Hedging Recessions Nicole Brangera

Linda Sandris Larsenb

Claus Munkc

Preliminary version March 10, 2014

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Abstract We analyze the life-cycle investment and consumption problem of an investor exposed to unemployment risk and business-cycle risk. Becoming unemployed is a jump event with a significant impact on the individual’s human capital. The typical counter-cyclical variation in unemployment risk and pro-cyclical variation in salary growth rates make human capital more stock-like. We find that the risk of unemployment and the consequences of unemployment for human capital lower consumption throughout life and reduces the portfolio share of the stock significantly. For individuals with high risk aversion or an income positively correlated with the stock market, a zero investment in stocks early in life can be optimal which can partly explain the observed limited stock market participation. The life-cycle pattern in consumption exhibits a hump several years before retirement as seen in the data. Other tings equal, an unemployed individual consumes less and invests a much smaller share of savings in the stock market than an otherwise similar employed individual. Keywords: Unemployment risk; Life-cycle model; Portfolio planning; Businesscycle JEL subject codes: G11 a Finance Center M¨ unster, University of M¨ unster, Universit¨ atsstrasse 14-16, D-48143 M¨ unster, Germany. E-mail: [email protected] b

Department of Business and Economics, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark. E-mail: [email protected] c

Department of Finance, Copenhagen Business School, Solbjerg Plads 3, DK-2000 Frederiksberg, Denmark. E-mail: [email protected]

Hedging Recessions

The paper contains graphs in color, use color printer for best results.

Abstract We analyze the life-cycle investment and consumption problem of an investor exposed to unemployment risk and business-cycle risk. Becoming unemployed is a jump event with a significant impact on the individual’s human capital. The typical counter-cyclical variation in unemployment risk and pro-cyclical variation in salary growth rates make human capital more stock-like. We find that the risk of unemployment and the consequences of unemployment for human capital lower consumption throughout life and reduces the portfolio share of the stock significantly. For individuals with high risk aversion or an income positively correlated with the stock market, a zero investment in stocks early in life can be optimal which can partly explain the observed limited stock market participation. The life-cycle pattern in consumption exhibits a hump several years before retirement as seen in the data. Other tings equal, an unemployed individual consumes less and invests a much smaller share of savings in the stock market than an otherwise similar employed individual. Keywords: Unemployment risk; Life-cycle model; Portfolio planning; Businesscycle JEL subject codes: G11

Hedging Recessions 1

Introduction How does the risk of recessions affect households and their consumption and invest-

ment decisions over the life cycle? Of course, households participating in the stock market are negatively influenced by the plunge in stock prices typically seen in recessions, but economic downturns also have detrimental effects on labor income and thus the human capital that constitutes an important part of the overall wealth of many (in particular, young) individuals. Indeed, one of the biggest fears of many individuals is the risk of becoming unemployed, and since the economy-wide unemployment rate obviously is higher in recessions than in booms, the unemployment risk for most individuals must also be counter cyclical. While unemployed, an individual typically receives some income in the form of welfare payments or insurance benefits, but much less than the salary earned when employed. In addition, a prolonged period of unemployment often entails a reduced salary when becoming reemployed due to, for example, the loss of training and confidence. Moreover, unemployment leads to lower pension savings and, thus, lower retirement income. In this paper we set up a model for the optimal consumption and investment decisions of an individual that incorporates the above-mentioned features. We calibrate the model to data from the U.S. over the period from January 1959 to July 2013. We assume that a single macroeconomic state variable captures the business cycle. The life time of the individual is dvivded into an active phase and a retirement phase. In the active phase the individual is either employed or unemployed, and the employment status is controlled by a jump process. The jump intensity depends on the macro state variable so that the probability of being fired is counter cyclical (higher in recessions than in booms), whereas the probability of finding a new job if unemployed is pro cyclical (higher in booms than in recessions). When employed the individual receives a salary that follows a diffusion process with a pro-cyclical expected growth rate that also depends on the individual’s age as observed in the date (higher growth rates when young). An employed individual pays a fraction of the salary earned as an insurance premium or tax, but in return the individual receives a low income (assumed deterministic) when unemployed. The salary that the unemployed would get upon reemployment varies over time similarly to the salary of the employed individual, but with a negative expected growth rate as motivated above. The investor can trade a stock (index) and a risk-free asset (a bond), but he cannot perfectly hedge business cycle (i.e., macroeconomic) risk, salary risk, and, in particular, investor-specific unemployment risk so the financial market is incomplete. In our calibrated model the expected excess return of the stock varies counter cyclically with the

1

macroeconomic state. Finally, the investor cannot directly observe the true state of the economy, but has to learn about it from observing the realized stock return and economic signals which, for concreteness, we take to be the aggregate unemployment rate and the growth rate of aggregate income in the economy. The individual maximizes expected life-time utility by optimally choosing his consumption rate and his portfolio share of the stock as functions of age and any other relevant variables. Because of the rich model with incomplete markets, portfolio constraints, and jumps we find the optimal consumption and investment strategies by a numerical method. We apply the recently introduced SAMS (Simulation of Artificial Market Strategies) method, suitably extended and adapted to our framework. The method is based on the closed-form solutions to a closely related consumption-investment problem in a set of artificial complete markets. Each such consumption-investment strategy is transformed into a feasible strategy in the true, constrained and incomplete, market. The life-time utility generated by each of the “feasibilized” strategies can be evaluated by Monte Carlo simulation and, by embedding this into a numerical optimization routine, we find the best of them. The method also delivers a measure of its precision in terms of the maximum wealth-equivalent utility loss suffered by following the consumption-investment strategy produced by the numerical approach instead of the unknown truly optimal strategy. Our numerical results show that both the unemployment risk in general and the business-cycle dependence of expected salary growth and un- and reemployment probabilities have significant effects on consumption and investments over the life cycle. The risk of unemployment and the consequences of unemployment for human capital lower consumption throughout life and reduces the portfolio share of the stock significantly. For individuals with high risk aversion or an income positively correlated with the stock market, a zero investment in stocks early in life can be optimal which can partly explain the observed limited stock market participation. In our model the life-cycle pattern in consumption exhibits a hump several years before retirement, which resembles the humped pattern seen in consumption data better than the consumption patterns generated by the theoretical models closely related to our where the hump occurs at retirement or not at all. When comparing the optimal decisions of an employed individual and an unemployed individual (other things equal), we find significant differences. The unemployed individual consumes less and invests a much smaller share of savings in the stock market than the employed individual. The remainder of the paper is organized as follows. Section 2 ties our paper to the existing literature. Section 3 formulates the model and the consumption-investment problem of the investor. Section 4 explains how we solve the problem. Section 5 calibrates parts of the model to existing data and motivates the assumed values of other parameters. 2

Section 6 presents and discusses numerical results based on the benchmark parameter values and considers various comparative statics. Finally, Section 7 concludes. Appendices provide details on the filtering aspects of our model, proofs of theorems, an explanation of a technique we employ for solving various partial differential equations, as well as information on the estimation procedure.

2

Related literature Our paper builds on the literature on human capital and life-cycle portfolio planning.

There are several distinctive features of income which one has to take into account. First, human capital is in general not tradable. Hence the investor cannot fully hedge the income stream, which amplifies its impact on the investment strategy. Second, the expected labor income growth rate depends on the business cycle. Third, investors are exposed to unemployment risk. Several papers have taken the first feature into account (e.g. Cocco, Gomes, and Maenhout 2005, Munk and Sørensen 2010, Koijen, Nijman, and Werker 2010, and Lynch and Tan 2011), a few papers have taken the second feature into account (e.g. Munk and Sørensen 2010, and Lynch and Tan 2011), whereas the last feature is less researched. Cocco, Gomes, and Maenhout (2005) and Lynch and Tan (2011) include unemployment risk in their robustness checks. They assume that the investor’s income level simply jumps down to a substantially lower level when he becomes unemployed, and jumps back to a normal level if the investor becomes employed again. Both papers show that unemployment risk has a significant additional impact on the investment strategy in particular for young investors, but does not change the overall conclusions from the papers. We also model the employment status of the investor by a jump process. However, we allow the intensity to depend on the business cycle. Furthermore, as seen in many European countries, the investor has an insurance that limits the income loss in case of becoming unemployed. Finally, the income that the investor can earn after a period of unemployment depends negatively on the length of that period. Cocco, Gomes, and Maenhout (2005) solve numerically for the optimal investment strategy of an investor receiving a non-hedgeable income stream in a setting with constant investment opportunities. For an empirically reasonable correlation between stock market shocks and labor income shocks, they find that labor income is a substitute for an investment in the risk-free asset so that the financial wealth should be directed to stocks. Hence, with a small financial wealth and a large value of the human capital, investors should put a large fraction of their financial wealth into the stock. This gives rise to an optimal investment strategy in which the allocation to risky assets indeed is increasing in the investor’s investment horizon as suggested by investment advisors.

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In order to investigate when human capital resembles a long-term bond and when it resembles cash and to assess the implications for the optimal bond/stock/cash portfolio choice Munk and Sørensen (2010) solve for the optimal investment strategy in a setting with stochastic interest rates. The stochastic labor income can be correlated with interest rates, bond prices, and stock prices. Furthermore, they allow the expected labor income growth rate to be affine in the short-term interest rate to incorporate business-cycle variations in the wage. They find that the business-cycle sensitivity of income is crucial for the valuation and riskiness of human capital and for the optimal investment strategy. Koijen, Nijman, and Werker (2010) also includes both stochastic interest rates and labor income. In their model the income process is assumed to have a constant expected growth and constant volatility so it cannot capture the business cycle variations in the income. On the other hand, they allow for a more realistic model of the short-term interest rate compared to Munk and Sørensen (2010), and hence allow for time-variation in bond risk premiums. Whereas the above-mentioned of the literature has tried to find theoretically support for the investment strategy given by investment advisors, the most recent literature tries to come up with an explanation for why investors do not to follow this advice. Lynch and Tan (2011) consider a model where the stock market dividend yield predicts both stock market returns and the expected growth of the labor income. This leads to a negative hedging demand for stocks that partially offsets the high speculative stock demand, and hence an investment strategy more consistent with the observed behavior of individual investors. As Munk and Sørensen (2010) and Lynch and Tan (2011) we allow the expected excess return of the stock to depend on the business cycle. However, our investor cannot directly observe the true state of the economy, but can learn about it from observing the realized stock return and an economic signal that may represent the aggregate unemployment rate, the overall growth rate, or the aggregate income of the economy. Other closely related papers include Bagliano, Fugazza, and Nicodano (2013), Benzoni, Collin-Dufresne, and Goldstein (2007), and Viceira (2001).

3

Model setup We consider an economy with a single consumption good which serves as the numeraire

so that prices, income, etc., are stated below in real terms. 3.1

Assets As in most models of life-cycle consumption and investments, we assume that the

investor can trade in a risk-free asset and a single risky asset. The risk-free asset is a money market account with a constant continuously compounded (real) rate of return 4

given by the short-term interest rate r. The risky asset represents the stock market index and for simplicity we refer to it as the stock in the following. We let St denote the time t stock price (with any dividends being reinvested in the stock when received) and assume dSt = St [(r + µ0 + µ1 Xt ) dt + σS dZSt ] ,

(1)

where ZS = (ZSt ) is a standard Brownian motion, and σS is the stock’s volatility which for simplicity is assumed constant. The excess expected return µ0 +µ1 Xt is assumed affine in the variable Xt which is interpreted as a measure of the overall state of the economy. To emulate real-life business cycle fluctuations, we assume that   q 2 dXt = −κx Xt dt + σX kSX dZSt + 1 − kSX dZXt ,

(2)

so that the state variable mean-reverts around zero. Here κx and σx are positive constants, and ZX = (ZXt )t≥0 is a standard Brownian motion independent of ZS so that kSX is the instantaneous correlation between the stock price and the state variable. A high negative value of Xt indicates that the economy is in a recession, whereas a high positive value of Xt indicates that the economy is in a boom. Hence, with µ1 < 0 the stock index in our model exhibits a counter-cyclical expected excess return and Sharpe ratio as found in the data (see, e.g., Harvey 2001; Lettau and Ludvigson 2010). 3.2

Income

We assume that the investor is in the labor force until a pre-determined retirement date T˜ and then lives on until time T ≥ T˜. At any time t < T˜, the investor is either employed (It = e) or unemployed (It = u). If employed at time t, the investor receives a salary at the rate yt of which a fraction −αe is paid as a contribution to an unemployment insurance. If unemployed at time t, the investor receives a deterministic unemployment benefit of αu (t) ≥ 0 (the time dependence allows a real growth rate in unemployment benefits over time). Hence the total income stream Yt of the investor at time t < T˜ is ( Yt = (1 + αe )yt 1{It =e} + αu (t)1{It =u} =

(1 + αe )yt if employed, αu (t)

if unemployed.

