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# LVaR (BIS paper - Bangia, 1999) - Cost of liquidity vs. Dowd's cost of liquidity

#### emilioalzamora1

##### Well-Known Member
Hi David,

I stumbled over the following two (confusingly) different calculations for the cost of liquidity (CoL): comparing the BIS equation (original paper: BCBS_wp19) on page 14 (http://www.bis.org/publ/bcbs_wp19.pdf) for exogenous liquidity and the formula used by K. Dowd (and in your spreadsheet with two different spread calculations):

Bangia (1999)proposes the formula:

CoL = V *{ (mu + deviate*sigma)/2 } where mu = (ask-bid)/mid price

However, looking at your spreadsheet where you simulate Dowd's LVaR it says the following for the CoL:

the CoL is simply 1/2 times the mean spread. Referring to the Bangia formula, this would mean: 1/2 * mu

and would therefore yield a completely different CoL compared to the Bangia equation.

This is more similar to the Bangia equation, but with the little difference of replacing the normal deviate with 'k'

CoL = V* { (mu + k*sigma)/2 } where k equals a random number (3 in your case)

Why is the normal deviate in Dowd's LVaR (exogenous random spread) replaced with a random number (k)?

For the exam, is it enough to know the difference between the two equation's for LVaR given in Jorion (attached): 1. simple LVaR where we add 1/2 of the spread 2. worst-case LVaR with the extended formula?

Any input and discussion is highly appreciated!

Thank you!

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#### emilioalzamora1

##### Well-Known Member
I would like to add the following: apparently 'k' stands for kurtosis here and 3 is chosen to simulate a normal distribution. But how can this be compared to the normal deviate in Bangia's equation? It would make sense if we say the spread is around 3 standard deviations away from it's mean (even if this is quite a large number).

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @emilioalzamora1 Right, great points about LVaR. Conceptually, I believe there are only two relevant forms to the incorporation of bid-ask spread into LVaR. I would prefer to call them both "exogenous LVaR" but Dowd parses them into constant spread versus exogenous spread, whereas in my opinion both of Dowd's are "exogenous" where the exogenous risk factor is the bid-ask spread. So in my view, we just have exogenous LVaR and/or endogenous LVaR (the "and" refers to the fact that not only are they not mutually exclusive but they do co-exist).

Okay, but in any case, the FRM only has the following two forms of incorporating bid-ask spread into the liquidity cost (LC) of LVaR and these are the only ones we care about for the exam:
• maybe you can see why I prefer to think of these as "exogenous constant spread" versus "exogenous volatile spread" approaches
Both Jorion and Dowd (and others like Carol Alexander) are using the same Bangia. You are correct about the slight difference in the deviate (my basic XLS actually has one sheet for each of these, I do show both actually!).

Dowd references k = 3, please see below (per your kurtosis point), while Jorion/FRM has treated k as a deviate, k = α, and further as a normal one-sided deviate (ie, 1.645 or 2.33). While not, to my knowledge, ever cited in this context, there is a bit of a tradition around k = 3 (eg. , Basel multiplier) because interestingly Chebyshev's inequality (https://en.wikipedia.org/wiki/Chebyshev's_inequality) implies that a multiplier of ~ 3.0 will adjust a 99% normal VaR to a 99% VaR for a non-normal (any!) distribution, such that 3.0 might have been a convenient proxy for adjusting for "model risk." I hope that's helpful!

Dowd's statement about k = 3:
"However, in practice, we might take a short-cut suggested by Bangia et al. (1999). They suggest that we specify the liquidity cost (LC) as:

LC

where k is some parameter whose value is to be determined. The value of k could be determined by a suitably calibrated Monte Carlo exercise, but they suggest that a particular value (k = 3) is plausible (e.g., because it reflects the empirical facts that spreads appear to have excess kurtosis and are negatively correlated with returns, etc.). The liquidity-adjusted VaR, LVaR, is then equal to the conventional VaR plus the liquidity adjustment (Equation (14.6)):

LVaR
= VaR + LC = P*(1 - exp[µ(R) - σ(R)*z(α)] + P/2 * [µ(spread) + k*σ(spread)]" -- Dowd Chapter 14

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Just to append, I was inaccurate above about k = 3. Chebyshev's give Pr(SE>k) < 1/K^2 such that k = sqrt[1/(2p)]. If we want a 95% non-normal confidence, then k = sqrt[1/(2*5%)] = 3.16 and if we want a 99% non-normal confidence, then k = sqrt[1/(2*1%)] = 7.07. So Jorion (appendix 5A) shows the Basel's k = 3 is a way of adjusting a 99% normal deviate of 2.326 into a 99% non-normal (any!) deviate with 2.326*3.0 = approx the 7.07. Contrast with the above, which is just 3*deviate such that we are ensuring only 88.8% non normal confidence = 1 - 1/3^2. Thanks,

#### emilioalzamora1

##### Well-Known Member
Excellent explanation, David! In particular, the details about Chebyshev's inequality. Again and again surprised how much subtleties are in and around one single concept like LVaR. Need some time to digest now.