Do Hedge Funds Have Enough Capital? A Value-at-Risk Approach

Motivation(s)

The rapid growth of the hedge fund industry has led to several studies that try to quantify hedge fund performance. Lamentably, there have been few studies on the risk profile of hedge funds despite the debacle of LTCM. Recent works have proposed a VaR approach to measure the maximum amount of assets a fund can lose over a certain time period with a specified probability, however, no study has applied VaR to address capital adequacy and risk estimation issues.

Proposed Solution(s)

Since most of the unexpected losses arise due to extreme events in financial markets, the authors assert the estimation of capital requirements can be considered as an extreme value problem. This assumption enables the use of Extreme Value Theory (EVT) to model and estimate tail-related risk. They analyzed both the VaR for each fund and its distribution across all funds and computed a VaR-based estimate of required equity capital for each fund. The proposed VaR methodology is more effective at capturing hedge fund risk than the mean-variance approach used by other works.

Evaluation(s)

The monthly return data came from the TASS hedge fund dataset. There are no data available on the position holdings of hedge funds since this constitutes proprietary trading information. Only the funds that have at least five years of return history were used to estimate the VaR. Despite this restriction, the remaining funds do overlap with some of the most turbulent times in financial markets (e.g. Asian currency crisis of 1997, the Russian debt crisis, LTCM, stock market crash from 2000 onwards).

The authors asserted that the target horizon should be longer than the banks’ ten-days requirement since hedge funds are far less regulated and can only use private funding. They used three times the 99% one-month VaR as the required equity capital for hedge funds. To avoid introducing another source of error into the VaR estimate, they use VaR relative to zero instead of estimated mean returns for hedge funds.

The analysis demonstrates that dead funds generally have higher relative VaRs than live funds, and there is a significant upward trend in VaR for dead funds starting two years before their death. Furthermore, all the investment styles defined by TASS exhibit high kurtosis, which indicates that hedge fund returns have nonnormal distributions, thus invalidating standard deviation as an appropriate measure for hedge fund risk. Another ineffective measure is the average leverage ratio since it ignores the inherent riskiness of the assets in the portfolios. Note that even though the correlation coefficient between the average leverage ratios and the proposed VaR-based Cap ratios are statistically significant, the magnitudes are economically trivial.

In order to verify the results, the authors compared the history of VaR estimates with actual realized returns. Since a large number of observations are needed to reliably backtest a VaR model, they used simulated hedge fund returns. Other limitations stem from restricting the analysis to a 60-month VaR window and ignoring the liquidity aspect of hedge fund risk.

Future Direction(s)

  • When backtesting is not possible, are Monte Carlo model simulations (e.g. Geometric Brownian Motion, jump diffusion) accurate enough to be used as an acceptance test versus a rejection test?

  • How effective is the rolling window of Cap ratios at capturing the dynamic nature of hedge funds?

  • Is EVT valid for different types of management styles?

Question(s)

  • The appendix only covers high level concepts related to estimating VaR and TCL using EVT, so it’s not enough for a newcomer to reproduce the derivations. Which papers to read first from the list of references?

  • Would hedge funds be willing to disclose their position holdings several years after the fact for research purposes?

  • What is the reason for using Geometric Brownian Motion and jump diffusion models in the simulations?

Analysis

The VaR methodology may give insight into whether a hedge fund is undercapitalized or not. The well-written analysis sounds reasonable, but the simulated data used in backtesting is dubious. The authors found about 3.7% (35 out of 942) of the live funds are undercapitalized while the corresponding fraction is 10.9% (54 out of 494) for the dead funds. These numbers illustrate capital inadequacy may not be the primary reason. Instead of looking at liquidity risk, they performed a multivariate analysis on capitalization by running a cross-sectional regression of Cap ratios on various fund characteristics and styles. While the results are interesting, they should have answered how much one should weigh VaR over the other valid indicators. Nonetheless, the application of EVT to the estimation of capital requirements is quite ingenious.

Notes

Value-at-Risk (VaR)

  • VaR is a measure of the minimum loss that can happen over a target horizon with a given confidence level.

    • It does not quantify how large the loss might be, if it occurs.

    • If \(c\) is the selected confidence level, VaR corresponds to the \(1 - c\) lower tail of the distribution, which should be minimized to reduce the probability of failure.

    • It is calculated in dollar amounts and is designed to cover most of the losses that a risky business might face.

    • Although the target horizon is often chosen arbitrarily, it should reflect the amount of time necessary to take corrective action if unplanned high losses occur, and should correspond to the time necessary to raise additional funds to cover losses.

  • VaR encompasses a broader concept of risk than leverage ratios.

    • Leverage ratios only depend on the liabilities sides of the balance sheet.

    • Leverage ratios does not account for the differences in the volatility of assets and liabilities, nor does it account for correlations.

Capital Adequacy

  • Capitalization (Cap) ratio is one approach to evaluate capital adequacy.

  • Tail Conditional Loss (TCL) measures the potential size of the expected loss if it exceeds the VaR.

  • The ratio of TCL to VaR can provide a more objective basis of determining the appropriate capital multiplier that should be used in conjunction with VaR and indicate how safe it is to use the standard multiplier of three recommended by BCBS.

Extreme Value Theory (EVT)

  • EVT models the limiting distribution of extreme returns i.e. fits only the tail of the distribution.

  • There are two typical implementations where the latter is preferred in applications with little available data.

    1. Fit one of three standard extreme value distributions (e.g. Fréchet, Weibull, or Gumbel) to block maxima values in a time series.

    2. Models the distribution of exceedances over a threshold as a generalized Pareto distribution (GPD) via Peak Over Threshold (POT) method.

References

GL05

Anurag Gupta and Bing Liang. Do hedge funds have enough capital? a value-at-risk approach. Journal of financial economics, 77(1):219–253, 2005.