1 Introduction
Reference Frederick, Levis, Malliaris and MeyerFrederick, Levis, Malliaris and Meyer (2018) demonstrated a laboratory phenomenon where participants systematically undervalued a pair of bets which perfectly “hedge” (eliminate) each other’s risk. Participants were first asked their maximum willingness to pay for a $10 bet that a given fair coin lands heads. Most participants answered less than $5, in line with risk aversion. Participants were then asked how much they would pay for $10 if the same coin lands tails, given that they already own a $10 bet on the coin landing heads. In isolation these bets are risky, but as a pair they perfectly hedge each other’s risk, and are equivalent to being given $10 for sure (the coin must land either heads or tails). But most participants in Frederick et al.’s studies valued the pair of bets at well under $10, effectively leaving free money on the table. These studies were, however, conducted in the laboratory, and only two studies involved non-hypothetical payoffs. Replications of Frederick et al.’s effect performed by Chaterjee and Mookerjee (2018) were also conducted with hypothetical payoffs. It is plausible that these instances of economic irrationality will disappear for more consequential decisions outside of the laboratory. Sports betting is a natural domain to investigate this question.
2 A real-life scenario: The 2015/16 English Premier League
The 2015/2016 English Premier League soccer season thrust a few sports bettors into some consequential potential hedging situations. The Premier League features 20 teams each year, but success is highly concentrated, with only six unique teams having won the Premier League since its inception in 1992. The previous season Leicester FC had barely survived in the League, finishing 14th out of 20 teams (teams finishing 18th to 20th in the Premier League are relegated and forced to play the next season in a lower division). A bet on Leicester at the start of the season to win the Premier League could earn £5,000 per £1 risked — an event that no soccer fan could imagine happening (Reference KhanKhan, 2016). Nonetheless, some people did bet on Leicester at these odds — presumably loyal Leicester fans (Reference Massey, Simmons and ArmorMassey, Simmons & Armor, 2011).
Leicester started the season well, winning their first match in August, and were even on top of the League on Christmas day. A defeat on December 26th saw Leicester slip to second place. In all expectation, Leicester would slip further behind, as the busy period of holiday season fixtures tends to favour the established clubs who can afford larger squads of more famous players. However, Leicester returned to the top on January 16th, and remained there until the end of the season, winning England’s top division for the first time in their 132-year history on May 2nd 2016 with two matches still to play.
This period from January 16th to May 2nd 2016 provides a window into the relevance of Frederick et al.’s results to consequential high-stakes decisions. As the remainder of the season played out, Leicester bettors began to contemplate that they could win some life-changing amounts. Table 1 summarizes three cases reported in the media, with potential payouts from £5,000 to £250,000.
Table 1: Cash-out versus hedging calculations
3 Opportunities to hedge
Modern sports betting markets create multiple opportunities to hedge risk. A bettor wishing to hedge the risk of Leicester winning could bet with a bookmaker on all the other teams that could still mathematically win the league. The odds on each team winning will vary, but there will always be a bet size for each contender which will guarantee a sure amount no matter what happens — a perfect “dutching” hedge (Reference Axén and CortisAxén & Cortis, 2019). A simpler alternative is to use a betting exchange, where bettors trade amongst themselves and the exchange takes a percentage commission (Reference Cortis and BriguglioCortis & Briguglio, 2017). A bettor can “lay” the odds on Leicester, a bet which pays-off only if Leicester do not win the League. The original bet on Leicester to win the League, plus a lay bet on Leicester not to win the league, is therefore a pair of hedged bets, just like the bets on heads and tails in Frederick et al.’s studies. We call this approach a lay-hedge for the sake of simplicity. A lay bet can be used to hedge the risk of Leicester winning, and calculation of the correct bet size will guarantee a sure amount no matter what happens (Reference Axén and CortisAxén & Cortis, 2019).