(3)

The salary level yt evolves according to q i h  2 − k 2 dZ , dyt = yt (ξ0 (t, It ) + ξ1 (It )Xt ) dt + σy kSy dZSt + kˆXy dZXt + 1 − kSy yt Xy (4)

5

where Zy = (Zyt ) is a standard Brownian motion independent ofqZS and ZX , and σy ≥ 0 2 , and k is the salary volatility. Furthermore kˆXy = (kXy − kSX kSy )/ 1 − kSX Sy and kXy are the instantaneous correlations of the salary with the stock price and the state variable, respectively. The expected salary growth rate can depend on the state of the economy, reflecting the intuition that salary increases tend to be more frequent and larger in booms than in recessions. To incorporate observed life-cycle variations in labor income (cf., e.g., Hubbard, Skinner, and Zeldes 1995; Cocco, Gomes, and Maenhout 2005), the salary growth rate can depend on time (age). Furthermore, the expected salary growth rate may depend on the employment status to capture the intuition that the salary level of the investor when becoming reemployed is lower the longer he has been unemployed. This effect is present if the drift rate is negative on average when the investor is unemployed. The employment status It is modeled by a two-state Markov chain jumping between employment e and unemployment u. Formally, the change between the two states is described by two jump processes N u and N e . If the investor is employed, a jump N u into unemployment occurs with jump intensity ηu (Xt ). Conversely, if the investor is unemployed, a jump N e into employment happens with a jump intensity ηe (Xt ). We allow the intensities to depend on the state Xt of the economy. With ηu (Xt ) being decreasing in Xt , the probability of getting fired is higher in a recession than in a boom. Similarly, with ηe (Xt ) being increasing in Xt , the probability of becoming employed again is higher in a boom than in a recession. Correspondingly, the expected duration of an unemployment period is longer in a recession than in a boom. 3.3

Assessing the state of the economy We assume that the investor cannot observe the state Xt of the economy, but has

to infer it from observations of quantities informative about the state. The stock price is informative as the drift depends on Xt . Various macro-economic variables are also likely to be informative. For concreteness, we assume that the investor applies two macro variables as signals, which in our applications are the aggregate unemployment rate and the growth rate of aggregate income. We let v1t , v2t denote the two macro signals and assume that dv1t = (ω1 + κ1x Xt − κ1v v1t ) dt + σv1 dZ1t ,

(5)

dv2t = (ω2 + κ2x Xt − κ2v v2t ) dt + σv2 dZ2t .

(6)

Here Z1 , Z2 are additional standard Brownian motions that may be correlated with each other as well as Zs , ZX , Zy . Based on observations of the stock price and the two macro signals, the investor forms his best guess xt about the state Xt of the economy and revises his guess over time 6

using Bayesian updating. More formally, xt is the expectation of Xt conditional on the realizations of the macro signals and the stock price up to, and including, time t. We assume that there is no information in the investor’s own salary in addition to the stock price and the macro signals. From Appendix A it follows that the filtered model (as seen by the investor) is given by dSt = St [(r + µ0 + µ1 xt ) dt + σS dzSt ] , (7)   q (8) dxt = −κx xt dt + σx ρSx dzSt + 1 − ρ2Sx dzxt , q i h  dyt = yt (ξ0 (t, It ) + ξ1 (It )xt ) dt + σy ρSy dzSt + ρˆxy dzxt + 1 − ρ2Sy − ρˆ2xy dzyt , (9) where zSq , zx , zy are independent standard Brownian motions, ρSy = kSy , ρˆxy = (ρxy − ρSx ρSy )/ 1 − ρ2Sx . Here ρSx and ρxy are instantaneous correlations, and the parameters of the two signal processes enter only via these two correlations, cf. Appendix A. Since jumps in employment status are investor-specific and thus unrelated to the macro economy, the jump intensities in the filtered model are ηu (x) and ηe (x). 3.4

The utility maximization problem We denote the financial wealth of the investor at time t by Wt . The investor has to

choose a consumption strategy c = (ct ) and an investment strategy π = (πt ). Here, ct > 0 is the rate at which goods are consumed at time t, and πt is the fraction of financial wealth invested at time t in the stock index with the remaining wealth Wt (1 − πt ) being invested in the money market account. We impose the standard borrowing and shortselling constraint πt ∈ [0, 1],

t ∈ [0, T ].

(10)

Note that the market is incomplete as the investor can trade only one risky asset but is exposed to several sources of risk, namely stock market risk, macro risk, and salary risk as captured by the three Brownian motions zS , zx , zy , as well as employment risk as captured by the two jump processes N u , N e . The investor is assumed to have a power utility function of consumption ct and possibly terminal wealth (bequest) WT with a constant relative risk aversion (CRRA) γ > 1.1 The indirect utility function of the investor is given by "Z J(t, W, x, y; i) = sup Et (c,π)

1−γ

T

e

−δ(s−t)

t

1

W c1−γ s ds + εe−δ(T −t) T 1−γ 1−γ

# .

(11)

The focus on γ > 1 rules out complications as those discussed in Kim and Omberg (1996), Korn and Kraft (2004), and Liu (2007). Empirical studies (e.g., Meyer and Meyer 2005) support this assumption.

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The expectation is conditional on Wt = W , yt = y, xt = x, and It = i (the employment status which is i = e or i = u). Furthermore, ε ≥ 0 defines the utility weight of terminal wealth relative to consumption, and δ is the investor’s subjective time preference rate. Because of incomplete markets and portfolio constraints, we are unable to solve the problem in closed form. In the next section we outline our numerical solution method.

4

Optimal consumption and investment decisions

4.1

Outline of the solution approach To solve for the optimal investment and consumption strategy we apply and generalize

the so-called SAMS (Simulation of Artificial Market Strategies) approach introduced by Bick, Kraft, and Munk (2013). The SAMS method is illustrated in Figure 1. The basic idea is to exploit the fact that we can derive a semi-explicit expression for the optimal strategy in each of various artificial markets. In any of the artificial markets the investor is unconstrained, has access to the same assets (with identical or higher returns) as in the true market plus additional assets completing the market. Consequently, the investor can obtain at least as high an expected utility in any such artificial market as in the true market.2 In the figure, the points marked to the right on the axis indicate the maximal utility in different artificial markets denoted by θ1 , θ2 , etc. The lowest expected utility among these artificial markets—indicated by θ¯ on the axis—is still at least as large as the unknown maximum in the true market. [Figure 1 about here.] The explicit, optimal strategy in any of the artificial markets is infeasible in the true market, but we can feasibilize it—that is, transform it into a feasible strategy in the true market—and then evaluate the expected utility it generates in the true market by standard Monte Carlo simulation. This procedure leads to the points on the left part of the axis in Figure 1. We then maximize over these feasibilized strategies and obtain the expected utility indicated by θ∗ in the figure. The corresponding near-optimal strategy is the strategy suggested by the SAMS approach. Just as with other numerical methods, the suggested strategy is unlikely to be identical to the unknown, truly optimal strategy so by using the suggested strategy the agent suffers a welfare loss. By comparing the expected utility generated by the near-optimal strategy to the expected utility in the worst of the artificial markets considered, we derive an upper 2

The artificial markets were introduced by Karatzas, Lehoczky, Shreve, and Xu (1991) and Cvitani´c and Karatzas (1992). Jin and Zhang (2012) also build on Cvitani´c and Karatzas (1992) to solve the portfolio planning problem in an incomplete market if asset prices follow a jump-diffusion.

8

bound on the welfare loss which is therefore a measure of precision of the approach. We show that, in wealth-equivalent terms, the loss bound corresponds to at most a few percent of the agent’s wealth, and in the examples studied by Bick, Kraft, and Munk the true loss is significantly smaller than the loss bound. Compared to alternative methods, this approach distinguishes itself by being relatively easy to implement, being based on closed-form consumption and investment strategies, and providing a measure of its accuracy. As our optimization problem (11), in addition to time, features three diffusion state variables (can be reduced to two after exploiting homogeneity of the utility function) and two jump processes, grid-based methods are cumbersome to implement and would involve very high computation times when used with the grid sizes necessary to bring us near the continuous-time solution. The following subsections provide details on the SAMS method applied to our problem. 4.2

A family of artificial markets Intuitively we are looking for artificial markets in which (i) we can solve the utility

maximization problem and (ii) the optimal consumption-investment strategy is close to what we expect to be a feasible and good strategy in the true market. Because the portfolio weight of the stock in an artificial market is unconstrained, whereas it has to satisfy the constraint (10) in the true market, we might want to increase the risk-free rate or the expected excess stock return to make the stock relatively less or more attractive. In an artificial market we thus replace the constant term µ0 in the expected stock return and the risk-free rate r by µ ˜0 = µ0 + νS ,

r˜ = r + νS− ,

(12)

where νS− = max(−νS , 0). Note that no matter the sign of νS , both the risk-free rate r˜ and the expected stock return r˜ + µ ˜0 + µ1 xt are at least as big in the artificial market as in the true market. To complete the market, we introduce assets not traded in the true market. We introduce a stylized business cycle derivative with price Bt evolving as    q 2 dBt = Bt (˜ r + νx0 + νx1 xt ) dt + σB ρSB dzSt + 1 − ρSB dzxt

(13)

and a stylized salary derivative with price Dt satisfying    q dDt = Dt (˜ r + νy0 + νy1 xt ) dt + σD ρSD dzSt + ρˆBD dzxt + 1 − ρ2SD − ρˆ2BD dzyt , where ρˆBD = (ρBD

(14) q − ρSB ρSD )/ 1 − ρ2SB with ρSB , ρSD , ρBD being pairwise instanta9

neous correlation coefficients, and where σB and σD are the volatilities of the artificial assets. These assets facilitate hedging against diffusion risks. Note that the excess expected returns on these artificial assets are allowed to depend on the perceived state of the economy, which intuitively is useful since hedging business cycle and salary risk may be more important in bad times than in good times. Finally, we introduce two assets allowing the investor to hedge against changes in employment status. Whenever employed the investor can trade an “unemployment insurance contract” with price Ut satisfying dUt = Ut− [(˜ r − νU ) dt + dNtu ] ,

(15)

and whenever unemployed the investor can trade an “employment insurance contract” with price Mt satisfying dMt = Mt− [(˜ r − νM ) dt + dNte ] .

(16)

As explained in Appendix B, νU and νM can be interpreted as the risk-neutral jump intensities in the artificial market. A given artificial market is characterized by the set of constant parameters
θ = νS , νx0 , νx1 , νy0 , νy1 , σB , σD , ρSB , ρSD , ρDB , νU , νM .

(17)

For future reference we let Mθ denote the artificial market corresponding to a given choice of θ. We focus on artificial markets with constant θ since we can then solve the utility maximization problem explicitly as shown below.3 Compared to the original SAMS approach of Bick, Kraft, and Munk (2013), we extend it to a jump-diffusion setting, and we also optimize over the correlations involving artificial assets, whereas Bick, Kraft, and Munk only consider a fixed correlation structure. In the artificial market Mθ the money market account and the stock provide at least as high returns as in the true market, the investor has access to additional assets, receives the same income from non-financial sources, and does not face borrowing and short-selling constraints. Letting J θ denote the indirect utility in the artificial market Mθ , we therefore have the inequality J(t, W, x, y; i) ≤ J θ (t, W, x, y; i) 3

(18)

We could also set up artificial markets in which θ is a stochastic process. The mathematical analysis of Karatzas, Lehoczky, Shreve, and Xu (1991), Cvitani´c and Karatzas (1992), and Cvitani´c, Schachermayer, and Wang (2001) shows that the consumption and investment strategy which is optimal in the worst of all artificial markets (corresponding to all well-behaved stochastic processes θ) is in fact identical to the optimal consumption and investment strategy in the true, constrained market. Here “worst” means the market leading to the lowest expected utility for the investor. However, it is generally not possible to determine the worst of all artificial markets and the corresponding strategy.

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and if we minimize the indirect utility J θ over some set of artificial markets represented by Θ, we obtain an upper bound ¯

J θ (t, W, x, y; i) = min J θ (t, W, x, y; i)

(19)

θ∈Θ

on the indirect utility obtainable in the true market. 4.3

Optimal decisions in the artificial markets Any artificial market Mθ is complete and therefore has a unique risk-neutral proba-

bility measure Qθ . In a complete market income is valued like the dividend stream of an asset so the human capital at any time t < T˜ is uniquely determined by θ

H θ (t, x, y; i) = EQ t

Z

T

 Ys e−˜r(s−t) ds ,

i ∈ {e, u}.

