A theory of underwriters’ risk management in a firm-commitment initial public offering

Abstract

A cynosure of the academic literature relating to initial public offerings (IPOs) is the question of why they are “mispriced” so frequently. The large and growing literature addressing this question is evidence as to its intractability. This paper develops a theory of underwriters’ behavior suggesting that they will exploit their private information to minimize the bilateral risks to themselves of firm-commitment IPOs. That minimization may cause them to knowingly underprice the issue. The main result in this paper is based, in part, on the premise that the random character of the investors’ demand for shares in the secondary market, given the spread, is governed by an estimable conditional probability distribution. The underwriters exploit their private knowledge of that probability distribution to influence the number of shares in the offering in such a way as to minimize their expected loss function.

Appendices

Appendix 1

Define the symbol θ ₁ to represent the underwriters’ spread earned on each share sold at the offering price: $\theta_{1} = P_{o} - P_{i}$.

Define the symbol θ ₂ to represent the loss per share, if any, incurred by the underwriters for each share sold in the secondary market expected to be at a price below the price paid to the issuer $\theta_{2} = P_{i} - \bar{P}$.

I assume that the members of the syndicate estimate fixed values of the two parameters θ ₁ and θ ₂. The expected loss function can be rewritten as:

$$E\left[ L \right] = \theta_{1} \mathop \int \limits_{S}^{\infty } (X - S)f\left( {X|P_{o} } \right)dX + \theta_{2} \mathop \int \limits_{0}^{S} \left( {S - X} \right)f\left( {X|P_{o} } \right)dX$$

(13)

The expected loss in (13) is minimized for the numerical value of S that satisfies the equality $\frac{dE[L]}{dS} = 0$ and the inequality $\frac{{d^{2} }}{{dS^{2} }}E\left[ L \right] > 0$.

Expression 13 can be decomposed into four integral functions:

$$E\left[ L \right] = \theta_{1} \mathop \int \limits_{S}^{\infty } Xf\left( {X|P_{o} } \right)dX - \theta_{2} \mathop \int \limits_{0}^{S} Xf\left( {X|P_{o} } \right)dX + S\left[ {\theta_{2} \mathop \int \limits_{0}^{S} f\left( {X|P_{o} } \right)dX - \theta_{1} \mathop \int \limits_{S}^{\infty } f\left( {X|P_{o} } \right)dX} \right]$$

(14)

Applying the well-known formula for the differentiation of a definite integral with respect to a parameter, we have the result¹⁴:

$$\frac{dE[L]}{dS} = - \theta_{1} \mathop \int \limits_{S}^{\infty } f\left( {X|P_{o} } \right)dX + \theta_{2} \mathop \int \limits_{0}^{S} f\left( {X|P_{o} } \right)dX$$

(15)

Expression 15 is simply a weighted sum of the dichotomized cumulative distribution function of f(X | P _o ). Thus, by the law of total probability, $\mathop \int \limits_{S}^{\infty } f\left( {X|P_{o} } \right)dX + \mathop \int \limits_{0}^{S} f\left( {X|P_{o} } \right)dX = 1$. The equality allows the substitution $\mathop \int \limits_{S}^{\infty } f\left( {X|P_{o} } \right)dX = 1 - \mathop \int \limits_{0}^{S} f\left( {X|P_{o} } \right)dX$ into expression 15. The result is:

$$\frac{dE\left[ L \right]}{dS} = - \theta_{1} + \theta_{1} \mathop \int \limits_{0}^{S} f\left( {X|P_{o} } \right)dX + \theta_{2} \mathop \int \limits_{0}^{S} f\left( {X|P_{o} } \right)dX$$

(16)

$\frac{dE\left[ L \right]}{dS}$ is equal to zero when

$$\mathop \int \limits_{0}^{S} f\left( {X|P_{o} } \right)dX = \frac{{\theta_{1} }}{{\theta_{1} + \theta_{2} }}$$

(17)

The definitions of θ ₁ and θ ₂ imply $\theta_{1} + \theta_{2} = P_{o} - \bar{P}$. Substituting these into Eq. 17 produces the result in the body of the text.

Appendix 2

Equation 17 can be written as:

$$Prob [X \le S | offering\,price = P_{0} ] = \frac{{\theta_{1} }}{{\theta_{1} - \theta_{2} }}$$

(18)

Taking the partial derivative of Eq. 18 with respect to θ ₁ results in the following inequality:

$$\frac{\partial }{{\partial \theta_{1} }}Prob\left[ \cdot \right] = \frac{{\theta_{2} }}{{\left[ {\theta_{1} + \theta_{2} } \right]^{2} }} > 0$$

(19)

An increase in θ ₁ is equivalent to an increase in the underwriting discount. If the price paid by the syndicate to the issuer is fixed and the syndicate has the contractual right to set the offering price, then we have the result:

$$sign\frac{\partial }{{\partial \theta_{1} }}Prob\left[ \cdot \right] = sign\frac{\partial }{{\partial P_{0} }}Prob\left[ \cdot \right]$$

(20)

The result in 20 proves Corollary 1.

Calculating the partial derivative of Eq. 18 with respect to θ ₂ results in:

$$\frac{\partial }{{\partial \theta_{2} }}Prob\left[ \cdot \right] = - \frac{{\theta_{2} }}{{\left[ {\theta_{1} + \theta_{2} } \right]^{2} }} < 0$$

(21)

For a fixed value of the offer price, an increase in θ ₂ is equivalent to a decrease in the conditional price expected in the public market if the issue is overpriced. Thus we infer that a decrease in the price expected in the after-market is associated with a decrease in the numerical value of the cdf governing the demand for shares in the market. If the underwriters’ gross spread is fixed, and if the underwriters expect the issue to be overpriced, the optimal numbers of shares in the offering will likewise diminish. This result proves Corollary 2.