Time series properties of liquidation discount

Abstract

This paper proposes an approach for quantifying liquidity risk. Urgent liquidation of a portfolio will entail a liquidation discount. This is the market impact discount in value yielded by the immediate sale of the portfolio relative to its in hand market value calculated from the prevailing market conditions.The proposed approach is to firstly construct the log liquidation discount rate using stock market data available from the order book. The behaviour of this empirical time series is modelled and subsequently used to predict future behaviour of the liquidity risk associated with the portfolio. This is achieved by constructing eight different sized portfolios, each corresponding to different numbers of shares from N stocks over two time periods (morning and afternoon). Each stock is to be liquidated on a daily basis. The bid side order book is used to price the immediate sale of a given stock at time t. The price differential between the bid value and market value of the stock is defined as the liquidation discount rate of the stock at time t. Replicating this process for N stocks produces a time series of portfolio liquidation discount rates. Specifically, there are total of eight time series which based on eight different scenarios, each consisting of a different number of shares for a given stock. These scenarios are represented by a which denotes differing proportions of all shares on issue for a given stock. A log transform is applied to the series and these are further segmented into two time periods to investigate liquidity behaviour over time. This paper proposes to model the time series properties of the log liquidation discount rate using the Autoregressive Fractional Integrated Moving Average (ARFIMA) - Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. The mean component of the series is modelled using the ARFIMA(r, d, s) model and contains both ARMA(r, 0, s) and ARIMA(r, 1, s) as special cases (d = 0 and d = 1 respectively). The GARCH(p, q) model is used to model the variance component. A number of models are tested under varying lag structures i.e. different values of p and q. Model performance is based on a model's ability to forecast future values of log liquidation discount rate. The forecast accuracy is measured using the mean square error (MSE). Optimal models resulting from differing values of a over two time periods are identified. The results indicate that the ARFIMA(p, d, q) - GARCH(p, q) model consistently produces the most accurate forecasts over both time periods. For practical purposes a simpler model (in terms of lag structure) is proposed. This model offers a more intuitive interpretation with only a marginal loss in performance. The parameter estimates pertaining to each model are averaged over all values of a for each time period. This produces a two final models each corresponding to a time period. Using these models one can forecast (over n horizons) the variance of the log liquidation discount rate. This forecast is interpreted as the future liquidity risk associated with the portfolio. The empirical results suggest that the variance converges to its long run value at a faster rate in the morning compared to the afternoon.