*Liquidity (or lack thereof) is a perennial issue in the hedge fund industry. How much should investors expect to be compensated for the lock-ups that hedge funds often require? And does such a premium really count as “alpha”? Pierre Laroche of Innocap Investment Management (see related posting on Innocap’s hedge fund replication work) tells us of a study he recently conducted that comes to a pretty definite conclusion about the illiquidity premium. Laroche is the co-author of three books on derivatives and risk management.*

**Special to AllAboutAlpha.com by:** Pierre Laroche, Managing Director – R&D Innocap Investment Management

Much has been done in recent years to better measure and price illiquidity. These developments had a positive impact on risk management practice. For example, some interesting liquidity-adjusted VAR models have been developed recently. There has also been quite a bit of discussion on this topic on the pages of AllAboutAlpha.com (e.g. “Liquidity Alpha“, “Liquidity Insurance“)

At Innocap, we recently finished an interesting study that proposes a way to quantify the cost of illiquidity and adjust the risk-return profile of an illiquid asset.

Our model aims at reproducing the trading methods and market environment of typical (median) CTA hedge funds. (We could have chosen other types of hedge funds. This one is used for illustrative purposes only). The model integrates the impact of liquidation delays, accrued bid-ask spread (BAS hereafter) and increased volatility (feedback effect). This is an improvement over other models, which only take into account one or two of these factors.

We implement our model by using a Monte Carlo simulation which assumptions are realistic but a bit conservative. They can be summarized as follows:

- The investor trades one commodity with a systematic daily trading rule based on Bollinger bands.
- Leverage and short selling are not allowed.
- The commodity return behaves randomly.
- The BAS is a bounded proportion of the true commodity price and is symmetrical around it.
- Market participants are atomistic: their trades do not impact the true price.
- Liquidity crisis occur on average once a year. They start at a random date and last a random number of days.
- To model the amplitude of the liquidity crisis, the BAS is increased by a random factor uniformly distributed between 1.5 and 3.
- We assume that if fully invested in the risky asset, the investor’s position can represent an important proportion of the daily transaction value. Under normal market conditions, the daily transactions value (DTV) is a positive Gaussian random variable with a mean equal to 1.5 times the portfolio’s value and a standard deviation equal to 0.6 times the mean.
- When a liquidity crisis occurs, the DTV is deduced by a factor uniformly distributed between 50 % and 90 %.
- During a liquidity crisis, when a transaction signal is obtained, the trader buys or sells as much as she can one day after the other unless a contrary signal is subsequently obtained.

We assume two regimes. The “exogenous volatility model” assumes that the true volatility is not influenced by a liquidity crisis (i.e. the stock’s volatility is an exogenous factor). The “endogenous volatility model” assumes that a liquidity crisis raises the stock’s volatility by a bounded factor uniformly distributed between 1.1 and 2.

The charts below illustrate a typical asset price daily path over a year under these two liquidity crisis regimes. We ran 5000 such simulations and eliminated around 400 that had no buy signals, no sell signals or more than 50 % of the time invested in cash. (red line=bid, blue line=ask)

The resulting return-risk profiles (average of the valid simulations results) are presented in the table below. Our exogenous model results suggest a liquidity premium of roughly 1 % per year, which is consistent with other findings. The endogenous model results suggest that the volatility feed-back effect may add about 0.50 % per year to the liquidity premium, bringing it to 1.5 %. These results are inline with those of other studies.

This research shows that there is a way to quantify the cost of illiquidity and adjust the risk-return profile of an illiquid asset accordingly. Our model is based on a Monte Carlo simulation that takes into account the time to liquidate and the impact of liquidity crisis on bid-ask spreads and volatility. This result suggest that subtracting 1.5 % from the average annual return of an illiquid portfolio is well justified.

*(ed: Laroche’s full paper can be downloaded **here**.)*

*The opinions expressed in this guest posting are those of the author and not necessarily those of AllAboutAlpha.com.*