Question:
A portfolio has an initial value of $1 million and a daily volatility of 1%. The portfolio’s average bid-ask spread is 0.3% (0.003).
(i) What is the 1-day liquidity-adjusted VaR (LVaR) at 99% confidence?
(ii) What is the 10-day LVaR?
(iii) What assumptions are required to extend (scale) the 1-day LVaR to 10-day LVaR?
(iv) What are the problems in using the bid-ask spread as a measure of liquidity?
Answer:
See this spreadsheet for calculations.
(i) Typical VaR = $ 1MM * 1% * NORMSINV(99%) = $23,263. Liquidity-adjusted VaR = VaR + [($1 MM)(0.003)/2] = $24,763
(ii)
10-day VaR = 1-day VaR * SQRT(10/1) = $73,566.
Then add liquidity adjustment: (1/2)*(0.003)*($1MM) = 1,500
10-day LVaR = $73,566 + 1500 = $75,066 10-day LVaR.
Note the liquidity adjustment is added after the VaR is scaled, as the spread is constant (hat tip: Michael)
(iii) Returns are normal and, importantly, independently and identically distributed (i.i.d.)
(iv) Culp gives three:
1. We unrealistically assume trades can be simultaneously crossed at the existing spread; i.e., the trades will dynamically impact the spread
2. The spread is not a stable function regardless of the transaction size; e.g., larger sales put more pressure on prices
3. Different spreads may exist and the quoted spread may be different from the effective spread
