This section formalises the prior argumentations and proves under which conditions an IPO is the dominant alternative for an investor and the firm. We furthermore present condi-tions that cause underpricing of an IPO. First, an analysis of the average prices in bilateral negotiations between a firm and a venture capitalist is conducted. It will be shown that this price is below the average valuationV. This means that even in bilateral negotiations we find underpricing. To understand this, we first calculate the average price resulting from bilateral negotiations.
Proposition 22. When bilateral negotiations between a firm and a venture capitalist are successful, the average deal price is
E(P|Success) = 1 (bb2−ss1)2
(
kb
[bb23
3s −b1
(b22
2 − b12s2 6b2
)]
+ (1−k)s
[b2b23
6s2 −b12 (
b2
2 − b1s 3b
)])
. Proof: See the Appendix.
Proposition 22 gives a formula for the average deal price in case of a bargaining success.
In section 2 we established optimal offer strategies (b, s) for both players in different bilateral negotiation settings. Using these strategies makes it possible to calculate the average deal price in the different bilateral negotiation settings discussed in section 2.
0 0.1 0.2 0.3 0.4 0.5
(a) Average deal prices in bilateral negotiations (k= 1/2)
(b) Average deal prices in bilateral negotiations with strategically behaving players
Fig. 16 – Average deal prices in bilateral negotiations
Figure 16 shows these prices as a function of valuation imprecision. When both players place their offers naively, the average deal price is V, according to proposition 22. Figure 16 (a) analyses the remaining bilateral negotiation alternatives10. When the firm is behaving strategically and the investor naively, there is some overpricing that increases with valuation imprecision. It reaches approximately 5% of V at the most. When the investor behaves strategically and the firm naively, then the investor reduces his offer, whereas the firm’s offer and reservation price are identical. This has the effect, that the average price is below V and decreasing in valuation imprecision. That price reaches down to approximately 86% of V, which is equivalent to 14% underpricing. When one player behaves strategically and the other naively, then the strategic player can shift the average price to his advantage. When both players behave strategically (this is indicated by the blue graph) the average deal price is decreasing in valuation imprecision and always below the average valuationV. In this case, there is significant underpricing of up to approximately 7% of V. This property of bilateral negotiations is unexpected, as one might reason that the price behaves symmetrically when both parties place their bids strategically. This effect has been explained in section 2, where it was shown that a buyer has the upper hand in bilateral negotiations. That section showed that a buyer’s profit is higher than that of a seller when they both bid strategically.
Figure 16 (b) analyses the influence of negotiation skills on the average deal price in bilateral negotiations with strategic individuals. The parameter k represents the relative
10Without loss of generality we setV = 1 in this figure
negotiation skills of the parties. Whenk = 0, the deal price is given by the firm’s offer. This means that the firm has superior negotiation skills compared to the investor. Whenk = 1 the opposite is true. Then the deal price is given by the investor’s offer and his negotiation skills exceed the firm’s. Ifk = 1/2, both parties share the same negotiation skills. The graph shows that when the investor’s negotiation skills are better than the firm’s, the deal price decreases, and vice versa. If the firm gains the upper hand in bilateral negotiations (for k = 0.25), the deal price is approximately constant at V. The average deal price even diminishes for a valuation imprecision above 20%. We conclude that in bilateral negotiations with strategic individuals, there is underpricing, even when the firm’s negotiation skills exceed that of the investor.
The next proposition shows that an IPO with asymmetric information dominates bilateral negotiations of a firm and a venture capitalist under full information. However, the investment banker needs to underprice an IPO in order to compete with the capital market which provides a spanning portfolio.
Proposition 23. LetE(PS)be the firm’s expected profit in bilateral negotiations with strategic players. Define
fmax :=α − 2 √αE(PS).
When the investment banker offers to buy a firm’s shares for Bd > V(1−fmax) and offers to sell the shares at the price Sd within the bounds Bd < Sd < V, then an IPO is Pareto efficient.
Proof: See the Appendix.
Proposition 23 develops a pricing strategy such that the IPO is Pareto efficient. The intuition behind the proposition is that, while the investment banker does not exaggerate his fee, the IPO is Pareto efficient over bilateral negotiations between a firm and a venture capitalist. We can infer from proposition 23 that the investment banker offers the IPO shares to investors below their average valuation V.
The next lemma shows an interesting property of an IPO which can be derived from proposition 23.
