In this chapter, we focus on liquidity risk, including the factors influencing liquidity, e.g., the bid-ask spread. Exogenous and endogenous liquidity will be differentiated. The chapter also gives a description of the challenges associated with the estimation of liquidity adjusted VaR (LVaR). The constant spread approach and the exogenous spread approach will both be described as ways of computing VaR.

By the end of the chapter, the reader should be able to describe endogenous price approaches to LVaR. The learner should again be able to compute the elasticity-based liquidity adjustment to VaR. Liquidity at risk (LaR) will be described and compared to VaR and LVaR, with the factors that affect future cash flows and challenges of LaR estimation and modeling being also studied. The chapter also provides lessons on the approaches to estimate liquidity risk, particularly during situations of crisis.

In general, the main focus areas of this chapter are:

- Market liquidity and illiquidity nature and associated risks and costs;
- VaR approximation in markets that illiquid or partially liquid – liquidity-adjusted VaR;
- LaR estimations; and
- An evaluation of crisis-related liquidity risks.

# Liquidity and Liquidity Risks

A trader’s ability to execute a trade or liquidate a position at little or no cost, risk, or inconvenience best depicts the notion of liquidity. This market function is affected by factors like the number of traders in the market, the frequency and size of the trades, the time taken for a trade to be executed, and transaction costs.

There are significant variations in the liquidity of markets, but no market can be said to be perfectly liquid. No matter how big and standardized they may be, their liquidity fluctuates over time and can, therefore, fall dramatically in a crisis situation.

Furthermore, the concept of the going price is reshaped by imperfect liquidity into two going market prices – an ask price (the selling price of traders) and a (lower) bid price (the buying price of traders). Liquidity cost is the difference between the bid price and the ask price.

Due to the fact that the bid-ask spread is a random variable itself, it has an associated risk. All factors remaining constant, a rise in the spread leads to a corresponding rise in the costs of closing-out a position. Therefore, the risk of the spread rising should always be reflected in the market price risk.

A further distinction should also be accounted for. The market price should be least affected by trading in the event of a small position in relation to the market size. Therefore, the bid-ask spread is regarded as exogenous, and the spread is assumed to be determined by uncontrollable markets.

Conversely, in the event of large positions relative to the market, the market itself will be affected by trading activities affecting both the market price and the bid-ask spread.

# Estimating Liquidity-Adjusted VaR

Liquidity-adjusted VaR can be estimated in very many ways which have various degrees of sophistication and ease of implementation. However, the sensitivity of the applied models should be evaluated and should answer the question “how does liquidity change following a change in confidence level, holding period, or any other aspect?”

## The Constant Spread Approach

Liquidity risk is simply incorporated into the computation of VaR-based on a spread that is assumed to be constant, and the best starting point is the bid-ask spread. With this assumption, the liquidity cost is equivalent to the product of half the spread and the size of the liquidated position.

The following liquidity cost (LC) can be added to a standard VaR:

$$ LC=\frac { 1 }{ 2 } { spread }^{ \ast }P $$

Spread in the equation is computes as the actual spread divided by the midpoint. We compare this to a benchmark conventional lognormal VaR lacking liquidity risk adjustment:

$$ VaR=P\left[ 1-exp\left( { \mu }_{ R }-{ \sigma }_{ R }{ z }_{ \alpha } \right) \right] \quad \quad \quad \quad \quad \left( i \right) $$

Midpoint prices of the bid-ask spread are used to compute returns. LVaR is therefore given by:

$$ LVaR=VaR+LC=P\left[ 1-exp\left( { \mu }_{ R }-{ \sigma }_{ R }{ z }_{ \alpha } \right) +\frac { 1 }{ 2 } spread \right] \quad \quad \quad \quad \left( ii \right) $$

Setting \({ \mu }_{ R }=0\),then LVaR to VaR ratio is given as:

$$ \frac { LVaR }{ VaR } =1+\frac { Spread }{ 2\left[ 1-exp\left( -{ \sigma }_{ R }{ z }_{ \alpha } \right) \right] } \quad \quad \quad \quad \quad \left( iii \right) $$

The liquidity adjustment can be shown to:

- Rise proportionally with the assumed spread;
- Fall with an increase in the confidence level; or
- Fall with an increase in the holding period.

This is an easily implemented approach with little information required. However, the constant spread assumption is highly implausible and does not account for any liquidity factors believed to be empirically plausible.

