1. (a) (1 pt.) First, you decide to use a hash table. You will make a has table that maps a piece of text to a bucket, then scrape the web for all English sentences, and hash each one to your table. Given a new sentence, you will check to see if it hashes to an empty bucket—if so, you will output option “fresh” otherwise you will output “potential plagiarism.” Suppose there are 1 billion unique sentences online. How many buckets will your hash table need to have to have the desired functionality?

   [We are expecting: A number (to the nearest order of magnitude) and one to two sentences of justification.]

2. (b) (2 pt.) You decide that is a little too much space usage, and consider the following approach: you choose 10 hash functions,h1,…,h10that each map sentences to the numbers 1 though 10 billion. You initialize an arrayAof 10 billion bits, initially set to 0. For each sentencesthat you encounter, you computeh1(s),h2(s),…,h10(s), and set the corresponding indices ofAto be 1 (namely you setA[h1(s)]?1, A[h2(s)]?1, . . .). Argue that after processing the 1 billion unique sentences, you expect a (1-1/(10 billion))10 billion˜0.37 fraction of the elements to be 0.

   For this part, feel free to assume that thehiare “idealized” hash functions that map each keys to a uniformly random bucket.
   [We are expecting: A paragraph with your argument.]

(c) (2 pt.) Now, given a sentences, to check if it might be plagiarized, you compute the 10 hashes of s, and check ifA[h1(s)] =A[h2(s)] =. . .=A[h10(s)] = 1. If so, you output “potential problem,” otherwise you output “fresh.” Prove that ifsis actually in your set of 1 Billion sentences, that you will output “potential problem” with probability 1, and that ifsisnotin your set of 1 Billion sentences, you will output “fresh” with probability˜1-(1-0.37)10˜0.99.

Again, feel free to assume that the hash functions are “idealized,” and that the claim of the previous part holds, namely that after processing the 1 Billion sentences, there are 3.7 billion indices in the arrayAwith value 0.
[We are expecting: Informal mathematical justifications for each of the bounds.]

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The plan called for KPE’s current shareholders to exchange their shares for 21% of the new firm’s common equity, or 204,898,750 shares. KKR’s current principals would retain the remaining (770,809,583) shares. To demonstrate their commitment to the long-term performance of KKR, the principals also proposed to grant Contingent Value Interests (“CVIs”) to KPE shareholders that would transfer up to an additional 6% of the new firm’s shares to KPE shareholders depending on the firm’s share price three years from closing. The relevant terms of the CVI (per KPE share turned in) are as follows:

If on the 3rd anniversary of the closing the new firm’s shares trade at or above $22.25 per share, the CVIs granted to the KPE shareholders would expire worthless.[2]

If the shares trade at or below $17.31 per share, KKR’s principals would transfer an additional 6% of the firm’s market value to the CVI shareholders.

At any price between $17.31 and $22.25 per share, the principals would transfer an additional percentage share of the firm’s market value such that each KPE shareholder would receive the difference between $22.25 and the firm’s share price.

1. (2pt) Assuming that KPE shareholders retain their initial KKR share grant and the CVIs attached to that grant, what is the maximum number of shares held after three years by KPE shareholders?

2. (2pt) What is the total payoff for a KPE shareholder who is granted one share of KKR stock and one unit of the CVI at closing if the KKR share price is $17.31 after three years?

3. (2pt) What is the total payoff for this KPE shareholder if the share price is $22.25 after three years?

4. (2pt) What is the total payoff for this KPE shareholder if the share price is $24.75 after three years?

5. (4pts) Draw and label the payoff diagram for a KPE shareholder who is granted one share of KKR stock and one unit of the CVI at closing. (see last page for graph template)

6. (3pts) On a separate graph, draw and label the payoff diagram for one unit of the CVI. (see last page for graph template)

7. (5pts) What combination of options, stock, and/or risk-free bonds would replicate the payoff from one unit of the CVI?

4. (10pts)Compare and contrast bookbuilding with Hambrecht’s Dutch auction for IPOs. Your response should include discussion of the fundamental incentive distortion they aim to resolve, how they go about doing so, and their relative strengths and weaknesses.

5. (5pts) Suppose “dirty” auctions became the norm for IPOs. What is a dirty auction and why might it be inferior to bookbuilding? You may wish to tie your response to your response to the preceding question.

6. (15pts)In what sense did Rural Metro’s board breach its fiduciary duty to shareholders? How did RBC “aid and abet” the Board’s breach of fiduciary duty? A strong response to these questions will evidence a good understanding of the economic tradeoffs and incentive conflicts facing sell-side boards and bankers as well as the legal standards for behavior.

7. (5pts)According to Holmstrom (2015), what is the primary purpose of the money markets and how is that purpose served by “over-collateralized” debt?

8. (10pts)Explain why repurchase agreements fit Holmstrom’s notion of over-collateralized debt and why repos were at the core of the 2008 financial crisis. Your response should evidence understanding of how repo financing contributed to a liquidity crisis that effectively led to a run on the shadow banking system.

9. (15pts) Several boutique investment banks already have gone public and Parella Weinberg Partners (PWP) has considered this option. Given that the founders of most of these firms have spoken of the benefits of being narrowly focused private partnerships, why would they want to go public? Your response should include a critical review of the costs and benefits from operating as a partnership, the tradeoffs facing a boutique that is considering going public, and should connect these ideas with PWP’s thoughts on organizational structure and culture.

[1] KPE was essentially a publicly-traded mutual fund that enabled unitholders (shareholders) to participatein some of KKR’s private equity investments.

[2]The payoff actually depended on the 30-day average price at the end of year three but we will ignore that for the purposes of this problem.

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