Question

Question 4. You own a telecommunications and equipment company looking to expand your business by identifying underperforming competitors, buying them, improving their performance and stock price, and then selling them. You have found such a prospect, Qcom. This company’s marketing department is mediocre; you believe that if you take over the company, you will increase its value by 80% of whatever it was before. But its accounting department is very good; it can conceal assets, liabilities, and transactions to a point where the company’s true value is hard for outsiders to identify. (But insiders know the truth perfectly.) You think that the company’s value in the hands of its current management is somewhere between $20 million and $120 million, uniformly distributed over range. The current management will sell the company to you if, and only if, your bid exceeds the true value known to them.

(a) If you bid $120 million for the company, your bid will surely succeed. Is your expected profit positive?

(b) If you bid $60 million for the company, what is the probability that your bid succeeds? What is your expected profit if you do succeed in buying the company? Therefore, at the point in time when you make your bid of $60 million, what is your expected profit? (Waring: In calculating this expectation, don’t forget the probability of your getting the company.)

(c) What should you bid if you want to maximize your expected profit? (Hint: Assume it is X million dollars. Carry on the same analysis as in part (b) above, and find an algebraic expression for your expected profit as seen from the point in time when you are making your bid. Then choose X to maximize this expression.)

Answer 1

a) Given that the value of the firm have a uniform distribution between 20 and 120,

Mean value of the firm = (Lower boundary + Upper boundary)/2 = (20 + 120)/2 = $70 million

Now, in case of turnaround, there is increase in value by 80%
Increased value = 70*180% = 126 million

Expected value = $126 million
Price paid to buy the company = $120 million
Expected profit = $126 million - $120 million = 6 million

Hence, if i buy company at 120$ million, expected profit is positive

b) Using the formula for probability density function for a uniform distribution (assuming continuous) to calculate the success rate of the bid
1/(Upper boundary - Lower boundary) = 1/(120 - 20) = 1%

Expected value = $126 million
Price paid to buy the company = $60 million

Expected profit = Probability of bid being successful * (Expected value - Price paid)
= 0.01 * (126 million - 60 million)
= 0.01 * 66million
= $0.66 million

c) Using the formula for probability density function for a uniform distribution (assuming continuous) to calculate the success rate of the bid
1/(Upper boundary - Lower boundary) = 1/(120 - 20) = 1%

Expected value = $126 million
Let the price paid to buy the company be $X million

Expected profit = Probability of bid being successful * (Expected value - Price paid)
= 0.01 * (126 million - X)

The above equation will be maximised when X is the lowest and in this case the lowest is $20 million
Substituting the values, we get,
Expected profit = 0.01 * (126 million - 20 million) = $1.06 million

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