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Quotes in gambling are driven by the probability of an outcome to occur and in a football match like Premier League’s Arsenal vs. Chelsea on Saturday, January 19th 2019 —at the time of writing this short essay about gambling on bad loans— bets can be placed at bookmaker William Hill on the game’s final result as is listed below.

result quote
ARSNL 3.0
draw 3.5
CHELS 2.3

Betting 100 euros (or pounds) on Chelsea would earn us 230 in return if Arsenal is beaten. Clearly, these quotes, or odds, imply probabilities (p, in percentages) that can be extracted by dividing 100 by the bookmaker’s quote.

result quote p
ARSNL 3.0 33.3
draw 3.5 28.6
CHELS 2.3 43.5

Sometimes, quotes are expressed in terms of odds which are the relative ratio of event probabilities: for each euro we would bet on Arsenal, the bookmaker would match it with 2 euros and the pot would add up to 3 euros; should Arsenal win the bookmaker would return us the pot of three euros: 2 euros of net profit plus our initial stake of 1 euro.

result quote p odds
ARSNL 3.0 33.3 2 / 1
draw 3.5 28.6 5 / 2
CHELS 2.3 43.5 13 / 10

It’s easy to notice the probabilities don’t add up to 100: 105.38. The reason is that the bookmaker charges a “vigorish” for its services. To obtain the real implied probabilities (pp) we have to divide the vigorish-full probabilities by their sum as follows.

result quote p odds pp
ARSNL 3.0 33.3 2 / 1 31.6
draw 3.5 28.6 5 / 2 27.1
CHELS 2.3 43.5 13 / 10 41.3

Thus, the real vigorish-free likelihood shows Chelsea is the favourite winner, then comes Arsenal and a draw is the least likely outcome. Having established some basic betting jargon and knowledge, it’s interesting and fun to explore lending as a game of gambling.

# Gambling and loan making

As we’ve seen, bookmakers’ quotes imply probabilities because quotes are based on probabilities. Let’s see how loan making can be characterized as a gambling game by starting with a simple example using sports betting jargon. “Bank vs. borrower” is a match in which a bank is challenged by a borrower’s default —default has already occured— that threatens the loan full amount recovery and is quoted by a bookmaker whose fair quotes and odds are listed below — where p denotes the likelihood (in decimals this time) of full recovery by the bank (the bank “wins”) and conversely the likelihood of no recovery (the borrower “wins”) is 1 – p (always in decimals).

case p quote odds payout
A 0.9 1.1 1 / 9 111
B 0.5 2.0 1 / 1 200
C 0.3 3.3 7 / 3 333
D 0.2 4.0 3 / 1 400

This example is clearly a simplistic one, though it lets us highlight the speculative act implicit in the lending market —especially for non performing loans. The bookmaker’s “fair” quote is obtained by dividing 1 by the probability p —quote = 1/p. Similarly, the probability of an outcome can be extracted by dividing 1 by the quote (and then by multiplying by 100 to transform it into a percentage). The payout is obtained by multiplying the quote by an initial bet (like 100 euros).

In case A the probability of full recovery is 90% (0.90 in decimals), thus a bookmaker’s fair quote is 1.11, i.e. if “the bank wins” (it fully recovers the due amount) the bookmaker pays 111 euros for a 100 euro stake — a profit of 11 euros. Likewise, if the probability of no recovery —in case of borrower A— is 10% (0.10 in decimals, or 1 – 0.90) a bookmaker’s “fair” quote is 10.0, i.e. if “the borrower wins” (the bank recovers nothing of the due amount) the bookmaker pays 1,000 euros for a 100 euro stake (with a 900 euro profit).

The odds are an equivalent measure to express quotes that matches the betting strategies in some sports, like horse races for example. In case A, the bookmaker will match with 1 euro every 9 euros of a punter’s bet; thus, should the punter win he or she will pocket a 10 euro pot broken down into 1 euro (the net profit) and 9 euros (the initial bet). In case C, the bookmaker will match with 7 euros every 3 euros of a punter’s bet; should the punter win, the bookmaker will return 10 euros: the initial bet of 3 plus the 7 euro profit.

