2. Dupont University Technology Fund (20 Points) Many universities have formed venture capital funds to help turn campus research discoveries into marketable products. Here we consider the fund of a hypothetical university, Dupont University, which is managed by a New York-based venture-capital firm, Wolfe Ventures. Wolfe Ventures serves as general partner for the fund and recruited additional investors who are limited partners in the fund. We will look at evaluating the risks and returns associated with this fund, after a few years of operation, when the fund's assets have been fully invested in start-up companies. Suppose the Dupont University Technology Fund (DUTF) has investments in 14 companies, as described in Table 1 below. (Note: This data is available for download on Canvas.) Company A B C D E F G H I J K L M N Table 1: All amounts are $ millions Probability of a Amount Liquidity Invested Event 19.7 0.90 8.2 0.90 8.6 0.80 1.5 0.25 2.9 0.65 19.4 0.90 1.6 0.35 0.2 0.15 7.3 0.65 4.0 0.50 0.9 0.50 5.9 0.80 0.4 0.10 0.1 0.20 Value of Liquidity Event: Lognormal with Mean Std. Dev. 30.00 20.00 10.00 2.50 15.00 7.50 8.50 6.00 4.40 1.26 20.00 10.00 5.70 2.30 2.20 1.10 12.50 12.50 7.50 2.50 2.00 2.25 13.40 7.50 5.00 3.75 1.00 2.25 Total 80.7 The second column in Table 1 shows how much DUTF has invested in each of these companies; these amounts have been committed to the companies and are now sunk costs. The third column shows the probability that each DUTF company has a "liquidity event" where DUTF's interest in the company is bought out by some other investors. If there is no liquidity event for a company, the payoff to DUTF is assumed to be zero and the entire amount invested by DUTF is lost. The payoff to DUTF associated with a liquidity event is quite uncertain. Some companies may show great promise and generate a large payoff while others may have a small payoff, perhaps less than DUTF's initial investment. The payoff from a liquidity event is modeled as a lognormal distribution where the means and standard deviations of the lognormal distribution are specific to the company. These means and standard deviations are listed in the fourth and fifth columns of Table 1. (The "locations" for the lognormal distributions are all equal to zero.) To simplify the model, assume that the occurrence of liquidity events for the different companies are independent and the (potential) payoffs associated with a liquidity event for the different companies are independent. 4 Please run your model for at least 10,000 trials to ensure a reasonable level of accuracy in your results. Build a simulation model to answer the following questions: a) (4 points) i) What is the expected total payoff of the DUTF portfolio? ii) Provide a 95% confidence interval for the expected total payoff of the DUTF portfolio: iii) What is the probability that the total payoff is more than the total invested? b) (2 points) We call a DUTF company a "success" if it has a liquidity event and the payoff exceeds the amount invested by DUTF in the company. i) What is the expected number of DUTF companies that will be successes by this standard? ii) What is the probability that 10 or more companies will be successful by this standard? Payoffs from the DUTF companies are distributed to the fund managers (Wolfe Ventures), to DU, and to the fund's limited partners. Wolfe Ventures will be paid a fee equal to 10% of all payoffs from the DUTF companies. The remaining proceeds are distributed to DU and the fund's limited partners (the outside investors). In recognition of DU's contribution of much of the intellectual property underlying the DUTF companies (as well as their financial investment), DU's claims to DUTF proceeds have been granted a higher priority than the limited partners' claims. Specifically, DU will receive the first $75 million generated by the fund, after Wolfe Ventures' fees. The limited partners will not receive any payment until after DU has received its $75 million. However, DU's share of the proceeds is capped at $75 million; if the fund generates more than that amount, DU will not further benefit and the limited partners will divide the proceeds. For example, if the total payoff of DUTF is $100 million, Wolfe Ventures would receive $10 million, DU would receive their full $75 million, and the limited partners would receive the remaining $15 million. If the total payoff is $80 million, Wolfe Ventures would receive $8 million, DU would receive $72 million and the limited partners would receive nothing. c) (4 points) i) What is the expected payment to DU from DUTF? ii) What is the probability that the limited partners receive any money at all? d) (4 points) One of the limited partners is a wealthy DU alumnus named Charlotte Simmons; she holds fifty percent (50%) of the limited partners' share. Simmons is risk averse and has an exponential utility function with a risk tolerance of $100 million. i) What is the expected payoff of Simmons's investment in DUTF? ii) What is Simmons's certainty equivalent for her investment in the fund? 5 e) (4 points) To better align incentives, the DUTF managers and investors are contemplating changing Wolfe Ventures' management fee structure from the current 10% fee on gross proceeds to a new fee that depends on the net proceeds for the fund, i.e., what they earn beyond the $80.7 million that was invested in the fund. The question is how to determine a new percentage that is equitable to Wolfe Ventures in that it gives them the same expected value as the original deal. For example, with this new fee structure, if the new fee is set at f% and the proceeds from the fund total $100 million, Wolfe Ventures would receive f% ($100$80.7). If the total proceeds fall short of $80.7 million, Wolfe Ventures would receive nothing. What fee f would be equitable to Wolfe Ventures in that it would lead them to have the same expected value as the original deal? (You can approximate f within 1%, if necessary.) f) (2 points) For this part of the question, assume that Wolfe Ventures' original fee structure is in place, not the one contemplated in part e. For simplicity, we have assumed that the occurrence of liquidity events for the different companies are independent and the (potential) payoffs associated with a liquidity event for the different companies are independent. Because the occurrence of a liquidity event is driven primarily by technological factors (e.g., does this new drug work?), it may be reasonable to assume that the liquidity events are independent. However, the assumption that the payoffs given a liquidity event are independent is not a good assumption: the payoffs for the portfolio companies may be affected by general market conditions and hence would likely be positively correlated. Suppose the payoffs given a liquidity event for individual companies retain the same distributions (and same means and standard deviations) as given in Table 1, but rather than being independent, they are all positively correlated with each other. How would this positive correlation affect: i) The expected payments to DU? ii) The expected payments to the limited partners? We are looking for a qualitative description (e.g., would the expected payments increase, decrease, or stay the same?) and a brief explanation. You do not need to build a new model to provide a precise answer for this