Value at Risk - Ch.13: Liquidity Risk



Reading Source: Textbook - Value at Risk

Topics Covered in this Reading:

  • Defining Liquidity Risk
    • Asset Liquidity Risk
    • Funding Liquidity Risk
  • Assessing Asset Liquidity Risk
    • Effect of Bid-Ask Spread
    • Incorporating Liquidity in Valuation
    • Effect of Price Impact
    • Trading Strategies
  • Assessing Funding Liquidity Risk
  • Lessons from LTCM
  • Conclusions

Value at Risk - Ch.13: Liquidity Risk


We are told that asset liquidity risk depends on:

  1. The price impact function
  2. The size of the position
  3. Prevailing market conditions

Can someone please clarify each of these points? I think an example would be helpful.




  1. This represents how much the price of an asset will change when we sell something (or buy something). If we sell, the price goes down; if we buy, the price goes up.

  2. The size of the position is linked to #1 very closely, to the point where it is hard to talk about them separately. Think of #1 as the derivative of the change.
    Putting them together, let’s say that for every $100 of an asset you sell, its price drops by $0.1. If the $0.1 gets larger, we have a larger price impact. If we sell more than $100, we are making the impact larger by moving a larger quantity.

  3. Again, this is linked to #1. If the market is booming, then that $0.1 might be only $0.01. But when the market is crashing, selling an asset is going to have a larger depressive effect on the price of that asset.


Hi there, I am having trouble understanding the concept of LVAR. Can someone please explain in words what these formulas mean and how they work? This reading is really technical and for now I want to focus on the main (and most important) concepts/formulas to know, and work my way from there.

Thanks so much!


My understanding is that LVAR just means that you are incorporating liquidity costs into the VaR calculation. When we compute VaR, we are saying what happens at the worst xth percentile which ignores any potential impact of liquidating assets, so LVAR takes this into account because you might actually be worse off than your regular VaR calculations due to liquidity impact. I don’t have the equations in front of me, but I hope that helps a bit conceptually.


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.


If you have questions about some of the other formulas, just let me know and we can try and work through them!


I’m having a bit of trouble understanding Asset vs Funding liquidity risk. Can anyone help explain this in simple terms, maybe using some examples? Thanks in advance!


I view Asset Liquidity Risk as more asset driven, and Funding Liquidity Risk as more liability driven.

Asset Liquidity Risk is the risk that the fair price of your assets can’t be achieved due to the impact of your sale has on the market price. For example: Let’s say I hold 1 million dollars of a certain stock, but the entire market is 1.5 million. However, if I try to sell all of my assets at once, I have really changed the supply/demand dynamics of the market since 2/3 of the entire market is for sale. Since supply has likely over saturated the demand, I might not be able to achieve the market price. Let’s say I’m only able to sell my stocks for 800K. The loss would be due to Asset Liquidity Risk. Other causes of loss could be due to non-liquidity of assets, bid-ask spreads, etc.

Funding Liquidity Risk is the risk that our liabilities come due, and we don’t have liquid assets available to pay them. For example, let’s say a number of policyholders choose to elect an early payment of a savings benefit. If we don’t have the liquid assets on hand to pay these obligations, we will need to find cash from another source, which can be costly.

Of course, they can compound and occur at the same time… funding liquidity risk may force us to sell non-liquid assets, which could cause us to sell them at a less-than-fair price (asset liquidity risk).


Thanks @Andrew for your explanation, very helpful!