The Mathematics of Hedge Fund Fees - RiskLab · The Mathematics of Hedge Fund Fees Ben Djerroud (Sigma Analysis & Management Ltd.), David Saunders ... A new investment paradigm 2

Hedge FundsThe First Loss Model

Consequences of the derivative pricing frameworkRecap

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

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



Outline

1 Hedge FundsDefinitionHedge Fund FeesA new investment paradigm

2 The First Loss ModelDefinitionOption pricing framework

3 Consequences of the derivative pricing frameworkGraphical analysisBeyond pricing

4 Recap




DefinitionHedge Fund FeesA new investment paradigm

A snow swap





Is this a good investment?

Consider a portfolio of two swaps:

A swap with the City, were $10M are exchanged as a function of snowprecipitation in the City.

A swap with the ski resort, where $10M are exchanged as a functionof snow precipitation at the resort.

We charge 10% of the flow ($200,000) as risk premium. We assume a50% correlation between snow fall in the two locations.






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 theresort

With 12.5% probability, we have to pay $1M to both the city and theresort






The investment parameters are:

The expected return is 10%.

The standard deviation is

σ =√

0.125 ∗ [(2M − 200k)2 + (2M + 200k)2] + 0.75 ∗ (200k)2

≈ 50%

Its Sharpe ratio is 0.2





The snow fund

We perform 100 similar swaps, with 100 different cities and ski resorts, forexample:

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.





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% ofthe AUM, and 20% of the net profits.





Hedge Funds





Hedge fund economics

Investor Hedge Fund

Gross Profit from Investments $200M 0

Management Fee - $20M $20M

Performance Fee -$36M $36M

Total 7.2% $56M





Hedge Fund Fees

Hedge Funds are pooled investment vehicles managed by a managementcompany.

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 profitsobtained by the fund

Management fees are usually paid quarterly or monthly, whereasperformance 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.





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?





A new investment paradigm

One of the major forces to restructure the finance world is the shift ofbalance between buy-side and sell-side.

The past: Sell side ruled the finance sector

The future: Investors (buy-side, asset owners), are gaining marketpower

The present: a battle of wits between buy and sell side

Hedge funds: we are moving towards a more symmetric compensationstructure.





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




DefinitionOption pricing framework

First loss models

A new investment model:

The investor provides an investment to the fund X0.Example $100M

The fund manager will absorb the first loss up to a fixed percentageamount c of the initial investment.Example 10 %.

The investor pays a management fee m and performance fee α to themanager.Example: 1% and 50%





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





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 optfor a first-loss model to achieve:

SizeTrack recordReputationAim for high grade, institutional investors

Also popular in trading houses, who are looking for young tradingtalent





A game-theoretic framework

He and Kou (2013) consider a liquidation barrier, adopting aninvestor viewpoint.

Hodder and Jackwerth (2007) adopt a liquidation decision from themanager’s viewpoint.

Goetzmann et al (2003) consider random liquidations andbarrier-driven liquidations





The first-loss value model

A new game-theoretic situation:

The fact that the manager risks the deposit c to compensate theinvestor for losses delivers value to the investor

The relationship between X0, c , m and α will determine whether theinvestor, or the manager, is the net winner of value add.

We will use the option pricing framework to determine the net businessvalue delivered to investor and manager.





Buy side vs. sell side

Question: Given values X0, c , m and α, and an investment horizon T , ...

...who wins?





The option pricing framework

The fund value Xt is split between the Investors It and the Manager Mt

Xt = It + Mt .

The manager receives a management fee equal to m · X0.

The performance fee is valued as a call option on the underlying valueαXt with strike price αX0, payable at time T .

The first-loss guarantee is valued as a (covered) put option, payableto the investor,





The option pricing framework

The value of the investment to the investor at time T is:

IT =

XT − α(XT − X0) −mX0 when XT ≥ X0

(1 −m)X0 when (1 − c)X0 ≤ XT ≤ X0

XT + (c −m)X0 when XT ≤ X0

and the value to the manager is MT = XT − IT .It is easy to check that

IT = XT −mX0 (pays a management fee)−α(XT − X0)+ (pays a performance fee)

+(X0 − XT )+ − ((1 − c)X0 − XT )+ (receives a guarantee)





Management Income

Equivalently, the manager’s income will be

MT = mX0 (receives a management fee)+α(XT − X0)+ (receives a performance fee)

−(X0 − XT )+ + ((1 − c)X0 − XT )+ (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 longerguaranteed to be positive.





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 anexchange 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 (theguarantee).

If c is high enough relative to m and α, it is possible that themanager pays the investor for the privilege of managing her money.





Pricing fees as a derivative

If we assume a fund value process for Xt given by

dXt = rXt dt + σ Xt dWt ,

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




Graphical analysisBeyond pricing

Payoff structure comparison

The investor owns 90% of the fund, the manager owns 10%.





Sensitivity to Volatility





Maturity sensitivity





Other topics of interest

Dynamic allocation

Investment redemption

Impact of the fee on portfolio construction

Fund shut-down




Recap

The world is changing: buy-side is gaining power with respect to thesell-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/benefitanalysis of a particular fee structure


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The Mathematics of Hedge Fund Fees - RiskLab · The Mathematics of Hedge Fund Fees Ben Djerroud (Sigma Analysis & Management Ltd.), David Saunders ... A new investment paradigm 2