(20)

t

Note that the time t employment status i affects the probability distribution of future incomes and thus the human capital. In the current version of the paper we assume a zero income from non-financial sources in retirement, i.e. Yt = 0 for t ∈ [T˜, T ]. Proposition 1 In the artificial market Mθ the human capital is given by H θ (t, x, y; i) = (1 + αe )yLθ (t, x; i) + αu (t)F θ (t, x; i),

i ∈ {e, u},

(21)

where Lθ (t, x; e), Lθ (t, x; u) satisfy the coupled PDEs (37)–(38) and F θ (t, x; e), F θ (t, x; u) satisfy the coupled PDEs (39)–(40) stated in Appendix B. See Appendix B for proofs. The term (1 + αe )yLθ (t, x; i) captures the present value of future salaries net of the unemployment insurance premium (L refers to “labor”), whereas αu (t)F θ (t, x; i) represents the present value of future insurance benefits received when unemployed (F refers to “fired”). The following proposition presents the indirect utility and the optimal strategies in the artificial market Mθ . Proposition 2 In the artificial market Mθ the indirect utility function is J θ (t, W, x, y; i) =

1 Gθ (t, x; i)γ (W + H θ (t, x, y; i))1−γ , 1−γ

i ∈ {e, u},

(22)

where the functions Gθ (t, x; e) and Gθ (t, x; u) solve the coupled partial differential equa-

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tions (43)–(44). The optimal consumption rate is ct =

Wt + H θ (t, xt , yt ; It ) . Gθ (t, xt ; It )

(23)

The optimal portfolio weight of the stock is  Λ0 + Λ1 xt Gθx (t, xt ; It ) σx φS Wt + H θ (t, xt , yt ; It ) + θ πt = γσS Wt G (t, xt ; It ) σS θ θ σx φS Hx (t, xt , yt ; It ) + yt σy ψS Hy (t, xt , yt ; It ) , − σS Wt 

(24)

where Λ0 , Λ1 , φS , and ψS are constants defined in Appendix B. As shown in (22) the indirect utility function is separable as seen in other complete market models with income and CRRA utility. Optimal consumption is an age- and state-dependent share of total wealth (the sum of financial wealth and human capital). The optimal portfolio weight of the stock involves the speculative term (Λ0 + Λ1 x)/γσS and a term hedging against changes in investment opportunities (Gx /G)(σx φS /σS ). Due to the human capital these terms are scaled by (W + H)/W . Furthermore, the term (σx φS Hx + yσy ψS Hy )/σS W adjusts for the implicit investment in the stock that is incorporated in the human capital through the instantaneous income-stock correlation and the joint dependence of income and expected stock returns on the state variable x. The proposition does not state the optimal investments in the artificial assets as they are unimportant for the strategy to be followed in the true incomplete market. We do not have exact, closed-form solutions for the pairs of coupled PDEs solved by the L, F , and G functions present in the above propositions. However, Appendix C develops approximate closed-form solutions of the form θ

Z

T

L (t, x; i) =

eA

L (t,s;i)+B L (t,s;i)x+C L (t,s;i)x2

eA

F (t,s;i)+B F (t,s;i)x+C F (t,s;i)x2

eA

G (t,s;i)+B G (t,s;i)x+C G (t,s;i)x2

ds,

t θ

Z

T

F (t, x; i) =

ds,

t θ

Z

G (t, x; i) =

T

1

˜G (t,T ;i)+B ˜ G (t,T ;i)x+C ˜ G (t,T ;i)x2

ds + ε γ eA

,

t

where the A, B, and C functions satisfy the system of ordinary differential equations stated in (52)-(54) in Appendix C. This system is easily solved with standard numerical methods.

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4.4

Consumption and investment decisions in the true market The consumption-investment strategy given by (23)–(24) is derived in a specific, un-

constrained, and artificially completed market and thus generally not feasible in the true market. We transform the strategy into a feasible strategy following the ideas of Bick, Kraft, and Munk (2013). In particular, in the true market financial wealth must stay non-negative because the salary may drop rapidly towards zero and the investor cannot hedge against that event. Due to this liquidity constraint, future income has a smaller present value when the current financial wealth of the investor is small. We capture this by multiplying the human capital H θ (t, x, y; i) by a dampening factor D(Wt ) = 1 − exp{−χWt }, where χ is a positive constant to be determined. Furthermore, in respect of the investment constraint (10), we project the stock portfolio weight from the artificial market upon the interval [0, 1]. Hence, the feasibilized strategy derived from the artificial market Mθ is Wt + H θ (t, xt , yt ; It )D(Wt ) , Gθ (t, xt ; It )    Λ0 + Λ1 xt Gθx (t, xt ; It ) σx φS Wt + H θ (t, xt , yt ; It )D(Wt ) min 1; + θ γσS Wt G (t, xt ; It ) σS !  + σx φS Hxθ (t, xt , yt ; It ) + yt σy ψS Hyθ (t, xt , yt ; It ) − D(Wt ) , σS Wt cθt =

πtθ =

(25)

(26)

where a+ = max{0; a}. Any parameter vector θ (together with a dampening coefficient χ) thus defines a specific feasible strategy (cθ , π θ ). The expected utility of any such strategy can be evaluated by Monte Carlo simulation of the dynamics of the stock price, the state variable, the salary, and the employment status. Embedding this evaluation in an optimization routine we can find the best of all such feasible strategies, and this strategy ∗

∗

(cθ , π θ ) is our suggested near-optimal strategy. ¯

An upper bound J θ (t, W, x, y; i) on indirect utility can be computed by minimizing the indirect utility in (22) over some set of artificial markets. Let Jcθ∗ ,πθ∗ (t, W, x, y; i) denote ∗

∗

the expected utility in the true market when following the suggested strategy (cθ , π θ ). We define ` = `(t, W, x, y; i) by Jcθ∗ ,πθ∗ (t, W, x, y; i) =

 h i1−γ 1 ¯ ¯ Gθ (t, x; i)γ [1 − `] W + H θ (t, x, y; i) 1−γ ¯

= (1 − `)1−γ J θ (t, W, x, y; i)

13

so that ¯

`=1−

J θ (t, W, x, y; i) Jcθ∗ ,πθ∗ (t, W, x, y; i)

!

1 γ−1

.

(27)

We can interpret ` as a welfare loss bound, i.e., an upper bound on the fraction of total wealth that the individual is willing to give up to get access to the unknown optimal strat∗

∗

egy instead of following the strategy (cθ , π θ ). Again refer to Figure 1 for an illustration. The parameter vector θ in (17) has 12 elements which makes the minimizations and maximizations over θ computationally challenging. A careful investigation of the expressions (22)–(24) reveals that σB and σD only show up in ratios νxj /σB and νyj /σD so we can fix σB = σD = 1 when searching through all constants νxj , νyj . We split the optimization over the remaining 10 artificial parameters into two. For a given choice of correlations ρSB , ρSD , ρDB (and the dampening constant χ when maximizing over the feasibilized strategies), we optimize over the seven parameters νS , νx0 , νx1 , νy0 , νy1 , νU , νM . Afterwards we optimize over various choices of ρSB , ρSD , ρDB (and χ when relevant).

5

Model calibration We calibrate our model of macro quantities to U.S. data from January 1959 to July

2013. The (real) risk-free rate is set to r = 1% as is standard in the literature. We assume the investor applies the economy-wide unemployment rate and the growth rate of aggregate income as signals about the state of the economy. Data on the unemployment rate is downloaded from the homepage of The Federal Reserve Bank of St. Louis.4 The growth rate of aggregate income is derived from the aggregate national personal income (of both employed and unemployed individuals), which is downloaded from the homepage of The Bureau of Economic Analysis.5 Finally, to determine the stock returns, we use the real stock prices of the S&P500, which are downloaded from the homepage of Professor Robert Shiller.6 Figure 2 depicts the time series of the stock price index, the unemployment rate, and the aggregate income growth rate. [Figure 2 about here.] 4

See http://research.stlouisfed.org/fred2/series/UNRATE/downloaddata?cid=32447. The unemployment rate represents the number of unemployed as a percentage of the labor force. Labor force data are restricted to people 16 years of age and older, who currently reside in 1 of the 50 states or the District of Columbia, who do not reside in institutions (e.g., penal and mental facilities, homes for the aged), and who are not on active duty in the Armed Forces. 5 See http://www.bea.gov/iTable/iTable.cfm?ReqID=9&step=1#reqid=9&step=3&isuri=1&903=76. The personal income includes wages and salaries, supplements to wages and salaries, personal income from financial assets, as well as government social benefits. 6 See http://www.econ.yale.edu/~shiller/.

14

The stock price, the signals, and the unobservable state variable x follow a Gaussian system, so we use Kalman Filtering and maximum likelihood to obtain a time series of x and our parameter estimates. The estimation routine is explained in Appendix D, and the annualized parameter estimates are presented in Table 1. [Table 1 about here.] The negative value of µ1 confirms that the equity premium varies counter cyclically. With our benchmark parameter values the equity premium averages 2.86%, but grows to 3.39% [drops to 2.33%] when the unobservable state variable is one standard deviation below [above] its average of zero. The estimated unemployment rate varies around ω1 /κ1v ≈ 5.4%, and the average level increases [decreases] by −κ1x σX /κ1v ≈ 5.76% when the state variable is one standard deviation below [above] average. The aggregate income growth rate exhibits some mean reversion around an average level of ω2 /κ2v ≈ 0.6% and tends to increase with the state of the economy in line with intuition; a one standard deviation increase in the state adds κ2x σX /κ2v ≈ 0.2% to the expected income growth rate. The volatility estimate of 1.81% is in line with existing literature (e.g., Munk and Sørensen 2010). Table 1 further shows that the unemployment rate and the aggregate income growth rate have very different mean reversion speeds. While the unemployment rate is rather persistent, changes in the income growth rate have a half life of less than a month. The correlation of the business cycle variable with aggregate income growth is close to zero, whereas its correlation with the unemployment rate is around 0.4. While the positive correlation may seem surprising, note that the volatility of the unemployment rate is very small and that large values of x lower its drift and thus the unemployment rate. The impact of the business cycle on the unemployment rate is mainly through its impact on the drift, not via contemporaneous changes. Next, we turn to the investor-specific parameters. Table 2 lists the benchmark values of these parameters that we assume when generating the results presented in the subsequent section. We assume that the individual salary level has zero correlation with the other variables. According to empirical studies, the correlation between labor income and the stock market index is indeed near zero for most individuals.7 In the existing literature, individual income is typically modeled as a pure diffusion with a volatility of 10-20% (cf., e.g., Munk and Sørensen 2010; Cocco, Gomes, and Maenhout 2005). In our setting, most 7

For example, Davis and Willen (2000) report that—depending on the individual’s sex, age, and educational level—the correlation between aggregate stock market returns and labor income shocks is between -0.25 and 0.3, while the correlation between industry-specific stock returns and labor income shocks is between -0.4 and 0.1. Heaton and Lucas (2000) find that the labor income of entrepreneurs typically is more highly correlated with the overall stock market (0.14) than with the labor income of ordinary wage earners (-0.07).

15

of the variability of personal income stems from unemployment risk, and not from the volatility of the salary level. We set the volatility of the individual salary level equal to the volatility of aggregate income, σy = 0.02, and comment on unemployment risk and total income risk below. Furthermore, we assume that the individual salary growth rate has the same state dependence as the aggregate income growth and assume that the state dependence of the potential salary when unemployed equals the state dependence of the actual salary when employed, i.e., ξ1e = ξ1u = 0.22. For now we ignore the age pattern in income growth (as documented by, e.g., Cocco, Gomes, and Maenhout 2005) and assume that the average income growth rate when employed equals 1% per year (equal to the risk-free interest rate), whereas the potential salary that an unemployed would receive when becoming reemployed drops by 10% per year, i.e., ξ0e = 0.01 and ξ0u = −0.1. [Table 2 about here.] We assume that the jump intensities are exponential-affine functions of the state variable x, ηe = eη0,e +η1,e x ,

ηu = eη0,u +η1,u x ,

(28)

which ensures positive intensities for all values of x. To calibrate the intensities, we rely on the unemployment rate again as well as the duration of unemployment for which we downloaded data from the homepage of The Federal Reserve Bank of St. Louis.8 It follows that, under the stationary distribution, the probability of an individual being unemployed is ηu /(ηe + ηu ). The time to find a job again follows an exponential distribution, and the median duration of unemployment equals (ln 2)/ηe . Given the time series of the unemployment rate and the median unemployment duration, we obtain a time series of values of ηe and ηu . By matching the means and volatilities, we find the values for the intensity parameters listed in Table 2. With these values the average intensity is e−η0,u ≈ 0.3 for a jump into unemployment and eη0,e ≈ 4.66 for reemployment jump, corresponding to an average unemployment duration of around 2.5 months. Consistent with intuition the estimated intensity of the unemployment jump is decreasing in the state, whereas the intensity of the reemployment jump is increasing in the state. When the state is one standard deviation above [below] its zero average, the unemployment intensity is approximately 0.23 [0.40], and the reemployment intensity is approximately 6.73 [3.23]. We assume that the consumer-investor pays αe = 20% of the salary to an unemployment insurance when being employed. We assume that the unemployment benefit received 8

See http://research.stlouisfed.org/fred2/series/UEMPMED/downloaddata?cid=12. The duration of unemployment represents the median weeks of unemployment.

16

starts at 40% of the initial salary level and then grows at the risk-free rate, αu (t) = 0.4 y0 ert . The possible unemployment with a significant drop in income contributes significantly to the income risk of the individual. A back-of-the-envelope computation of the instantaneous variance rate of the income at time 0 is to add the diffusion variance rate of (0.02)2 to the product of the unemployment intensity of 0.3 (assuming x0 = 0) and the squared income drop of (0.6)2 , leading to a total variance rate of 0.1084 per year or, equivalently, √ an income volatility of 0.1084 ≈ 0.33. A sizable share of this risk is transitory since a fairly swift reemployment of the individual is likely. However, a prolonged unemployment period is also possible and would have an enduring backlash on income through the drop in the attainable salary when becoming reemployed. Finally, we assume that our benchmark investor is employed at t = 0, works for 20 years, and is retired for 10 years before he dies. The benchmark length of both the active phase and the retirement phase will be increased in a later version—some results with longer periods are shown below. The investor’s initial salary level equals his initial financial wealth, in particular we assume y0 = W0 = 1. The investor is assumed to have a relative risk aversion of γ = 6, a time preference rate of δ = 0.02, and applies a relative weight of ε = 1 to terminal wealth.

