Fund Manager s Portfolio Choice


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1 Fund Manager s Portfolio Choice Zhiqing Zhang Advised by: Gu Wang September 5, 2014 Abstract Fund manager is allowed to invest the fund s assets and his personal wealth in two separate risky assets, modeled by binomial tree pricing model, and a risk free asset. This project aims at an optimal portfolio choice for fund manger to maximize his expected utility on personal wealth in a finite horizon. 1
2 1 Introduction The criteria that fund managers use to select their assets vary widely according to the individual manager. The investors must look closely at the fund manager s investing style to ensure it fits his or her riskreturn profile. A fund manger s investing style is largely affected by the amount of his personal wealth, how he gets compensated, and whether he is allowed to invest his personal wealth. To start with, fund manger with higher personal wealth tends to exert less effort on driving fund flows. The intuition behind this is a decline in the marginal utility of wealth as manager keeps accruing wealth. Jensen s free cash flow hypothesis (see [1]) also states that firms with high levels of cash flow will waste it on negative net present value projects. In real life, fund mangers usually have long tenure, and due to the increase of his personal wealth, he become less and less motivated to generate excess returns. However, wealthier managers are usually more experienced and more skilled. Therefore, a manger s strategy even varies through the course of his career. Furthermore, fund manager charges fees from clients: management fee and performance fee. Management fees, covering advisory services and administrative services, is calculated as a proportion (generally 1%2%) of assets under management. Performance fee is generally calculated as 20% of the increase in the net asset value (NAV) of the fund. To prevent managers getting paid large sums for poor performance, the fund must hit high water mark, the highest NAV of a fund to date, before it can charge a performance fee. So if a fund loses money over a period, the manager must get the fund above the high water mark before receiving a performance bonus. Guasoni and Obloj researched on the incentives of fees and high water marks at 2013 (see [2]). Since fund manager bears no potential loss of a highrisk portfolio, he can always devise a portfolio with high returns at the price of taking high risks. Some clients require fund manager to invest his personal wealth in the same fund so that he will incur losses when fund s wealth goes down. Finally, fund manager can also hedge personal investments with fund s investments. The fund manager usually has more information about the market than the clients have. Also, he is able to make decisions on behalf of the clients. The manager thus has an incentive 2
3 to act inappropriately if there is a conflict of interest. Guasoni and Wang did a research on this problem in a long horizon at 2014 (see [3]) and concluded that fund manager makes fund s portfolio choice and private portfolio choice independently in certain situation. Based on the assumption that manager is allowed to invest personal wealth, and that manager receives ϕ (0 < ϕ < 1) of assets value as management fees, this project will try to find an optimal strategy for both manager s personal wealth and the fund s wealth to maximize utility in a finite horizon. 2 Model A fund manager invests in fund s wealth as well as personal wealth. His goal is to maximize expected utility of management fees from fund s asset plus earnings from personal investments. To spread the risk, the fund manager will allocate fund s assets F in a risk free asset with constant return r and a risky asset S F, whose price fluctuation follows binomial model. The manager will also allocate his personal wealth X in a risk free asset with the same constant return r and a risky asset S X. 2.1 Binomial Model We use Binomial Model (see [4]) for asset prices. In our model, the price of risky asset S F follows a binomial model. We call the beginning of a period time zero and the end of the period time one. At time zero, we denote the price of risky asset S F as S0 F, which is a positive number known at time zero. At time one, the price of S F will be S1 F, which is one of two positive values u F S0 F and d F S0 F. We can imagine that the result of a tossed coin determines the price at time one but the coin is not necessarily fair. We assume that the probability of result u F S0 F is p F (p F > 0) and the probability of result d F S0 F is 1 p F (1 p F > 0). Similarly, S X yields gross return u X with probability p X and d X with probability 1 p X. 3
