A5.4 RECOMMENDED READING
8.2 ESTIMATING LIQUIDITY-ADJUSTED V A R AND ETL
8.2.1 A Transactions Cost Approach
One approach is to adjust VaR and ETL for liquidity effects through an examination of the impact of transactions costs on our P/L. Given our earlier discussion of liquidity, we can plausibly assume that transactions costs rise with the size of the transaction relative to the market size for the instrument concerned (i.e., because of adverse market reactions due to limited liquidity) and with the bid–ask spread. We can also assume that transactions costs fall with the length of time taken to liquidate the
2There is an extensive financial economics literature on this subject, but broadly speaking, the literature suggests two reasons why market prices might move against the trader. The first is the liquidity effect already alluded to, namely, that there is a limited market, and prices must move to induce other traders to buy. The other reason is a little more subtle: large trades often reveal information, and the perception that they do will cause other traders to revise their views. For example, a large sale may encourage other traders to revise downwards their assessment of the prospects for the instrument concerned, and this will further depress the price.
3There is in fact no need to say much about adjusting ETL estimates. Once we can estimate the VaR, we can easily estimate the ETL by using the average tail VaR approach outlined in Chapter 3: we adjust the VaR for liquidity factors, and then estimate the liquidity-adjusted ETL as the tail average of liquidity-adjusted VaRs. There is therefore no need to say anything further about ETL estimation in the presence of liquidity adjustments.
position, because we can expect to get better prices if we are prepared to take longer to complete our transactions.4A functional form with these properties is the following:
TC=[1+PS/MS]λ1(AL×spread/2) exp (−λ2hp) (8.1) whereTCare transactions costs,PSandMSare the position size and market size (so thatPS/MS is an indicator of position size relative to the market),ALis the amount liquidated at the end of the holding periodhp,spreadis the bid–ask spread (sospread/2 is the difference between the quoted mid-spread price and the actual transaction price), andλ1 andλ2 are positive parameters. We can easily show thatλ1is closely related to the elasticity of transactions costs,TC, with respect to relative position size,PS/MS, and this is helpful because we can apply some economic intuition to get an idea of what the value of this elasticity might be (e.g., a good elasticity of this sort might be in the range from a little over 0 to perhaps 2). For its part,λ2has the interpretation of a rate of decay — in this particular case, the rate of decay (measured in daily units) ofTCashprises — and this is useful in putting a plausible value to this second parameter (e.g.,λ2might be, say, 0.20 or thereabouts).
The first square-bracketed term in Equation (8.1) gives us an indicator of the effect of relative position size on transactions costs: this term will generally be bigger than 1, but goes to 1 asPS/MS goes to zero and our relative position size becomes insignificant. The second term gives the effect of the bid–ask spread on transactions costs, scaled by the amount liquidated at the end of the holding period. The third term in Equation (8.1) gives us the impact of the holding period on transactions costs, and says that this impact declines exponentially with the length of the holding period, other things being equal. Equation (8.1) thus captures our transactions cost story in a way that allows us to quantify transactions costs and assign plausible values to the parameters concerned.
This framework now gives us the market risk measures (i.e., theLVaRand, using obvious termi- nology, theLETL) associated with any chosen holding period taking into account the transactions costs involved. Noting that if we make a loss equal to theLVaR, the amount liquidated (AL) at the end of the holding period will be equal to the initial position size (PS) minus theLVaR, we get:
TC =[1+PS/MS]λ1(AL×spread/2) exp(−λ2hp)
=[1+PS/MS]λ1[(PS−LVaR)×spread/2] exp(−λ2hp) (8.2) which gives usTCin terms ofLVaR. TheLVaR, in turn, is equal to theVaRwe would have obtained in the absence of transactions costs plus the transactions costs themselves:
LVaR=VaR+TC (8.3)
We now solve these two equations to obtain our expression forLVaR, i.e.:
LVaR= VaR+kPS
1+k (8.4)
4This transactions cost story is similar to that of Lawrence and Robinson (1995b,c), which was the first published analysis of LVaR. However, they did not report a precise (i.e., operational) specification for the transactions cost function, which makes their results impossible to reproduce, and they also used transactions costs as only one element of a broader notion of liquidation costs, the other elements of their liquidation costs being exposure (or capital) costs and hedging costs. I do not believe that these latter costs really belong in a liquidation cost function (i.e., it is best to focus more narrowly on transactions costs) and I believe that their including these costs, the former especially, leads them to the mistaken notion that there is an
‘optimal’ liquidation period which minimises liquidation costs. This is inplausible, as liquidation costs should continue to fall indefinitely as the holding or liquidation period continues to rise, until they become negligible. I also have serious doubts about some of the results they report (see Lawrence and Robinson (1995b, p. 55; 1995c, p. 26)): of their four main sets of results, two involve VaRs that are less than the LVaRs, which makes no sense given that liquidation or transactions costs must be non-negative; and one set of results gives LVaR estimates that are about 10 times their corresponding traditional (or unadjusted) VaRs, which seems excessively high.
