Economic agents face risks of many kinds, which may mutually covary. A stock broker, for example, is likely to earn a salary bonus that is positively related to the performance of the stock market; if that broker also has personal stock investments, his ﬁnancial wealth and labor income will be positively correlated.

The ﬁrst part of this handout presents a convenient (and empirically realistic) formulation in which a consumer faces two shocks (which can be interpreted as a shock to noncapital income and a shock to the rate of return) that are distributed according to a multivariate lognormal that allows for correlation between them. The second part describes a computationally simple and convenient method for approximating that joint distribution.

1 Theory

and a risky log rate-of-return that is aﬀected by following factors: the riskless rate $r$ ; a risk premium $φ$ ; an additional constant $ζ$ (whose purpose will become clear below); a component that is linearly related to $𝜃 1,t+1$ ; and an independent shock $2 2 𝜃2 ∼ 𝒩 (− 0.5σ 2,σ 2)$ :

for some constant $ω$ . Since $(σ2 ∕σ1 )ω𝜃1,t+1$ is the only component of $rt+1$ that covaries with $𝜃 1,t+1$ ,

Equation (2) yields a description of the return process in which the parameter $ω$ controls the correlation between the risky log return shock and the risky log labor income shock. If $ω = 0$ the processes are independent.

Now we want to ﬁnd the value of $ζ$ such that the mean risky return is unaﬀected by $2 σ 1$ (so that we will be able to understand clearly the distinct eﬀects of labor income risk, the independent component of rate-of-return risk $2 σ 2$ , and the correlation between labor income risk and rate-of-return risk, $ω$ ). Thus, we want to ﬁnd the $ζ$ such that

Using standard facts about lognormals (cf. MathFacts), and for convenience deﬁning $ˆω = ( σ ∕σ )ω 2 1$ , we have

2 Computation

A key step in the computational solution of any model with uncertainty is the calculation of expectations. Writing $˜ ˜ Θ1 ≡ Θ1,t+1$ and $˜ R ≡ Rt+1$ and $𝔼 [∙ ] = 𝔼t [∙t+1 ]$ , the expectation of some function $h$ that depends on the realization of the risky return $˜ R$ and the labor income shock is:

where $F (˜Θ1, ˜R )$ is the joint cumulative distribution function. Standard numerical computation software can compute this double integral, but at such a slow speed as to be almost unusable. Computation of the expectation can be massively speeded up by advance construction of a numerical approximation to $F (˜Θ1, R˜ )$ .

Such approximations generally take the approach of replacing the distribution function with a discretized approximation to it; appropriate weights $wi,j$ are attached to each of a ﬁnite set of points indexed by $i$ and $j$ , and the approximation to the integral is given by:

where the $ˆ Θ1$ and $ˆ R$ matrices contain the conditional means of the two variables in each of the ${i,j }$ regions. Various methods are used for constructing the weights $w [i,j]$ and the nodes (the $i$ and $j$ points for $Θ1$ and $R$ ).

Perhaps the most popular such method is Gauss-Hermite interpolation (see Judd (1998) for an exposition, or Kopecky and Suen (2010) for some alternatives). Here, we will pursue a particularly intuitive alternative: Equiprobable discretization. In this method, $m = n$ and boundaries on the joint CDF are determined in such a way as to divide up the total probability mass into submasses of equal size (each of which therefore has a mass of $n − 2$ ). This is conceptually easier if we represent the underlying shocks as statistically independent, as with $𝜃 1,t+1$ and $𝜃 2,t+1$ above; in that case, each submass is a square region in the $Θ1$ and $Θ2$ grid. We then compute the average value of $Θ 1$ and $R$ conditional on their being located in each of the subdivisions of the range of the CDF. Since, in this speciﬁcation, $R$ is a function of $Θ1$ , the $R$ values are indexed by both $i$ and $j$ , but since we have written $Θ 1$ as IID, the representation of the approximating summation is even simpler than (7):

Details can be found in the Mathematica notebook associated with this handout. A particular example, in which $σ22 = σ21$ and $ω = 0.5$ , is illustrated in ﬁgure 1; the red dots reﬂect the height of the approximation to the CDF above the conditional mean values for $Θ1$ and $R$ within each of the equiprobable regions.

Figure 1:‘True’ CDF With Approximation Points in Red for $ω = 0.5$

References

Judd, Kenneth L. (1998): Numerical Methods in Economics. The MIT Press, Cambridge, Massachusetts.

Kopecky, Karen A., and Richard M.H. Suen (2010): “Finite State Markov-Chain Approximations To Highly Persistent Processes,” Review of Economic Dynamics, 13(3), 701–714, http://www.karenkopecky.net/RouwenhorstPaper.pdf.