Lemma 2. When the investment banker applies a fee strategy as in proposition 23, IPOs are underpriced.
Proof: See the Appendix.
Lemma 2 states that IPOs are underpriced. In order to attract investors for the IPO, the investment banker needs to compete with an efficient market, according to proposition
23. Prices in the efficient market are exact and therefore the spanning portfolio for the IPO shares has the exact price V. To attract investors, the investment banker needs to price the IPO shares below that value V. In other words, the investment banker underprices the IPO shares. Otherwise investors will reject the investment banker’s offer.
Note that when a spanning portfolio for the firm’s shares does not exist, the investor looses option (c), the exactly priced duplicating portfolio. Then the investment banker’s strategy to attract an investor and a firm to the IPO is simpler.
Proposition 24. Assume there is no spanning portfolio for the firm’s shares. LetE(PB)and E(PS) be an investor’s and a firm’s expected profits in bilateral negotiations with strategic players, respectively. Define
fb :=α − 2 √αE(PB) fs :=α − 2 √αE(PS).
When the investment banker offers to buy a firm’s shares for Bd > V(1−fs) and offers to sell the shares at price Sd, with Bd< Sd < V(1 +fb), then an IPO is Pareto efficient.
Proof: See the Appendix.
When there is no spanning portfolio for the firm’s shares, it is simpler for the investment banker to install a fee structure such that an investor and a firm are in preference of an IPO.
Dropping the spanning portfolio assumption, IPOs are still underpriced on average, but to a smaller extent.
Lemma 3. Assume there is no spanning portfolio for the firm’s shares. When the investment banker applies a fee scheme as in proposition 24, IPOs are underpriced on average.
Proof: See the Appendix.
In general it is possible that IPOs may be overpriced if the assumption of the existence of a spanning portfolio is dropped. Lemma 3 however states, that on average they are under-priced. Furthermore, section 2 proved that a firm’s profit is lower than that of an investor in bilateral negotiations. Section 4 discussed that the investment banker may therefore install an asymmetric fee structure to make his market Pareto efficient over bilateral negotiations.
In fact, to successfully compete with bilateral negotiations, the investment banker may offer the firm a lower profit than an investor, because a firm’s gain in bilateral negotiations is lower than an investor’s. This asymmetric offer strategy is responsible for the IPO to be underpriced on average. We conclude that IPO underpricing is a robust property. However, an IPO is deterministically underpriced when a spanning portfolio exists.
The underpricing of IPOs is illustrated in figure 17. Using propositions 23 and 24 we computed feasible IPO prices for different valuation imprecision.
0 0.1 0.2 0.3 0.4 0.5
(a) Feasible prices with spanning portfolio
0 0.1 0.2 0.3 0.4 0.5
(b) Price borders without spanning portfolio
Fig. 17 – Feasible price strategies of the dealer
Figure 17 (a) shows feasible IPO prices with the spanning portfolio hypothesis, that is an investor’s option (c). The blue area in this figure represents feasible investment banker’s price strategies. It can be seen that IPOs are always underpriced. The intensity of the underpricing is the dealer’s choice and depends on several factors. For example, when a firm is more inclined to sell its shares, it is easier for the investment banker to satisfy the firm with a lower price.
As a result, the IPO can be underpriced to a greater extent, if the investment banker does not change his fee. When a firm needs more encouragement for an IPO, the investment banker may offer a higher price. While the investment banker’s earnings are constant, the initial shares become more costly. They are, however, still underpriced in order to dominate the investor’s spanning portfolio alternative.
Figure 17 (b) shows price bounds without the option of a spanning portfolio. The minimum price the firm demands is illustrated by the red line. It is equal to the lower price bound in figure 17 (a). The investment banker needs to offer the firm a higher price than this lower bound. When there is no spanning portfolio, the investment banker may charge the investor a price higher thanV. The maximum price he may charge the investor is illustrated by the black line in figure 17 (b). The investment banker has more freedom in pricing and thus possibly more fee earnings when imprecision increases. In this case however, IPOs are not necessarily underpriced: the investment banker may set an IPO price from the whole spectrum between 1−fs(the red line) and 1+fb (the black line). Actual pricing may be dependent on numerous parameters, such as the necessity for sellers to raise capital and investors to buy those shares.
The size of the IPO market and the supply of investment capital, negotiation skills and the investment banker’s minimum fee requirement are further factors that determine the actual IPO price.