Incorporating Liquidity Risk

In the case of exogenous liquidity risk, one approach is to use the bid-ask spread to directly adjust the metric. Please note: Risk models are different than valuation models and this method assumes there are observable bid/ask prices.

Let's illustrate with value-at-risk (VAR). Assume the daily volatility of a $1,000,000 position is 1.0%. The position has positive expected return, also referred to as drift, but as our horizon is daily, we bring our tiny daily expected return down to zero. This is a common practice. So let the expected daily return equal zero. If the returns are normally distributed, then the one-tailed deviate at 5.0% is 1.65. That is, the 5% left tail of normal distribution is 1.65 standard deviations to the left of mean. In excel, we get this result with =NORM.S.INV(5%) = -1.645.

The 95% value at risk (VAR) is given by:

$1,000,000 * 1.0% volatility * 1.65 = $16,500

Under these assumptions, we can say "only 1/20 days (5% of the time) do we expect the daily loss to exceed $16,500." But this does not adjust for liquidity.

Let's assume the position is in a single stock where the ask price is $20.40 and the bid price is $19.60, with a midpoint of $20. In percentage terms the spread (%) is:

($20.40 - $19.60) ÷ $20 = 4.0%

The full spread represents the cost of a round trip: Buying and selling the stock. But, as we are only interested in the liquidity cost if we need to exit (sell) the position, the liquidity adjustment consists of adding one-half (0.5) the spread. In the case of VaR, we have:

• Liquidity cost (LC) = 0.5 x spread
• Liquidity-adjusted VaR (LVaR) = position ($) * [-drift (%) + volatility *deviate + LC], or
• Liquidity-adjusted VaR (LVaR) = position ($) * [-drift (%) + volatility *deviate + 0.5 * spread].

In our example,

LVaR = $1,000,000 * [-0% + 1.0% * 1.65 + 0.5 * 4.0%] = $36,500

In this way, the liquidity adjustment increases the VaR by one-half the spread ($1,000,000 * 2% = +$20,000).

The Bottom Line

Liquidity risk can be parsed into funding (cash-flow) or market (asset) liquidity risk. Funding liquidity tends to manifest as credit risk, or the inability to fund liabilities produces defaults. Market liquidity risk manifests as market risk, or the inability to sell an asset drives its market price down, or worse, renders the market price indecipherable. Market liquidity risk is a problem created by the interaction of the seller and buyers in the marketplace. If the seller's position is large relative to the market, this is called endogenous liquidity risk (a feature of the seller). If the marketplace has withdrawn buyers, this is called exogenous liquidity risk—a characteristic of the market which is a collection of buyers—a typical indicator here is an abnormally wide bid-ask spread.