4 We assume that u > d for both S F and S X, if d > u, we can achieve u > d by relabeling the symbol; if d = u, the stock price at time one is not random and the model is not interesting. Thus it is helpful to think of u as the up factor and always greater than one, and d as the down factor and less than one. 2.2 Preference To maximize the manager s welfare, we maximize terminal expected utility of manager s investment. In Finance, we face the tradeoff between return and risk. One can always devise investments with arbitrarily high return, but at the price of taking arbitrarily high risks. Suppose we invest x dollars at time zero, and at time one we collect some random payoff X. The question is, given several random payoff, how do we decide which one is better? In other words, instead of simply choosing the random payoff with highest expected value, we need to take account of variance. That is why we maximize the expected utility which accounts for both risks and return. To model people s preference, a utility function possesses two properties: increasing and concave. Firstly, it is increasing because people are nonsatiated and prefer more to less. Secondly, the concavity of utility function has two related consequences (see [5]). One is the decreased marginal utility. As we add 1 dollar to a payoff, the increase in expected utility is lower and lower. The last bite is never like the first bite. The other consequence is Jensen s inequality: U(E[X]) E[U(X)], which shows most people are risk averse. Risk aversion (green line) imply that people may refuse to play a fair gamble even though the expected value is zero, since the joy from winning is less than the 4
5 sorrow from losing. On the other hand, risk loving individuals (red line) may choose to play the same fair gamble, and risk neutral individuals (blue line) are indifferent between playing or not. We will use concave down utility function in this project since most people are risk averse. The concavity of U(X) measures one s risk aversion level. Suppose random payoff X has expected value x and variance σ 2. A person is willing to pay a risk premium, RP to get rid of that variance. Thus U(x RP ) = E[U(X)]. Using Taylor s expansion, U(X) U(x) + U (x)(x x) U (x)(x x) 2 (1) Taking expectation of (1), E[U(X)] U(x) U (x) σ 2 (2) Plug in X = x RP to (1), U(x RP ) U(x) U (x) RP (3) Set (2) and (3) equal, RP σ 1 U (x) 2 2 U (x) We call U (x) U (x) the absolute risk aversion. Define the risk premium as a proportion of total wealth, we can get the relative risk aversion, U (x) x. For detailed discussion, please see [5]. U (x) We will use utility function U(X) = lnx and U(X) = 1 1 γ X1 γ in this project, where γ allow to parametrize preferences. We can actually prove γ is directly proportional to the 5
6 relative risk aversion by plugging in U(X) = 1 1 γ X1 γ and simplifying U (x) x U (x). 2.3 One Period Model U (X) = X γ U (X) = γ X γ 1 (4) Relative Risk Aversion = U (X) X U (X) = γ (5) To model the way how a fund manger allocates fund s wealth and personal wealth. We suppose the fund manager will invest a percentage π F of fund s wealth in asset S F and the rest in a risk free asset. He will also invest a percentage π X of personal wealth in asset S X and the remaining in a risk free asset. With such allocation between two assets, the return on fund s wealth of one period time is a weighted average between S F s return and the risk free rate r, which is [π F (S F 1 /S F 0 )+(1 π F ) (1+r)]. Therefore, the fund s wealth at time one is F 1 = F [π F (S F 1 /S F 0 )+(1 π F ) (1+r)]. The fund manager collects a proportion ϕ of fund s wealth at the end of one period as management fees ϕ F 1 = ϕ F [π F SF 1 S F 0 + (1 π F ) (1 + r)] (6) The fund manager s personal wealth, including management fees and personal wealth from previous period, grows in a different way. X 1 = ϕ F 1 } {{ } management fees + X [π X SX 1 + (1 π X ) (1 + r)] S0 } X {{ } personal wealth from previous period (7) Plug in equation (6), we get X 1 = ϕ F [π F SF 1 S F 0 + (1 π F ) (1 + r)] + X [π X SX 1 S X 0 + (1 π X ) (1 + r)] (8) When faced with choices about random wealth X 1, the individual acts to maximize his expected utility, E[U(X 1 )]. Our goal is to determine π F and π X which maximize E[U(X 1 )]. 6
