## The Twelfth Camel

In a way, this post should write itself. For those of you with context, the title should be self explanatory. And you need not read further.

For the rest I’ll write a rather small essay.

The story is of the old Arab who died leaving his eldest son half his wealth, the second a fourth of his wealth and the youngest one sixth. The wealth in question turned out to be 11 camels.

With 11 being a prime number, how could this will be executed without any of the camels being executed? An ingenious neighbour came in and lent his camel. Now there were twelve. The three sons respectively received 6 ($\frac{12}{2}$), , 3 ($\frac{12}{4}$) and 2 ($\frac{12}{6}$) camels respectively. One camel was left over – the neighbour’s, who took it back.

This is mathematically inaccurate, since the sons received fractions of their father’s wealth slightly different from what he had intended. However, in general in life, this parable of the twelfth camel offers a useful metaphor.

In engineering, this is rather common – you have systems such as a choke, for example, to enable systems to get started from a “cold start process”. The choke comes in only at the time of startup – once the thing has started, it plays no role.

However, it has its role in normal life and business as well. For example, after a bad breakup, you might rebound to a “stop gap partner”. You know that this is not going to be a long term relationship, but this partner helps you tide over the shock of the bad breakup, and by the time this relationship (inevitably) breaks up, it has achieved its purpose of getting you back on track. And you get on with life, finding more long term partners.

Then, when the company is in deep trouble, you have specialists who come in to take over with the explicit goal of cleaning things up and getting the company ready for new ownership. For exanple, John Ray III has recently taken over as CEO of FTX. His previous notable appointment was as CEO of Enron, soon after that scandal had broken. He will not stay for a long term – he will just clean things up and move on.

And sometimes the role of the twelfth camel is rather more specific. Apart from “generic cleaning”, the temporary presence of the twelfth camel can be used to get rid of people who had earlier been hard to get rid of.

In sum, the key thing about the twelfth camel theory is that the neighbour knew all along that he was going to get back his camel. In other words, it is a deliberate temporary measure intended to achieve a certain set of specific outcomes. And the camel itself may not know that it is being “lent”!

## Valuing loan deals for football players

Initial reports yesterday regarding Radamel Falcao’s move to Manchester United mentioned a valuation of GBP 6 million for the one year loan, i.e. Manchester United had paid Falcao’s parent club AS Monaco GBP 6 million so that they could borrow Falcao for a year. This evidently didn’t make sense since earlier reports suggested that Falcao had been priced at GBP 55 million for an outright transfer, and has four years remaining on his Monaco contract.

In this morning’s reports, however, the value of the loan deal has been corrected to GBP 16 million, which makes more sense in light of his remaining period of contract, age and outright valuation.

So how do you value a loan deal for a player? To answer that, first of all, how do you value a player? The “value” of a player is essentially the amount of money that the player’s parent club is willing to accept in exchange for foregoing his use for the rest of his contract. Hence, for example, in Falcao’s case, GBP 55M  is the amount that Monaco was willing to accept for foregoing the remaining four years they have him on contract.

Based on this, you might guess that transfer fees are (among other things) a function of the number of years that a player has remaining on his contract with the club – ceteris paribus, the longer the period of contract, the greater is the transfer fee demanded (this is intuitive. You want more compensation for foregoing something for a longer time period than for a shorter time period).

From this point of view, let us now evaluate what it might take to take Falcao on loan for one year. Conceptually it is straightforward. Let us assume that the value Monaco expects to get from having Falcao on their books for a further four years is a small amount less than their asking price of GBP 55M – given they were willing to forego their full rights for that amount, their valuation can be any number below that; we’ll assume it was just below that. Now, all we need to do is to determine how much of this GBP 55M in value will be generated in the first year, how much in the second year and so on. Whatever is the value for the first year is the amount that Monaco will demand for a loan.

Now, loans can be of different kinds. Clubs sometimes lend out their young and promising players so that they can get first team football in a different club – something the parent club would not be able to provide. In such loans, clubs expect the players to come back as better players (Daniel Sturridge’s loan from Chelsea to Bolton is one such example) and thus with a higher valuation. Given this expectations, loan fees are usually zero (or even negative – where the parent club continues to bear part of the loanee’s wages).

Another kind of loan is for a player who is on the books but not particularly wanted for the season. It could happen that player’s wages are more than what the club hopes to get in terms of his contribution on the field (implying a negative valuation for the player). In such cases, it is possible for clubs to loan out the player while still covering part of the player’s salary. In that sense, the loan fee paid by the target club is actually negative (since they are in a sense being paid by the parent club to loan the player out). An example of this kind was Andy Carroll’s loan from Liverpool to West Ham United in the 2012-13 season.

Falcao is currently in the prime of his career (aged 29) and heavily injury prone. Given his age and injury record, he is likely to be a fast depreciating asset. By the time he runs out his contract at Monaco (when he will be 33), he is likely to be not worth anything at all. This means that a lion’s share of the value Monaco can derive out of him would be what they would derive in the next one year. This is the primary reason that Monaco have demanded 30% of the four year fee for one year of loan.

Loaning a player also involves some option valuation – based on his performance on loan his valuation at the end of the loan period can either increase or decrease. At the time of loaning out this is a random variable and we can only work on expectations. The thing with Falcao is that given the stage of his career the probability of him being much improved after a year is small. On the other hand, his brittleness means the probability of him being a lesser player is much larger. This ends up depressing the expected valuation at the end of the loan period and thus pushes up the loan fee. Thinking about it, this should have pushed up Falcao’s loan fee above GBP 16M but another factor – that he has just returned from injury and may not be at peak impact for a couple of months has depressed his wages.

Speaking of option valuation, it is possibly the primary reason why young loan signings to lesser clubs come cheap – the possibility of regular first team football increases significantly the expected valuation of the player at the end of the loan period, and this coupled with the fact that the player is not yet proven (which implies a low “base sale price”) drives the loan valuation close to zero.

Loaning is thus a fairly complex process, but players’ valuations can be done in rather economic terms – based on expected contribution in that time period and option valuation. Loaning can also get bizarre at times – Fernando Torres’s move to Milan, for example, has been classified by Chelsea as a “two year loan”, which is funny given that he has two years remaining on his Chelsea contract. It is likely that the deal has been classified as a loan for accounting purposes so that Chelsea do not write off the GBP 50M they paid for Torres’s rights in 2010 too soon.