Pastor-Stambaugh Model

The Pastor-Stambaugh model (shorthand PS-model) is a multifactor asset pricing model proposed by Lubos Pastor and Robert f. Stambaugh in 2003. The Pastor-Stambaugh model improves the Fama and French three-factor model by considering a liquidity factor as an additional factor to explain stock returns.

Intuitively, it makes sense that investors that hold illiquid assets should earn a risk-premium. This is because they may have a hard time to sell their securities when a crisis occurs. Thus, they should be compensated for taking that risk and providing financing to companies that become illiquid in times of financial distress.

On this page, we discuss how we can measure liquidity, how to construct the liquidity factor, and finally present the full Pastor-Stambaugh model formula.

Liquidity

There a several ways to measure market liquidity. For example one can have a look at the bid-ask spread. More liquid securities typically have smaller spreads. Alternatively, investors can look at the average daily volume. Pastor and Stambaugh rely on a measure that uses the same inputs as the Amihud illiquidity measure. Moreover, alternative definitions of the liquidity factor may actually use Amihud illiquidity measure to sort stocks based on liquidity. This is because the measure is easy to calculate and only requires daily stock returns and the daily turnover (volume multiplied by the price).

The main insight of Pastor and Stambaugh is that stock returns are related to market-wide liquidity. Thus, companies whose stocks are more sensitive to aggregate market liquidity tend to have a premium compared to stocks whose liquidity is less sensitive to market-wide liquidity. Market-wide liquidity is calculated as the equal-weighted average of the liquidity of all the stocks in the stock market.

Since the market-wide liquidity factor cannot be traded, Pastor and Stambaugh also propose a long-short portfolio that is tradable. In particular, and similar to other equity risk factors, the authors suggest to buy the decile of stocks most sensitive to liquidity shocks and to sell the stocks that are least sensitive to liquidity shocks. Thus, the liquidity factor is constructed by putting the assets in order of liquidity. This is the liquidity factor in the Pastor-Stambaugh model. Now that we have discussed the construction of the liquidity factor, we can discuss the model itself.

Pastor-Stambaugh Model Formula

The PS model is very similar to the Fama-French 3 factor model. In particular, the PS model consists of 4 factors:

    $$E(r_t) = r_f + \beta_1 r_m + \beta_1 HML + \beta_1 SMB + \beta_1 ILLiQUID + \epsilon_t$$

where SMB is the small-minus-big (size) factor, HML is the high-minus-low (value) factor, and ILLIQUID is the illiquidity factor.

Summary

We discussed the Pastor-Stambaugh model, a model that argues that more illiquid stocks tend to earn higher returns because they are exposed to shocks in aggregate market liquidity.

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