LO 52.5: Describe endogenous price approaches to LVaR, their motivation and

LO 52.5: Describe endogenous price approaches to LVaR, their motivation and limitations, and calculate the elasticity-based liquidity adjustment to VaR.
Both the constant spread approach and the exogenous spread approach assume that prices do not change in response to trading (i.e., prices are exogenous). This may not always be the case, and it may be necessary to make a liquidity adjustment for endogenous prices. In the case of selling for example, there may be downward pressure on prices, which causes a loss. VaR should include an adjustment for the possibility of this loss. The adjustment should be larger if the market prices are more responsive to trades. 1. Bangia, A.E Diebold, T. Schuermann, and J. Stroughair. (1999). Liquidity on the outside.
Risk 12 (June): 68-73.
2018 Kaplan, Inc.
Page 197
Topic 52 Cross Reference to GARP Assigned Reading – Dowd, Chapter 14
Of the various ways to include an adjustment, a relatively simple method uses the concept of elasticity, E. In this case, it is the proportional change in price divided by the proportion of the market traded:
AP/P AN/N
where: AN/N = size of the trade relative to the entire market
Generally, it is the case that E < 0 and AN/N > 0. A researcher can estimate values for E and AN/N and input them into an expression for LVaR as follows:
LVaR = VaR x
1 Ex AP) = VaRx f 1 Ex p
^ A N j
N
, ^ AN LVaR ——- = 1 E x —— N VaR
The approach is very convenient because the adjustment is independent of the computation of VaR and its assumptions, and the ratio of LVaR to VaR is a function of only two inputs. The obvious limitation is its narrow focus and that it entirely ignores bid-ask spreads and transactions costs. On the other hand, a researcher can easily combine this adjustment with one of the other liquidity adjustments by simply multiplying the effects:
LVaR VaR combined
LVaR VaR
exogenous
x
LVaR VaR endogenous
Example: Endogenous price approach
A trader has a position worth 10% of the size of the market (i.e., AN/N = 0.1) and estimates that E = 0.4 so that AP/P = E x AN/N = 0.4 x 0.1 = 0.04. Compute the ratio of LVaR to VaR based only on endogenous factors and the combined LVaR to VaR ratio assuming the ratio for the exogenous approach is 1.89.
Answer:
LVaR VaR endogenous
= 1 ( 0.04) = 1.04
Thus, the adjustment for endogeneity will increase the total adjustment for liquidity by 4%. Using the liquidity adjustment for the exogenous approach yields the following combined result:
LVaR VaR combined
= 1.89×1.04 = 1.97
Page 198
2018 Kaplan, Inc.
Topic 52 Cross Reference to GARP Assigned Reading – Dowd, Chapter 14
Jarrow and Subramanian (1997)2 offer a more sophisticated method called the liquidity discount VaR, where the trader maximizes expected utility by liquidating the position within a certain period of time. It incorporates both exogenous and endogenous market liquidity, spread cost, spread risk, an endogenous holding period, and an optimal liquidation policy. It does so with three modifications: (1) uses an optimal holding period based on the traders expected-utility optimization problem, (2) adds the average liquidity discount to the traders losses, and (3) has the volatility measure include the volatility of the time to liquidation and the volatility of the liquidity discount factor, as well as the volatility of the underlying market price.
L i q u i d a t i o n , T r a n s a c t i o n s C o s t s , a n d M a r k e t P r i c e I m p a c t
As with most ancial activities, there are tradeoffs to consider when executing a trade. Attempting to sell quickly will usually increase the transactions costs and may have an unfavorable impact on the selling price. Taking more time to sell can increase the exposure to exogenous and unfavorable price changes. A trader should recognize the tradeoff and identify a set of efficient trading strategies that produce the minimum remaining risk exposure at any given point in time, for any given expected cost. The trader should choose the strategy that best fits his risk aversion. A more (less) risk averse trader would choose a strategy that executes more (less) quickly. A more (less) quick execution will reduce (increase) price uncertainty with higher (lower) transactions costs.
L i q u i d i t y a t R i s k

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