The Mathematics of Hedge Fund Fees

Transcription

1 The Mathematics of Hedge Fund Fees Ben Djerroud (Sigma Analysis & Management Ltd.), David Saunders (University of Waterloo), Mohammad Shakourifar (Sigma Analysis & Management Ltd.), Luis Seco. May 10, 2016

2 Outline 1 Hedge Funds Hedge Fund Fees A new investment paradigm 2 Option pricing framework 3 Graphical analysis Beyond pricing 4

3 Hedge Fund Fees A new investment paradigm A snow swap

4 Hedge Fund Fees A new investment paradigm Is this a good investment? Consider a portfolio of two swaps: A swap with the City, were $10M are exchanged as a function of snow precipitation in the City. A swap with the ski resort, where $10M are exchanged as a function of snow precipitation at the resort. We charge 10% of the flow ($200,000) as risk premium. We assume a 50% correlation between snow fall in the two locations.

5 Hedge Fund Fees A new investment paradigm Is this a good investment? Our cashflows are: We post $2M as collateral We receive $200,000 as a fee With 75% probability, the two swaps cancel each other s cash flows With 12.5% probability, we receive $1M from both the city and the resort With 12.5% probability, we have to pay $1M to both the city and the resort

6 Hedge Fund Fees A new investment paradigm Is this a good investment? The investment parameters are: The expected return is 10%. The standard deviation is σ = [(2M 200k) 2 + (2M + 200k) 2 ] (200k) 2 50% Its Sharpe ratio is 0.2

7 Hedge Fund Fees A new investment paradigm The snow fund We perform 100 similar swaps, with 100 different cities and ski resorts, for example: Blue Mountain (Toronto) Mountain Creek (New Jersey) Panorama Mountain Village (Calgary) Snowshoe Mountain (West Virginia) Steamboat Ski Resort (Hayden, Denver) Stratton Mountain Resort (Vermont) Tremblant (Montreal) Whistler Blackcomb (Vancouver) etc.

8 Hedge Fund Fees A new investment paradigm The snow fund With $2Bn, we can then create a very interesting hedge fund: 10% expected return 5% standard deviation Sharpe ratio of 2. Since we do not like to work for free, we will charge our investors 1% of the AUM, and 20% of the net profits.

9 Hedge Fund Fees A new investment paradigm Hedge Funds

10 Hedge Fund Fees A new investment paradigm Hedge fund economics Investor Hedge Fund Gross Profit from Investments $200M 0 Management Fee - $20M $20M Performance Fee -$36M $36M Total 7.2% $56M

11 Hedge Fund Fees A new investment paradigm Hedge Fund Fees Hedge Funds are pooled investment vehicles managed by a management company. Hedge funds pay the management company two types of fees (2/20): Fixed management fees, ranging from 1% to 2% of assets or higher Performance fee, most commonly equal to 20% of net profits obtained by the fund Management fees are usually paid quarterly or monthly, whereas performance fees are usually paid annually. This fee structure is asymmetric: only the investor has downside. The business value delivered to the manager is strictly positive.

12 Hedge Funds Hedge Fund Fees A new investment paradigm The CalPers syndrome Calpers announced in 2014 that they were exiting hedge fund investments. One of the reasons quoted was high fees Are hedge fund fees too high? Ben Djerroud (Sigma Analysis & Management Ltd.), David Saunders The Mathematics (University of of Waterloo), Hedge Fund Mohammad Fees Shakourifar (Sigma

13 Hedge Fund Fees A new investment paradigm A new investment paradigm One of the major forces to restructure the finance world is the shift of balance between buy-side and sell-side. The past: Sell side ruled the finance sector The future: Investors (buy-side, asset owners), are gaining market power The present: a battle of wits between buy and sell side Hedge funds: we are moving towards a more symmetric compensation structure.

