Fund flow with a lower bound

It is worth noting that many models in the literature assume that there is no lower bound of fund value (For example, Taylor [2003]), which is not realistic as fund wealth can never go below 0, and fund managers will never receive negative compensation. To account for this, we modify the piecewise fund flow function by adding a lower bound for the fund value. We assume that the fund flow will not go below the year-end wealth, i.e., F_i ≥ −W_i,2, where W_i,2= max{m_i[θ_i,1(x − r) + 1 + r], 0}. Thus, the fund flow function can be expressed as follows.

F_i =

Figure 4.1 depicts the fund flow function with a lower bound, which will be used for the remainder of the paper. In order to illustrate the tournament behavior, we solve numerically for the optimal portfolios of managers who aim to maximize the expected fund flow. We will frequently refer to a standard set of parameters as displayed in Table 4.1. Given the values of the parameters, the manager’s relative interim performance is always smaller than the threshold

−_2a(1+r)^b = 2.72 and thus the sign of the first term of Equation 4.8 is always positive.

Figure 4.2 shows that this adjustment leads to a very different risk taking choice when the fund has poor interim performance. When the year-end fund flow approaches its lower bound, the fund manager has an incentive to increase the variance of his relative performance as he is essentially facing a call option. The

Table 4.1: Standard Parameters

Mean µ 0.05

Volatility σ 0.10

Interest rate r 0.02

Industry mid-year performance m_j 1.054 Individual mid-year performance m_i [0.1,2]

Industry risk-taking θ_j [0,5]

Risk Aversion γ 4

The coefficient of squared relative performance a -0.196 The coefficient of relative performance b 1.088

The average net fund flow c 0.0568

Management fee k 0.02

Incentive fee y 0.2

Initial fund value W₀ 1

Current HWM H 1

Liquidation boundary B 0.5

Outside compensation C_out 0.005

Note: we assume the risky asset has a mean return of µ = 0.05 and a standard deviation of σ = 0.10. The riskless asset yields r = 0.02.

Based onChen[2011]’s empirical findings, we assume that a = −0.196, b = 1.088. According toGetmansky[2012], the average net flows into hedge funds is 5.68% and the average monthly return of the industry is 0.9%. Thus we set c = 0.0568 and the average industry mid-year performance of hedge funds as mj = 1.054. The basic management fee and incentive performance fee are set as the standard rate of 2%

and 20% respectively. The risk aversion coefficient of the manager’s power utility is γ = 4. In addition, the starting fund value of 1 equals the current high water mark and the liquidation boundary is set as B = 0.5.

Figure 4.1: Manager i’s fund flow function

fund flow function thus exhibits both concavity and convexity features as shown in Figure 4.1. For this reason, hedge fund managers’ risk taking choice depends on the interplay of these two opposing effects. Specifically, when a fund is far behind the industry, convexity dominates concavity and the fund manager would optimally choose to increase the tracking error variance. When the concavity of the fund flow function dominates over convexity, managers would mimic the benchmark to decrease the variance of their performance relative to the industry average.

In Figure 4.2, we identify two regions where managers would opt for com-pletely different risk shifting strategies. Given the value of m_j, when m_i drops below a threshold M , managers would try to maximize the tracking error vari-ance, and we label the region where m_i < M as ‘contrarian’ region. As we know, the volatility of the relative performance is σ_w = |θ_i − ^m_m^j

iθ_j|m_iσ. To maximize σ_w, managers would take either the highest risk θ when ^m_m^j

iθ_j is small or no risk when ^m_m^j

iθ_j is too high. When m_i > M , managers would mimic the benchmark, and we call it ‘mimicking’ region. To better illustrate the mimicking behavior of

Figure 4.2: Risk-neutral manager i’s optimal portfolio choice to maximize the expected fund flow (with lower fund flow boundary)

managers, we also solve the case when there is no risk premium.(See Figure18in Appendix C.2). Basak, Pavlova, and Shapiro [2007] provide similar insights- the better-performing fund follow the benchmark while the under-performing funds choose to deviate away from the benchmark.

Since relative performance plays a critical part in the manager’s risk shifting decision, we alter the value of mid-year industry average performance m_j to see how it would change the manager’s optimal risk choice. Figure 4.3 reveals that with increasing m_j (decreasing interim relative performance m_i − m_j), the

‘contrarian’ region widens, which means that with poor interim performance, managers are more likely to allocate all the money to the riskless asset in order to maximize their tracking error variance instead of minimizing it. This is because with lower relative performance m_i − m_j, they are closer to the lower bound of the fund flow function. On the other hand, the optimal risk level is more likely to hit the upper bound θ when m_j is higher in the mimicking region. This is

Figure 4.3: Sensitivity analysis of the industry’s interim performance m_j

because managers need to take on more risk in order to mimic the benchmark as

mj

mi increases.

In Figure4.4, we show the effect of concavity of the fund flow function. As we know, the concavity feature of fund flow controls manager’s mimicking behavior.

When the concavity is dominant, the more concave of the fund flow function is, the closer the optimal risk θ_i^∗ is to ^m_m^j

iθ_j, as the manager would have a stronger incentive to minimize the tracking error variance and exhibit stronger mimicking propensity. However, when the convexity of the fund flow is dominant, parameter a does not affect manager’s risk choice. As shown in Figure 4.4, the ‘contrarian’

region remains unchanged when we increase the value of a.

Proposition 5 The convexity feature of the fund flow function dominates over the concavity when a fund’s relative performance is close to the lower bound of

Figure 4.4: Sensitivity analysis of the concavity, a, of the fund flow function.

the money flow and the hedge fund manager would shift risk to increase tracking error variance. As the relative performance improves, the concavity of the fund flow function becomes dominant and the manager would mimic the industry to minimize the tracking error variance. Based on the relative interim performance, hedge fund managers choose one of these two opposite risk shifting strategies.