6

Results We investigate in the following how the optimal consumption rate and the optimal

share of wealth invested in the stock vary over the life cycle of an individual. As derived in Section 4, the optimal consumption rate and portfolio weight at a given point in time depend on the investor’s financial wealth, employment status, salary, and age, as well as the overall state of the economy. While we can easily compute the optimal decisions for any given combination of these variables from the optimization of (25) and (26), we must acknowledge that the ranges of typical values for wealth and salary vary over life. We simulate 10,000 paths of the dynamics of wealth, salary, employment status, and the macro state variable using the optimal strategies. We take the average over these paths as a representative life-cycle profile in consumption, investment, and wealth. Unless otherwise mentioned we employ the parameter values listed in Tables 1 and 2 and assume that the initial state of the economy is x0 = 0.

17

6.1

Consumption and investment conditional on the state Table 3 illustrates how the portfolio share of the stock and the consumption rate

depend on the employment status of the investor and the macro-economic state variable at various ages of the investor. The values shown for time 0 are computed using the initial values of salary and financial wealth y0 = W0 = 1, whereas the values for later dates are for the average salary and wealth based on the simulations starting at time 0. Quite naturally the consumption rate increases with the macro state, and consumption is higher when employed than when unemployed. The portfolio share of the stock is much lower in good macro states than in bad, which can be explained by the counter-cyclical variation in expected stock returns. The stock weight is considerably smaller for the unemployed than the employed investor in cases where the upper bound of 1 is not binding. This can be explained by the drop in human capital when becoming unemployed. Note that the impact of the employment status diminishes with the age of the investor. [Table 3 about here.] 6.2

Life-cycle results for benchmark parameter values The life-cycle patterns of wealth, consumption, and the portfolio share of the stock

generated by our full model are shown by the blue curves in Figures 3 and 4. These patterns are based on the average paths of relevant variables, cf. the explanations above. As in similar models, the investor accumulates financial wealth while being active in the labor market in order to finance consumption in retirement. [Figure 3 about here.] [Figure 4 about here.] Consumption tends to increase over life but is fairly smooth. Interestingly, our model produces a hump in consumption some years before retirement. Standard life-cycle models of the Merton-Samuelson type (also with labor income) generally lead to an increasing consumption pattern over the entire life both because precautionary savings diminish over life and because the return on investments exceeds the time preference rate. Empirical studies have documented that the consumption of individuals typically increases up to age 45-50 years and then decreases over the remaining life time (see, e.g., Thurow 1969; Gourinchas and Parker 2002). Cocco, Gomes, and Maenhout (2005) obtain a consumption hump exactly at retirement presumably resulting from a combination of increasing mortality risk (and thus an increasing effective time preference rate) and borrowing constraints

18

(limiting consumption smoothing over time). Our model produces a consumption hump before retirement, which is more in line with empirical findings. In our benchmark case the investor is initially fully invested in stocks, but then the stock weight is gradually reduced until retirement. This behavior is explained by the decline in human capital over life. Human capital increases the desired speculative investment in the stock which dominates any negative income-induced hedging demand for the stock as long as the salary-stock correlation is small. Therefore, human capital leads to an increase in the stock weight compared to the case without income, but this increase declines over time along with human capital. Similar results are reported by, e.g., Cocco, Gomes, and Maenhout (2005). A different life-cycle pattern in investments may arise for other parameter values, cf. the comparative statics discussed below. By comparing the different curves in Figures 3 and 4, we can see the impact of unemployment risk and state-dependent income on the wealth, consumption, and stock weight. Take as starting point the model without unemployment risk and no state dependence of income growth which is represented by the red curves. By adding unemployment risk, we increase the total income risk and also lowers the expected future income and thus human capital. As a consequence, the investor accumulates less financial wealth and reduces the consumption rate significantly, and thus increases the savings rate, but the stock’s share of the investments is roughly unchanged (green curves). Further adding state dependence of both the unemployment intensity and the expected salary growth rate (blue curves) further lowers wealth and consumption though only very little, but reduces the stock weight in the portfolio significantly. Figure 5 disentangles the effects by separately turning off the state dependence of the unemployment intensity and of the salary growth rate. The stock is highly positively correlated with the macro state and, hence, a positive relation between the salary growth rate and the state implies that the human capital is more stock-like. Consequently, the investor should invest less of the financial wealth in the stock. If the unemployment intensity is higher in bad macro states, the investor cuts back stock investments to reduce the risk of a simultaneous unemployment shock and severe drop in the value of financial asset holdings. [Figure 5 about here.] 6.3

Comparative statics First, we consider the impact of extending the number of years until retirement from 20

to 35 years and the number of years in retirement from 10 to 15 years. With the longer time horizons, Figure 6 shows that unemployment risk motivates the investor to save and build 19

up more wealth in the early years in the work force. Eventually, the wealth of the investor facing unemployment risk falls below the wealth of the investor with no unemployment risk mainly due to the lower income caused by periods of unemployment. Figure 7 depicts the optimal consumption and portfolio share of the stock with the extended time horizons, and we see that same general patterns as in the benchmark case with shorter horizons, compare Figure 4. [Figure 6 about here.] [Figure 7 about here.] Figure 8 investigates the role of the risk aversion parameter γ. Not surprisingly, a higher risk aversion leads to a lower portfolio weight of the stock. In our setting the investor with a relative risk aversion of 10 starts off by investing all savings in the risk-free rate and then gradually includes more and more stocks in the portfolio at least up to a few years before retirement. With a high risk aversion, the increase in speculative stock demand caused by human capital is more than offset by negative hedge terms, so that the total effect of human capital is to lower the stock investment. As human capital becomes smaller over life, the associated reduction in the stock weight declines so that the stock weight increases. The lower stock investment of the highly risk-averse investor generates low average wealth which also leads to lower average consumption throughout life as shown in Panel A of Figure 8. [Figure 8 about here.] The impact of the level of unemployment risk on life-cycle consumption and investment is illustrated by Figure 9. Increasing the parameter ηu0 and thus the unemployment intensity leads to a higher total income risk and lower expected future income, so the investor decreases his consumption and shifts some savings from stocks to the risk-free asset, at least early in life. [Figure 9 about here.] Figure 10 studies the importance that the degree of state dependence of the unemployment risk—captured by the parameter ηu1 in our model—has on life-cycle consumption and investments. The benchmark value of ηu1 = −2.34 corresponds to a modestly countercyclical unemployment risk and produces the blue curves in the figure. The red curves multiplies the benchmark value by 5 and thus represent a strongly counter-cyclical unemployment risk. Then severe simultaneous drops in income and returns on stock investments are more likely, which lowers the value of the income stream and thus consumption 20

(slightly) and induces the investor to reduce the portfolio share of stocks. Conversely, the green curves assume a modestly pro-cyclical unemployment risk (i.e., higher unemployment risk in good times than in bad), which could be appropriate for a minority of individuals. Compared to the benchmark, this leads to a slightly higher human capital and consumption as well as a marginally higher stock weight in the portfolio at least in the early years. [Figure 10 about here.] A distinct feature of our model is that the length of the unemployment period affects the obtainable salary when becoming reemployed through the parameter ξ0u . Our benchmark assumption is a 10% drop per year of unemployment. Figure 11 investigates the implications for life-cycle consumption and portfolio decisions. The red curve assumes that the unemployment has no adverse effect on reemployment salary, which leads to a higher expected future income and thus higher human capital than in the benchmark case. Consequently, the investor consumes more and invests a higher share of wealth in stocks. Conversely, when we assume a 20% drop in salary per year of unemployment. [Figure 11 about here.]

7

Conclusion This paper shows that both unemployment risk in general and the business-cycle depen-

dence of expected salary growth and un- and reemployment probabilities have significant effects on individuals’ consumption and investments over the life cycle. The risk of unemployment and the consequences of unemployment for human capital lower consumption throughout life and reduces the portfolio share of the stock significantly. Individuals who are highly risk averse or receive a salary positively correlated with the stock market do optimally not even participate in the stock market in early years as observed for many real-life investors. In our model the life-cycle pattern in consumption exhibits a hump several years before retirement, which resembles the humped pattern seen in consumption data better than the consumption patterns generated by the theoretical models closely related to our where the hump occurs at retirement or not at all. When comparing the optimal decisions of an employed individual and an unemployed individual (other things equal), we find significant differences. The unemployed individual consumes less and invests a much smaller share of savings in the stock market than the employed individual.

21

A

Filtering To get the filtered model as stated in equation (7)-(9) we rely on Theorem 12.7 in

Liptser and Shiryaev (2001). First, we rewrite the system of the unobservable and the observable state variables. The dynamics of the unobservable state variable is written as dXt = −κx Xt dt + Σ0X dZt where ΣX = (σX , 0, 0, 0)0 , whereas the system of observable state variables is written as 

dSt /St

  

dv1t dv2t







r + µ0



µ1







Σ0S



         =  ω1 − κv v1t  +  κx  Xt  dt +  Σ0  dZt , 1  v1    1     Σ0v2 κx2 ω2 − κv2 v2t

q  0 2 , 0, 0 , and Σ = (Σ , Σ )0 is a (4 × 2)-dimensional where ΣS = σS kSX , σS 1 − kSX v2 v v1 matrix. Σv is chosen such that the volatility of the first signal v1 equals σv1 , the volatility of the second signal v2 equals σv2 , and the correlation between the two signals equals kv : Σ0v Σv

=

σv21

σv1 σv2 kv

σv1 σv2 kv

σv22

! .

Furthermore, the correlation between X and v1 , and X and v2 equal kXv1 and kXv2 , respectively, i.e., Σ0X Σv = (kXv1 σx σv1 , kXv2 σx σv2 ). From Theorem 12.7 in Liptser and Shiryaev (2001), it now follows that the filtered model (as seen by the investor) is given by the system ˆ 0S dzt + Kv0 Σ ˆ 0v dzt , dxt = −κx xt dt + KS Σ h i ˆ 0 dzt , dSt = St (r + µ0 + µ1 xt ) dt + Σ

(29)

ˆ 0 dzt , dv1t = (ω1 − κv1 v1t + κx1 xt ) dt + Σ v1

(31)

ˆ 0 dzt , dv2t = (ω1 − κv2 v2t + κx2 xt ) dt + Σ v2

(32)

S

(30)

where z is a four-dimensional standard Brownian motion, describing the perceived innovations to the stock price, the two signals, and the salary level. That is, the investor relies on ˆ 0v dzt and Σ ˆ 0 dzt in the signals and the stock price to learn about the perceived innovations Σ S

ˆ v is a (4 × 2)-dimensional matrix, and Σ ˆ S is a four-dimensional vector, which we X. Σ define below. The constant KS and the constant two-dimensional vector Kv = (Kv1 , Kv2 )0

22

are given by


−1 KS Kv0 = b B 0 + k A01 B B 0 , where, following notation in Liptser and Shiryaev (2001), we have that b = Σ0X ,


B 0 = Σ0S , Σ0v1 , Σ0v2 ,

A01 = (µ1 , κx1 , κx2 ) .

Furthermore, k is the steady-state variance of the estimation error and is the positive solution to the equation k2 k 2 + k1 k + k0 = 0, where
0 b B 0 − b b0 ,

−1 k1 = 2κx + 2 b B 0 B B 0 A1 ,
−1 A1 . k2 = A01 B B 0

k0 = b B 0




B B0


−1

The variance of the estimation error is generally a deterministic function of time but, for simplicity, we assume that it has already converged to its long-run level.9 The dynamics of the investor’s salary level in the filtered model equals h i 0 ˆ dyt = yt (ξ0 (t, It ) + ξ1 (It )xt ) dt + Σy dzt

(33)

ˆ y is a four-dimensional vector. Σ ˆ v, Σ ˆ S , and Σ ˆ y follow from the fact that filtering where Σ does not change the variance-covariance matrix of the observed variables v, S, and y, i.e., ˆS, Σ ˆ v, Σ ˆ y )0 (Σ ˆS, Σ ˆ v, Σ ˆ y ) = (ΣS , Σv , Σy )0 (ΣS , Σv , Σy ). (Σ The system (7)-(9) now follows directly from rewriting the above system of equations, and noting that ρSx =

ˆ0 Σ ˆ Σ x S , σS σx

ρxy =

ˆ0 Σ ˆ Σ x y , σx σy

ρSy = kSy ,

(34)

2 ˆ 0 = KS Σ ˆ 0 + K0 Σ ˆ0 ˆ0 ˆ where Σ x v v and σx = Σx Σx . S 9

The same assumption has been made by Scheinkman and Xiong (2003) and Dumas, Kurshev, and Uppal (2009), among others.

