Question

Horus has developed successfully and needs additional funding. An IPO seems to be a natural step forward. After the bookbuilding phase, it is decided that 1,000,000 stocks would be sold in the IPO at a per-unit price of $10. You think that there is a 0.55 probability that the price will go up to $13 (underpricing) and 0.45 probability it will go down to 8 (overpricing) . You also know that there are two types of investor in the market (in addition to you): - Uninformed (e.g., individual investors): these investors always buy, and their aggregate demand is 800,000 stocks. - Informed (e.g., hedge funds): these investors know if the issue is under- or overpriced and they have the capacity to demand up to n stocks in aggregate. If the IPO is oversubscribed (i.e., if total demand is higher than the 1,000,000 stocks offered), every investor is rationed on a pro-rata basis. For instance, if total demand is 1,250,000 stocks, every investor receives 1, 000, 000/1, 250, 000 = 0.8 times the number of stocks he asked for. You plan to buy 100 stock in this IPO, what is the largest informed investor’s capacity (i.e., largest n) such that you do not lose money in expectation?

Answer 1

To determine the maximum number of shares the institutional investors will get, we need to use the following equation:

Number of Shares We Will Get from IPO = 100*(1*10^(6))/(800,000 + n) where n is the maximum number of shares the institutional investors will get.

Now, we need to arrived at the number of shares we will get from IPO with the use of following Payoff equation:

Payoff = Probability that the Price will go to $13*(13 - IPO Price)*Number of Shares We will Get from IPO + Probability that the Price will go to $8*(8 - IPO Price)*100

Solving the equation, we get,

Payoff = .55*(13 - 10)*Number of Shares We will Get from IPO + .45*(8 - 10)*100

The payoff equation will be set off to zero in order adjust for any possible losses as follow:

0 = .55*(13 - 10)*Number of Shares We will Get from IPO + .45*(8 - 10)*100

0 = 1.65*Number of Shares We will Get from IPO - 90

Solving for Number of Shares We will Get from IPO,

Number of Shares We will Get from IPO = (90/1.65) = 54.55 shares

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Now, we can arrive at the number of shares the investors will get as below:

54.55 = 100*(1*10^(6))/(800,000 + n)

Solving for n, we get,

54.55*(800,000 + n) = 100,000,000

43636364 + 54.55n = 100,000,000

n = (100,000,000 - 43,636,364)/54.55 = 1,033,333.33 shares (answer)

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Notes:

There can be a slight difference in final answer on account of rounding off values.