Cutting edge: Liquidity risk A liquidity haircut for hedge funds Investors in hedge funds have learned to be cautious when making decisions due to problems of survivorship bias, autocorrelation and hidden optionality. Here, Hari Krishnan and Izzy Nelken show how to quantify such caution. By analysing the incentive structure of hedge fund managers using an option pricing approach, they derive a liquidity haircut to compensate for lockup periods, and an illiquidity premium that effectively increases volatility ince the 1950s, mean variance optimisation has been widely used to Second, we assume that an investor holds a portfolio consisting of a tra- construct portfolios of traditional assets, such as stocks and bonds. ditional fund and a hedge fund. While the hedge fund manager wants to SOver the past 10 years, alternative investments such as hedge funds maximise the value of his incentive clause, the investor wants to maximise have risen in prominence. There are more than 6,000 registered hedge the return/risk ratio of his portfolio. Thus, the hedge fund manager and in- funds worldwide. Many have performed well in bull and bear markets and vestor are playing a cat and mouse game. If the investor had unrestricted there has been a significant flow of assets into hedge funds. liquidity, he could rebalance between the hedge fund and traditional as- It is natural to ask the question: how much of an investor’s portfolio sets in response to a change in leverage or the probability of a blow-up. should be allocated to a specific hedge fund or a portfolio of hedge funds? Since he cannot rebalance, he needs to discount the return or increase the The usual approach has been to incorporate hedge funds in a mean vari- volatility of the hedge fund before he makes a decision to invest. We adapt ance framework. However, many hedge funds have outperformed stocks Longstaff’s method (1999) to calculate the illiquidity premium. and bonds (in a risk-adjusted sense) over the past few years. Thus, a naive We conclude the article with a numerical example, where we show that optimiser would allocate nearly all assets to these funds rather than tradi- the volatility of a hedge fund should be increased by a factor of about 10% tional assets. Many funds of funds have capped the allocation to hedge over its historical value, to account for illiquidity. funds at 20% to 30% in an ad hoc way. This reflects the idea that there are subtle risks in hedge fund investing. Some of these risks are: Valuing the hedge fund manager’s contract
The ‘true’ volatility of a hedge fund may be much larger than its histori- Goetzmann, Ingersoll & Ross (2001) have modelled the evolution of the cal volatility. Managers who trade illiquid assets tend to receive dispropor- high water mark and calculated the expected value of the contract to a tionately large allocations according to mean variance theory. Since the assets manager. The high water mark is the highest asset level reached, net of do not trade much, portfolio returns tend to have a large serial correlation withdrawals, since initial investment. Suppose that a fund’s asset level is from month to month. This lowers the calculated historical volatility. given by S(t) at time t and that S(t) evolves according to a discrete log-
Fung & Hseih (1997) have noted that the returns of trend-following fu- normal diffusion with drift µ – c – W and volatility σ. The asset level drift tures managers can be extremely non-normal and use an options pricing µ (assumed to be greater than zero) and management fee c are constant. approach to model them. Many managers are natural sellers of credit or Funds are added or redeemed at the rate W = W(t, S). If an investor re- liquidity premiums. This means that they generate attractive Sharpe ratios deems, then W > 0, otherwise W ≤ 0. Thus: under normal conditions while retaining a small probability of a large neg- St()+∆ t= St()+−−()()µσξ W cSt∆+ t St() ∆ t ative return.