23

B

Proofs In this appendix we consider a given artificial market as represented by a fixed param-

eter set θ, cf. (17). For simplicity we notationally suppress θ. B.1

Setting and notation

In an artificial market the invest has access to a risk-free asset with a constant rate of return r˜ and three financial assets with prices Pt = (St , Bt , Dt )> having dynamics dPt = diag(Pt ) [(˜ r1 + µ(xt )) + Σ dzt ] , where z is a three-dimensional standard Brownian motion (with independent components), and 







σS 0 0 q     2 0 Σ =  σB ρSB σB 1 − ρSB  q   σD ρSD σD ρˆBD σD 1 − ρ2SD − ρˆ2BD

µ ˜ 0 + µ1 x    µ(x) =  νx0 + νx1 x , νy0 + νy1 x with ρˆBD = (ρBD − ρSB ρSD )/

q 1 − ρ2SB .

In addition, the investor has access to unemployment and employment insurance contracts. When employed he can invest in a contract with price Ut satisfying dUt = Ut− [(˜ r − νU ) dt + dNtu ] , and when unemployed he can invest in a contract with price Mt satisfying dMt = Mt− [(˜ r − νM ) dt + dNte ] . The dynamics of the state variables are dxt = −κx xt dt + Σ> x dzt ,   dyt = yt (ξ0 (t, It ) + ξ1 (It )xt ) dt + Σ> y dzt , where  > q 2 Σx = σx ρSx , 1 − ρSx , 0 ,

q  > Σy = σy ρSy , ρˆxy , 1 − ρ2Sy − ρˆ2xy

24

and ρˆxy = (ρxy − ρSx ρSy )/ 

q 1 − ρ2Sx . For future use, we note that

1 σS − √ρSB 2 σS 1−ρ √SB ρSB ρˆBD −ρSD 1−ρ2SB

  Σ−1 =  

σB

0

0



√1

0

  , 

1−ρ2SB

BD √ 2 √ 1 − √ 2 ρˆ√ 1−ρSD −ˆ ρ2BD σB 1−ρSB 1−ρ2SD −ˆ ρ2BD σD 1−ρ2SD −ˆ ρ2BD   λSS λSB λSD (˜ µ + µ x) + (ν + ν x) + (ν + ν x) 0 1 x0 x1 y0 y1 2 σS σB σS σD  σS   λSB  λBD λBB > −1 µ0 + µ1 x) + σ2 (νx0 + νx1 x) + σB σD (νy0 + νy1 x)  , (ΣΣ ) µ(x) =  σB σS (˜ B   λBD λDD λSD (ν + ν x) (˜ µ + µ x) + (ν + ν x) + y0 y1 0 1 x0 x1 2 σD σS σD σB σD     −1 −1 Σ> = σσx φS σσx φx 0 , Σ> = σσy ψS σσy ψx σσy ψy , xΣ yΣ S B S B D

σS

√

1−ρ2SB

where λSS =

1 − ρ2BD , (1 − ρ2SB )(1 − ρ2SD − ρˆ2BD )

λSB = −

ρSB − ρSD ρBD , (1 − ρ2SB )(1 − ρ2SD − ρˆ2BD )

1 − ρ2SD ρSD − ρSB ρBD , λ = , BB (1 − ρ2SB )(1 − ρ2SD − ρˆ2BD ) (1 − ρ2SB )(1 − ρ2SD − ρˆ2BD ) 1 ρBD − ρSB ρSD , λDD = , λBD = − 2 2 2 2 (1 − ρSB )(1 − ρSD − ρˆBD ) 1 − ρSD − ρˆ2BD q 1 − ρ2Sx φx = q , φS = ρSx − ρSB φx , 1 − ρ2SB q 1 − ρ2Sy − ρˆ2xy ρˆxy − ρˆBD ψy ψy = q , ψx = q , ψS = ρSy − ρSB ψx − ρSD ψy . 1 − ρ2SD − ρˆ2BD 1 − ρ2SB λSD = −

B.2

Proof of Proposition 1

The artificial market is complete so a unique risk-neutral measure Q exists and the human capital can be valued as if the income was a dividend stream. We write the human capital at time t as H e (t, x, y) if employed and as H u (t, x, y) if unemployed, where H i (t, x, y) = EQ

"Z

T˜

# Ys e−˜r(s−t) ds | xt = x, yt = y, It = i ,

i ∈ {e, u}.

t

We need the risk-neutral dynamics of x and y. The market price of (diffusion) risk in the market is Σ−1 µ(xt ) so ztQ

t

Z = zt +

Σ−1 µ(xs ) ds

0

25

defines a standard Brownian motion under Q. Therefore
Q −1 > dxt = −κx xt − Σ> x Σ µ(xt ) dt + Σx dzt , h i
Q −1 > dyt = yt ξ0 (t, It ) + ξ1 (It )xt − Σ> y Σ µ(xt ) dt + Σy dzt , We also need to identify the intensities of the jumps in employment status under Q. The ¯tu = dNtu − ηu dt. Let compensated (zero-expectation) version of the jump dNtu is dN ηuQ denote the Q-intensity of N u . Then the change of measure corresponds to replacing
¯ u by dN ¯ u,Q = dN ¯ u − η Q − ηu dt. Consequently, the price dynamics (15) of the dN t t u t unemployment insurance contract can be rewritten as dUt = Ut− [(˜ r − νU ) dt + dNtu ]   ¯tu = Ut− (˜ r − νU + ηu ) dt + dN h    i ¯ ν,u + ηuQ − ηu dt = Ut− (˜ r − νU + ηu ) dt + dN t i h  ¯ u,Q . = Ut− r˜ − νU + ηuQ dt + dN t Under the risk-neutral measure the drift equals the risk-free rate so we see that νU = ηuQ . Similarly, νM = ηeQ for the employment derivative. Hence, νU and νM are the jump intensities under Q. Recalling the specification of income in (3), it follows from standard asset pricing theory that H e and H u solve the coupled PDEs
e
−1 e e > −1 e r˜H e = Hte + −κx x − Σ> x Σ µ(x) Hx + ξ0 (t) + ξ1 x − Σy Σ µ(x) yHy 1 1 e e e u e + σx2 Hxx + σy2 y 2 Hyy + Σ> x Σy yHxy + (H − H ) νU + y(1 + αe ), 2 2
u
−1 u u > −1 u r˜H u = Htu + −κx x − Σ> x Σ µ(x) Hx + ξ0 (t) + ξ1 x − Σy Σ µ(x) yHy 1 1 u u u e u + σx2 Hxx + σy2 y 2 Hyy + Σ> x Σy yHxy + (H − H ) νM + αu (t) 2 2

(35)

(36)

with terminal values H e (T˜, x, y) = H u (T˜, x, y) = 0. We can split H i (t, x, y) into two terms since H i (t, xt , yt ) =

EQ t

"Z t

T˜

# Ys e−˜r(s−t) 1{Is =e} ds +

= (1 + αe )yt EQ t

"Z t

T˜

EQ t

"Z t

T˜

# Ys e−˜r(s−t) 1{Is =u} ds

# "Z ˜ # T ys −˜r(s−t) α (s) u e 1{Is =e} ds + αu (t) EQ e−˜r(s−t) 1{Is =u} ds t yt αu (t) t

= (1 + αe )yt Li (t, xt ) + αu (t)F i (t, xt ).

26

We substitute this into (35)–(36) and collect terms involving y and the remaining terms. We find that Le , Lu satisfy the coupled PDEs
e −1 e 0 = − r˜ − ξ0e (t) − ξ1e x + Σ> y Σ µ(x) L + Lt
e 1 2 e −1 > u e + −κx x − Σ> x Σ µ(x) + Σx Σy Lx + σx Lxx + (L − L ) νU + 1, 2
u −1 u 0 = − r˜ − ξ0u (t) − ξ1u x + Σ> y Σ µ(x) L + Lt
u 1 2 e −1 > e u + −κx x − Σ> x Σ µ(x) + Σx Σy Lx + σx Lxx + (L − L ) νM , 2

(37)

(38)

with the terminal values Le (T˜, x) = Lu (T˜, x) = 0, and F e , F u satisfy the coupled PDEs 
e αu0 (t) −1 F e + Fte + −κx x − Σ> 0 = − r˜ − x Σ µ(x) Fx αu (t) 1 2 e + σx Fxx + (F u − F e ) νU , 2  
u αu0 (t) −1 F u + Ftu + −κx x − Σ> 0 = − r˜ − x Σ µ(x) Fx αu (t) 1 2 u + σx Fxx + (F e − F u ) νM + 1, 2 

(39)

(40)

with the terminal values F e (T˜, x) = F u (T˜, x) = 0. Note that here Σ> x Σy = ρxy σx σy , 

   φS µ ˜0 φx νx0 φS µ1 φx νx1 + + Σx Σ µ(x) = σx + σx x, σS σB σS σB     ψS µ1 ψx νx1 ψy νy1 ψS µ ˜0 ψx νx0 ψy νy0 > −1 + + σy + + + σy x. Σy Σ µ(x) = σS σB σD σS σB σD >

B.3

−1

Proof of Proposition 2

B.3.1

The HJB equation

The fractions of financial wealth invested in these three risky assets are gathered in the vector Πt = (πt , πBt , πDt )> . When employed (unemployed) a fraction πU t (πM t ) of financial wealth is invested in the contract with price Ut (Mt ). The wealth dynamics when employed is thus h i > u dWt = Wt (˜ r + Π> µ(x ) − π ν ) dt + Π Σ dz + π dN t t Ut U Ut t t t − ct dt + (1 + αe )yt dt, and when unemployed it is h i > e dWt = Wt (˜ r + Π> µ(x ) − π ν ) dt + Π Σ dz + π dN t t Mt M Mt t t t − ct dt + αu (t) dt.

27

We let J e (t, W, x, y) and J u (t, W, x, y) denote the indirect utility function in the artificial market when the investor is employed and unemployed, respectively. It follows from standard dynamic optimization results that the Hamilton-Jacobi-Bellman (HJB) equation for J e is δJ e = L1 J e + L2 J e + L3 J e + L4 J e ,

(41)

where L1 J e = sup



c

1 1−γ e c − cJW 1−γ

 ,

e L2 J e = sup {J u (t, W [1 + πU ], x, y)ηU − πU JW (t, W, x, y)W νU } , πU   1 2 e e > > > e > e > e L3 J = sup JW W Π µ(x) + W JW W Π ΣΣ Π + JW x W Π ΣΣx + yJW y W Π ΣΣy , 2 Π e L4 J e = Jte + r˜W JW 1 e 2 + Jxx σx + 2

e + (1 + αe )yJW − κx xJxe + yJye (ξ0e + ξ1e x) 1 2 e 2 e e y Jyy σy + yJxy Σ> x Σy − η U J . 2

In the corresponding HJB equation for J u the terms L1 J u and L3 J u are unchanged, L2 J e is replaced by u (t, W, x, y)W νM } , L¯2 J u = sup {J e (t, W [1 + πM ], x, y)ηe − πM JW πM

and L4 J e is replaced by u u L¯4 J u = Jtu + r˜W JW + αu (t)JW − κx xJxu + yJyu (ξ0u + ξ1u x) 1 u 2 1 2 u 2 u > + Jxx σx + y Jyy σy + yJxy Σx Σy − ηM J u . 2 2

The terminal conditions are J e (T, W, x, y) = J u (T, W, x, y) =

ε 1−γ . 1−γ W

We conjecture a solution of the form J i (t, W, x, y) =


1−γ 1 Gi (t, x)γ W + H i (t, x, y) , 1−γ

28

i ∈ {e, u}.

It turns out to be useful to express the relevant derivatives in terms of J itself: (1 − γ)J i i , JW W W + Hi   γ Gix Hxi i i i Jx = (1 − γ)J , JW + x 1 − γ Gi W + Hi Hyi i i i Jy = (1 − γ)J , JW y W + Hi   γ Git Hti i i i Jt = (1 − γ)J , Jyy + 1 − γ Gi W + H i

i JW =

i Jxx i Jxy

B.3.2

γ(1 − γ)J i , (W + H i )2  i  1 Hxi i Gx = γ(1 − γ)J , − Gi W + H i (W + H i )2 Hyi i = −γ(1 − γ)J , (W + H i )2 " # i i )2 H γ(H yy y = (1 − γ)J i − , (W + H i ) (W + H i )2 =−

 i γ Gixx (Gix )2 Gix Hxi (Hxi )2 Hxx , = (1 − γ)J − γ i 2 + 2γ i −γ + 1 − γ Gi (G ) G W + Hi (W + H i )2 W + H i " # i Hxi Hyi Hxy Gix Hyi i = (1 − γ)J γ i −γ + . G W + Hi (W + H i )2 W + H i i



Computation of L1 J i

i )−1/γ which again implies that L J e = The first-order condition for c leads to c = (JW 1 γ i 1−1/γ . 1−γ (JW )

With our conjecture for J i we get W + Hi , Gi L1 J i = γJ i (Gi )−1 . c=

B.3.3

Computation of L2 J e and L¯2 J u

The first-order condition for πU leads to u e JW (t, W [1 + πU ], x, y)ηU = JW (t, W, x, y)νU .

