Here, we provide numerical examples that illustrate the change in the cost of capital that results from a change in information asymmetry while holding the average precision of investors' information fixed. In order to change the degree of information asymmetry while holding investors' average precision fixed, at least two exogenous variables must change simultaneously while also calibrating the magnitude of these changes. As this is computationally challenging, we illustrate this with a numerical example. To explain these examples, \( \lambda \) and \( \Pi_d \), as endogenous variables, could be thought of as functions of 6 exogenous variables: \( N \), the number of informed investors; \( r_I \), the risk tolerance of informed investors; \( x \), the aggregate weight of uninformed investors; \( \sigma^2_v \), the a priori variance of the risky asset's cash flow; \( \sigma^2_{\epsilon} \) , the variance in the error in informed investors' private information; and \( \sigma^2_z \), the variance in the supply of shares of the risky asset. Thus, first,we specify values for the 6-tuple \( \Omega = \left(N, r_I, \omega, \sigma^2_v, \sigma^2_{\epsilon}, \sigma^2_z \right), \) then we solve for \( \lambda \) and \( \Pi_d \), and finally, we compute investors' average precision and the cost of capital using Equations (19) and (18), respectively.
N <- 4
w <- 1
sig_v <- sig_z <- sig_e <- 1
rI <- 1
To start, consider an economy with 4 informed investors, who have a risk tolerance of 1. Let the aggregate weight of the uninformed investors converge to 1 as their number becomes large (i.e., \( \lim_{M \to \infty} M r_U =: \omega = \) 1). We assume that the a priori variance of the risky asset's cash flow is \( \sigma^2_v = \) 1. We begin with the assumptions that \( \sigma^2_{\epsilon} = \) 1 and \( \sigma^2_z = \) 1.
Pi_v <- 1/sig_v
Pi_e <- 1/sig_e
Pi_i <- Pi_v + Pi_e
These assumptions imply that the precision of the information available to informed investors is \( \Pi_i = \Pi_v + \Pi_e = \) 2.
eq_A5 <- function(lambda, N, rI, w, sig_v, sig_e, sig_z) {
rI^3 * w * sig_z * (sig_v + sig_e)^3 * lambda^4 + rI^2 * sig_v * sig_z *
(sig_v + sig_e)^2 * (rI * (sig_v + sig_e) * (N - 2) + 3 * w * sig_e) *
lambda^3 + rI * sig_v^2 * (sig_v + sig_e) * ((3 * sig_z * sig_e^2 +
N^2 * rI^2 * (sig_v + sig_e)) * w + rI * sig_z * sig_e * (sig_v + sig_e) *
(2 * N - 5)) * lambda^2 + sig_v^3 * sig_e * ((N^2 * rI^2 * (sig_v +
sig_e) + sig_z * sig_e^2) * w + rI * (sig_v + sig_e) * (sig_z * sig_e *
(N - 4) + N^2 * rI^2 * (N - 2))) * lambda - sig_v^4 * sig_e^2 * (sig_z *
sig_e + N^2 * rI^2)
}
Using Equation (A.5), we compute \( \lambda \) as follows.
sol <- uniroot(eq_A5, c(0, 1), N = N, rI = rI, w = w, sig_v = sig_v,
sig_e = sig_e, sig_z = sig_z)
lambda <- sol$root
This yields a value of \( \lambda \) of 0.1541. From this we can use Equation (A.6) to calculate \( \Pi_{\delta} \):
Pi_d <- (N^2 * rI^2 * sig_v^2)/(rI^2 * sig_z * (sig_v + sig_e)^2 *
lambda^2 + 2 * rI * sig_v * sig_e * sig_z * (sig_v + sig_e) * lambda + sig_v^2 *
sig_e * (N^2 * rI^2 + sig_e * sig_z))
Pi_u <- Pi_v + Pi_d
Pi_avg <- (N * rI * Pi_i + w * Pi_u)/(N * rI + w)
The precision of the information an uninformed investor gleans from price is \( \Pi_{\delta} = \) 0.9034, and the (total) precision of the information available to uninformed investors is \( \Pi_u = \Pi_v + \Pi_d \), which gives 1.9034.
Using Equation (19), investors' average precision computes to \( \Pi_{avg} = \) 1.9807.