7 3 Result 3.1 Utility Function U(X) = lnx To find the optimal strategy that maximize the expected utility function, E[U(X 1 )] = E[ln(X 1 )], we take its derivative with respect to π F and π X respectively and set them to zero. Take the derivative with respect to π F. E[ln(ϕ F 1 )] = p F ln{ϕ F [π F u F + (1 π F ) (1 + r)]} π F π F +(1 p F ) ln{ϕ F [π F d F + (1 π F ) (1 + r)]} = p F [π F u F +(1 π F ) (1+r)] (uf 1 p 1 r) + F [π F d F +(1 π F ) (1+r)] (df 1 r) (10) Set the result equal to zero, p F (u F 1 r) π F u F + (1 π F ) (1 + r) = (1 pf ) (1 + r d F ) π F d F + (1 π F ) (1 + r) p F (u F 1 r) π F (u F 1 r) r = (1 pf ) (1 + r d F ) π F (d F 1 r) r π F (u F 1 r) (d F 1 r) = (1 + r) [(1 p F ) (1 + r d F ) + p F (1 + r u F )] (13) (9) (11) (12) Similarly, we can get π F = (1 + r) [(1 pf ) (1 + r d F ) + p F (1 + r u F )] (u F 1 r) (d F 1 r) π X = (1 + r) [(1 px ) (1 + r d X ) + p X (1 + r u X )] (u X 1 r) (d X 1 r) (14) (15) 3.2 Utility Function U(X) = 1 1 γ X1 γ E[ 1 1 γ (ϕ F 1) 1 γ ] π F = π F p F 1 1 γ {ϕ F [πf u F + (1 π F ) (1 + r)]} 1 γ +(1 p F ) 1 1 γ {ϕ F [πf d F + (1 π F ) (1 + r)]} 1 γ (16) = p F ϕ F [π F u F +(1 π F ) (1+r)] γ ϕ F (u F 1 r) 1 p F + ϕ F (d F 1 r) ϕ F [π F d F +(1 π F ) (1+r)] γ (17) 7
8 Set the result equal to zero, p F (u F 1 r) [π F u F + (1 π F ) (1 + r)] = (1 p F ) (1 + r d F ) (18) γ [π F d F + (1 π F ) (1 + r)] γ [ πf u F + (1 π F ) (1 + r) π F d F + (1 π F ) (1 + r) ]γ = p F (u F 1 r) (1 p F ) (1 + r d F ) π F (u F 1 r) + (1 + r) π F (d F 1 r) r = [ p F (u F 1 r) (1 p F ) (1 + r d F ) ] 1 γ (20) (u F 1 r) [(1 p F ) (1 + r d F )] 1 π F γ [p F (u F 1 r)] 1 γ = (1 + r) (d F 1 r) [p F (u F 1 r)] 1 γ [(1 p F ) (1 + r d F )] 1 γ (19) (21) π F (1 + r) {[p F (u F 1 r)] 1 γ [(1 p F ) (1 + r d F )] 1 γ } = (u F 1 r) [(1 p F ) (1 + r d F )] 1 γ (df 1 r) [p F (u F 1 r)] 1 γ Similarly, (22) π X (1 + r) {[p X (u X 1 r)] 1 γ [(1 p X ) (1 + r d X )] 1 γ } = (23) (u X 1 r) [(1 p X ) (1 + r d X )] 1 γ (dx 1 r) [p X (u X 1 r)] 1 γ We can see from the solution of π F and π X that fund manager s portfolio choice in personal wealth and in fund s wealth are independent in one period. The reason is that there is no correlation between S F and S X. Also, ϕ does not appear in the solution, which means that ϕ does not affect fund manger s choice. This is because we only considered fund manager s portfolio choice in one time period. If the fund manger needs to reinvest the fund s wealth, he might want to take less money out from the fund at the beginning so that the increase of fund s wealth would be more significant in the future. ϕ will affect his portfolio choice in multiple period. 4 Future Work 4.1 Correlation between S F and S X In this project, there is no correlation between two risky assets that the manager is investing in. Thus the strategies are relatively independent. If two assets are positively or negatively correlated, the manager can use this information to develop a different strategy. Moral hazard arises in this situation. 8
9 4.2 High Water Mark As we said in the introduction, high water mark is the highest NAV of a fund to date that a fund must hit to charge a performance fee. If high water mark is used as a criteria, fund manager has an incentive to take arbitrarily high risks in order to hit high water mark. At the same time, he is free from any loss that might caused by the high risks. References [1] Jensen, Michael C., The Performance of Mutual Funds in the Period , Journal of Finance 23, , 1968 [2] Guasoni, Paolo and Obloj, Jan, The Incentive of Hedge Fund Fees and HighWater Marks, Mathematical Finance, [3] Guasoni, Paolo and Wang, Gu, Hedge and Mutual Funds Fees and the Separation of Private Investments, Working paper, [4] Shreve, Steven., Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, New York, SpringerVerlag, 2005 [5] Ingersoll, Jonathan E. Jr., Theory of Financial Decision Making, Totowa, Rowman and Littlefield,
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