With our conjecture for J e and J u , this leads to Gu (t, x) (W + H e (t, x, y)) W [1 + πU ] + H (t, x, y) = e G (t, x) u

and thus Gu (t, x) W + H e (t, x, y) πU = e G (t, x) W



29

νU ηU

− 1 γ

−



νU ηU

− 1

W + H u (t, x, y) . W

γ

By use of the preceding two expressions, we get 1 W νU πU (Gu )γ (W [1 + πU ] + H u )1−γ ηU − (1 − γ)J e 1−γ W + He  1− 1 u 1−γ γ 1 νU u γ (G ) e 1−γ = ηU (G ) (W + H ) 1−γ (Ge )1−γ ηU " #  − 1 u W + He u γ W G ν W + H U − (1 − γ)J e νU − W + He Ge W ηU W  − 1   1 γ νU Gu Gu νU − γ W + Hu νU = e Je νU − (1 − γ)J e e νU + (1 − γ)J e G ηU G ηU W + He ( )  − 1 u u γ γ G ν W + H U = (1 − γ)J e νU + νU 1 − γ Ge ηU W + He ( )   1 γ Gu νU − γ Hu − He e = (1 − γ)J νU + νU + νU . 1 − γ Ge ηU W + He

L2 J e =

By symmetry, we have

πM

Ge (t, x) W + H u (t, x, y) = u G (t, x) W

and

( L¯2 J = (1 − γ)J u

B.3.4

u

γ Ge 1 − γ Gu



νM ηM



νM ηM

− 1 γ

−

− 1 γ

νM

W + H e (t, x, y) W

He − Hu + νM + νM W + Hu

) .

Computation of L3 J i

The first-order condition for Π leads to Π=−

i i i yJW JW JW y > −1 > −1 x (ΣΣ ) µ(x) − (Σ ) Σ − (Σ> )−1 Σy x i i i W JW W J W J W WW WW

and after lengthy computations we get L3 J i = − −

2 i 2 i )2 i )2 1 (JW 1 y (JW y ) 2 1 (JW > > −1 2 x µ(x) (ΣΣ ) µ(x) − σ − σy x i i i 2 JW 2 JW 2 JW W W W i Ji i Ji i Ji yJW yJW JW Wy x Wy > > > −1 > > −1 Wx µ(x) (Σ ) Σ − µ(x) (Σ ) Σ − Σx Σy . x y i i i JW J J W WW WW

Applying our conjecture for J i we arrive at 1 W + Hi −1 (ΣΣ> ) µ(x) + Π= γ W



Gix W + H i Hxi − Gi W W

30



(Σ> )−1 Σx −

yHyi > −1 (Σ ) Σy (42) W

and (

  1 (Hxi )2 Gix Hxi γ 2 (Gix )2 > > −1 L3 J = (1 − γ)J + −2 i µ(x) (ΣΣ ) µ(x) + σx 2γ 2 (Gi )2 (W + H i )2 G W + Hi   i γ 2 y 2 (Hyi )2 Gx Hxi µ(x)> (Σ> )−1 Σx + σy + − 2 (W + H i )2 Gi W + Hi ) ! i i Hi i yHyi yH yH G y x y x − µ(x)> (Σ> )−1 Σy − γ − Σ> x Σy . W + Hi Gi W + H i (W + H i )2 i

B.3.5

i

Computation of L4 J e and L¯4 J u

With our conjecture for J e we get ( L4 J e = (1 − γ)J e

yHye γ Get Hte r˜H e (1 + αe )y + + r ˜ − + + (ξ e + ξ1e x) 1 − γ Ge W + H e W + He W + He W + He 0 !   e y 2 Hyy y 2 (Hye )2 1 2 γ Gex Hxe −γ − κx x + + σy 2 W + He (W + H e )2 1 − γ Ge W + He   e γ Gexx (Gex )2 Gex Hxe (Hxe )2 Hxx 1 2 −γ + 2γ e −γ + + σx 2 1 − γ Ge Ge G W + He (W + H e )2 W + H e )  e  e yHxe Hye yHxy Gx yHye ηU > + Σx Σy γ e −γ + − . G W + He (W + H e )2 W + H e 1−γ

Adding up L3 J e and L4 J e numerous terms cancel, and we are left with ( e

e

L3 J + L4 J = (1 − γ)J

e

 e  1 Gx Hxe > > −1 µ(x) (ΣΣ ) µ(x) + − µ(x)> (Σ> )−1 Σx 2γ Ge W + He  
yHye γ Gex Hxe − κx x + + ξ0e + ξ1e x − µ(x)> (Σ> )−1 Σy e e e 1−γ G W +H W +H e γ Get Hte r˜H e (1 + αe )y 1 2 y 2 Hyy + + r ˜ − + + σ 1 − γ Ge W + H e W + He W + He 2 y W + He )   e e yH 1 2 γ Gexx Hxx η xy U + σx + + Σ> − . x Σy 2 1 − γ Ge W + He W + He 1 − γ

+

31

Similarly, we get ( L3 J + L¯4 J = (1 − γ)J u

u

u

  u Gx Hxu 1 −1 µ(x)> (Σ> )−1 Σx µ(x)> (ΣΣ> ) µ(x) + − 2γ Gu W + H u  
yHyu γ Gux Hxu − κx x + + ξ0u + ξ1u x − µ(x)> (Σ> )−1 Σy u u u 1−γG W +H W +H u γ Gut Htu r˜H u αu (t) 1 2 y 2 Hyy + + r ˜ − + + σ 1 − γ Gu W + H u W + Hu W + Hu 2 y W + Hu )   u u yHxy Hxx ηM 1 2 γ Guxx > + + Σ x Σy − . + σx 2 1 − γ Gu W + Hu W + Hu 1 − γ

+

B.3.6

Implications for Gi and H i

Next, we add up L1 J e , L2 J e , and L3 J e + L4 J e . After dividing through by (1 − γ)J e , we arrive at   1 δ γ γ Gu νU − γ Hu − He e −1 = (G ) + ν + νU + νU U 1−γ 1−γ 1 − γ Ge η U W + He  e  1 Gx Hxe > > −1 + µ(x) (ΣΣ ) µ(x) + − µ(x)> (Σ> )−1 Σx 2γ Ge W + He  
yHye Hxe γ Gex e e > > −1 + + ξ + ξ x − µ(x) (Σ ) Σ − κx x y 0 1 1 − γ Ge W + He W + He e γ Get Hte r˜H e (1 + αe )y 1 2 y 2 Hyy + + + r ˜ − + + σ 1 − γ Ge W + H e W + He W + He 2 y W + He   e e yHxy 1 γ Gexx Hxx ηU > + + Σ Σ − + σx2 y x e e e 2 1−γ G W +H W +H 1−γ Collecting terms involving 1/(W + H e ), we have

0 = (H u − H e )νU + Hx −κx x − µ(x)> (Σ> )−1 Σx + yHye ξ0e + ξ1e x − µ(x)> (Σ> )−1 Σy 1 1 e e e + Hte − r˜H e + (1 + αe )y + σy2 y 2 Hyy + σx2 Hxx + Σ> x Σy yHxy , 2 2 which we know is satisfied from (35). Multiplying all the remaining terms by Ge (1 − γ)/γ we then obtain the following PDE for Ge (t, x):  δ γ−1 ηU γ−1 > > −1 0=1+ − + (˜ r + νU ) + + µ(x) (ΣΣ ) µ(x) Ge γ γ γ 2γ 2    − 1 γ 1 νU γ−1 + −κx x ˆ− µ(x)> (Σ> )−1 Σx Gex + σx2 Gexx + νU Gu γ 2 ηU Get



32

(43)

1

with terminal condition Ge (T, x) = ε γ . Note that here >

>

−1

µ(x) (Σ )

 Σx = σx

φS µ ˜0 φx νx0 + σS σB



 + σx

φS µ1 φx νx1 + σS σB

 x

and µ(x)> (ΣΣ> )

−1

µ(x) =

λSS λBB λDD (˜ µ0 + µ1 x)2 + 2 (νx0 + νx1 x)2 + 2 (νy0 + νy1 x)2 σS2 σB σD λSD λSB (˜ µ0 + µ1 x) (νx0 + νx1 x) + 2 (˜ µ0 + µ1 x) (νy0 + νy1 x) +2 σS σB σS σD λBD +2 (νx0 + νx1 x) (νy0 + νy1 x) . σB σD

Going through the same steps for J u , we end up with the PDE (36) for H u (t, x, y) and the following PDE for Gu (t, x):  δ γ−1 ηM γ−1 > > −1 0=1+ − + (˜ r + νM ) + + µ(x) (ΣΣ ) µ(x) Gu γ γ γ 2γ 2   1   νM − γ 1 2 u γ−1 > > −1 u µ(x) (Σ ) Σx Gx + σx Gxx + νM Ge + −κx x ˆ− γ 2 ηM Gut



(44)

1

also with terminal condition Gu (T, x) = ε γ . Note that the PDEs (43) and (44) are coupled and thus have to solved jointly. B.3.7

Optimal investments

The optimal investment strategy is given by (42) with the first component being the optimal fraction of wealth invested in the stock, π. Applying the computations at the end of Section B.1, we obtain  i  yHyi σy ψS Gx W + H i Hxi σx φS 1 W + H i Λ0 + Λ 1 x πS = + − − γ W σS Gi W W σS W σS   i i i σx φS Hx + yσy ψS Hyi Λ 0 + Λ 1 x Gx σ x φ S W + H = + i − , γσS G σS W σS W where Λ0 =

λSS λSB λSD µ ˜0 + νx0 + νy0 , σS σB σD

Λ1 =

33

λSS λSB λSD µ1 + νx1 + νy1 . σS σB σD

(45)

For the business-cycle derivative we find  i  Gx W + H i Hxi σx φx yHyi σy ψx 1 W + H i ΛB0 + ΛB1 x + − πB = − γ W σB Gi W W σB W σB   i i i σx φx Hx + yσy ψx Hyi ΛB0 + ΛB1 x Gx σx φx W + H = + i − , γσB G σB W σB W where ΛB0 =

λBB λBD λSB µ ˜0 + νx0 + νy0 , σS σB σD

ΛB1 =

λSB λBB λBD µ1 + νx1 + νy1 . σS σB σD

Likewise, the optimal fraction of wealth invested in the salary derivative is πD =

1 W + H i ΛD0 + ΛD1 x yHyi σy ψy − , γ W σD W σD

where ΛD0 =

C

λSD λBD λDD µ ˜0 + νx0 + νy0 , σS σB σD

ΛD1 =

λSD λBD λDD µ1 + νx1 + νy1 . σS σB σD

Approximate Solution of PDEs The values Le , Lu of future salary satisfy the coupled PDEs
e −1 e 0 = 1 − r˜ − ξ0e (t) − ξ1e x + Σ> y Σ µ(x) + νU L + Lt
e 1 2 e u −1 > + −κx x − Σ> x Σ µ(x) + Σx Σy Lx + σx Lxx + νU L , 2
u −1 u 0 = 0 − r˜ − ξ0u (t) − ξ1u x + Σ> y Σ µ(x) + νM L + Lt
u 1 2 e −1 > e + −κx x − Σ> x Σ µ(x) + Σx Σy Lx + σx Lxx + νM L , 2

with terminal condition Le (T˜, x) = Lu (T˜, x) = 0. The values of the unemployment benefits F e , F u satisfy the coupled PDEs   αu0 (t) 0 = 0 − r˜ − + νU F e + Fte αu (t)
e 1 2 e −1 u + −κx x − Σ> x Σ µ(x) Fx + σx Fxx + νU F , 2   αu0 (t) 0 = 1 − r˜ − + νM F u + Ftu αu (t)
u 1 2 u −1 e + −κx x − Σ> x Σ µ(x) Fx + σx Fxx + νM F , 2

34

with terminal condition F e (T˜, x) = F u (T˜, x) = 0. Finally, the values Ge , Gu which go into the indirect utility satisfy the coupled PDEs  δ γ−1 ηU γ−1 > > −1 0=1+ − + (˜ r + νU ) + + µ(x) (ΣΣ ) µ(x) Ge γ γ γ 2γ 2    − 1 γ 1 2 e γ−1 νU e > > −1 νU Gu + −κx x ˆ− µ(x) (Σ ) Σx Gx + σx Gxx + γ 2 ηU   δ γ−1 ηM γ−1 u > > −1 0 = 1 + Gt − + (˜ r + νM ) + + µ(x) (ΣΣ ) µ(x) Gu γ γ γ 2γ 2    1  γ−1 1 2 u νM − γ u > > −1 + −κx x ˆ− νM Ge µ(x) (Σ ) Σx Gx + σx Gxx + γ 2 ηM Get



1

with terminal condition Ge (T, x) = Gu (T, x) = ε γ . These three systems of coupled PDEs all have the same structure and can in general be written as e

e e 0 = αe + fte (t, x) + α1e (t, x)f e (t, x) + αxe (t, x)fxe (t, x) + αxx fxx (t, x) + eαN (t,x) f u (t, x) u

u u 0 = αu + ftu (t, x) + α1u (t, x)f u (t, x) + αxu (t, x)fxu (t, x) + αxx fxx (t, x) + eαN (t,x) f e (t, x)

with boundary conditions f e (T, x) = εe and f u (T, x) = εu . An educated guess for the solution is given by i

Z

f (t, x) =

T

f˜i (t, s, x)ds + εi g˜i (t, T, x)

(46)

t

with boundary condition g˜i (T, T, x) = 1 for i = e, u. Plugging this guess into the system of PDEs gives Z

T

f˜ti (t, s, x)ds + εi g˜ti (t, T, x) − f˜i (t, t, x) Z T  + α1i (t, x) f˜i (t, s, x)ds + εi g˜i (t, T, x) t Z T  Z T  i i i i i i i i + αx (t, x) f˜x (t, s, x)ds + ε g˜x (t, T, x) + αxx f˜xx (t, s, x)ds + ε g˜xx (t, T, x) t t Z T  αiN (t,x) i+∆i i+∆i ˜ +e f (t, s, x, i + ∆i)ds + εi g˜ (t, T, x, i + ∆i) i

0=α +

t

t

for i ∈ {e, u}, where i + ∆i = u for i = e and i + ∆i = e for i = u. This equation holds

35

true if f˜e and f˜u solve for every s the coupled PDEs i i 0 = f˜ti (t, s, x) + α1i (t, x)f˜i (t, s, x) + αxi (t, x)f˜xi (t, s, x) + αxx (t, x)f˜xx (t, s, x) i

+ eαN (t,x) f˜i+∆i (t, s, x)

(47)

with boundary condition f˜i (s, s, x) = αi and if g˜e and g˜u solve the coupled PDEs i i 0 = g˜ti (t, T, x) + α1i (t, x)˜ g i (t, T, x) + αxi (t, x)˜ gxi (t, T, x) + αxx (t, x)˜ gxx (t, T, x) i

+ eαN (t,x) g˜i+∆i (t, T, x)

(48)

with boundary condition g˜e (T, T, x) = g˜u (T, T, x) = 1. In the following, we focus on the solution of the coupled PDEs for f˜, since the solution for g˜ is completely analogous. Due to the jump-component in (47) we cannot find a closed-form solution. However, by using a log-linear approximation as Wu and Zeng (2006) we can come up with an approximate solution. A guess for an solution of the coupled PDEs for f˜ is given by i i i 2 f˜i (t, s, x) = eA (t,s)+Bx (t,s)x+Bxx (t,s)x .

