucas, D., 2000b. Portfolio choice in the presence of background risk. The Economic Journal 110 (460), 1–26. URL http://www.jstor.org/stable/2565645 25

MaCurdy, T., 1982. The Use of Time Series Processes to Model the Error Structure of Earnings in a Longitudinal Data Analysis. Journal of Econometrics 18 (1), 83–114. Mankiw, N., 1986. The Equity Premium and the Concentration of Aggregate Shocks. Journal of Financial Economics 17 (1), 211–219. Mehra, R., Prescott, E., 1985. The Equity Risk Premium: A Puzzle. Journal of Monetary Economics 15 (2), 145–61. Merton, R., 1971. Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3 (4), 373–413. Moskowitz, T., Vissing-Joregessen, A., 2002. The Returns to Entrepreneurial Investment: A Private Equity Premium Puzzle? American Economic Review. Schwartz, E., Tebaldi, C., 2006. Illiquid Assets and Optimal Portfolio Choice. NBER Working Paper. Vissing-Jorgensen, A., 2002. Limited Asset Market Participation and the Elasticity of Intertemporal Substitution. Journal of Political Economy 110 (4), 825–853.

7

Appendix

7.1 Pt

A.1: Pricing of Cash-Flows =

Et [

∞ X

exp(−jρ − γ(μc + σc εt+j ))Xt+j ]

j=0

Pt Xt Pt Xt Pt Xt

=

Et [

∞ X

exp(−jρ − γ(μc + σc εt+1 )) exp(μ + GZt+j σ εt+j+1 )]

j=0

=

∞ X

Et [exp(−jρ − γ(μc + σc εt+1 )) exp(μ + GZt+j σ εt+j+1 )]

j=0

=

∞ p X t,j j=0

Xt

26

Where pt,j denotes the price at t of a claim on the cash-flow above j periods ahead, and can be recursively written as:

pt,j Xt pt,j Xt pt,j Xt

pt+1,j−1

=

Et [exp(−ρ − γ(μc + σc εt+1 ))

=

Et [exp(−ρ − γ(μc + σc εt+1 ))

=

Et [exp(−ρ − γ(μc + σc εt+1 )) exp(μ + GZt + σ εt+1 )

Xt

]

pt+1,j−1 Xt+1 Xt+1

Xt

] pt+1,j−1 Xt+1

]

Where the first step comes from the fact that no cash-flow is paid from t to t+1 since the claim is just on t+j payoff, and in the third step I just substitute the cash-flow growth process. Now as standard in term-structure literature we guess that

pt,j Xt

=

pt,j Xt

is of the form:

exp(0 (j) + 1 (j)Zt )

Plugging in the above expression we get:

exp(0 (j) + 1 (j)Zt )

exp(0 (j) + 1 (j)Zt )

exp(0 (j) + 1 (j)Zt )

=

Et [exp(−ρ − γ(μc + σc εt+1 )) exp(μ + GZt σ εt+1 ) ×

×

exp(0 (j − 1) + 1 (j − 1)Zt+1 )]

=

[exp(−ρ − γμc + μ + GZt + 0 (j − 1)) ×

×

Et [exp(−γσc εt+1 + σ εt+1 + 1 (j − 1)(AZt + Cεt+1 )]

=

[exp(−ρ − γμc + μ + GZt + 0 (j − 1) + 1 (j − 1)AZt ) ×

×

Et [exp((−γσc + σ + 1 (j − 1)C)εt+1 )]

Using the assumed normality and collecting term I get: 0 (j)

=

−ρ − γμc + μ + 0 (j − 1) +

1 (j)

=

G + 1 (j − 1)A

27

1 2

| − γσc + σ + 1 (j − 1)C|2

pt,j=0 Xt

Now I use that

= 1, to pin down that 0 (0) = 1 (0) = 0. This

give us that: 1 (j)

=

( − Aj )( − A)−1 G

and: 0 (j) − 0 (j − 1)

