Yes, davidwpg is right. Considering LVaR is just a way to consider the impact that liquidity has on your position. The first two formulas include the impact of bid-ask spreads to the VaR calculation, and the third formula includes the impact of price effects.
1: LVaR=VaR+L1=Wασ + 0.5WS
This is the normal VaR + the liquidity term. The normal VaR is just the starting position multiplied by alpha standard deviation terms. The liquidity impact is: 0.5WS. Here, spread is the bid-ask spread and is defined as: [P(ask)-P(bid)]/P(mid). When calculating the normal VaR, we assumed the prices of stocks were P(mid) or in the middle of the bid-ask spread. In reality, if we needed to sell our stocks, we would be be selling at the P(bid) price, or something that is lower than the P(mid) assumed in the original calculation. In fact, for each dollar we would sell, we would lose half of the spread (since P(mid) is half way between the P(ask) and P(bid)). So that is where the 0.5WS comes from. In the event we needed to sell our position, we would lose that much additional more than assumed in our original calculation: Position times half the spread.
2: LVaR=VaR+L2=Wασ + 0.5W(S bar+α’σs) Note it’s a little hard to write the formulas here so follow along with the textbook.
This formula is similar to the first one. We are still adjusting for additional losses due to the fact that we would need to sell our position at the P(bid) price instead of the P(mid) price. However, in the first formula, we assumed the bid-ask spread, S, was always fixed. Here, we are assuming the bid-ask spread also varies. For the VaR calculation, we assume that the spread has become larger. It is alpha x spread volatility away from the mean. So we are losing more now than we would have at the average spread value, because the bid-ask spread has gotten larger in a tail event.
3: LVaR= α(V(W))^0.5+C(W)
Here, C(W) represents that costs due to market impacts from selling our stock. If we own a large position in the market, then the price in the market can be impacted by our selling position. This happens because the price drops when we sell our stocks, and so we receive less for them. The two C(W) formulas reflect formulas to determine how much cost will occur due to these price effects. This is added on at the end of the formula because it is an additional known cost. The V(W) is the volatility of the dollar amount of the position. When all stocks are immediately sold, there is no volatility to the dollar amount of the position, so V(W)=0. We have sold the entire position, so we will have no wealth volatility, only fixed costs. When we sell position over time, the value of our stocks can vary over time, so that is what V2(W) is calculating. At the end of the day, this is the volatility of the value of our stock (as a sort of standard deviation) multiplied by alpha: which gives us something that is essentially the same as our usual VaR calculation PLUS the costs incurred due to the prices in the market being impacted by our sale.