Here we will demonstrate how to calculate a lay-hedge for the original bet, using the case reported on 12th February 2016 (Table 1) as an example. This bettor placed £1 at odds of £5,000. That is if Leicester won the league in May, the bettor would have received £5,000. When looking at betting markets, the odds for Leicester to win the league on 12th February were approximately 4. That is, any bettor placing a new bet on 12th February 2016 on Leicester to win the league (that ends in May) would have received £4 for every £1 bet. The lay-hedge bet size that results in an equal payoff irrespective of outcome can be calculated as follows:
This equation is derived by equating the potential wins from Leicester winning net of the cost of placing a lay bet, and the potential gains from the lay bet if Leicester does not win. Our calculations indicate that the bettor could have hedged the original bet’s risk by laying (1−5%)× 500/4−0.05 = £3,797.47 on the betting exchange. This creates a perfect hedge as follows. If Leicester win the league, the £5,000 profit from the original bet, minus a loss of £3,797.47 from the hedge, results in a profit of £1,202.53. If Leicester failed to win the league, the bettor would have not received anything from the original bet, but still have received £1,202.53 from the lay-hedge. In either case, the profit is the same. Table 1 reports these calculations for each of the three bettors.
Bookmakers offer another way to exit bets. Bettors are offered sure amounts that they can “cash-out” a risky bet for (Reference Lopez-Gonzalez and GriffithsLopez-Gonzalez & Griffiths, 2017). This offer is much simpler to understand than a hedge. Table 1 lists the sure amounts that each reported case cashed-out for. For example, the first bettor who could have gotten £1,202.53 from lay-hedging, but accepted a cash-out of £1,100.22 from the bookmaker, resulting in a loss of £102.31. Bettors cost themselves up to £8,173.47 by cashing-out instead of lay-hedging.
However, there are two noteworthy economic differences between lay-hedging and cashing-out. First, hedging requires additional funds for the offsetting bet. This additional money may not be trivial; Table 1 calculates these amounts as from £3,797.47 to £169,491.50. However, bettors without this additional money on hand should be able to borrow the required money from a range of sources, construct the hedge, and still gain more after repaying interest on the loan than they received from cashing-out. They could have borrowed the funds at an Annual Percentage Rate (APR) of between 11.1% to 28.8% APR, and would have stilled earned more by lay-hedging than by cashing-out (see Table 1). Any financially savvy lender would have been able to lower the interest rate charge used to lay-hedge given the risk-free circumstances of the loan.
Second, hedging means receiving the sure amount after a delay, whereas cashing-out is instant. Time preference is therefore a potential economic reason to cash-out now rather than hedge with a delay. However, the percentage profits of a lay-hedge over a cash-out ranged from 9.3% to 11.3% of the amount cashed-out for, which translate to annual equivalent rates (AERs) of 42.3% to 77.7% (see Table 1). These rates of return are much higher than returns on conventional investments, and were risk free. Only very impatient decision makers should prefer to cash-out now than to hedge at those rates of return (Reference Frederick, Loewenstein and O’DonoghueFrederick, Loewenstein, & O’Donoghue, 2002).
In conclusion, at least some of these observed cases of cashed-out Leicester bets may be due to an inability to comprehend the logic of hedging — rather than alternative economic explanations. It is unclear why the bettors did not seek professional advice when faced with such large sums. We conclude that a misunderstanding of hedging logic, shown in experimental studies (Chaterjee & Mookerjee, 2018; Reference Frederick, Levis, Malliaris and MeyerFrederick et al., 2018), extends to consequential high-stakes decisions. And in fact, when considering Frederick et al.’s studies 1a-1g, we note that the largest undervaluation of the hedged portfolio occurred for the highest-stakes non-hypothetical study (study 1f, with one $100 bet played out for real). This misunderstanding of hedging logic appears robust to payoff size.
Open access
1 Introduction
Reference Frederick, Levis, Malliaris and MeyerFrederick, Levis, Malliaris and Meyer (2018) demonstrated a laboratory phenomenon where participants systematically undervalued a pair of bets which perfectly “hedge” (eliminate) each other’s risk. Participants were first asked their maximum willingness to pay for a $10 bet that a given fair coin lands heads. Most participants answered less than $5, in line with risk aversion. Participants were then asked how much they would pay for $10 if the same coin lands tails, given that they already own a $10 bet on the coin landing heads. In isolation these bets are risky, but as a pair they perfectly hedge each other’s risk, and are equivalent to being given $10 for sure (the coin must land either heads or tails). But most participants in Frederick et al.’s studies valued the pair of bets at well under $10, effectively leaving free money on the table. These studies were, however, conducted in the laboratory, and only two studies involved non-hypothetical payoffs. Replications of Frederick et al.’s effect performed by Chaterjee and Mookerjee (2018) were also conducted with hypothetical payoffs. It is plausible that these instances of economic irrationality will disappear for more consequential decisions outside of the laboratory. Sports betting is a natural domain to investigate this question.