For example, we might believe that the spread is normally distributed:

$$ Spread-N\left( { \mu }_{ spread },{ \sigma }_{ spread }^{ 2 } \right) \quad \quad \quad \quad \quad \left( iv \right) $$

Where \({ \mu }_{ spread }\) is the mean spread and \({ \sigma }_{ spread }^{ 2 }\) is the volatility spread.

Some heavy-tailed distribution might alternatively be applied for excess kurtosis in the spread to be accommodated.

Monte Carlo simulations could be used for LVaR evaluation; both \(P\) and the spread could be simulated, the spread incorporated into \(P\) to get liquidity-adjusted prices, and the liquidity adjusted VaR inferred from the distribution of simulated liquidity-adjusted prices.

However, according to Bangia et. al. (1995):

$$ LC=\frac { P }{ 2 } \left( { \mu }_{ spread }+k{ \sigma }_{ spread } \right) \quad \quad \quad \quad \quad \left( v \right) $$

A suitably calibrated Monte Carlo exercise can be used to determine the value of \(k\), but as suggested, a particular value \(\left( k=3 \right) \) is plausible.

LVaR is then equal to a sum of the conventional VaR and the liquidity adjustment, as shown below:

$$ LVaR=VaR+LC=P\left[ 1-exp\left( { \mu }_{ R }-{ { z }_{ R }\sigma }_{ R } \right) +\frac { P }{ 2 } \left( { \mu }_{ spread }+3{ \sigma }_{ spread } \right) \right] \quad \quad \quad \left( vi \right) $$

Note that equation (\(ii\)) is incorporated into VaR as a special case when \({ \sigma }_{ spread }=0\).Therefore, the ratio of LVaR to VaR is:

$$ \frac { LVaR }{ VaR } =1+\frac { LC }{ VaR } =1+\frac { 1 }{ 2 } \frac { \left( { \mu }_{ spread }+3{ \sigma }_{ spread } \right) }{ \left[ 1-exp\left( -{ \sigma }_{ R }{ z }_{ \alpha } \right) \right] } $$

This implies that the spread volatility \({ \sigma }_{ spread }\) increases the volatility adjustment as compared to the earlier case.

## Endogenous-Price Approaches

The assumption in the previous approaches is that prices are exogenous and the possibility of market prices responding to our trading is ignored. However, a liquidity adjustment reflecting the market response to the trading has to be made.

An additional loss relative to the case where the market price is exogenous is created if a sell is executed and the price is reduced by the sell. The extra loss is then added to our VaR. Furthermore, the market prices responsiveness to our trade affects the liquidity adjustment such that more responsive markets lead to bigger losses.

Various other ways can be used to evaluate the extra loss, but the application of some elementary economic theory is the simplest way. The notion of price elasticity of demand, \(\eta \),is defined as the ratio of proportional change in price divided by the proportional change in quantity demanded, and is expressed as:

$$ \eta =\frac { { \Delta P }/{ P } }{ { \Delta N }/{ N } } <0;\frac { \Delta N }{ N } >0 $$

Where \(N\) is the market size and \({ \Delta N }\) is the size of our trade relative to the market size. \({ \Delta N }/{ N }\) is thus the size of our trade relative to the market size. Therefore, the impact of the trade on the market price is:

$$ \frac { \Delta P }{ P } =\eta \frac { \Delta N }{ N } $$

Hence, \(\frac { \Delta P }{ P } \) can be estimated based on information about \(\eta\) and \(\frac { \Delta N }{ N }\). Thus:

$$ LVaR=VaR\left( 1+\frac { \Delta P }{ P } \right) =VaR\left( 1-\eta \frac { \Delta N }{ N } \right) $$

The liquidity adjustment given is consequently a small one and depends on two easily-calibrated parameters and does not rely on VaR.

The LVaR to VaR ratio is dependent entirely on the demand elasticity, \(\eta\), and the size of our trade relative to the market size, \(\frac { \Delta N }{ N }\).

This is an easily implemented approach whose application is considerable in situations where the effect on VaR of endogenous market responses to our trading activity is our main concern. However, the focus of the approach is narrow and entirely ignores the bid-ask spreads and costs of the transaction.