(49)

i follow from the boundary condition f˜i (s, s, x) = The boundary conditions for Ai , Bxi and Bxx i (s, s, i) = 0. For αi ≤ 0, we αi . For αi > 0, we get Ai (s, s, i) = ln αi , Bxi (s, s, i) = Bxx represent f˜i by

f˜i (t, s, x) = f˜i,1 (t, s, x) − f˜i,2 (t, s, x) where f˜i,1 and f˜i,2 both solve the system of coupled PDEs (47) with the same coefficient functions α as f˜i and boundary conditions f˜i,1 (s, s, x) = αi,1 and f˜i,2 (s, s, x) = αi,2 . αi,1 and αi,2 are chosen such that αi,1 − αi,2 = αi , αi,1 > 0 and αi,2 > 0. Plugging the guess for f˜ into Equation (47) gives the following system of coupled PDEs i (t, s) ∂Bxx ∂Ai (t, s) ∂Bxi (t, s) + x+ x2 ∂t ∂t ∂t   i + αxi (t, x) Bxi (t, s) + 2Bxx (t, s)x   i i i i + αxx (t, x) Bxi (t, s)2 + 2Bxx (t, s) + 4Bxi (t, s)Bxx (t, s)x + 4Bxx (t, s)2 x2

0 = α1i (t, x) +

i

+ eαN (t,x) eA

i+∆i (t,s)+B i+∆i (t,s)x+B i+∆i (t,s)x2 −Ai (t,s,i)−B i (t,s)x−B i (t,s)x2 x xx x xx

(50)

.

Due to the jump-component we do not know a exact solution of the system, but under the i (t, x) = αi i assumption that the log intensities are affine functions of x, i.e. αN N,1 + αN,x x,

36

the last term can be approximated by n  i  i exp αN,1 + Ai+∆i (t, s) − Ai (t, s) + αN,x + Bxi+∆i (t, s) − Bxi (t, s) x  i+∆i  o i + Bxx (t, s) − Bxx (t, s) x2 n o i ≈ exp αN,1 + Ai+∆i (t, s) − Ai (t, s)   i   i+∆i   i × 1 + αN,x + Bxi+∆i (t, s) − Bxi (t, s) x + Bxx (t, s) − Bxx (t, s) x2 i i i =α ˜ N,0 (t, s) + α ˜ N,x (t, s)x + α ˜ N,xx (t, s)x2 .

With this approximation, we get i (t, s) ∂Ai (t, s) ∂Bxi (t, s) ∂Bxx + x+ x2 ∂t ∂t ∂t   i + αxi (t, x) Bxi (t, s) + 2Bxx (t, s)x  i  i i i i + αxx Bx (t, s)2 + 2Bxx (t, s) + 4Bxi (t, s)Bxx (t, s)x + 4Bxx (t, s)2 x2

0 = α1i (t, x) +

(51)

i i i +α ˜ N,0 (t, s) + α ˜ N,x (t, s)x + α ˜ N,xx (t, s)x2 .

The coefficient function α1i (t, x) is the sum of a quadratic function in x and – when it includes jump intensities – some exponentially affine functions of x, which we approximate by quadratic functions in x, so that α1i (t, x) is approximately a quadratic function of x. The coefficient function αxi (t, x) is an affine functions of x. In more detail, we have that i i i α1i (t, x) ≈ α1,1 (t) + α1,x (t)x + α1,xx (t)x2 i i αxi (t, x) = αx,1 (t) + αx,x (t)x.

Plugging this into (51) yields  i  ∂Ai (t, s) i i i i i + αx,1 (t)Bxi (t, s) + αxx Bx (t, s)2 + 2Bxx (t, s) + α ˜ N,0 (t, s) 0 ≈ α1,1 (t) + ∂t h ∂Bxi (t, s) i i i + α1,x (t) + + 2αx,1 (t)Bxx (t, s) ∂t i i i i i + αx,x (t)Bxi (t, s) + 4αxx Bxi (t, s)Bxx (t, s) + α ˜ N,x (t, s) x

h i i (t, s) ∂Bxx i i i i i i + 2αx,x (t)Bxx (t, s) + 4αxx Bxx (t, s)2 + α ˜ N,xx (t, s) x2 . + α1,xx (t) + ∂t Since this must hold for all values of x and t, we obtain a system of coupled ODEs which

37

i (t, s) have to solve for every s: the functions Ai (t, s), Bxi (t, s), and Bxx

0=

0=

0=

D

∂Ai (t, s) i i + α1,1 (t) + αx,1 (t)Bxi (t, s) ∂t  i  i i+∆i (t,s)−Ai (t,s) i i + αxx Bx (t, s)2 + 2Bxx (t, s) + eαN,1 +A

(52)

∂Bxi (t, s) i i i i i i + α1,x (t) + 2αx,1 (t)Bxx (t, s) + αx,x (t)Bxi (t, s) + 4αxx Bxi (t, s)Bxx (t, s) ∂t (53)  i i+∆i (t,s)−Ai (t,s)  i + eαN,1 +A αN,x + Bxi+∆i (t, s) − Bxi (t, s) i (t, s) ∂Bxx i i i + α1,xx (t) + 2αx,x (t)Bxx (t, s) ∂t  i i+∆i (t,s)−Ai (t,s)  i i i+∆i i + 4αxx Bxx (t, s)2 + eαN,1 +A Bxx (t, s) − Bxx (t, s) .

(54)

Estimation We use a Kalman Filter and maximum-likelihood to obtain a time series of the unob-

servable state process x and our parameter estimates. In order to use the Kalman filter, first we need to discretize our continuous-time model10 Rt+∆t

  1 2 = r + σS λ0 − σS ∆t + σS λ1 b(∆t)xt + εR,t+∆t 2

(55)

vt+∆t = vt + (θv − κv vt ) ∆t + κvx b(∆t)xt + εv,t+∆t

(56)

xt+∆t = e−κx ∆t xt + εx,t+∆t

(57)

where Rt+1 = ln St+1 − ln St is the log return, b(τ ) =

1 κx

(1 − e−κx τ ), and εR , εv , and εx

are normally distributed error terms given by Z

t+∆t

εR,t+∆t = t

(σS λ1 b(t + ∆t − u)Σx + ΣS )0 dzu ! Z

εv1 ,t+∆t

εv,t+∆t =

εv2 ,t+∆t Z

εx,t+∆t =

t+∆t

t+∆t

=

(κv b(t + ∆t − u)Σx + Σv )0 dzu

t

e−κx (t+∆t−u) Σ0x dzu .

t

Second, we need to set up the model on a state space form. That is we need to define the state equation as well as the observation equation.11 Including the stock return and 10

When discretizing the process of v, we use an Euler discretization for the mean-reversion term to simplify the resulting system. 11 See for example Hamilton (1994).

38

signals as state variables it follows from (55)–(57) that the state equation is given by 

Rt+∆t

  v  1,t+∆t   v2,t+∆t  xt+∆t





r + σS λ0 − 12 σS2

      =    





0 0 0 σS λ1 b(∆t)

    0 1 0   ∆t +     0 0 1   0 0 0

θv1 θv2 0



Rt

  κv1 b(∆t)    v1,t   κv2 b(∆t)    v2,t e−κx ∆t

xt





εR,t+∆t

    ε   v1 ,t+∆t +   εv2 ,t+∆t   εx,t+∆t

Including the stock return and signals as state variables implies the following very simple observation equation 



Rt

  v1,t  v2,t

R  t     v1,t  =  0 1 0 0      v2,t 0 0 1 0  xt 



1 0 0 0





   .  

The variance of the disturbances in the state equation is given by Var(εR,t ) = σS2 ∆t + σS2 σx2 λ21 B2 (∆t) + 2σS2 σS ρSx B1 (∆t) Var(εvi ,t ) = κ2vi σx2 B2 (∆t) + σv2i ∆t + 2κvi σx σvi ρxvi B1 (∆t)
1 Var(εx,t ) = 1 − e−2κx ∆t σx2 , 2κx with i = 1, 2, and 1 (∆t − b(∆t)) κx 1 1 B2 (∆t) = 2 (∆t − b(∆t)) − b(∆t)2 . κx 2κx

B1 (∆t) =

The covariance between the disturbances is given by Cov(εR,t , εvi ,t ) = σS σvi ρvi S ∆t + σx2 σS λ1 κvi B2 (∆t) + (λ1 σvi ρxvi + ρSx κvi ) σS σx B1 (∆t) 1 Cov(εR,t , εx,t ) = σS λ1 σx2 b(∆t)2 + ρSx σx σS b(∆t) 2 Cov(εv1 ,t , εv2 ,t ) = κv1 κv2 σx2 B2 (∆t) + σv1 σv2 ρv ∆t + (κv1 σv2 ρxv2 + κv2 σv1 ρxv1 ) σx B1 (∆t) κv Cov(εvi ,t , εx,t ) = i b(∆t)2 σx2 + σx σvi ρxvi b(∆t). 2 Again i = 1, 2.

39

    .  

References Bagliano, F. C., C. Fugazza, and G. Nicodano (2013). Optimal Life-Cycle Portfolios for Heterogeneous Workers. Discussion Paper 06/2013-025, Netspar. Review of Finance, forthcoming. Benzoni, L., P. Collin-Dufresne, and R. S. Goldstein (2007). Portfolio Choice over the Life-Cycle when the Stock and Labor Markets are Cointegrated. Journal of Finance 62(5), 2123–2167. Bick, B., H. Kraft, and C. Munk (2013). Solving Constrained Consumption-Investment Problems by Simulation of Artificial Market Strategies. Management Science 59(2), 485–503. Cocco, J., F. Gomes, and P. Maenhout (2005). Consumption and Portfolio Choice over the Life Cycle. Review of Financial Studies 18(2), 491–533. Cvitani´c, J. and I. Karatzas (1992). Convex Duality in Constrained Portfolio Optimization. Annals of Applied Probability 2(4), 767–818. Cvitani´c, J., W. Schachermayer, and H. Wang (2001). Utility Maximization in Incomplete Markets with Random Endowment. Finance and Stochastics 5(2), 259–272. Davis, S. J. and P. Willen (2000, March). Using Financial Assets to Hedge Labor Income Risks: Estimating the Benefits. Working paper, University of Chicago and Princeton University. Dumas, B., A. Kurshev, and R. Uppal (2009). Equilibrium Portfolio Strategies in the Presence of Sentiment Risk and Excess Volatility. Journal of Finance 64(2), 579–629. Gourinchas, P.-O. and J. A. Parker (2002). Consumption Over the Life Cycle. Econometrica 70(1), 47–89. Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press. Harvey, C. R. (2001). The Specification of Conditional Expectations. Journal of Empirical Finance 8(5), 573–637. Heaton, J. and D. Lucas (2000). Portfolio Choice and Asset Prices: The Importance of Entrepreneurial Risk. Journal of Finance 55(3), 1163–1198. Hubbard, R. G., J. Skinner, and S. P. Zeldes (1995). Precautionary Saving and Social Insurance. Journal of Political Economy 103(2), 360–399. Jin, X. and A. X. Zhang (2012). Decomposition of Optimal Portfolio Weight in a JumpDiffusion Model and Its Applications. Review of Financial Studies 25(9), 2877–2919.