14 Hedge Fund Fees A new investment paradigm Innovation in hedge fund economics Back in our snow-fund example: In 2008, Intrawest filed for bankruptcy 26 of our swap counterparties therefore are in default Assume a cold, snowy winter: $260M of lost swap revenue Investor Hedge Fund Gross Profit from Investments -$60M 0 Management Fee -$20M $20M Performance Fee $0M $0M Total -4% $20M

15 Option pricing framework First loss models A new investment model: The investor provides an investment to the fund X 0. Example $100M The fund manager will absorb the first loss up to a fixed percentage amount c of the initial investment. Example 10 %. The investor pays a management fee m and performance fee α to the manager. Example: 1% and 50%

16 Option pricing framework First loss economics Back in our snow-fund example: Investor ND HF ND Investor D HF D Investment Profit $200M 0 0 -$60M Management Fee -$20M $20M 0 0 Performance Fee -$90M $90M 0 0 Total 4.5% $110M 0 -$60M

17 Option pricing framework A start-up model Hedge fund start-ups have become more difficult in recent times New comers, and funds who seek to introduce new strategies, can opt for a first-loss model to achieve: Size Track record Reputation Aim for high grade, institutional investors Also popular in trading houses, who are looking for young trading talent

18 Option pricing framework A game-theoretic framework He and Kou (2013) consider a liquidation barrier, adopting an investor viewpoint. Hodder and Jackwerth (2007) adopt a liquidation decision from the manager s viewpoint. Goetzmann et al (2003) consider random liquidations and barrier-driven liquidations

19 Option pricing framework The first-loss value model A new game-theoretic situation: The fact that the manager risks the deposit c to compensate the investor for losses delivers value to the investor The relationship between X 0, c, m and α will determine whether the investor, or the manager, is the net winner of value add. We will use the option pricing framework to determine the net business value delivered to investor and manager.

20 Option pricing framework Buy side vs. sell side Question: Given values X 0, c, m and α, and an investment horizon T,......who wins?

21 Option pricing framework The option pricing framework The fund value X t is split between the Investors I t and the Manager M t X t = I t + M t. The manager receives a management fee equal to m X 0. The performance fee is valued as a call option on the underlying value αx t with strike price αx 0, payable at time T. The first-loss guarantee is valued as a (covered) put option, payable to the investor,

22 Option pricing framework The option pricing framework The value of the investment to the investor at time T is: X T α(x T X 0 ) mx 0 when X T X 0 I T = (1 m)x 0 when (1 c)x 0 X T X 0 X T + (c m)x 0 when X T X 0 and the value to the manager is M T = X T I T. It is easy to check that I T = X T mx 0 (pays a management fee) α(x T X 0 ) + (pays a performance fee) +(X 0 X T ) + ((1 c)x 0 X T ) + (receives a guarantee)

23 Option pricing framework Management Income Equivalently, the manager s income will be M T = mx 0 (receives a management fee) +α(x T X 0 ) + (receives a performance fee) (X 0 X T ) + + ((1 c)x 0 X T ) + (provides a guarantee) The performance fee is a call option. The guarantee is a (short) covered put option. The net income to the management company is now no longer guaranteed to be positive.

24 Option pricing framework The price of investing in a hedge fund An interesting array of new possibilities: When the investor subscribes to a first-loss fund structure, there is an exchange of options between the investor and the manager: The investor pays a call option to the manager (the performance fee) and the investor provides a covered put option to the investor (the guarantee). If c is high enough relative to m and α, it is possible that the manager pays the investor for the privilege of managing her money.

25 Option pricing framework Pricing fees as a derivative If we assume a fund value process for X t given by dx t = rx t dt + σ X t dw t, we can obtain a valuation of the investor fee expenses or manager income, in a risk-neutral valuation framework. Issues Lock-ups Frequency of fee payments

26 Graphical analysis Beyond pricing Payoff structure comparison The investor owns 90% of the fund, the manager owns 10%.

27 Graphical analysis Beyond pricing Sensitivity to Volatility

28 Graphical analysis Beyond pricing Maturity sensitivity

29 Graphical analysis Beyond pricing Other topics of interest Dynamic allocation Investment redemption Impact of the fee on portfolio construction Fund shut-down

30 The world is changing: buy-side is gaining power with respect to the sell-side Hedge fund start-ups are becoming more difficult Some managers offer a first loss structure to investors The option pricing framework allows us to perform a cost/benefit analysis of a particular fee structure