Why does hedging lower EV?

Hedging bets will reduce your variance, or the up and down fluctuations of your bankroll, but doing so will lower your long term expected value. Below is a simplified example to illustrate this.

Suppose each day you get a 30% profit boost and use it on a three leg parlay, where each leg is -110/-110. For simplicity, we'll assume the fair value of each leg is +100, and we stake $10 on each bet. We can look at the expected results after 1,000 days with different strategies.

Strategy 1: No hedging

Since the chance of winning each leg is 50%, the probability of all three hitting is 12.5%

0.5 × 0.5 × 0.5 = 0.125

The parlay builds to +595 pre-boost, and is +774 after the boost. Our profit on each win is $77.40

Profit = ($10 × 774) / 100
Profit = $7,740 / 100
Profit = $77.40

We can then find the total expected profit by multiplying our $77.40 by the amount of bets we expect to win, and subtracting $10 for the days we expect to lose.

Total profit = (1,000 × 0.125 × $77.4) + (1,000 × 0.875 × -$10)
Total profit = (125 × $77.4) + (875 × -$10)
Total profit = $9,675 + -$8,750
Total profit = $925

The total profit is $925 for an ROI of 9.25%. The devig calculator confirms this by showing that it is a 9.2% EV bet.

Strategy 2: Hedge the last leg

Now let's calculate the ROI if we hedge every parlay that reaches its final leg with a -110 bet.

25% of bets will make it to the last leg:

0.5 × 0.5 = 0.25

Of those 25%, the arbitrage calculator shows that we can lock in roughly $31.6 of profit with a $46 wager on a -110 hedge line. So:

Total profit = (1,000 × 0.25 × $31.6) + (1,000 × 0.75 × -$10)
Total profit = (250 × $31.6) + (750 × -$10)
Total profit = $7,900 - $7,500
Total profit = $400

The total profit is $400, which is significantly less than the unhedged strategy.

Conclusion

Why does this happen? When you hedge the last leg in this scenario, you're trading a 50% chance to win $77.4 for a 100% chance to win $31.6

Unhedged EV = 0.5 × $77.4 - 0.5 × $10
Unhedged EV = $38.7 - $5
Unhedged EV = $33.7

Hedged EV = 1 × $31.6
Hedged EV = $31.6

So while you smooth out variance with hedging, you are forfeiting $2.1 in expected value on each of those 250 parlays that make it to the last leg. $2.1 × 250 = $525, which is the difference in profit between the two strategies above.