=

−ρ − γμc + μ +

1 2

| − γσc + σ + ( − Aj−1 )( − A)−1 GC|2

The difference 0 (j) − 0 (j − 1) gives us the rate at which time t cash-flows are discounted between time periods t +j and t +j−1 if the state variables were in it’s means at time t. The difference between 1 (j) − 1 (j − 1) capture the adjustment for the predictability in the cash-flow growth. The term structure of expected returns can be recovered by subtracting from the discounting terms the pieces due to growth and due to risk-free discounting:

E[Rej ]

=

γσc (σ + ( − Aj−1 )( − A)−1 GC)0

Price-dividend ratios are given by: Pt Xt

7.2

(Zt )

=

∞ X

exp(0 (j) + 1 (j)Zt )

j=0

A.2:The stochastic discount Factor

Gross returns in the factors are given by: ƒt+1 = exp(rƒ + μƒ + ƒ ξt+1 )

(11)

Where μƒ = E[n(ƒ ) − rƒ ] − 21 |ƒ |2 . This is just a variance log-normal correction.Them non-arbitrage imposes that: 28

1

=

Et [exp(Δst+1 )Rt+1 ]

(12)

=

Et [exp(−μs − bƒ ξt+1 ) exp(rƒ + μƒ + ƒ ξt+1 )]

(13)

=

exp(−μs + rƒ + μƒ )Et [exp((−bƒ + ƒ )ξt+1 )] 1 exp(−μs + rƒ + μƒ + | − bƒ + ƒ |2 ) 2 1 1 exp(−μs + rƒ + |bƒ |2 +μƒ + |ƒ |2 − bƒ 0ƒ ) 2{z 2 | }

(14)

= =

(15) (16)

=μs

=

exp(μƒ +

1 2

|ƒ |2 − bƒ 0ƒ )

(17) (18)

Where in the last step I use that the drift in the sdf is calibrate to nail the risk free rate (plus the log-normal adjustment for the variance term). Now recognize that we estimate b using the following moment condition:

E[(1 − bƒ )(n(R) − n(Rƒ )]

=

Let r e = n(R) − n(Rƒ ) then:

E[(1 − bƒ )r e ]

=

E[re ] = bE[ƒ r e ] E[re ] = bco(ƒ , r e )

If the factors are tradable excess returns:

E[ƒ ] = bco(ƒ , ƒ ) b = E[ƒ ]co(ƒ , ƒ )−1 b = E[ƒ ](0ƒ ƒ )−1

29

0

0

Plugging b back in the pricing equation (17):

=

exp(μƒ +

=

exp(μƒ +

=

1

1 2 1 2

|ƒ |2 − E[ƒ ](0ƒ ƒ )−1 ƒ 0ƒ ) |ƒ |2 − E[ƒ ])

Where the last equality hods by definition of μƒ .

7.3

A.3:Realized returns and price dividend ratios

Given the pricing function derived in the appendix (7.1) and presented in equation (11) we can construct the realized path of pricedividend ratios and returns as follows. In time zero we observe the state variables Z0 , we can use the pricing function to determine time zero price-dividend ratio. Given the realized growth of the cash-flows and new realization of the state variable at time 1, we can compute the new price dividend ratio. I then use the accounting return identity to back out the realized returns: • Z0 →

P0 D0

• Z1 →

P1 D1 , D1 D0

•

P0 P1 D1 , , D0 D1 D0

→ R1 =

P1 D1 D1 D1 D0 + D0 P0 D0

• repeat

7.4

A.4:From VAR estimates to the state space representation

From the estimates in table (6) I construct the state space representation specified in equation (6).

30



[cmkt

csmb

chm

cp ]

0

0

0

0



σ(e)   0

=

 

Where, 0

=

co([Rmkt Rsmb Rhm Rp ])

(20)

G

=

[ 1

(21)

σ

=

A

=

G×C   B     1 0

C

0 ]

(19)

(22) (23) (24)

31

Figure 1: Valuation ratios and gains of diversification:Proprietors vs. Non-Proprietors In the top figure the difference between valuation ratios from nonproprietors and proprietors as a function of the risk aversion coefficient. In the bottom figure the valuation ratios for both groups. The valuation ratios are computed from equation (2) as described in the appendix. Both consumption series are constructed from the the CEX as described in the data section. For each value of γ the time discount factor ρ is choose to pin down the real risk free rate in 1%.