The approach can be combined with one of the others due to the fact that it only focuses on liquidity and the earlier ones focus on exogenous liquidity. Hence, for the production of an adjustment that addresses both exogenous and endogenous liquidity risk, the two simple approaches are applied, whose combination is given by:

$$ \frac { LVaR }{ VaR } { | }_{ combined }=\frac { LVaR }{ VaR } { | }_{ exogenous }\frac { LVaR }{ VaR } { | }_{ endogenous } $$

## The Liquidity Discount Approach

The liquidity discount approach is quite complicated and was suggested by Jarrow Subramanian (1997). Supposing a trader faces an optimal liquidation challenge, he/she must, therefore, liquidate his/her position in a certain timeframe for the maximization of expected liquidity.

The liquidity discount approach is best as it encompasses both exogenous and endogenous market liquidity, spread cost and spread risk, an endogenous holding period, and an optimal liquidation policy.

The traditional VaR should be modeled in three ways:

- Instead of some arbitrary holding period being applied, an optimal holding period should be used, determined by the solution of the trader’s expected-utility optimization problem. This problem accounts for liquidity considerations and the possible effect of the trading strategy of the trader on the obtained prices.
- The average liquidity discount should also be added to the trader’s losses, or deducted from our prices, to account for the expected losses from the process of selling.
- The volatility of time to liquidation, the volatility of the liquidity discount factor, and the underlying market price’s volatility should all be accounted for by the volatility term.

Supposing that prices between trades follow a geometric Brownian motion with parameters \(\mu \) and \(\sigma\), with the current time being \(0\) and the price at time \(t\) being \(p\left( t \right) \), then the geometric returns \(log\left( { p\left( t \right) }/{ p\left( 0 \right) } \right) \) are normally distributed.

The prices obtained from trading are discounted from \(p\left( t \right)\). A random execution lag \(\Delta \left( S \right) \) is the basis of any order placed at time \(t\) and, thus, happens at time \(t+\Delta \left( s \right) \). Also, all other factors remaining constant, a rise in the execution lag \(\Delta \left( s \right) \) is followed by a corresponding rise in \(s\) such that bigger orders usually take longer to carry out.

Consider a trader who has \(S\) shares and wishes to maximize his/her current position’s present value under the assumption that by the end of some time horizon \(t\), the position will be liquidated while considering all relevant factors.

The following liquidity-adjusted VaR will be produced after a solution for the above problem has been determined:

$$ LVaR=P\left\{ E\left[ ln\left( \frac { p\left( \Delta \left( s \right) \right) c\left( s \right) }{ p\left( 0 \right) } \right) \right] +std\left[ ln\left( { p\left( \Delta \left( S \right) \right) c\left( s \right) }/{ p\left( 0 \right) } \right) \right] { z }_{ \alpha } \right\} $$

$$ =P\left\{ \left( \mu -\frac { { \sigma }^{ 2 } }{ 2 } \right) { \mu }_{ \Delta \left( s \right) }+{ \mu }_{ lnc\left( s \right) }+\left[ \sigma \sqrt { { \mu }_{ \Delta \left( s \right) } } +\left( \mu -\frac { { \sigma }^{ 2 } }{ 2 } \right) { \sigma }_{ \Delta \left( s \right) }+{ \sigma }_{ lnc\left( S \right) } \right] { Z }_{ \alpha } \right\} $$

The conventional VaR is different from the above expression in the following three ways:

- The expected execution \(lag\) \( { \mu }_{ \Delta \left( s \right)}\) replaces the liquidation horizon \(t\) is the conventional VaR;
- The expected discount \({ \mu }_{ lnc\left( s \right) }\) on the shares to be sold is taken into account by the LVaR; and
- Additional terms related to \( { \sigma }_{ \Delta \left( s \right)}\) and \({ \sigma }_{ lnc\left( S \right) }\) supplement the volatility in the conventional VaR, thus reflecting the execution time’s volatility and the quantity discount.

It should also be noted that, in the event that our liquidity imperfections disappear, then \( { \mu }_{ \Delta \left( s \right)}=t\), \( { \sigma }_{ \Delta \left( s \right)}=0\),and \(c\left( S \right) =1\). The LVaR collapses to a conventional VaR as a special case.

Estimates of the usual Brownian motion parameters and the liquidity parameter are necessary for the LVaR expression to be applied, and all of them can be easily obtained.

# Estimating Liquidity-at-Risk (LaR)

LaR is sometimes called the cash-flow-at-risk (CFaR), and it is linked to the risk attached to prospective cash flows over a given time period. It can be defined in terms analogous to VaR.

