Question

An investment bank has been asked to underwrite an issue of 10 million shares by a company.

The bank is trying to decide between a best-efforts deal where it charges a fee of $0.2 for each share sold and a firm-commitment deal where it buys the shares for $10 per share.

For the latter deal, the bank considers that the selling price per share is either $10.8 or $9.8.

What are the break-even probabilities of the two selling prices so that the bank is indifferent to the two deals (Assume that all 10 million shares are sold out)?

Answer 1

Answer-

In the case of best efforts deal the fee charged = $ 0.2 for each share
Number of shares issued = 10 million shares

Total amout earned by the investment bank in the best efforts deal = fee charged per share x number of shares issued

Total amout earned by the investment bank in the best efforts deal = $ 0.2 x 10 million = $ 2 million

In the firm-commitment deal the buying price per share = $ 10 per share

The selling price per share = $ 10.8 or $ 9.8 per share
If the selling price is $ 10.8 the profit = $ 10.8 - $ 10 = $ 0.8 per share
If the selling price is $ 9.8 the loss = $ 9.8 - $ 10 = - $ 0.2

Let the probabilities of profit and loss are p and (1-p)

Given that the bank is indifferent between the two deals means the probabilities should be such that the profit should be same in both deals

profit in best-efforts deal = profit in firm-commitment deal

Profit in best efforts deal = $ 2 million
Profit per share in best efforts deal = $ 2 million / 10 million = $ 0.2 / share

Therefore

( profit per share x probability of profit + loss per share x profitability of loss ) = $ 0.2
substituting the values we get

$ 0.8 x p + (- $ 0.2) ( 1-p) = $ 0.2
$ 0.8 p + ( - $ 0.2 ) + $ 0.2 p = $ 0.2
$ 0.8 p + $ 0.2 p = $ 0.2 + $ 0.2
$ p = $ 0.4
p =0.4

1- p = 1- 0.4 = 0.6

Therefore the break-even probabilities of the two selling prices so that the bank is indifferent to the two deals

= p = 0.4 = 40 % for selling price of $ 10.8 and

= 1-p = 0.6 = 60 % for selling price of $ 9.8

Therefore it is 40 % for selling price of $ 10.8 / share and 60 % for selling price of $ 9.8 / share