32

Figure 2: Valuation ratios and gains of diversification:Proprietors vs. Representative agent In the top figure the difference between valuation ratios for the representative agent and proprietors as a function of the risk aversion coefficient. In the bottom figure the valuation ratios for both groups. The valuation ratios are computed from equation (2) as described in the appendix. The proprietors consumption series is constructed from the the CEX as described in the data section. The Aggregate consumption series is deflated non-durable consumption from the NIPA. For each value of γ the time discount factor ρ is choose to pin down the real risk free rate in 1%.

33

Figure 3: Valuation ratios and gains of diversification:Proprietors vs. Representative agent In the top figure the difference between the value of proprietors’ income for the representative agent and proprietors as a function of the risk aversion coefficient. This wealth loss is plotted as a proportion of United States total consumption expenditures in the year of 2005. The valuation ratios are computed from equation (2) as described in the appendix. The proprietors consumption series is constructed from the the CEX as described in the data section. The Aggregate consumption series is deflated non-durable consumption from the NIPA. For each value of γ the time discount factor ρ is choose to pin down the real risk free rate in 1%.

34

Figure 4: Lifetime Consumption increase for proprietors. The figure plots the perceptual increase in lifetime consumption if aggregate proprietors’ income risk was broadly held. Difference in valuations are computed as in figure 2. This difference is multiplied by the sum of proprietors’ income of all agents identified as proprietors’ in the 2006 CEX extract. It is transformed in a life time annuity by using the average real risk free rate. And this number is divided by the total consumption of agents identified as proprietors’ in the 2006 CEX extract.

35

A:Realized one period returns and cash flow growth

B:Evolution of price-dividend ratios

Figure 5: Realized returns and price-dividend ratios Panel (A) present realized returns and panel (B) evolution of the pricedividend ratio of aggregate proprietors’ income. Both are computed for a relative risk aversion of γ = 5. The Panel A , however is independent of the risk aversion parameter, since returns are demeaned. I use the function in equation (3), the realized evolution of the state variables to construct the realized path of price-dividend ratios. I then use the realized proprietors’ income growth and the price-dividend ratio to construct realized returns. Returns are demeaned.

36

37

Panel A presents parameter estimates for specification(A) and Panel B for specification B.   mkt  R     smb    R  Δt − μ  mkt smb hm p   + σεt+1 Δt+1 − μ = B  c c c ]  + [c   Rhm  Δt−1 − μ   Rp

Figure 6: Histogram for the parameter estimates of the VAR

Panel a

Panel B

Figure 7: Mean reversion and the term structure of expected excess returns The figure plots the rate at which cash-flows j years ahead are discounted.

38

39 D:Term structure slope

B:Price-dividend ratio

T-aggregate proprietors’ income is traded (risk return properties from factor mimicking portfolio),NT-aggregate proprietors’ income is not traded.

(CZ) proprietors’ income. The plot is for the two different specifications of returns on aggregate proprietors’ income (A and B). And for two different trading arrangements:

proprietors’ consumption volatility changes with risk aversion. These measures are computed by constraining the investor to hold 50% of his wealth in commuting zone

The figure plots how commuting zone proprietors’ income return(A), price-dividend ratio (B) and term-structure slope(D) change with risk aversion. In (C) we see how

Figure 8: Under-diversification effects on valuation

C:Consumption volatility

A:Return volatility

40 Figure 9: Hedging demands

HmL

Market

factor mimicking portfolio),NT-aggregate proprietors’ income is not traded.

returns on aggregate proprietors’ income (A and B). And for two different trading arrangements: T-aggregate proprietors’ income is traded (risk return properties from

zone (CZ) proprietors’ income, and by subtracting what would be the optimal portfolio without the portfolio constraint. The plot is for the two different specifications of

The figure plots how hedging demands changes with risk aversion.Hedging demands are computed by constraining the investor to hold 50% of his wealth in commuting