Funds that blow up tend to disappear from the radar screen. Spurgin where ξ ~ N(0, 1) is a normally distributed random variable. (2000), among others, have documented how hedge fund returns can Suppose that, over time, the manager collects a percentage p of prof- be upwardly skewed by survivorship bias. Many composite measures of its above a high water mark H(t). In particular, suppose a manager col- hedge fund performance only include funds that have survived up to lects a performance fee of p max(S(t) – H(t – ∆t),0) at time t ≥∆t. Then, the present. Thus, these composites tend not to include the funds that the value of the manager’s performance fee is equal to the value of p call closed or did not perform well in the past. Many funds, such as Long- options on S(t) struck at H(t). The value of the call can be expressed as Term Capital Management, have had periods of excellent performance the solution to a Black Scholes-type partial differential equation over some before blowing up. time horizon T. In this article, we focus on the liquidity risk that a hedge fund investor For our purposes, it is not necessary to specify the precise value of the faces. An investor faces liquidity risk because capital is typically locked up high water mark contract. However, we can use the Goetzmann, Ingersoll in a fund for a year or two before it can be taken out.1 & Ross (2001) model to determine a manager’s optimal use of leverage. During the lockup period, a manager can vary his strategy, increase his Our leverage model is somewhat stylised. In our opinion, however, it cap- risk exposure or, worse, blow up. To our knowledge, a liquidity haircut tures a manager’s typical response to varying asset levels. Suppose that we for hedge funds has not yet been quantified and we make a calculation denote a fund’s leverage by l and assume that l is capped by lmax. (Usu- here. The haircut can be used when making an allocation decision be- ally, a fund’s offering documents set a strict limit on leverage.) If the man- tween a hedge fund and a traditional asset or fund. Our method consists ager’s strategy is scalable, then µ = µ(l), σ = σ(l) increase linearly in l. This of two main steps. First, we extend the paper of Goetzmann, Ingersoll & means that slippage does not increase as more capital is put to work. Since Ross (2001) and study the value of the incentive clause to a hedge fund 1 manager. We qualitatively describe how a hedge fund manager can opti- If a hedge fund has a one year lockup, funds can typically only be taken out at the end of the calendar year following the year of investment. Thus, an investor who allocates mally vary his leverage to maximise his expected profit, using an option money in January 2002 can only take the money out in December 2003 and the effective pricing approach. lockup period is two years

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the high water mark gives the manager a long call position, the manag- er’s expected profit should be increasing in µ and σ. Naively, a manager 1. Manager profit and loss profile would want to maximise his leverage in order to make his expected pay- 6 High vega zone, Manager collects 1.5% out as large as possible. Survival zone, leverage is high of assets + 20% of profits This is a reasonable assumption whenever S(t) is close to H(t), since in leverage is low over high water mark this region the high water mark contract increases sharply as a function of 4 volatility. However, a manager will typically lower his volatility at the ex- tremes when S(t) and H(t) are far apart. When S(t) << H(t), the fund may 2 be close to liquidation. Although a manager would like to increase the ex- Manager collects 1.5% of assets pected value of his performance fee, he doesn’t want to risk his regular 0 management fee. Increasing leverage increases the probability of default; Fund has defaulted, if this were to occur, the manager would no longer receive the percentage –2 manager loses future c.2 When S(t) >> H(t), a manager may wish to lock in a profit for the year coupon payments on $100 million for three Coasting zone, by reducing leverage and ‘coasting’. It is widely known that many man- –4 years agers who are outperforming their return targets for a year will coast. If Payout to manager ($ million) leverage is moderate Payout to manager they finish a year ahead of their stated target, investors may conclude that –6 the fund manager is unable to accurately gauge the risk in his portfolio. $70 $72 $75 $77 $80 $82 $84 $87 $89 $92 $94 $96 $99 104 106 111 113 While Goetzmann, Ingersoll and Ross (2001) think of the incentive $101 $ $ $108 $ $ clause as a call option type structure, we prefer to compare it with a risky Asset value ($ million) convertible bond. So long as a manager stays in business, he continues to receive the ‘coupon’ payment cS. In the meantime, he can collect a per- funds, since the expected return and volatility of a hedge fund depend formance fee according to whether p max(S(t) – H(t – ∆t), 0) is in-the- on the level of assets relative to the high water mark. If the hedge fund’s money. If the fund defaults, he collects nothing. We use a knock-out barrier return and volatility were constant over time, an investor would increase to model the probability of default. If, at any time t, S(t) < LH(t), where 0 his allocation to the hedge fund after a drawdown. This is not consistent < L < 1 is a constant, then many investors will withdraw and the fund will with the behaviour of most investors. Investors usually decrease their al- be forced to liquidate its positions. For example, if L = 0.85, then the fund location to a hedge fund that has suffered a negative return, for the fol- will be forced to liquidate as soon as assets decline 15% from the high lowing reasons. water mark. After liquidation, we further assume that the investor is only
The hedge fund investment is now considered riskier than before. able to recover some percentage of LH(t), say 50%.