2 A real-life scenario: The 2015/16 English Premier League
The 2015/2016 English Premier League soccer season thrust a few sports bettors into some consequential potential hedging situations. The Premier League features 20 teams each year, but success is highly concentrated, with only six unique teams having won the Premier League since its inception in 1992. The previous season Leicester FC had barely survived in the League, finishing 14th out of 20 teams (teams finishing 18th to 20th in the Premier League are relegated and forced to play the next season in a lower division). A bet on Leicester at the start of the season to win the Premier League could earn £5,000 per £1 risked — an event that no soccer fan could imagine happening (Reference KhanKhan, 2016). Nonetheless, some people did bet on Leicester at these odds — presumably loyal Leicester fans (Reference Massey, Simmons and ArmorMassey, Simmons & Armor, 2011).
Leicester started the season well, winning their first match in August, and were even on top of the League on Christmas day. A defeat on December 26th saw Leicester slip to second place. In all expectation, Leicester would slip further behind, as the busy period of holiday season fixtures tends to favour the established clubs who can afford larger squads of more famous players. However, Leicester returned to the top on January 16th, and remained there until the end of the season, winning England’s top division for the first time in their 132-year history on May 2nd 2016 with two matches still to play.
This period from January 16th to May 2nd 2016 provides a window into the relevance of Frederick et al.’s results to consequential high-stakes decisions. As the remainder of the season played out, Leicester bettors began to contemplate that they could win some life-changing amounts. Table 1 summarizes three cases reported in the media, with potential payouts from £5,000 to £250,000.
Table 1: Cash-out versus hedging calculations
Note: The assumptions are lay odds of 4 in February and 3 in March (Reference InnesInnes, 2016b; Reference SchoolerSchooler, 2016), and a 5% betting exchange commission. Both of these are conservative estimates, which ensure we do not overestimate the possible gains from hedging. An explanation of how betting exchanges work can be found in Reference Cortis and BriguglioCortis and Briguglio (2017). “Reported date” is the day that the bet was reported in the national press. “Bet size” is the reported amount that the bettor placed on Leicester to win the Premier league. “Potential win if Leicester are champions” is the amount of money the bettor could win from this bet size if Leicester win the Premier League. “Amount cashed-out for” is the sure amount of money that the bettor accepted in return for cancelling their bet. “Certain amount available via laying” is the guaranteed win from the combination of the original bet and the lay hedge. “Gain from laying” is the difference between “Certain amount available via laying” and “Amount cashed-out for.” “Percentage gain from laying” compares the gain from laying to the amount cashed-out for. “Assumed available lay odds” are the betting odds on a bet for Leicester to not win the league, which is required to hedge the original bet’s risk. “Amount required to lay” is the amount of money required to bet on Leicester to not win the league to hedge the original bet’s risk.
3 Opportunities to hedge
Modern sports betting markets create multiple opportunities to hedge risk. A bettor wishing to hedge the risk of Leicester winning could bet with a bookmaker on all the other teams that could still mathematically win the league. The odds on each team winning will vary, but there will always be a bet size for each contender which will guarantee a sure amount no matter what happens — a perfect “dutching” hedge (Reference Axén and CortisAxén & Cortis, 2019). A simpler alternative is to use a betting exchange, where bettors trade amongst themselves and the exchange takes a percentage commission (Reference Cortis and BriguglioCortis & Briguglio, 2017). A bettor can “lay” the odds on Leicester, a bet which pays-off only if Leicester do not win the League. The original bet on Leicester to win the League, plus a lay bet on Leicester not to win the league, is therefore a pair of hedged bets, just like the bets on heads and tails in Frederick et al.’s studies. We call this approach a lay-hedge for the sake of simplicity. A lay bet can be used to hedge the risk of Leicester winning, and calculation of the correct bet size will guarantee a sure amount no matter what happens (Reference Axén and CortisAxén & Cortis, 2019).