SmB

CZ proprietors’ income

Table 1: Commuting zone proprietors’ income:descriptive statistics Data is form NIPA and ranges from 1969 to 2005. There are 733 commuting zones. X cz means commuting zone proprietors’ income. Y cz means commuting zone total income. X s means US aggregate proprietors’ income.Ncz denotes number of proprietors’ in the commuting zone

Mean

Percentile 5

Median

Percentile 95

0.01

-0.01

0.01

0.04

0.09

0.06

0.08

0.12

0.08

0.05

0.08

0.12

-0.01

-0.03

-0.01

0.00

0.19

-0.07

0.21

0.40

-0.06

-0.28

-0.07

0.18

0.11

-0.10

0.11

0.31

0.59

-0.10

0.60

0.82

PT

Δ ln(Xtcz ) t=1 T b (Δ ln(X cz )) σ t PT cz / Y cz ) (X t t t=1 T PT cz Δ ln(Xt / Ncz ) t t=1 T ) ρ(Δ ln(Xtcz ), Rmkt t−j ρ(Δ ln(Xtcz ), Rsmb ) t−j ρ(Δ ln(Xtcz ), Rhm ) t−j ρ(Δ ln(Xtcz ), Δ ln(Xts ))

41

Table 2: Proprietors’ income dynamics   

Δyt+1 et+1 − yt+1

  =μ+

2 X j=1

  Aj 

Δyt+1−j et+1−j − yt+1−j

   + Cεt+1

I estimate the above expression using quarterly NIPA data on proprietors’ income and corporate earnings. The sample ranges from 1948 to 2007. t-stats are in brackets and are computed using Newey-west standard errors with 6 leads and lags.



0.33

0.12

   (4.47) (6.82) A1 =    −0.07 1.04  (−0.30) (12.88)   0.30     R2 =     0.99

       



 −0.12      (2.25) (−6.88)   A2 =     −0.08 −0.04    (−0.34) (−0.55)   0.015 0.000   C=  −0.006 0.057

42

0.13

T = 240

Table 3: Consumption of proprietors’ vs non-proprietors’ In the first row we can see the consumption growth standard deviation for each group, in the second row the correlation with aggregate proprietors’ income growth, in the third row the correlation with the permanent component of innovations of proprietors’ income as identified by the VAR of last section, and in the fourth row the transitory innovations. The construction of the Consumption series are discussed in detail in the data section. Non-Proprietors are individuals with zero proprietary business income, Proprietors* (second column) are individuals with more than ten thousand dollars in proprietary business income and Proprietors** are individuals for which proprietary business income is more than 50% of total income.

Non-Proprietors’

Proprietors’*

Proprietors’**

σ(Δct )

0.016

0.033

0.041

ρ(Δct , Δt )

0.179

0.186

0.080

ρ(Δct , ξp )

0.065

0.205

0.188

ρ(Δct , ξt )

-0.231

-0.137

-0.066

43

Table 4: State Space Representation The first column of the C matrix of shock exposures is identified by projecting the innovations from the VAR in the consumption growth series normalized by it’s own standard deviation.

The last two

columns are given by a Cholesky decomposition of the piece that is orthogonal to consumption growth. Non-Proprietors are individuals with zero proprietors’ income, Proprietors are individuals with more than ten thousand dollars in proprietary business income and aggregate consumption is aggregate per capita consumption growth from NIPA.

Non-Proprietors 

 0.0014 C= −0.0123

0.0132

0

−0.0113

0.0537

  

Aggregate Consumption 

 0.0019 C= 0.0152

0.0147

0

−0.0059

0.0570

  

Proprietors 

 0.0029 C= −0.0058

0.0132

0

−0.0113

0.0537

44

  

45

and I estimate:

J-stat p-value J − Jr p-value

Prop

HmL

SmB

Mkt

47.65 0.0012 4.65 0.0310

4.93 (4.67) -2.25 (-1.57) 8.75 (6.86)

(i)

