An Equiprobable Approximation to the Bivariate Lognormal

$©$ February 20, 2012, Christopher D. Carroll Equiprobable

Economic agents face risks of many kinds, some of which may be correlated with each other. A stock broker, for example, is likely to earn a salary bonus that is positively correlated with the performance of the stock market; if the 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 convenient method for approximating that joint distribution.

1 Theory

Consider a consumer who faces both a risk to transitory noncapital income¹

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 $θ$ ; and an independent shock $ξ ~ N (- 0.5σ2ξ,σ2ξ)$ :

r ≡ log R = r + ϕ + ζ + ϱ θ(σξ∕ σθ) + ξ (2)

for some constant $ϱ$ . Since $(σ ∕σ )ϱ θ ξ θ$ is the only component of $r$ that is correlated with $θ$ ,

cov (θ,r) = cov (θ, (σ ∕σ )ϱ θ)
ξ θ
= ϱ (σξ∕ σθ) cov (θ,θ )
◟---◝◜--◞
= σ2θ
= ϱ σξσ θ

corr (θ,r) ≡ cov (θ, r)∕ σξσ θ

= ϱ.

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

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

regardless of the values of $σ2 θ$ and $σ2 ξ$ . We therefore need:

ζ+(σξ∕σθ)ϱθ+ ξ
E [e ] = 1. (4)
log E [e ζ+(σξ∕σθ)ϱθ+ ξ] = 0. (5)

Using standard facts about lognormals (cf. MathFacts), and for convenience defining $ˆϱ = (σ ξ∕σ θ)ϱ$ , we have

0. = ζ - 0.5ϱˆσ2 - 0.5σ2 + 0.5 ˆϱ2σ2 + 0.5σ2 (6)
θ ξ θ ξ
= ζ - 0.5σ2θˆϱ(1 - ˆϱ ) (7)
2 2 2 2
ζ = 0.5 (ˆϱ - ϱˆ )σθ = 0.5 (ϱσ ξσθ - ϱ σξ ). (8)

One way to represent the joint distribution would be to stack the two variables $θ$ and $ξ$ into a vector $⃗q$ and to write

where $μ = { μ ,μ } ′ θ ξ$ and the covariance matrix $Σ$ is diagonal with variances $σ2 θ$ and $2 σ ξ$ .²

2 Computation

A key step in the computational solution of any model with uncertainty is the calculation of expectations. The expectation of some function $h$ that depends on the realization of the risky return $R$ and the labor income shock is:

∫ ∫
Θ¯ R¯
E [f(Θ, R )] = h (Θ, R )dF (Θ, R ) (10)
Θ- R-

where $F (Θ, R )$ 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. But computation of the expectation can be massively speeded up by advance construction of a numerical approximation to $F (Θ, R )$ .

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

∑ n ∑m
E[f(Θ, R )] ≈ h (Θ ,R )w (11)
i,j i,j i,j
i=1 j=1

where various methods are used for constructing the weights $wi,j$ and the nodes (the $i$ and $j$ points for $Θ$ 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 $θ$ and $ξ$ above; in that case, each submass is a square region in the $Θ$ and $Ξ$ grid (where $Ξ = exp (ξ)$ ). We then compute the average value of $Θ$ 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 $Θ$ , the $R$ values are indexed by both $i$ and $j$ , but since we have written $Θ$ as IID, the representation of the approximating summation is even simpler than (11):

∑n ∑ n
E[f(Θ, R )] ≈ n - 2 h(Θi, Ri,j ). (12)

i=1 j=1

Details can be found in the Mathematica notebook associated with this handout. A particular example, in which $σ2 = σ2 ξ θ$ and $ϱ = 0.5$ , is illustrated in figure 1; the red dots reflect the height of the approximation to the CDF above the conditional mean values for $Θ$ and $R$ within each of the equiprobable regions.

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

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.