The investor is worried that other investors will withdraw. If the with- For example, suppose that the high water mark is $100 million, the an- drawal amounts are large enough, the hedge fund manager may have to nual management fee is 1.5% and the performance fee is 20%. The man- unwind positions at unfavourable prices. This would compound the neg- ager’s profit and loss profile and use of leverage are illustrated in figure ative return. 1. If the fund reaches the default threshold, we assume that the manager
Since most hedge funds have high water marks, the investor is worried not only loses this year’s performance fee, but also a performance fee for about organisational risk. If the key employees in the fund do not believe the next three years (we assume a discount rate of zero). Here, we have they will get reasonable bonuses for the next few years (bonuses are usu- assumed that it will take some time for the manager to increase his assets ally taken from the performance fee), they may leave. to $100 million again. We have incorporated investor behaviour into our model by using vari- able leverage and the knock-out feature to model the evolution of a fund’s Longstaff’s method assets over time. Now that we have a picture of the hedge fund manager’s strategy, we can Before we can apply a variant of Longstaff’s method to a concrete ex- study the appropriate rebalancing strategy for an investor. We want to ample, we need to specify the investor’s profit and loss function, as know how much the investor should be compensated (in volatility terms) follows: for his inability to move funds out of the hedge fund. Our approach is to
If S(t) >> H(t), the value of the investor’s allocation is (1 – p)(S(t) – H(t)) use a variation of Longstaff’s method (1999). + H(t) and l (the strategy’s leverage) is moderate. The manager is coast- Longstaff has developed a technique for calculating the illiquidity pre- ing, since there is only a small amount of vega in the management and mium of an asset, such as a stock. The idea is to construct a portfolio con- performance contract. Thus, µ and σ are also moderate. sisting of a reference asset (for example, a money market account) and
If S(t) is close to H(t), then l is large, since the value of the incentive the restricted asset. First, it is assumed that there are no liquidity con- clause to the manager increases sharply in l. Here, vega is large and straints. This is the benchmark case. The investor tries to maximise his ex- positive. pected utility by rebalancing his portfolio over time, as necessary. Next,
If S(t) is close to a threshold default level, then l is small, since the man- the investor isn’t allowed to transfer money between the liquid asset and ager wants to continue to collect a percentage c of assets. As volatility in- the restricted one. Here, he chooses static weights to maximise the ex- creases, the probability of default increases sharply, and so the manager pected utility of his portfolio. keeps l at a relatively low level. Vega is large and negative. It is clear that the expected utility is at least as large in the case where
If S(t) drops below the default level, the fund liquidates. The value of the investor does not have any liquidity constraints. In general, it should the investor’s allocation is then some fraction of the default level. In the be larger. We then ask, how much extra return should the restricted asset example in the next section, we assume that the investors receive 50% of have to make the expected utility functions the same? According to the threshold amount. Longstaff, this is the appropriate illiquidity premium. For example, sup- In our model, drift and volatility of assets are level-dependent and the pose that the investor’s utility function is a simple information ratio, such leverage l is a step function in the argument S(t) – H(t). as portfolio return/portfolio volatility. Further, an investor is able to achieve 2 Our anonymous referee has correctly pointed out that some managers will increase an information ratio of 1.1 if both assets are liquid and a ratio of one if leverage when S(t) << H(t), to increase the expected value of the performance fee. These one of the assets is restricted. Then, he should haircut the illiquid asset’s tend to be small managers who need a large performance fee to stay in business. In our return by 10% or increase its volatility by 10% before equating it with a experience, however, funds with relatively large assets under management tend to take a conservative posture near the default zone. Thus, they decrease leverage when S(t) << more liquid asset. H(t). The liquidity haircut is larger than in our article if we assume a manager will We need to modify Longstaff’s method slightly when dealing with hedge increase his leverage near default