43.00 0.0031

4.72 (4.18) -1.43 (-0.87) 8.04 (5.34) 11.63 (1.25) 47.65 0.0012 1.56 0.2111

46.09 0.0012

(ii) 4.93 4.69 (4.67) (4.09) -2.25 -1.91 (-1.57) (-1.21) 8.75 8.37 (6.86) (5.64) 7.38 (0.58) 47.65 0.0012 5.98 0.0145

41.68 0.0046

(iii) 4.93 5.96 (4.67) (4.67) -2.25 -2.06 (-1.57) (-1.43) 8.75 10.47 (6.86) (6.20) -25.14 (-1.70)

R ] = 0, ∀ = 1..25 E[Δs∗ t t

52.66 0.0003 11.14 0.0008

41.53 0.0048

(iv) 5.21 6.46 (4.81) (4.94) -2.32 -1.85 (-1.60) (-1.17) 9.38 10.56 (7.16) (6.49) -22.00 (-2.25)

Table 5: Stochastic discount factor estimation Second stage efficient GMM estimates. r p denotes log returns on aggregate proprietors’ income and is constructed in four different ways: in column(iv) returns are from the valuation exercise of section (3) and presented in figure (5), in p column (i) returns are as constructed by Heaton and Lucas (2000a), that is rt = og(Xt−1 + Xt−2 ) − og(Xt−3 + Xt−4 ), in column (ii) I use as returns the proprietors’ income log growth rate lagged one period, in column (iii) I use the contemporaneous log growth rate. The data is quarterly, the sample ranges from 1947 to 2007. Data on aggregate proprietors’ income is from NIPA. Data on portfolio returns (three Fama-French factors and 25 size/book-to-market sorted portfolios) are from professors French website. The test assets are the excess returns of the 25 portfolios sorted on Bookto-market and size. Quarterly returns are constructed by compounding monthly returns. t-stats are in brackets. Let, mkt−rƒ p Δs∗ = 1 − bmkt (rt − E[r mkt−rƒ ]) − bsmb (rtsmb − E[r smb ] − bhm (rthm − E[r hm ]) − bp (rt − E[r p ]) t

Table 6: Proprietors’ income growth dynamics: Pooled commuting zones

The data on proprietors’ income is from NIPA and is in yearly frequency 1969 to 2005 and for 733 commuting zones.  Rmkt t+1  smb    Rt+1 Δt − μ + [ cmkt csmb chm cp ]  Δt+1 − μ = B  Rhm Δt−1 − μ  t+1 p Rt+1

ranging from     + σεt+1  

The estimates are averages of commuting zone time-series regressions and the standard errors are constructed from the standard deviation from the coefficient estimates. t-stats are in brackets. Proprietors’ income growth is adjusted by the mean growth in the number of proprietors. The regression includes two lags of commuting zone proprietors’ income growth, the current year return on three Fama-French factors and the log return of aggregate proprietors’ income. In Panel (A) I construct aggregate proprietors’ income returns assuming constant price-dividend ratios by compounding quarterly smoothed growth rates: p

rt =

4 X

[og(Xt−j/ 4−1/ 4 + Xt−j/ 4−2/ 4 ) − og(Xt−j/ 4−3/ 4 + Xt−j/ 4−4/ 4 )]

j=1

In panel (B) I construct returns assuming a constant discount rate as in section (3) and described in appendix (7.3).



 0.011 B= (1.67)

Panel A 



−0.100  mkt smb hm p  0.012 c c c ]= [c (−16.97) (23.25)

−0.005

0.009

(−8.53)

(18.00)

 0.010   (85.42)

b = 0.088 R2 = 0.28, T = 34, N = 733 , σ(e)

Panel B 

 0.205 B= (29.84)





−0.141  mkt smb hm p  0.013 c c c ]= [c (−24.46) (25.46)

−0.007

0.017

(−11.35)

(37.00)

b = 0.093 R2 = 0.19, T = 34, N = 733 , σ(e)

46



0.008   (60.81)

