Betting parlays is a very popular form of sports betting these days, but usually the odds are stacked in the house’s favor. If the sportsbook has a 5% edge on each leg, then for a two-leg parlay their total edge is roughly 10%. Even worse, if you win the first leg then you’re paying vig on a much larger amount of money, since you essentially are rolling over your winnings from the first leg onto the second.
On the other hand, if you find two legs that are both positive EV (either by handicapping, line shopping, or from a promo), you can use a parlay to help grow your bankroll – if you don’t risk too much. Even +EV plays can have negative expected growth (EG) if you overbet them, so overbetting is one of the pitfalls that sharp bettors must avoid. But, is there ever a time when overbetting is the better play? Yes. If you expect to be able to hedge against the second leg of a parlay at a decent price before it starts, you can essentially overbet and stake much more initially to take advantage of your overall edge.
Let’s look at an example to see how this works. Say you’re awarded a 33% profit boost for an NFL parlay of at least two legs. For standard odds of -110 on sides and totals, that would bump up the odds on a two-legger from a juiced +264 to a juicy +350. If you bet on Sunday morning and choose one afternoon game and the featured Sunday night football game, the huge liquidity in this efficient market should make the probability of each of your bets right around 50% (p1 = p2 = 50%). Given those probabilities, the chance of winning your parlay (p) is 25%, like so:
p = p1 * p2
p = 50% * 50%
p = 25%
With your odds (b) boosted to +350, however, the edge for this parlay is a whopping 12.5%:
edge = bp – q
edge = bp – (1 – p)
edge = 3.5*0.25 – 0.75
edge = 0.875 – 0.75
edge = 0.125 or 12.5%
Despite this large edge, the simple Kelly fraction (f*) to bet is only 3.6% of your bankroll, like so:
f* = p – q/b
f* = 0.25 – 0.75/3.5
f* = 0.036 or 3.6%
If you convert the +350 parlay odds into odds for each of the two legs, you get about +112 for each game. The difference between these effective odds and the vig-free line of +100 tells you how much value (sometimes referred to as closing line value, or CLV, if it lasts until the market closes) you could expect. But, with so much CLV, why is this spot not even a play for 4% of your bankroll? The answer is risk. 75% of the time you’ll lose, so you do best by limiting your stake size to keep from losing too much of your bankroll too often.
Let’s also say you’re not a pro bettor, or even a high-rolling rec, so your bankroll is relatively small and the max bet for this promo is 10% of it. Clearly you should bet within your roll and stake your parlay at 3.6% (or even less) to avoid overbetting, right? On the other hand, maybe you can make more profit by trying a gambit to increase your initial bet and then make a neutral EV hedge later to reduce your risk. That way, you can realize most, or all, of your closing line value on the second leg of your parlay before the game starts, and you won’t have to endure the risk associated with it. Remember that when you have more at stake, you benefit more from hedging (even if the hedge is slightly -EV). The key is being able to line shop to get a low-vig price on your hedge.
I call this strategy the Neutral Hedge Gambit (NHG), and I worked out a formula to help calculate how much is optimal to use for your initial stake:
f* = f*i / f*p
Where:
f* = the optimal fraction of your bankroll for this play
f*i = the simple Kelly fraction for an independent bet
f*p = the “parlay Kelly fraction”, i.e., the optimal Kelly fraction to bet if the parlay had the same probability of winning as the second leg only
Expressing your optimal fraction this way makes it easier to calculate by running a Kelly calculator twice, and it also illustrates how this neutral hedge play concentrates your edge into the first leg by removing the risk of the second. If the second leg is already a sure thing, then there’s no risk associated with it and the theoretical parlay Kelly fraction would equal 1 (meaning don’t adjust your simple Kelly fraction at all, because you can’t gain anything by hedging). The riskier the second leg is, the smaller the parlay Kelly fraction will be and the more you gain by removing that risk from the equation.
How can we use the NHG to reduce the risk of betting here so that you win a guaranteed amount half of the time instead of a larger amount 25% of the time? Let’s take a look. First, we’ll calculate the parlay Kelly fraction to see how much risk reduction you can get by hedging. For that, we use the full odds of the parlay (b = 3.5) and the probability that leg 2 will win (p2 = 50%) to plug into our formula:
f*p = p2 – q2 / b
f*p = 0.5 – 0.5 / 3.5
f*p = 0.36
Given that result, the optimal fraction for you to bet initially is:
f* = f*i / f*p
f* = 0.036 / 0.36
f* = 0.10 or 10%
And, if you win the first leg, then the optimal hedge bet against your second will be 22.5% of your current bankroll. Hold on a minute, that can’t be right. If you can actually get down that much, then your optimal play would be to overbet about three times the Kelly fraction on a parlay that will only hit 1 out of 4 times, and then somehow hedge over 20% of your bank against your second leg? I know it seems crazy, but it’s true. That’s how you maximize your expected growth and turn your CLV on the last leg of the parlay into real money. Suspend your disbelief for a few minutes and look at how and why that’s the case.
Because you no longer care which way leg #2 falls, you’d only lose the parlay outright 50% of the time. In fact, if you can find +100 odds on your hedge, you’d lock up all your CLV for the second leg and get three times the EG by using the NHG than you would by staking your parlay according to the simple Kelly fraction. This increase is so large because losing 10% of your bankroll half of the time (and earning all of your CLV the other half of the time) is far better than losing 3.6% of your bankroll three out of four times and winning 12.6%, but only at a 25% clip.
Obviously, even the best line shopping app can’t promise to find you a vig-free line for a specific market, so maybe it’s better to have a backup plan. If you’re confident that you can find a low-vig line for your hedge like -105 instead, then the optimal play is to stake 8.5% of your bankroll initially and prepare to hedge with 17% or so of your starting bankroll (i.e., before you bet the 8.5%). Hopefully, those numbers are a little more palatable, especially since you’ll still gain twice the EG as when making a naked bet that you plan to let ride. If you can only hedge with a square line like -110, though, you can have problems. If you stake the parlay with 10% initially, then your optimal hedge will net you less growth than if you didn’t try the NHG at all.
The bottom line is that if you can line shop effectively for your hedge, then you can earn much more theoretical profit than you previously thought. But, be careful. Staking the full amount of an NHG is somewhat risky and will cause large swings in your bankroll, so practically speaking it’s better to use a fraction of the theoretical amount (just like when using the simple Kelly formula). Even so, betting at a half of the 8.5% NHG stake is still twice as profitable for your bankroll as simply betting “half Kelly” and letting it ride!