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# Cost of liquidation formula

#### clement

##### New Member
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.

Clement

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
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

##### New Member
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?