47 5 17.71 0.17 0.13 0.093 0.090 0.048 0.116 0.23

5 16.36 0.16 0.13 0.096 0.090 0.053 0.125 0.30

FD 26.05 0.26 0.20 0.091 0.090 0.030 0.107 0.00

FD 30.29 0.29 0.24 0.093 0.090 0.025 0.117 0.00

γ E[ XP ] σts ( XP ) σcs ( XP ) σts (R) σts (ΔX) E[R] σ(Δc) b

γ E[ XP ] σts ( XP ) σcs ( XP ) σts (R) σts (ΔX) E[R] σ(Δc) b

20 7.65 0.07 0.06 0.099 0.090 0.122 0.056 1.10

20 6.63 0.07 0.05 0.104 0.090 0.142 0.061 1.20

A/T 10 15 12.64 9.97 0.12 0.10 0.09 0.07 0.094 0.096 0.090 0.090 0.071 0.092 0.071 0.058 0.50 0.75 B/T 10 15 11.73 8.70 0.11 0.08 0.09 0.07 0.098 0.101 0.090 0.090 0.077 0.107 0.075 0.063 0.55 0.85

25 5.48 0.05 0.05 0.107 0.090 0.174 0.058 1.50

25 6.34 0.06 0.05 0.101 0.090 0.149 0.054 1.40

FD 30.29 0.29 0.24 0.093 0.090 0.025 0.110 0.00

FD 26.05 0.26 0.20 0.091 0.090 0.030 0.105 0.00

5 17.37 0.17 0.14 0.096 0.090 0.050 0.121 0.26

5 18.37 0.18 0.14 0.092 0.090 0.046 0.114 0.20

B/NT 10 15 11.73 8.33 0.11 0.08 0.09 0.07 0.098 0.101 0.090 0.090 0.077 0.112 0.075 0.066 0.55 0.90

A/NT 10 15 12.00 8.83 0.12 0.08 0.09 0.06 0.095 0.097 0.090 0.090 0.075 0.105 0.073 0.063 0.55 0.90

20 6.63 0.07 0.05 0.104 0.090 0.142 0.060 1.20

20 6.73 0.06 0.05 0.101 0.090 0.140 0.060 1.30

25 5.17 0.05 0.04 0.108 0.090 0.185 0.060 1.60

25 5.68 0.05 0.04 0.103 0.090 0.167 0.056 1.60

Table 7: Results Results are obtained using the parameter estimates of table (19) and evaluating equation (9) as described in appendix (7.1).σcs (·) denotes time series means of cross-sectional standard deviation of variable (·), σts denotes cross-sectional mean of time-series variation. XP is the analogous of a price-dividend ratio for proprietors’ income.σ(Δc) denotes the implied yearly volatility of consumption growth. b denotes the shadow price of risk for the commuting zone (CZ) proprietors’ income. The first column (FD) presents the full-diversification results(portfolio holdings are unconstrained ). For each different risk aversion value the shadow price of risk(b ) is set so that 50% of the portfolio allocation is in the commuting zone proprietors’ income.All computations are presented under two different trade arrangements: (T)aggregate proprietors’ income is traded12 , (NT)- aggregate proprietors’ income is not traded. And the computations are performed for two different constructions of the returns on aggregate proprietors’ income: (A)-returns are constructed by compounding smoothed quarterly log growth rates(column (2) of table (5)), (B) -returns are constructed by assuming a constant discount rate (column (1) of table (5)).

48

5 0.51 0.47 0.02 0.43 0.03 -0.46

5 0.49 0.42 0.04 0.43 0.39 -0.78

FD 0.06 0.51 0.02 0.45 0.22 -0.25

FD -0.12 0.46 -0.01 0.50 0.71 -0.54

γ CZ prop. inc. Mkt SmB HmL Ag. prop. inc. Rƒ

γ CZ prop. inc. Mkt SmB HmL Ag. prop. inc. Rƒ

20 0.52 0.08 0.01 0.08 -0.18 0.48

20 0.52 0.08 0.05 0.06 -0.13 0.43

A/T 10 15 0.52 0.50 0.21 0.13 0.01 0.01 0.20 0.12 -0.10 -0.14 0.16 0.38 B/T 10 15 0.49 0.50 0.20 0.12 0.04 0.04 0.19 0.10 0.07 -0.06 0.02 0.29 25 0.50 0.06 0.05 0.03 -0.16 0.52