where:
k=[1+PS/MS]λ1(spread/2) exp(−λ2hp) (8.5) which can be interpreted as a (positive) transactions cost rate (i.e., it gives transactions costs per unit position involved). For low values ofhp,khas an order of magnitude of around half the spread rate, and ashpgets high,kgoes towards zero. The impact of transactions costs on VaR can readily be appreciated by considering theLVaR/VaRratio:
LVaR
VaR = 1+kPS/VaR
1+k (8.6)
The impact of transactions therefore depends critically on the transactions cost ratekand on the ratio PS/VaR, which should also be greater than 1.
Ifkis very low (e.g.,k≈0), thenLVaR/VaR≈1 and transactions costs have a negligible effect on VaR — a result that makes sense because the transactions costs themselves will be negligible. On the other hand, ifkis (relatively) high (e.g.,k=0.025 andPSis high relative toVaR(e.g.,PS/VaR≈ 20?), thenLVaR/VaR≈1.46. So if we takek≈0 as one extreme, andk=0.025 andPS/VaR=20 as a plausible characterisation of the other, we might expect a transactions cost adjustment to alter our VaR estimate by anything in the range from 0 to nearly 50%.
The impact of transactions costs onLVaRalso depends on the holding period, and this impact is illustrated in Figure 8.1, which plots theLVaR/VaRratio against the holding period for a fairly reasonable set of parameter values. In this case, a holding period of 1 day leads to anLVaR/VaRratio of about 1.22, but the ratio falls withhpand is under 1.01 by the timehpreaches 20 days. Clearly,
1.2
1.15
1.1
1.05
10 2 4 6 8 10 12 14 16 18 20
Holding period
LVaR / VaR
Ratio of LVaR to VaR
Figure 8.1 The impact of holding period on LVaR.
Note: Based on normal P/L with assumed values ofµ=0,σ=1,spread=0.05,λ1=1,λ2=0.1, relative position size 0.05, and initial position size 10 times VaR at the 1-day holding period.
if we take a holding period long enough, we can effectively eliminate the impact of transactions costs onLVaR.
Box 8.1 Liquidation Strategies
A trader who wishes to liquidate a position over a certain period has a number of ways to do so. Typically, a strategy of selling quickly will involve high transactions costs — the more rapid the sale, the more pressure the trader puts on the market, and the worse the price he/she gets, and so on — but it also means that the trader rapidly reduces his/her exposure to loss from adverse price movements. On the other hand, a more leisurely strategy generally involves lower transactions costs, but a greater exposure over a longer period. There is therefore a trade-off between transactions costs and exposure costs.
A solution to the optimal liquidation problem is suggested by Almgren and Chriss (1999, 2000). They suggest that we begin by identifying this trade-off and estimating the set of efficient trading strategies that produce the minimum remaining risk exposure at any given point in time, for any given expected (mainly transactions) cost. Once we have identified the efficient trading strategies, we choose one that best fits our risk-aversion. If we are risk averse, we would choose a strategy that rapidly reduces our exposure, but at the cost of accepting a high expected cost;
and if we are less risk averse, we would choose a strategy that leaves us more exposed, but is not expected to be so expensive.