Therefore, LaR is the maximum likely cash outflow over the horizon period whose confidence level is specified. For a positive LaR, an outflow of cash is the worst possible outcome. Conversely, a negative LaR implies that an inflow of cash is the worst case scenario.

The focus of LaR is the risk of cash flows whereas for VaR, it is the risk of losses. Therefore, LaR is the cash flow equivalent of VaR. The amounts involved in both cases are likely to be different from one another.

Assuming that there exists a large market risk position hedged with a futures hedge of a similar amount, then for a good hedge, the remaining net risk should be fairly small, with VaR estimates reflecting the low basis risk.

The netted or hedged position is largely relied upon by the VaR, whereas LaR is affected by the larger gross position. For a good hedge, there will be a low VaR in relation to the hedge position’s gross risk and, therefore, LaR can easily qualify as an order of magnitude greater than the VaR.

Conversely, most market risk positions have a positive VaR, with little or no cash flow risk, thereby causing the VaR to overshadow the LaR. The LaR is potentially sensitive to the following factors:

- Borrowing or lending, since they both affect future cash flows;
- Necessities of margins on market positions subject to daily marking-to-market;
- Collateral obligations generating cash inflows or outflows based on market movements;
- The exercise of options leading to unexpected cash flows;
- Risk management policy changes.

Often, the obligation to make cash payments happens at not good times for the companies in question as they are often triggered by bad events. Furthermore, apparently similar positions from the perspective of a market risk might differ in cash flow risks. This difference occurs due to the positions having different credit risk features, rather than different market risk characteristics.

Most methods applied in the measurement of VaR can be applied to evaluate LaR. One such strategy is the application of existing VaR estimation tools to evaluate the VaRs of marginable securities, thereby inferring a LaR directly from VaR.

However, it is often advisable that a liquidity risk evaluation model be created from scratch by setting out the basic types of cash flow to be considered, including:

- Certain or near certain cash flows that are known;
- Unconditional uncertain cash flows; and
- Conditional uncertain cash flows.

An appropriate engine to carry out our evaluation, whose choice relies on the types of cash flow risks dealt with, can be created after the above-mentioned factors have been specified.

# Estimating Liquidity in Crises

The occurrence of some events leads to a large fall in prices, thereby triggering a huge number of sell orders, with traders becoming reluctant to buy leading to a dramatic rise in the bid-ask spread. Conversely, a sell order flood can overwhelm the market leading to drastic slow-down of the time taken to execute orders thereby causing lower prices to be obtained.

During market crises, markets assumptions can break down. Crisis liquidity risks should be estimated using methods that account for the distinctive features of a crisis.

The application of a CrashMetrics is a good way of executing such an exercise.

The following delta-gamma expression gives the profit/loss \(\pi\) on a single derivatives instrument:

$$ \pi =\delta \Delta S+\frac { \gamma }{ 2 } { \left( \Delta S \right) }^{ 2 } $$

Where \(\Delta S\) is the stock price change, with the maximum loss happening when \(ds=\frac { -\delta }{ \gamma } \), which is equal to:

$$ { L }^{ max }={ \pi }^{ min }=\frac { { \delta }^{ 2 } }{ 2\gamma } $$

The worst case cash outflow is \(\frac { { m\delta }^{ 2 } }{ \left( 2\gamma \right) } \), with \(m\) being the margin or collateral requirement.

A more complicated model should take into account sophisticated factors like:

- How discrete are the credit events;
- The level of interdependence of the credit events;
- Credit and market risk factors interaction; and
- Complications of applying credit enhancement methods.

# Practice Questions

1) Assuming that we are provided with a position worth 1.2% of the market, and are also informed that the price elasticity of demand η=-0.32. Based on this data, what is the ratio of LVaR to VaR?

- 0.9625
- 0.0387
- 1.00384
- 1.0375

The correct answer is **C**.

Recall that:

$$ \frac { LVaR }{ VaR } =1-\frac { \Delta P }{ P } $$

But:

$$ \frac { \Delta P }{ P } =\eta \frac { \Delta N }{ N } =-0.32\times 0.012 $$

$$ \frac { \Delta P }{ P } =-0.00384 $$

Therefore:

$$ \frac { LVaR }{ VaR } =1+0.00384 $$

$$ =1.00384 $$