25 0.51 0.06 0.01 0.06 -0.19 0.55

FD 0.01 0.56 -0.02 0.55 -0.10

FD 0.16 0.52 -0.02 0.49 -0.15

Portfolio weights

5 0.50 0.49 0.03 0.47 -0.48

5 0.48 0.48 0.01 0.44 -0.41

B/NT 10 15 0.50 0.52 0.21 0.11 0.04 0.05 0.19 0.09 0.07 0.24

A/NT 10 15 0.51 0.51 0.20 0.11 0.03 0.04 0.18 0.09 0.08 0.26

20 0.50 0.06 0.05 0.05 0.35

20 0.51 0.06 0.05 0.04 0.34

25 0.50 0.03 0.05 0.02 0.40

25 0.48 0.03 0.05 0.02 0.42

Table 8: Results:Portfolio allocations Results are obtained using the parameter estimates of table (19) and evaluating equation (9) as described in appendix (7.1). The first column (FD) presents the full-diversification results(portfolio holdings are unconstrained ). For each different risk aversion value the shadow price of risk is set so that 50% of the portfolio allocation is in the commuting zone proprietors’ income.All computations are presented under two different trade arrangements: (T)-aggregate proprietors’ income is traded13 , (NT)- aggregate proprietors’ income is not traded. And the computations are performed for two different constructions of the returns on aggregate proprietors’ income: (A)-returns are constructed by compounding smoothed quarterly log growth rates(column (2) of table (5)), (B) -returns are constructed by assuming a constant discount rate (column (1) of table (5)).

Table 9: Consumption volatility Proprietors are households with more than $10,000 dollars in proprietors’ income in a given year. Stockholders are households with more than $ 5,000 dollars in stocks. Non-Proprietors are households with zero proprietors’ income. The other categories are analogous defined. The Data is from the CEX and is constructed as described in the data section (2). The standard deviation is computed in the quarterly frequency and reported in annual frequency.

σ(Δc) Proprietors

0.037

Non-Proprietors

0.027

Proprietors/Stockholders

0.068

Non-Proprietors/Stockholders

0.031

Proprietors/Non-Stockholders

0.040

49

Table 10: Correlation of proprietors’ consumption Growth and permanent shocks to proprietors’ income. Permanent shocks to proprietors income are identified from the VAR with aggregate log proprietors income growth and the deviations of log Corporate Earnings to log Proprietors income (specification discussed in section (3). Consumption growth is from the Consumption Expenditure Survey constructed as described in (2). Individuals are pooled by level of proprietors income, level of liquid wealth and level of stock holdings.Yp stays for proprietors income, W for liquid wealth and S for dollar value of stock holdings.

Permanent Shock

Transitory Shock

Proprietors Income growth

Yp > 1000

0.207

0.160

-0.267

Yp > 5000

0.234

0.220

-0.177

Yp > 10000

0.194

0.213

-0.134

Yp > 1000/ S = 0

0.167

0.176

-0.255

Yp > 5000/ S = 0

0.198

0.228

-0.171

Yp > 10000/ S = 0

0.128

0.160

-0.153

Yp > 1000/ S > 5000

0.103

0.052

-0.061

Yp > 50000/ S > 5000

0.140

0.090

-0.024

Yp > 10000/ S > 5000

0.193

0.196

-0.007

Proprietors

Proprietors Non-Stockholders

Proprietors Stockholders

50

Table 11: Consumption idiosyncratic risk Proprietors are households with more than $10,000 dollars in proprietors’ income in a given year. Stockholders are households with more than $ 5,000 dollars in stocks. Non-Proprietors are households with zero proprietors’ income. The other categories are analogous defined. The Data is from the CEX and is constructed as described in the data section. The cross-sectional dispersion is the time-series average of the dispersion in quarterly consumption growth within each group. σcs (Δc) Proprietors

0.262

Non-Proprietors

0.263

Proprietors/Stockholders

0.262

Non-Proprietors/Stockholders

0.273

Proprietors/Non-Stockholders

0.259

51