Financing a Project using Venture Capital

A Signalling Game

An entrepreneur E is seeking financing for a certain venture which he cannot finance with his own funds. The project is widely known to be viable, but only the entrepreneur knows precisely the value of the project. The entrepreneur knows that the project can either be worth H or L after an investment of I dollars. Since the entrepreneur does not have funds to finance the project he needs to get a venture capitalist to fund it. In return for investing in the project the entrepreneur can offer the capitalist an equity stake of e, where e can vary from 0 to 1.

We assume that this is a two period operation. The investement takes place in period 1 and return is realized in the second period. The rate of return i on the investement can be thought of as the current rate of interest or as the opportunity cost of capital.

The venture capitalist whom we will call player C, can either accept the offer of equity e or reject it. At the time the offer is made, player C knows only that the project is worth H dollars with probability p and L dollars with probability 1- p. The number p is nature's choice and is treated by the players as a parameter. Therefore, the venture capitalist makes his decision of whether or not to accept the offer on the basis of the offer e and his knowledge of the chances of success of the project. Thus, the game that the two players are engaged in is a signaling game in which the entrepreneur (the sender of the signal) sends the signal e, and the venture capitalist (the receiver of the signal) reacts on the basis of the signal.

The signaling game, which is a game of imperfect information without subgames, is shown in the figure. If

H - I > L - I > (1+i)I,

then the venture capitalist, player C, who knows that the project is worth H with probability p and L with probability 1 - p, will accept an offer e only if

(**) p(eH - I) + (1 - p)(eL - I) >= (1 + i)I

That is, once player C has to choose at his information set, he realizes that he is at node X with probability p and at node Y with probability 1 - p. Given this belief, it is rational for player C to accept the offer only if e satisfies (**). Further, given the information that player C has, the beliefs are sequentially rational. Knowing this, player E, the entrepreneur, then offers player C an e that satisfies (**). This leads to a sequential equilibrium in which:

player C has beliefs given by p and 1 - p at the nodes in his information set,
player E offers an equity stake e which is a little more than the smallest equity state e*<1 that makes (**) an equality independently of whether he observes H or L, and
player C accepts the offer e since it is sequentially rational.

Therefore, in case e*<1, the project is financed and the venture capitalist gets an equity stake e which is a little more than e*.

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