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