The bookmaker’s “fair” quotes for respectively the outcome the bank fully recovers the due amount (or “the bank wins”) and the bank recovers nothing (or “the borrower wins”) for cases A to C in the previous table would be the following:

case bank borrower
A 1.11 10.00
B 2.00 2.00
C 3.33 1.43
D 4.00 1.33

Similarly, the “fair” odds would be:

case bank borrower
A 1 / 9 9 / 1
B 1 / 1 1 / 1
C 7 / 3 3 / 7
D 3 / 1 1 / 3

Let’s assume then that the bookmaker’s actual quotes (vigged or vigorish-full) are the following:

case bank borrower
A 1.07 9.97
B 1.91 1.92
C 3.25 1.30
D 3.75 1.12

This would imply that the probabilities of “winning” —by reversing the quote computation— would be:

bank borrower sum
93.5 10.0 103.5
52.4 52.1 104.4
30.8 76.9 107.7
26.7 89.3 116.0

As just mentioned the bookmaker would quote less than what is fair by charging a “vigorish” —or “vig— that would come in a smaller pay-out (1.07 instead of 1.11, in case A for example) and thus the sum of the implied probabilities would exceed 100%. The”actual” implied probabilities of winning would be the following:

bank borrower sum
90.3 9.7 100
50.1 49.9 100
28.6 71.4 100
23.0 77.0 100

In other words, the bookmaker makes a profit by overestimating the real probabilities of the events which it accepts bets for. So who would be the bookmaker? A CDS trader? A hedge fund? An NPL servicer? In the originate-to-distribute age there’s obviously a huge potential market for loans —especially of the non-performing kind— in Italy that entails gigantic transactions whereby several institutions involved buy and sell credit risk, thus betting on recovery capacities.

# The market for Italian bad loans

Expected credit losses on a loan come from a mix of factors that can be listed as follows:

• default (PD);
• recovery (LGD);

The greater these factors, the larger is credit risk. Default is a gauge for if anything can be recovered, recovery and exposure can be seen as factors regarding how much can be recovered.

In the introductory example it was assumed that recovery was a binary event with a given probability: either full recovery or no recovery at all. In reality, recovery is a random variable that needs to be estimated on a continuous scale ranging from 0 (no recovery at all) to 1 (or 100% full recovery). Recovery is the complement of loss: if an investor recovers 100% it has a loss given by default of 0, and if it recovers 40% it incurs a loss given by default of 60% (LGD).

Recovery is one factor of credit risk that is compounded by another factor: default. Recovery is an issue when the borrower defaults and default is a random event of binary type whose likelihood has to be estimated. In the bookmaker example, default was given — we were handling a bad loan, after all. When dealing with bad loans default probabilities PD are approximately 1 (100%).

Exposure is another issue that has to be faced. When default occurs, what if exposure is large or small? Some debt contracts are amortizing term loans and thus exposure can be large (if defaults occurs early) or small (if defaults occurs late), whereas some contracts like revolving facilities imply that drawn amounts can vary substantially notwithstanding the timing of defaults — in revolving facilities exposure risk is usually deemed to be larger than amortizing loans because borrowers tend to draw down more as default approaches.

Credit losses (expected) = LGD × PD × EAD

Assuming that PD and EAD are both 1 (default has already occured and exposure is given by the then current outstanding loan balance + recovery expenses in unit terms) and that recovery (RR) is the complement to one of LGD, conversely we can approximate credit “wins” in terms of:

Credit wins (expected) = 1 − LGD = RR (recovery)

Transactions on the Italian market for mixed, secured, unsecured and consumer exposures are priced at around 28%, 33%, 5% and 7%, respectively (source Banca Ifis, Market Watch NPL – the Italian Scenario). With no methodological accuracy, we can use these market prices as proxies for full recovery probabilities then as follows.

case p quote odds payout
Mixed 0.28 3.57 18 / 7 357
Secured 0.33 3.03 67 / 33 303
Unsecured 0.05 20.00 19 / 1 2,000
Consumer 0.07 14.29 93 / 7 1,429

If these prices can be thought of as fair bookmaker’s quotes, I wonder how much it would charge in terms of vigorish for accepting bets on “bank wins” given the odds.

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