Here we will demonstrate how to calculate a lay-hedge for the original bet, using the case reported on 12th February 2016 (Table 1) as an example. This bettor placed £1 at odds of £5,000. That is if Leicester won the league in May, the bettor would have received £5,000. When looking at betting markets, the odds for Leicester to win the league on 12th February were approximately 4. That is, any bettor placing a new bet on 12th February 2016 on Leicester to win the league (that ends in May) would have received £4 for every £1 bet. The lay-hedge bet size that results in an equal payoff irrespective of outcome can be calculated as follows:
This equation is derived by equating the potential wins from Leicester winning net of the cost of placing a lay bet, and the potential gains from the lay bet if Leicester does not win. Our calculations indicate that the bettor could have hedged the original bet’s risk by laying (1−5%)× 500/4−0.05 = £3,797.47 on the betting exchange. This creates a perfect hedge as follows. If Leicester win the league, the £5,000 profit from the original bet, minus a loss of £3,797.47 from the hedge, results in a profit of £1,202.53. If Leicester failed to win the league, the bettor would have not received anything from the original bet, but still have received £1,202.53 from the lay-hedge. In either case, the profit is the same. Table 1 reports these calculations for each of the three bettors.
Bookmakers offer another way to exit bets. Bettors are offered sure amounts that they can “cash-out” a risky bet for (Reference Lopez-Gonzalez and GriffithsLopez-Gonzalez & Griffiths, 2017). This offer is much simpler to understand than a hedge. Table 1 lists the sure amounts that each reported case cashed-out for. For example, the first bettor who could have gotten £1,202.53 from lay-hedging, but accepted a cash-out of £1,100.22 from the bookmaker, resulting in a loss of £102.31. Bettors cost themselves up to £8,173.47 by cashing-out instead of lay-hedging.
However, there are two noteworthy economic differences between lay-hedging and cashing-out. First, hedging requires additional funds for the offsetting bet. This additional money may not be trivial; Table 1 calculates these amounts as from £3,797.47 to £169,491.50. However, bettors without this additional money on hand should be able to borrow the required money from a range of sources, construct the hedge, and still gain more after repaying interest on the loan than they received from cashing-out. They could have borrowed the funds at an Annual Percentage Rate (APR) of between 11.1% to 28.8% APR, and would have stilled earned more by lay-hedging than by cashing-out (see Table 1). Any financially savvy lender would have been able to lower the interest rate charge used to lay-hedge given the risk-free circumstances of the loan.
Second, hedging means receiving the sure amount after a delay, whereas cashing-out is instant. Time preference is therefore a potential economic reason to cash-out now rather than hedge with a delay. However, the percentage profits of a lay-hedge over a cash-out ranged from 9.3% to 11.3% of the amount cashed-out for, which translate to annual equivalent rates (AERs) of 42.3% to 77.7% (see Table 1). These rates of return are much higher than returns on conventional investments, and were risk free. Only very impatient decision makers should prefer to cash-out now than to hedge at those rates of return (Reference Frederick, Loewenstein and O’DonoghueFrederick, Loewenstein, & O’Donoghue, 2002).
In conclusion, at least some of these observed cases of cashed-out Leicester bets may be due to an inability to comprehend the logic of hedging — rather than alternative economic explanations. It is unclear why the bettors did not seek professional advice when faced with such large sums. We conclude that a misunderstanding of hedging logic, shown in experimental studies (Chaterjee & Mookerjee, 2018; Reference Frederick, Levis, Malliaris and MeyerFrederick et al., 2018), extends to consequential high-stakes decisions. And in fact, when considering Frederick et al.’s studies 1a-1g, we note that the largest undervaluation of the hedged portfolio occurred for the highest-stakes non-hypothetical study (study 1f, with one $100 bet played out for real). This misunderstanding of hedging logic appears robust to payoff size.