In addition, using Equation (18) in Theorem 2:
CoC <- (((1 + rI * Pi_i * lambda)^-1 * (N * rI * Pi_i) + w * Pi_u))^-1
the risky asset's cost of capital computes to 0.1247.
Now consider a circumstance where an informed investor acquires more private information, but the
“average” precision of information remains at the same level. For example, suppose \( \sigma^2_{\epsilon} \) falls
sig_e <- 0.9
Pi_e <- 1/sig_e
Pi_i <- Pi_v + Pi_e
These assumptions imply that the precision of the information available to informed investors is \( \Pi_i = \Pi_v + \Pi_e = \) 2.1111.
In order for average precision of information to remain at the same level, we now require the variance of the liquidity shock to increase. We calculate the required \( \sigma^2_z \) using the following code:
avg_precision <- function(sig_z, N, rI, w, sig_v, sig_e, Pi_avg) {
Pi_v <- 1/sig_v
Pi_e <- 1/sig_e
Pi_i <- Pi_v + Pi_e
sol <- uniroot(eq_A5, c(0, 1), N = N, rI = rI, w = w, sig_v = sig_v, sig_e = sig_e,
sig_z = sig_z)
lambda <- sol$root
Pi_d <- (N^2 * rI^2 * sig_v^2)/(rI^2 * sig_z * (sig_v + sig_e)^2 * lambda^2 +
2 * rI * sig_v * sig_e * sig_z * (sig_v + sig_e) * lambda + sig_v^2 *
sig_e * (N^2 * rI^2 + sig_e * sig_z))
Pi_u <- Pi_v + Pi_d
(N * rI * Pi_i + w * Pi_u)/(N * rI + w) - Pi_avg
}
sol <- uniroot(avg_precision, c(0, 100), N = N, rI = rI, w = w, sig_v = sig_v,
sig_e = sig_e, Pi_avg = Pi_avg)
sig_z <- sol$root
We find that \( \sigma^2_z \) needs to rise to 14.0353.
sol <- uniroot(eq_A5, c(0, 1), N = N, rI = rI, w = w, sig_v = sig_v,
sig_e = sig_e, sig_z = sig_z)
lambda <- sol$root
This yields a value of \( \lambda \) of 0.1618 (note this is higher than the value calculated above). From this we can use Equation (A.6) to calculate \( \Pi_{\delta} \):
Pi_d <- (N^2 * rI^2 * sig_v^2)/(rI^2 * sig_z * (sig_v + sig_e)^2 *
lambda^2 + 2 * rI * sig_v * sig_e * sig_z * (sig_v + sig_e) * lambda + sig_v^2 *
sig_e * (N^2 * rI^2 + sig_e * sig_z))
Pi_u <- Pi_v + Pi_d
Pi_avg <- (N * rI * Pi_i + w * Pi_u)/(N * rI + w)
The precision of the information an uninformed investor gleans from price is \( \Pi_{\delta} = \) 0.4589, and the (total) precision of the information available to uninformed investors is \( \Pi_u = \Pi_v + \Pi_{\delta} \), which gives 1.4589. Using Equation (19), investors' average precision computes to \( \Pi_{avg} = \) 1.9807.
In addition, using Equation (18) in Theorem 2:
CoC <- (((1 + rI * Pi_i * lambda)^-1 * (N * rI * Pi_i) + w * Pi_u))^-1
the risky asset's cost of capital computes to 0.129.
Now if I take values from the appendix:
sig_z <- 14.034
sol <- uniroot(eq_A5, c(0, 1), N = N, rI = rI, w = w, sig_v = sig_v,
sig_e = sig_e, sig_z = sig_z)
lambda <- sol$root
Pi_d <- (N^2 * rI^2 * sig_v^2)/(rI^2 * sig_z * (sig_v + sig_e)^2 *
lambda^2 + 2 * rI * sig_v * sig_e * sig_z * (sig_v + sig_e) * lambda + sig_v^2 *
sig_e * (N^2 * rI^2 + sig_e * sig_z))
Pi_u <- Pi_v + Pi_d
Pi_avg <- (N * rI * Pi_i + w * Pi_u)/(N * rI + w)
The value of \( \Pi_{\delta} \) equals 0.459, while \( \lambda \) equals 0.1618. Using Equation (19), investors' average precision computes to \( \Pi_{avg} = \) 1.9807, which the target value.