LO 52.4: Describe and calculate LVaR using the constant spread approach and the

LO 52.4: Describe and calculate LVaR using the constant spread approach and the exogenous spread approach.
The constant spread approach, as the name implies, calculates LVaR assuming the bid-ask spread is constant. This makes the liquidity cost equal to half the spread multiplied by the size of the position to be liquidated. The liquidity cost (LC) to add on to the initial VaR estimate is then:
LC = 0.5 x V x spread
where: V value of the position = value of the position
, (ask price bid price) spread = ——– ;——- ————– (ask price + bid price) / 2
Recall that VaR quantifies the maximum loss for a given confidence level over a particular holding period. For example, a typical VaR calculation may indicate a 1% probability of losses exceeding $ 10 million over a five-day holding period. LVaR is calculated using the following formula assuming a constant spread:
LVaR = (V x z x a) + [0.5 x V x spread] V -A /
LVaR = VaR + LC
where: V = asset (or portfolio) value za = confidence parameter a = standard deviation of returns
Professors Note: Notice that VaR in this example is dollar VaR as opposed to percentage VaR.
The confidence level of the estimate is 1 o l (e.g., 5% level of significance (a) = 95% confidence level). Note that the larger the spread, the larger the calculated LVaR. Since liquidity risk incorporates selling the asset, not a full round trip, only half of the spread is used.
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Topic 52 Cross Reference to GARP Assigned Reading – Dowd, Chapter 14
Example: Computing LVaR
Suppose that ABC Company has a current stock price of $100 and a daily standard deviation of 2%. The current bid-ask spread is 1%. Calculate LVaR at the 95% confidence level. Assume a constant spread.
Answer:
LVaR = (100 x 1.65 x 0.02) + (0.5 x 100 x 0.01) = $3.80
The previous discussion involved the use of normal VaR (i.e., VaR assuming asset prices are normally distributed). In practice, asset prices are lognormally distributed as was illustrated in the FRM Part I curriculum when we examined the Black-Scholes-Merton option pricing model. In this assigned reading, the author uses lognormal VaR to calculate the liquidity- adjusted VaR. The conventional lognormal VaR, with no adjustment for liquidity risk, is calculated in the following fashion:
Lognormal VaR = [1 exp(p a x z )] x V
where: p = mean return
The liquidity-adjusted VaR is then calculated as follows:
LVaR = VaR + LC = [1 exp(p a x z ) + 0.5 x spread] x V
Using the simplifying assumption of p = 0, the ratio of LVaR to VaR becomes:
^
LVaR VaR
spread
2 x [1 exp(a x za )] This expression indicates that the liquidity adjustment will increase (decrease) when there is an increase (decrease) in the spread, a decrease (increase) in the confidence level, and a decrease (increase) in the holding period.
Professors Note: Notice that the calculation o f lognorm al VaR and normal VaR w ill be similar when we are dealing with short-tim e periods and practical return estimates.
Example: Computing LVaR to VaR ratio (constant spread)
Assume the following parameters: p = 0, a = 0.012, spread = 0.02, and a 95% confidence level. Compute the LVaR to VaR ratio.
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Topic 52 Cross Reference to GARP Assigned Reading – Dowd, Chapter 14
Answer:
, LVaR ——– H ——–r——————————- i VaR 2 x [l – exp(0.012 x 1.65)] 0.02
1.51
The increase from VaR to LVaR is just over 50%, from only a 2% spread. This demonstrates that even a small spread can translate into a surprisingly large liquidity adjustment to VaR.
LVaR can also be calculated given the distribution characteristics of the spread. This is the foundation underlying the exogenous spread approach. If you are given the mean and standard deviation of the spread, you would apply the following formula:
LVaR = VaR + 0.5 x [(ps + zfi x as)] x V
P ro fesso rs N ote: We a d d th e co n fid en ce p a ra m eter tim es th e v o la tility o f th e sp rea d to th e m ean o f th e sp rea d sin ce th e liq u id ity a d ju stm en t in crea ses th e v a lu e a t risk. Also, n o tice th a t th e co n fid e n ce p a ra m eter (or z -score) u sed f o r th e u n certa in ty o f th e sp rea d is la b eled d ifferen tly. T he co n fid en ce p a ra m eter, in th is case, is a v a lu e to b e d eterm in ed .
The exogenous spread approach assumes that the spread is stochastic and that the trades of a single trader do not affect the spread. The spread could follow one of many distributions; for example, the normal distribution or a more leptokurtic distribution (historically, the distribution of the spread has been highly non-normal with excess amounts of kurtosis). Once having assumed a distribution, the researcher can estimate the LVaR using Monte Carlo simulation by simulating values for both Vand the spread, incorporating the spread into Vto get liquidity-adjusted prices, and then infer the liquidity-adjusted VaR from the distribution of simulated liquidity-adjusted prices.
Example: Computing LVaR (assuming normal VaR)
Suppose that ABC Company has a current stock price of $100 and a daily standard deviation of 2%. The mean of the bid-ask spread is 2%, and the standard deviation of the bid-ask spread is 1%. Calculate LVaR at the 95% confidence level assuming the confidence parameter of the spread is equal to 3.
Answer:
LVaR = (100 x 1.65 x 0.02) + -100 x (0.02 + 3 x 0.01) = $5.8
2
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Topic 52 Cross Reference to GARP Assigned Reading – Dowd, Chapter 14
The researcher can determine the optimal value of t! using some suitably calibrated Monte Carlo exercise [Bangia et al. (1999)1 assume a value of three for z; ]. Applying lognormal assumptions, the LVaR using the exogenous spread approach is the lognormal VaR plus the liquidity adjustment:
LVaR = VaR + LC = V x {[1 exp(p a x za)] + [0.5 x (ps + z’a x as)]}
It is worth noting that if a s equals zero, then this expression becomes the LVaR formula for the constant spread approach where ps = spread. Thus, this approach is simply the constant spread approach with an added expression to allow for a stochastic spread.
We can now apply the familiar LVaR to VaR ratio:
LVaR _ ^ j LC VaR VaR
(p>s+ZqXas )
2 x [l exp(a x za )] Example: Computing LVaR to VaR ratio (exogenous spread)
A researcher estimates the mean and standard deviation of the spread to be 0.02 and 0.005, respectively. He also estimates that p = 0 and a = 0.012 for the underlying returns distribution. Using a 95% confidence level, compute the ratio of LVaR to VaR. Assume the confidence parameter for the spread, t! , is equal to 3.
Answer:
LVaR _ , , I —————- —- 1 ————- r———————————————————–7 l . o y VaR
2 x [1 – exp(0.012 x 1.65)] (0.02 + 3 x 0.005)
The result here, when compared to the previous answer, demonstrates how including the possibility of the spread being random (stochastic) can increase the liquidity adjustment. In this case, it almost doubles from 51% to 89%.
Endogenous Price Approaches