*Illiquidity premiums are often misidentified as alpha. So with their lock-ups and “gates”, hedge funds have recently come under the scrutiny of academics who wonder how much of their returns are simply a fair compensation for having to lock-up your investments for up to a couple of years. In a **1998 paper** on liquidity and security prices, UCLA’s Francis Longstaff said “The extent to which liquidity affects security prices has itself become a controversial in asset pricing…it may take an extended period of time to accumulate or unwind a specific portfolio position.”*

*In a 1995 article for the Journal of Finance, Longstaff also predicted the appropriate illiquidity premium for a security given its volatility and the number of days it would take to unwind a position. Now this general approach has been applied to hedge funds by Pierre Laroche, head of R&D at Innocap, a subsidiary of the National Bank of Canada and a joint venture partner with BNP Paribas. *

**A simple model to adjust funds of funds returns for their illiquidity**

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

We all know that the lack of liquidity can impact the performance of an investment in several ways:

- Liquidity constraints usually increase the bid-ask spread, hence lowering the expected return.
- A liquidity crisis usually raises the volatility.
- A longer time required to liquidate the position increases the uncertainty on the total amount that will finally be obtained.

There are several models that adjust the risk-return profile of an investment to take into account the two first elements of the above list. But I’d like to address the third impact by using the model first proposed by Alan Malz of Risk Metrics in 2003.

Malz (2003) proposes the following liquidity-adjusted VAR (LVAR) formula if the investor is assumed to liquidate her/his position at the market price in equal portions over next T periods (instead of 1 period which corresponds to the VAR horizon) due to market liquidity constraints:

Let’s call the term in parentheses the Malz Factor (MF). If T = 1, then the MF is equal to 1.0: the illiquidity-adjusted VAR is not different from the unadjusted VAR. But it T is greater than 1, then the MF term is always greater than 1.0 as well: the liquidity-adjusted VAR is higher than the unadjusted VAR.

For example, if we are interested in calculating the daily VAR of an asset that has a 3-month liquidity, then T = 3. In this example, if the unadjusted monthly VAR is 2 % of the estimated asset value, then the illiquidity-adjusted VAR will roughly be 2.5 % (i.e. a MF of 1.25 corresponds to a 25 % increase).

The table below contains the values of the MF for different values of parameter T.

Since the volatility is approximately a linear function of the VAR, Malz’s model can be applied to adjust the asset’s volatility as a function of its time to liquidate. The table below shows the volatility adjustments for different liquidity lags (in months) and monthly volatilities.

We can adjust the Sharpe ratio in a similar manner. Sharpe ratios unadjusted for the investment’s liquidity are overstated. To correct this situation, one has to substitute the unadjusted volatility for the (higher) liquidity-adjusted one. This causes the illiquidity-adjusted Sharpe ratio to be *lower* than the unadjusted one.

The differences are quite significant for liquidity of 3 months and more. For example, typical above-median fund of hedge funds that generate an unadjusted annual average excess return of 6 % with a unadjusted annualized volatility of 5.2 % (1.5 % x 12^0.5) see their Sharpe ratios significantly drop from 1.15 to 1.03 after adjusting for a 2-month liquidity, to 0.93 after adjusting for a 3-month liquidity, and to 0.73 after adjusting for a 6-month liquidity.

In order to “price” illiquidity, we now simply need to determine the additional excess return required to bring the liquidity-adjusted Sharpe ratio in line with its unadjusted counterpart. This additional excess return can be thought of as a liquidity risk premium (*rp*). Algebraically, the problem consists in calculating the value of *rp* such that:

As you can see below, the annualized risk premia (*rp*) for a fund with a 1.5% monthly standard deviation (5.2% annual s.d.) and a 3-month lock-up is approximatley 1.5% per annum.

If we add the typical requirement for two months prior notice of redemption, the resulting possibility of *5-month* illiquidity raises the illiquidity price tag to 2.90 % per year! These results suggest that the cost liquidity erases much – if not all- of the additional average return often attributed to illiquid FoHFs (vs. their liquid or managed account counterparts).

Illiquidity artificially boosts performance measures by understating the true volatility of an investment. The above analysis, which is based on a conservative estimate of the impact of illiquidity on an investment’s risk, suggests that it is well justified to subtract at least 1.5 % from the average annual return of typical classical FoHFs offering a 3-month liquidity.

In fairness, some investors with long investment horizons may argue that they do not care, since they do not need short-term liquidity. While this may be true, it does not mean that one should ignore liquidity altogether. After all, institutional investors always compare risk-adjusted returns no matter their risk appetite. The same reasoning applies to illiquidity – *one should adjust returns for illiquidity no matter what his/her current liquidity needs are*. Not doing so is like assuming that each building in a “bricks-and-mortar” real estate portfolio is as liquid as a REIT share.

The hedge fund industry broadly defines and sells itself as a provider of absolute (i.e. LIBOR+) returns. Since LIBOR is highly liquid, investors should adjust absolute return investments for their incremental illiquidity, no matter what their investment horizon is. Not doing so would significantly boas their decision making.

*– P. Laroche, September 10, 2008*

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

## One Comment

September 16, 2008 at 9:42 am## Sean