Von Tim Friederich (15.10.2010)

Hedge funds belong to the "great mysteries" of the financial world. In the media they are celebrated for generating extraordinary returns on one day, condemned for exploiting industries only driven by generating even higher returns the other day (isn’t that what they have just been celebrated for?), and laughed at the next day when the by now notorious Bernard L. Madoff has to sheepishly admit how he kept the wheel turning – or the returns growing.

One of the characteristics of hedge funds is that only little is known about their assets. Yet, it’s the assets which drive their performance. We model the value of a hedge fund as a down-and-out call option on its assets with its liabilities as strike price and barrier. This means that, in the end, the investor receives the difference between the value of the assets and the value of the liabilities – but only if the hedge fund has not defaulted in the meantime, that is if the value of the assets is always above the value of the liabilities.

Modeling the asset process of the hedge fund as a Geometric Brownian Motion (which is the "standard") has one major drawback: The Black-Scholes model assumes the volatility of the asset process to be constant over time. This characteristic feature is called homoscedasticity. However, even the stock market suggests that the volatility is not constant over time. For example, it can be observed that those times when the stock markets take a hit the volatility is higher than in "peaceful" times of strong markets. Therefore, the asset process is modeled with stochastic volatility. This allows for a more realistic mapping of reality, yet requires a process not only for the assets but also a process for the volatility itself. Thus, it increases the parameters which are describing the process. In order to find those parameters, estimators are derived which enable to fit the parameters to the asset process.

A lot of hedge funds claim to be independent of market developments (especially those following so-called market-neutral strategies). Yet, for example August 2007 proved this claim wrong, when, according to the CFO magazine, more than three quarters of the hedge funds were suffering losses. Thus, a model should not look at two separate hedge funds developing independently. To be more precise: Not the hedge funds themselves, but their assets obviously seem to be dependent. Therefore, we also allow for correlation between the hedge funds’ asset processes and also provide a method to fit that correlation parameter.

This model not only allows to describe the hedge fund and its assets. It also enables to simulate multiple scenarios of possible paths how the hedge fund would evolve over time according to the model. These scenarios can be analyzed under different aspects.

As actual hedge funds only provide monthly data, we apply this model to listed companies. We examine two cases which can be regarded as two key events of the financial crises: the decay of the two US mortgage lenders Fannie Mae and Freddie Mac. The model analyzes the risks and examines the default probabilities of these corporations. For example, we can compare their default probabilities at two points in time: January 2007 and July 2008. Their probabilities to survive the next 100 years were 24.8% (Fannie Mae) and 28.7% (Freddie Mac) in 2007. They collapsed to 4.7% and 12.4% in 2008. The probability to default within the next three years exploded from 6.1% to 39.3% for Freddie Mac and from 7.0% to an extraordinary 57.3% for Fannie Mae.

What can also be observed from the case of Fannie Mae and Freddie Mac is that the events of a default are more likely to coincide than to occur independently, which goes in line with the positive correlation of their asset processes which has been estimated. For example, the probability that the default of both companies would occur within no longer than one year difference is 14.1% (or 26.7% for July 2008). The same conclusion can be drawn when looking at the conditional probabilities, i.e. the probability for one company to default given that the other one also goes bankrupt. For example, the unconditional probabilities for Fannie Mae and Freddie Mac in July 2008 to default within the next three years amounted to 57.3% and 39.3%. Given that one company defaults the other would default with a higher probability of 71.3% (Fannie Mae) or 49.0% (Freddie Mac). Thus, the already severe situation of Fannie Mae and Freddie Mac before the crisis deteriorates when only comparing January 2007 to July 2008, when a near end of the two Government Sponsored Enterprises was already foreseeable, considering the dramatic increase in the default probabilities.

**seit März 2009**- Doktorand am HVB-Stiftungsinstitut für Finanzmathematik und Analyst bei risklab GmbH, München
**Juni 2008 - Dez. 2008**- Forschungsaufenthalt am RiskLab Toronto im Rahmen von FIM, University of Toronto, Kanada
**Okt. 2006 - Febr. 2009**- Studium im Rahmen des Elitestudiengangs Finance and Information Management, TU München und Universität Augsburg. Bachelor Thesis: "Private Equity as an Asset Class", Master Thesis: "Credit Modeling of Hedge Funds"
**Okt. 2004 - Sept. 2006**- Studium der Finanz- und Wirtschaftsmathematik (Vordiplom), TU München

**2007 - 2009**- Online-Stipendium von e-fellows.net, Stipendium der Allianz Global Investors und Stipnedium im Rahmen des Max Weber-Programms
**2008**- DZ-Bank Karrierepreis: 1. Preis für die Bachelorarbeit "Private Equity as an Asset Class"

**März 2009**- Allianz Global Investors Client Institute, Kempfenhausen bei München
**Dez. 2008**- Konferenz "Enterprise Risk Management", Wuhan University, China
**Juni 2008**- Allianz Global Investors IDO Absolute Return Workshop, Frankfurt
**März 2008**- Allianz Global Investors Knowledge Camp, Würzburg

- *Philipp Aigner, Stefan Albrecht, Georg Beyschlag, Tim Friederich, Markus Kalepky, Rudi Zagst: "What Drives PE? Analyses of Success Factors for Private Equity Funds". In: Journal of Private Equity, Fall 2008.
- *Philipp Aigner, Georg Beyschlag, Tim Friederich, Markus Kalepky, Rudi Zagst: "Optimal Risk-Return Profiles for Portfolios including Stocks, Bonds, and Listed Private Equity". Submitted for publication, 2008.