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2. Histogram of return/risk over 40,000 simulations, 3. Histogram of return/risk over 40,000 simulations, no with monthly rebalancing rebalancing

1.8 1.6 1.6 1.4

1.4 1.2 1.2 1.0 1.0 0.8 0.8 0.6 Probability (%) Probability (%) 0.6 0.4 0.4 0.2 0.2 0 0 0.30 0.75 1.20 1.65 2.10 2.55 3.00 3.45 3.90 4.35 4.80 5.25 5.70 0.30 0.75 1.20 1.65 2.10 2.55 3.00 3.45 3.90 4.35 4.80 5.25 5.70 –1.50 –1.10 –0.60 –0.20 –1.50 –1.10 –0.60 –0.20 Annual return/volatility Annual return/volatility

Simulating the illiquidity premium If, over time, the level of assets goes down, a more complicated cal- Here, we present a numerical example where an investor’s illiquidity pre- culation needs to be made. For example, if assets drop below 88.75, the mium can be directly simulated, using Longstaff’s method. expected return is 5%. However, the effective asset volatility is larger than Suppose there is an investor who wants to invest in a hedge fund and 5%, since there is a positive probability of default at the next time step. a traditional mutual fund. The investor’s utility function is specified by port- We approximate the effective volatility by taking a probability weighted folio return/portfolio volatility and he wishes to maximise his expected average, as follows. utility over two years. Initially, the mutual fund and hedge fund have iden- Suppose that we are at time t and asset level S(t). We know that: tical risk/reward characteristics. St()+∆ t= St()+−()()µσξ cSt∆+ t St() ∆ t
The historical annual return and volatility of the mutual fund and hedge fund are 10% and 10%, respectively. We assume that the mutual fund re- (assuming no withdrawals) and wish to calculate the probability that S(t + turns evolve according to a random walk. ∆t) ≤ 85, where ∆t = 0.083 years. Solving for ξ, we find that this proba-
The mutual fund and the hedge fund are uncorrelated. bility is the same as the probability p that: However, for simplicity, we assume that the investor cannot move 85 − St()−−()()µ cSt∆ t money in or out of the hedge fund over time. ξ ≤ σSt() ∆ t In turn, the hedge fund manager varies his leverage according to the following schedule. which can be calculated explicitly since ξ ~ N(0, 1). As the asset level moves
The current level of assets is 100 (expressed in millions of dollars) and close to default, p approaches 50%, assuming that the drift is zero. If: is equal to the current high water mark. 85 − St()−−()()µ cSt∆ t
The manager can vary his leverage once a month. ξ ≤ σSt() ∆ t
If assets drop below 85, the fund liquidates and investors receive 50%, or 42.5. The manager loses his management fee. The number 85 is cho- the fund defaults and we set: sen because it is 1.5 standard deviations below the initial asset value, as- 42. 5 − St()−−()()µ cSt∆ t suming moderate leverage. ξ = σSt() ∆ t
If assets rise above 110, the manager uses moderate leverage. His ex- pected annual return is 10% with 10% volatility. The volatility of S(t) can then be approximated by:
If assets fall below 88.75 (75% of the way to default), the manager low- var()() 1− ppξξ+ ers his leverage (to half the moderate level). His expected annual return 12 ξ σ2 is 5% with 5% volatility where 1 ~ N(0, ) and:
If assets are between 88.75 and 110, the manager uses maximal lever- age to increase the value of the high water mark contract. His expected  42. 5 − St()−−()()µ cSt∆ t  ξ ~ N  ,0 annual return is 20% with 20% volatility. 2  σSt() ∆ t  We now create two sets of simulations, a benchmark set where the in- vestor is allowed to rebalance once a month and another set where no re- We have simulated the investor’s utility 40,000 times by calculating the re- balancing is allowed. In each case, we approximate the investor’s expected alised annual return and volatility for each simulation. A histogram of in- utility. We summarise the results below. formation ratios appears in figure 2. In the histogram, we have partitioned the x-axis in 0.025 increments. The expected utility over these simulations Simulation 1 (monthly rebalancing) is 1.5. The investor is aware of the manager’s variable leverage and wants to max- imise his expected utility over the next two years. At each time step, the Simulation 2 (no rebalancing) manager’s leverage is known. For example, at the start, the manager’s as- In this case, we are not allowed to rebalance the portfolio for the entire sets are 100 so his expected return is 20% with 20% volatility. Thus, the two years. Beforehand, we do not know the optimal static allocation to investor initially optimises his expected utility by allocating 67% to the tra- the traditional fund and the hedge fund. By performing a separate simu- ditional mutual fund and 33% to the hedge fund. lation for different weights (in increments of 1%), we have found the op-

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timal allocation to the traditional fund should be 55%, with 45% to the fund to fund, the volatility adjustment in our numerical example should hedge fund. We can then plot the information ratio histogram in the same only be viewed as a guideline. Typically, funds that trade in more liquid way as simulation 1. The results appear in figure 3. markets (for example, interest rate futures) offer more liquidity to an in- Here, the distribution is clearly bimodal, since there is a much larger vestor and should not be haircut as severely. probability that the hedge fund will blow up before an investor can trans- We have also made assumptions about the way in which a hedge fund fer funds to the traditional fund. The expected utility over 40,000 simula- manager varies his leverage according to the level of assets in the fund. tions is 1.3, or about 10% smaller than the benchmark case. We can either From experience, we have found that these assumptions are qualitatively express the 10% difference as a return premium or in volatility terms. We correct. In a companion paper, we plan to develop a way to determine a choose the latter. Thus, the historical 10% volatility for the hedge fund should hedge fund manager’s optimal use of leverage more precisely.
really be considered to be 11% (again, assuming moderate leverage). In certain instances, we can use the liquidity haircut to decide how Hari Krishnan is vice-president in the investment management division much we should allocate to a given hedge fund. of Morgan Stanley. Izzy Nelken is president of Super Computer Consult-
As we have pointed out, it can be dangerous to characterise a hedge ing. e-mail: [email protected], [email protected] fund purely by its historical mean, variance and correlation with other funds. However, many funds of funds already use mean variance optimisers to make asset allocation decisions for traditional assets and are reluctant to References develop entirely new models for alternatives. The result in this article pro- vides a ‘correction’ term that can be applied before the optimisation step.
Fung W and D Hseih, 1997 Fund of funds managers often make incremental changes to their port- Survivorship bias and investment style in the returns of CTAs folios using a ranking system. New funds are ranked according to a few Journal of Portfolio Management 24, pages 30–41 summary statistics, such as historical mean, variance and maximum draw- Goetzmann W, J Ingersoll and S Ross, 2001 down. When a change is made, a new fund is chosen from the list. Our High water marks liquidity adjustment puts funds that trade in liquid markets (and usually Working paper have shorter lockups) on a more equal footing with funds that ‘sell liq- Longstaff F, 1999 uidity’ for premium and should result in a different ranking from the one Optimal portfolio choice and the valuation of illiquid securities that relies purely on historical returns. Forthcoming in Review of Financial Studies

Spurgin R, 1999 Conclusion A study of survival: commodity trading advisors, 1988–1996 We have developed a technique for calculating the illiquidity haircut for a Journal of Alternative Investments, winter, pages 16–22 hedge fund. Since the lockup period and redemption schedules vary from

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