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Karatzas, I., J. Lehoczky, S. Shreve, and G. Xu (1991). Martingale and Duality Methods for Utility Maximization in an Incomplete Market. SIAM Journal on Control and Optimization 29(3), 702–730. Kim, T. S. and E. Omberg (1996). Dynamic Nonmyopic Portfolio Behavior. Review of Financial Studies 9(1), 141–161. Koijen, R. S. J., T. Nijman, and B. J. M. Werker (2010). When Can Life-Cycle Investors Benefit from Time-Varying Bond Risk Premia? Review of Financial Studies 23(2), 741–780. Korn, R. and H. Kraft (2004). On the Stability of Continuous-Time Portfolio Problems with Stochastic Opportunity Set. Mathematical Finance 14(3), 403–414. Lettau, M. and S. C. Ludvigson (2010). Measuring and Modeling Variation in the RiskReturn Tradeoff. In Y. Ait-Sahalia and L. P. Hansen (Eds.), Handbook of Financial Econometrics, Volume 1, pp. 618–682. North Holland. Liptser, R. S. and A. N. Shiryaev (2001). Statistics of Random Processes. Volume I: General Theory. Springer. Liu, J. (2007). Portfolio Selection in Stochastic Environments. Review of Financial Studies 20(1), 1–39. Lynch, A. W. and S. Tan (2011). Labor Income Dynamics at Business-Cycle Frequencies: Implications for Portfolio Choice. Journal of Financial Economics 101(2), 333–359. Meyer, D. J. and J. Meyer (2005). Relative Risk Aversion: What Do We Know? Journal of Risk and Uncertainty 31(3), 243–262. Munk, C. and C. Sørensen (2010). Dynamic Asset Allocation with Stochastic Income and Interest Rates. Journal of Financial Economics 96(3), 433–462. Scheinkman, J. A. and W. Xiong (2003). Overconfidence and Speculative Bubbles. Journal of Political Economy 111(6), 1183–1219. Thurow, L. (1969). The Optimum Lifetime Distribution of Consumption Expenditures. American Economic Review 59(3), 324–330. Viceira, L. M. (2001). Optimal Portfolio Choice for Long-Horizon Investors with Nontradable Labor Income. Journal of Finance 56(2), 433–470. Wu, S. and Y. Zeng (2006). The Term Structure of Interest Rates under Regime Shifts and Jumps. Economics Letters 93(2), 215–221.

41

Monte Carlo evaluation

loss bound loss

maximize θ 3 θ2

θ1

θ¯

θ∗

Computable artificial markets minimize

Unknown optimal

θ¯ θ1 θ∗

θ2

θ3

Expected utility

“feasibilization” Figure 1: Illustration of our solution technique. The axis measures the expected utility of the investor. The point marked “unknown optimal” represents the indirect utility in the true market, which is therefore the expected utility generated by the unknown optimal feasible consumption and investment strategy. Each of the points indicated to the right corresponds to the indirect utility in an artificial market with deterministic modifiers characterized by some parameter set θ. The corresponding strategy is transformed into a feasible strategy in the true market which generates an expected utility on the left part of the axis. The best of these strategies is derived from the optimal strategy in an artificial market characterized by some parameter set θ∗ . The arrows above the axis indicate the unknown utility loss and a computable upper bound on the loss that the investor suffers by following the best of the considered feasible strategies instead of the unknown optimal strategy.

42

Panel A: The real stock price 2000

1000

0 1960

1970

1980

1990

2000

2010

2000

2010

2000

2010

The unemployment rate, % 10 5 0 1960

1970

1980

1990

The aggregate income growth rate, % 5

0

−5 1960

1970

1980

1990

Figure 2: Historical evolution of stock market, the aggregate unemployment rate, and the aggregate income growth rate. The stock market is represented by the S&P500 stock price index. The unemployment rate represents the number of unemployed as a percentage of the labor force in the U.S. The income growth rate represents the national aggregated income level.

43

10 ηu = 0, ξ1 = 0 η1,u=0, ξ1 = 0

9

Full model

8

Financial Wealth

7

6

5

4

3

2

1

0

0

5

10

15 Time

20

25

30

Figure 3: The effect of unemployment risk on wealth over the life cycle. The graphs show the average financial wealth at different ages derived from 10,000 simulated paths. The red curve is for the case in which the unemployment risk is zero and the expected income growth is independent of the state of the economy. The green curve adds state-independent unemployment risk. The blue curve is for our full model in which both the unemployment risk and the expected income growth rate are state dependent.

44

Panel A: Consumption

Panel B: Portfolio weight in Stock

1.5

1.5 η = 0, ξ = 0 u

1

η1,u=0, ξ1 = 0 Full model

1

1

0.5

0.5

0

0

10

20

0

30

0

10

20

30

Time

Time

Figure 4: The effect of unemployment risk on consumption and stock investment. Panel A shows the average consumption rate and Panel B the average fraction of financial wealth invested in the stock over the life cycle. The average is based on 10,000 simulated paths. The red curve is for the case in which the unemployment risk is zero and the expected income growth is independent of the state of the economy. The green curve adds state-independent unemployment risk. The blue curve is for our full model in which both the unemployment risk and the expected income growth rate are state dependent.

Panel A: Consumption

Panel B: Portfolio weight in Stock

1.5

1.5 Full model η1,u = 0 ξ =0 1

1

1

0.5

0.5

0

0

10

20

0

30

0

10

20 Time

Time

Figure 5: The effect of unemployment risk on consumption and stock investment. Panel A shows the average consumption rate and Panel B the average fraction of financial wealth invested in the stock over the life cycle. The average is based on 10,000 simulated paths. The solid blue curve is for our full model in which both the unemployment risk and the expected income growth rate are state dependent. The dotted blue curve is for the case where the state dependence of the unemployment risk is turned off. The dashed blue curve is for the case where the state dependence of the expected income growth rate is turned off.

45

30

15 ηu = 0, ξ1 = 0 η1,u=0, ξ1 = 0 Full model

Financial Wealth

10

5

0

0

5

10

15

20

25 Time

30

35

40

45

50

Figure 6: The effect of unemployment risk on wealth for a longer time horizon. The graphs show the average financial wealth at different ages derived from 10,000 simulated paths. The red curve is for the case in which the unemployment risk is zero and the expected income growth is independent of the state of the economy. The green curve adds state-independent unemployment risk. The blue curve is for our full model in which both the unemployment risk and the expected income growth rate are state dependent. In contrast to the benchmark, this figure assumes T˜ = 35 and T = 50.

46

Panel A: Consumption

Panel B: Portfolio weight in Stock

2

1.5 ηu = 0, ξ1 = 0 η

=0, ξ = 0

1,u

1

Full model

1.5 1 1 0.5 0.5

0

0

10

20

30

40

0

50

0

10

20

Time

30

40

50

Time

Figure 7: The effect of unemployment risk on consumption and stock investment for a longer time horizon. Panel A shows the average consumption rate and Panel B the average fraction of financial wealth invested in the stock over the life cycle. The average is based on 10,000 simulated paths. The red curve is for the case in which the unemployment risk is zero and the expected income growth is independent of the state of the economy. The green curve adds stateindependent unemployment risk. The blue curve is for our full model in which both the unemployment risk and the expected income growth rate are state dependent. In contrast to the benchmark, this figure assumes T˜ = 35 and T = 50.

Panel A: Consumption

Panel B: Portfolio weight in Stock

1.5

1.5 γ=2 γ=6 γ=10

1

1

0.5

0.5

0

0

10

20

0

30

0

10

20 Time

Time

Figure 8: The effect of risk aversion on consumption and stock investment. Panel A shows the average consumption rate and Panel B the average fraction of financial wealth invested in the stock over the life cycle. The average is based on 10,000 simulated paths. The blue curve is for our benchmark case with a risk aversion parameter of γ = 6. The red curve is for a risk aversion of γ = 2, and the green curve for a risk aversion of γ = 10.

47

30

Panel A: Consumption

Panel B: Portfolio weight in Stock

1.5

1.5 Half intensity Benchmark Double intensity

1

1

0.5

0.5

0

0

10

20

0

30

0

10

20

30

Time

Time

Figure 9: The effect of the unemployment intensity on consumption and stock investment. Panel A shows the average consumption rate and Panel B the average fraction of financial wealth invested in the stock over the life cycle. The average is based on 10,000 simulated paths. The three curves in each panel represent different values of the parameter ηu0 . The blue curve is based on the benchmark value, whereas the red curve and the blue curve are based on a parameter value equal to half and double the benchmark value, respectively.

Panel A: Consumption

Panel B: Portfolio weight in Stock

1.5

1.5 Highly counter−cyclical unemployment risk Benchmark Procyclical unemployment risk

1

1

0.5

0.5

0

0

10

20

0

30

Time

0

10

20 Time

Figure 10: The effect of the state dependence of the unemployment intensity on consumption and stock investment. Panel A shows the average consumption rate and Panel B the average fraction of financial wealth invested in the stock over the life cycle. The average is based on 10,000 simulated paths. The three curves in each panel represent different values of the parameter ηu1 . The blue curve is for the benchmark value ηu1 = −2.34, the red curve is for a value five times as high, ηu1 = −11.7, and the green curve is for ηu1 = +2.34 and thus a pro-cyclical unemployment risk.

48

30

Panel A: Consumption

Panel B: Portfolio weight in Stock

1.5

1.5 ξ

=ξ

0,u

0,e

ξ0,u=−0.1 ξ0,u=−0.2

1

1

0.5

0.5

0

0

10

20

0

30

0

10

20 Time

Time

Figure 11: The effect of reemployment income consumption and stock investment. Panel A shows the average consumption rate and Panel B the average fraction of financial wealth invested in the stock over the life cycle. The average is based on 10,000 simulated paths. The three curves in each panel represent different values of the parameter ξ0u that determines the expected salary growth rate during unemployment. The blue curve is based on the benchmark value of −10% per year. The red curve assumes the same salary growth rate in unemployment and employment, whereas the green curve assumes an expected salary growth rate of −20% per year while unemployed.

49

30

Parameter

Description

Estimate

Benchmark

r µ0 µ1 σS

Financial assets Interest rate Average excess stock return State sensitivity of stock drift Stock volatility

pre-set 0.0278 -0.0429 0.1265

0.01 0.0286 -0.0442 0.13

ω1 κ1x κ1v σv1

Unemployment rate Constant level in drift State sensitivity of drift Mean reversion speed Volatility

0.0137 -0.1237 0.2549 0.0055

0.01 -0.12 0.25 0.0055

ω2 κ2x κ2v σv2

Aggregate income growth rate Constant level in drift State sensitivity of drift Mean reversion speed Volatility

0.0762 0.2159 13.3693 0.0181

0.08 0.22 13.50 0.02

κx σX

Unobservable state Mean reversion speed Volatility

0.7300 0.1227

0.73 0.12

kSv1 kSv2 kSX kv1 v2 kXv1 kXv2

Correlations Stock index and unemployment rate Stock index and aggregate income growth Stock index and unobservable state Unemployment rate and agg. income growth Unobservable state and unemployment rate Unobservable state and agg. income growth

0.0012 0.0279 0.5985 -0.0295 0.3820 -0.0890

0.0 0.0 0.6 0.0 0.38 -0.09

Table 1: Estimates and benchmark values of macro parameters. The estimates are based on monthly data on the aggregated unemployment rate, the growth rate of the aggregated income level, and real stock prices of the S&P500. The sample period goes from January 1959 to July 2013.

50

Parameter

Description

Benchmark

σy ξ0e ξ0u ξ0e ξ0u

Salary dynamics Volatility Constant in drift when employed Constant in drift when unemployed State sensitivity of drift when employed State sensitivity of drift when unemployed

0.02 0.01 -0.10 0.22 0.22

η0,e η1,e η0,u η1,u αe αu (0)

Employment status Constant in employment jump intensity State sensitivity of employment jump intensity Constant in unemployment jump intensity State sensitivity of unemployment jump intensity Fraction of salary paid to unemployment insurance Initial value of unemployment benefits

1.54 3.06 -1.20 -2.34 0.2 0.4y0

T˜ T − T˜ γ δ ε W0 y0

Other parameters Years until retirement Years in retirement Relative risk aversion Time preference rate Utility weight on bequests Initial financial wealth Initial level of salary (per year)

20 10 6 0.02 1 1 1

Table 2: Benchmark values of investor-specific parameters.

Portfolio weight of stock

Consumption rate

x = −σx

x=0

x = σx

x = −σx

x=0

x = σx

t=0 Employed Unemployed

1.0000 1.0000

1.0000 1.0000

0.9042 0.3821

0.5713 0.5508

0.5911 0.5747

0.6141 0.6021

t=5 Employed Unemployed

1.0000 0.8962

0.7957 0.6077

0.4951 0.3015

0.5873 0.5685

0.6058 0.5904

0.6277 0.6164

t = 10 Employed Unemployed

0.6893 0.6028

0.5343 0.4430

0.3686 0.2690

0.6014 0.5840

0.6154 0.6009

0.6313 0.6206

t = 15 Employed Unemployed

0.4880 0.4474

0.3978 0.3518

0.3013 0.2544

0.6092 0.5942

0.6183 0.6051

0.6267 0.6174

Table 3: Portfolio and consumption as a function of the macro state and the employment state. The values of salary and financial wealth are y = 1, W = 1 for t = 0; y = 1.0187, W = 2.2033 for t = 5; y = 1.0376, W = 3.4474 for t = 10; y = 1.0558, W = 4.7647 for t = 15.

51