Cost of liquidation formula

[clement]

Hello,

I was just wondering something that may be fairly basic: why do we express the cost of liquidation of an entire book as s*alpha/2, when this could be simplified? Namely, the mid-market price components cancel each other out, and we could just have 0.5*(offer-bid)*number of positions. What am I missing here?

Sorry if the question has already been answered, but I couldn't find it.

Many thanks in advance!
Clement

 

[David Harper CFA FRM]

Hi @clement I don't think it quite works because (n) is just an index in the summation. To illustrate with ridiculously rounded spreads, say we two positions:

• #1 position: 10 shares of $9.00 / $11.00 with position value of #10 * $10.00 = $100.00; spread, s = (11 - 9)/10 = 20.0%
• #2 position: 20 shares of $19.00 / $21.00 with position value of #20 * $20.00 = $400.00; s = (21 - 19)/20 = 10.0%

Per the liquidation cost (LC) formula, the LC = 20%*$100.00/2 + 10%*400/2 = $30.00.

But we cannot use (11 - 9)*0.5 + (21 - 19)*0.5 = [(11 - 9) + (21 - 19)]*0.5 = $2.00 because it doesn't incorporate the size of each position. So we need to include the number of shares: [(11 - 9)*10 + (21 - 19)]*20]*0.5 = $30.00. So I think we're back to where we started; each position has a different offer/bid and spread, of course (otherwise we could simplify!). Let me know if i missed something, I hope that's helpful!

 

[clement]

Many thanks David for the answer. What I meant was that we can simply compute the liquidation costs as:
(11-9)/2*10 + (21-19)/2*20
and we get the same answer.

In other words, the liquidation cost formula can be written as the sum{ (offer-bid)/2 * shares} and there is no need for the middle step to calculate the mid-market price, the mid-market value, nor the spread. So, expressing the LC formula as sum{0.5*s*alpha} just complicates it for not much... and I was just wondering why (especially as the it can be useful to save some time in the exam)? I guess that computing s and alpha are probably standards and that's why we ended up for this LC formula; but maybe there is something else at play here?

 

[Varun Momaya]

Hi
I wanted to understand the ÷2 logic in this formulae. As in why do we make the value half?

 

[David Harper CFA FRM]

@Varun Momaya If you immediately buy and sell an asset (aka, round trip) your loss the bid-ask spread, by definition. The cost of liquidation, however, assumes that you already own the asset such that your cost is one-half the round trip. Hope that's helpful!

 

[Dominik Hosefelder]

Hi
I wanted to understand the ÷2 logic in this formulae. As in why do we make the value half?

The way that I like to think about this is that if you own the asset you will mark it to market in your books at the mid-market price. That is, in your companies book / IT system it will show the position at the assets mid-market price.
If you now try to sell the asset, however, you will have to do so at the prevailing bid price, i.e. you will get less than your book had you think. The difference is exactly:
Loss = Mid-market price - bid pirce
Since Mid-market price = (ask price + bid price) / 2, we have:
Loss = (ask price + bid price) / 2 - bid price = (ask price - bid price) / 2

 

[RPate2114]

1

Many thanks David for the answer. What I meant was that we can simply compute the liquidation costs as:
(11-9)/2*10 + (21-19)/2*20
and we get the same answer.

In other words, the liquidation cost formula can be written as the sum{ (offer-bid)/2 * shares} and there is no need for the middle step to calculate the mid-market price, the mid-market value, nor the spread. So, expressing the LC formula as sum{0.5*s*alpha} just complicates it for not much... and I was just wondering why (especially as the it can be useful to save some time in the exam)? I guess that computing s and alpha are probably standards and that's why we ended up for this LC formula; but maybe there is something else at play here?

I have the same question and have not found the answer to.

 

[Dominik Hosefelder]

1

I have the same question and have not found the answer to.

You are perfectly right that there is no need to calculate a spread or mid market price, if you are only interested in calculating the cost of liquidation. This is because the pair (offer pirce, bid price) and the pair (mid market price, liquidity spread) contain the exact same information. To save time in an exam, you should probably skip the extra step.

So why would you even care for the mid market price and the liqudity spread? Let me give you two reasons.

1) Interpretability

Spreads and mid-market-prices are a lot more intuitive and easier to interpret than offer and bid prices. The spread is a really easy to understand measure to compare liquidity of fianancial instruments across time and products. The mid market price gives you a good idea of direction and relative performance over time of a single product. If you try to interpret the combination of offer and bid prices themselves - well, it's possible but it's really messy.
The representation via mid market price and spread also decouples two different risk dimensions: the value of the financial product and it's market liquidity.

2) Usage in practice

Think about how you use these numbers in practice. One example is a standard VaR calculation for market risk. Consider, or example, a bond; it's value depends on an interest rate as a single risk factor and you use a standard discounted cash flow model to calculate the price. You usually calibrate the valuation model to reflect the mid market price. You would then calculate the cost of liquidation for each scenario by applying the liquidity spreads from your historical data sets. If you would use offer and bid prices instead, you would need to calibrate your model twice (increasing your model risk) and compute twice (doubling the necessary computations).
Different but relate is the use of stress testing. You would not stress offer and bid prices seperately, but stress the mid market price and shift the liquidity spread of your positions, to reflect a declining market and a worsening of market liquidity.

 

[RPate2114]

You are perfectly right that there is no need to calculate a spread or mid market price, if you are only interested in calculating the cost of liquidation. This is because the pair (offer pirce, bid price) and the pair (mid market price, liquidity spread) contain the exact same information. To save time in an exam, you should probably skip the extra step.

So why would you even care for the mid market price and the liqudity spread? Let me give you two reasons.

1) Interpretability

Spreads and mid-market-prices are a lot more intuitive and easier to interpret than offer and bid prices. The spread is a really easy to understand measure to compare liquidity of fianancial instruments across time and products. The mid market price gives you a good idea of direction and relative performance over time of a single product. If you try to interpret the combination of offer and bid prices themselves - well, it's possible but it's really messy.
The representation via mid market price and spread also decouples two different risk dimensions: the value of the financial product and it's market liquidity.

2) Usage in practice

Think about how you use these numbers in practice. One example is a standard VaR calculation for market risk. Consider, or example, a bond; it's value depends on an interest rate as a single risk factor and you use a standard discounted cash flow model to calculate the price. You usually calibrate the valuation model to reflect the mid market price. You would then calculate the cost of liquidation for each scenario by applying the liquidity spreads from your historical data sets. If you would use offer and bid prices instead, you would need to calibrate your model twice (increasing your model risk) and compute twice (doubling the necessary computations).
Different but relate is the use of stress testing. You would not stress offer and bid prices seperately, but stress the mid market price and shift the liquidity spread of your positions, to reflect a declining market and a worsening of market liquidity.

Thanks this is helpful