## - How do you know how much a bet is worth?

## - Understand that variance has a real cost

## - Is the risk of variance costing you money?

## Introduction to the "Swap Equivalent" for Astute Bettors

Recently, I've introduced a nifty concept called the “Swap Equivalent” that's quite handy for folks who take their betting seriously. It's designed to quantify how you should equate the Expected Value (EV) of your risky bets with what's known as Certainty Equivalents (CE). Simply put, multiply your EV by the Swap Equivalent to get the CE—this is essentially the cash amount that would make you feel just as content as holding your open wager. Moreover, this formula helps you gauge the Cost of Variance.

For many, the notion of variance is a bit vague and puzzling. However, for savvy sports bettors, it symbolizes the regular fluctuations in profits as you pursue that ultimate treasure at the end of your betting journey. Variance is more than just a bothersome hurdle; it carries a tangible cost. Why? Because if it didn’t, the Certainty Equivalent for any bet would match its Expected Value right off the bat. This concept has been explored in several of my previous write-ups, highlighting their distinction.

Let’s define the actual Cost of Variance (CoV) as the gap between your EV and your CE. Although it might seem minuscule per bet, it accumulates significantly over time:

CE = s * EV

CoV = EV - CE

This relationship shows that the real Cost of Variance is essentially your EV minus a portion determined by the Swap Equivalent:

CoV = EV - s * EV

CoV = EV * (1-s)

Take, for instance, a sportsbook offering today’s cricket match with odds like Team A +130/Team B -150. Using the odds from 7x7Bets, you reckon Team B has a 60% winning chance. Theoretically, betting on Team B at these odds should mean your EV is zero, suggesting that over time, the outcome would be as good as having that bet amount in your pocket.

Yet, these figures don’t reveal everything. They only cover one aspect of your betting strategy—the value aspect. There’s another critical dimension to consider: risk. By placing any bet on Team B, irrespective of the EV, you expose your money to risk and face variance to reclaim it. So, what’s the cost of this variance? Let’s find out.

Imagine you have a bankroll of ₹1,000 and decide to bet ₹50 on Team B, considering there’s no EV loss. You would win 60% of the time (returning ₹83.33) and lose 40% of the time (returning nothing). The expected value of your bankroll post-match would be:

0.6 * ₹83.33 + 0.4 * ₹0 + ₹950 = ₹50 + ₹950 = ₹1,000

Now, to calculate the Swap Equivalent post-bet:

s = ((1 + w) ^ p - 1) / (p * w)

s = ((1 + 0.088) ^ 0.6 - 1) / (0.6 * 0.088)

s = (1.052 - 1) / 0.053

s = 0.985 or 98.5%

Here, w represents the payout as a percentage of your bankroll, and p is the probability of your bet winning. In this case, w is ₹83.33/₹950 = 0.088. While the EV of your ticket stands at ₹50, the CE is only ₹49.25 due to the Swap Equivalent. This shows that the variance cost incurred is:

CoV = EV * (1 - s)

CoV = ₹50 * (1 - 0.985)

CoV = ₹50 * 0.015

CoV = ₹0.75

This example clearly illustrates the financial impact of variance on your betting outcomes, essential for any serious bettor in India to understand.

## A Slippery Slope for Your Bankroll

It might look like a small sum, but consistently making this wager could slightly erode your theoretical growth each time, potentially leading to bankruptcy. In fact, simulating this bet 10,000 times results in bankruptcy 81.6% of the time (illustrated by the plot of five typical simulation runs provided below).

To grasp this more intuitively, consider the state of your bankroll after wins versus losses. If you win, your bankroll climbs to ₹1,033, making your next ₹50 bet just 4.8% of your total funds. Conversely, a loss drops your funds to ₹950, where the same ₹50 bet then constitutes 5.3% of your bankroll. This dynamic results in you wagering a smaller fraction of your bankroll when you win and a larger fraction when you lose. This disparity can quickly snowball, taking large bites out of your remaining funds during a losing streak, which is hardly a strategy for wealth accumulation or even breaking even.

You might think, as long as you don't bet your entire bankroll at once, you can't possibly go broke, right? This theory sounds solid but is it foolproof? Consider proportional betting — betting 5% of your current bankroll each time. This approach means you bet more following wins and less after losses, presumably evening things out. Moreover, since you never wager 100% of your funds, going broke seems unlikely. But what does "broke" really mean? Technically, you never lose all your money with proportional staking, but what if your bankroll dwindles to just ₹10? That's practically broke, isn't it? Let's examine another simulation under these conditions, but this time, you're considered broke if you fall below ₹10.

The results are even more dire. The increased stakes after winning amplify the impact of any subsequent losses, leading to steeper downswings. Even starting with a stroke of luck (probably the only way to avoid going broke after 10,000 bets), you typically end up broke more than 88% of the time, as shown in the chart below (Y-axis displayed on a logarithmic scale for clarity).

Not surprisingly, given a substantial bet percentage with no betting advantage, your Expected Growth (EG) for a single bet is -0.083%. While it might not seem significant, after 5,600 bets, you would typically see your ₹1,000 bankroll reduced to under ₹10. Calculating your expected ROI with the same odds but a 3.3% edge gives you a full Kelly fraction of 5% for betting on Team B, with an EG of +0.083%—exactly offsetting the negative EG in my scenario, indicating equal disappointment for a neutral EV bet and equal satisfaction for a profitable one.

Imagine if, rather than being a regular bettor, you were Jeff Bezos with a staggering $100 billion at your disposal. In such a scenario, the Swap Equivalent for your betting ticket would virtually hit 100%, implying zero economic cost from your wagers. The calculations to demonstrate this are straightforward but interesting due to the sheer scale of the numbers involved.

The Swap Equivalent (s) and Cost of Variance (CoV) equations would be adjusted as follows due to the massive bankroll size:

s = ((1 + w) ^ p - 1) / (p * w)

s = ((1 + 0.00000000083) ^ 0.6 - 1) / (0.6 * 0.00000000083)

s ≈ (1.0000000005 - 1) / 0.0000000005

s = 1 or 100%

Here, the wager amount (w) is minuscule relative to the bankroll, leading to an s value of virtually 1, or 100%. This results in the Cost of Variance calculation:

CoV = EV * (1 - s)

CoV = 50 * (1 - 1)

CoV = $0

With such an immense bankroll, the economic implications of typical betting variances become negligible. This illustrates how variance, a critical factor for average bettors, fades into insignificance with sufficient financial backing, highlighting the unique position of high-net-worth individuals in high-stake gambling scenarios.

## Conclusion

Realizing the tangible cost of variance helps understand why focusing solely on +EV bets while ignoring -EV or neutral ones is short-sighted. Variance acts like a hidden fee or commission, similar to stock trading, costing you money. Reducing this risk might mean making smaller initial bets, but even optimal staking can't always prevent your bet's EV from exceeding its certainty equivalent significantly.

In situations where the EV of a bet drastically outstrips the certainty equivalent, hedging your risks (either by betting the opposing side at a low-margin bookmaker like 7x7Bets or managing your exposure on an exchange) serves as a financial safeguard. If the cost of this insurance is lower than the cost of your variance, then it becomes the more economically sensible choice.

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