Apr 24, 2023
Apr 24, 2023

# To Bet High Volume or High Value? That is the Question

## Understanding Expected Value

Along with topics like closing line value, distinguishing luck from skill, and what makes a good return on investment, there is one other debate that comes up frequently amongst aspiring sports bettors: should I make a large number of low value bets, or a small number of more selective high value bets?

### To Bet High Volumes or High Value? That is the Question

Often this question divides bettors into two camps. Those who believe value is the most important aspect of a bet will invariably seek to maximise it, rejecting opportunities of lower value since they return less profit.

In contrast, the other camp will insist that it’s far more important to control the influence of variance or luck, and the best way to manage that is to have a large number of bets, even if their value is smaller. Who is right? That is the topic of my latest article.

Let’s briefly recap: value in betting is a concept used to describe a bettor’s expected profit. The expected value (or EV) of a bet shows us how much we can expect to win or lose (on average) per bet.

The simplest way to calculate it is to divide the bookmaker’s odds by the true odds and subtract 1.

This is equivalent to dividing the true outcome probability by the bookmaker’s implied outcome probability and subtracting 1.

Of course, the hardest task in betting is knowing how to figure out the true outcome probability, but we won’t worry about that in this article.

Suppose the true odds of something are 2. If the bookmaker mistakenly offers 2.1, then the expected value is 2.1/2 – 1 = 0.05 or 5%. After 1,000 1-unit bets of the same expected value, on average the bettor should have made a profit of 50 units.

If they offered 2.25, the expected value is then 12.5%. Usually, it is the bettor who makes the mistake. If they accepted odds of 1.95, their expected value would be -2.5%.

Of course, on average means most likely, but luck, both good and bad, can result in different outcomes over a series of bets. The expected profit from 1,000 bets with odds of 2 holding 5% value might be 50 units, but we’ll finish with less if we are unlucky, more if we’re lucky.

I’ve discussed at length how the possible outcomes will distribute in a previous article looking at the range of possible betting returns. Below, I have shown the distribution of possible yields we can expect from this scenario.

For those interested in the mathematics behind this distribution, my website has a simple Excel calculator explaining the methodology, whilst my latest book Monte Carlo or Bust - Simple Simulations for Aspiring Sports Bettors offers a lot more detail.

You can see from the distribution that the most likely outcome, assuming our expected value of 5% is correct, is a yield of 5%. From 1,000 bets of each of unit stake this would be equivalent to 50 units of profit.

You can also visually see the range of possibility from terribly bad luck to exceptionally good luck. It’s highly unlikely that we’ll finish with a yield of less than -5%, or more than +15%.

And we can also visually assess the likelihood of making a loss – it’s simply the area under the bell-curve to the left of the vertical axis at yield = 0%. Mathematically, this can be calculated, and it’s about 6%.

Look at what happens to the distribution of possible outcomes if we change the number of bets. Below I’ve repeated the exercise for 100 bets, and the distribution is compared to the first one.

I’ve increased the scale of the x-axis to allow both curves to visually fit, but the blue curve is exactly the same as before. What’s important here is the comparison.

Two things stand out. Firstly, there is a much broader range of possibilities, with a relatively greater likelihood of both larger losses and larger profits, but a relatively smaller likelihood of seeing the most expected outcome.

In statistical terms, the variance has increased. This is simply another way of saying that luck plays a bigger role.

Secondly, and as a consequence, there is now a greater area under the bell curve sitting in negative-yield territory. In fact, it’s about 31%.

This means that despite having a value of 5% in these 100 bets, there’s a chance of nearly a third that we will make a loss. More bets will reduce the variance in possible outcomes by reducing the influence of luck, and the likelihood of losing if you hold positive expected betting value.

It’s 1-0 to those who advocate having a higher betting volume.

But wait, people who bet more selectively do so precisely because they are targeting higher-value bets. This means their expected yield will be higher. We need to adjust the orange curve above to reflect this.

Let’s suppose our high-value bettor with only 100 bets holds an average expected value of 20% instead. How will they now compare to the first high-volume bettor? Take a look.

The high-value bettor’s yield distribution still has the same shape, but it’s been shifted to the right by 15 percentage points with the expected (average) yield now at 20%. Variance in outcome is still just as high, but there are now far fewer unprofitable outcomes.

In fact, the figure is just 2%, fewer than the high-volume bettor with 1,000 bets and 5% expected value.

A convenient way to compare these two bettors’ distributions is a metric known as the z-score. This divides the expected yield by the standard deviation of the distribution. In effect, it’s a measure of expected yield per unit of variance, and the bigger the score, the better it is.

In the financial industry, this metric goes by the name of the Sharpe Ratio. The standard deviation, σ, is a measure of the spread of the distribution. Its square is simply the variance.

For a series of bets with the same odds, o, it can be calculated using the following equation:

p is the true win probability of the bet, and n is the number of bets. The equation can be reliably used in situations where bets have different odds, provided stake sizes are the same. In this case, o is simply the average odds.

To calculate p, simply add 1 to the expected yield (giving you the expected return) and divide by the odds. Hence, for our high-volume bettor with an expected yield of 5% (or 0.05), p = (1+0.05)/2 = 0.525 or 52.5%, and therefore σ = 0.0316 (or 3.16%).

Roughly two-thirds of all possible yields in the high-volume bettor’s distribution will be found within 3.16% above or below the expected value of 5%. Finally, we can now calculate the z-score: 5% / 3.16% = 1.58.

Doing the same for the high-value bettor gives a z-score of 2.04, superior to the score of the high-volume bettor.

The score is now 1-1 in the competition between volume and value.

More volume reduces the variance in outcomes, but being more selective and achieving a higher expected yield can improve your z-score and expected yield per unit of variance.

So far, I’ve just arbitrarily picked expected values and number of bets. Our high-value bettor had four times the expected value of the high-volume bettor, and one 10th of the number of bets, but how realistic are these figures?

Bets of 20% expected value are very hard to come by when betting odds are 2. Can we really expect to be able to find 100 of them for every 1,000 with an expected value of 5%? We need a way to find out.

A proxy method of assessing how much value exists in a betting market is to study the price movements.

If we first hypothesise that a market’s closing odds, on average, represent a fair reflection of the true odds - a topic I covered previously - the size of the preceding movements can be used to calculate a rough measure of any preceding expected value that might have existed in the odds.

The bigger the move, the larger the expected value. This is not to argue that closing odds never hold value, but merely that on average, the difference between the two sets of odds is a reasonable reflection of the true underlying value available.

Taking a large dataset of opening and closing Pinnacle soccer match betting odds, I’ve calculated the relative availability of expected value that might realistically be available. This is shown below.

In this betting market, there was roughly 20 times the available number of betting opportunities where expected value was at least 5%, compared to where it was at least 20%.

Hence, if our high-volume bettor was able to find 1,000 betting opportunities with expected value of 5%, our high-value bettor in the same market, and over the same timeframe, would realistically be expected to find no more than 50 opportunities, half of what I had previously assumed.

Taking this information, let’s now recalculate the z-score for the high-value bettor with this smaller sample size.

It is now 1.44, smaller than the number for the high-volume bettor. The probability of loss after those 50 bets is now 7.5%. Consequently, given the relative availability of different sizes of expected value, in this scenario the high-turnover bettor would appear to have the superior strategy in terms of risk management.

Volume takes a 2-1 lead over value.

Thus far, I’ve just considered betting yields, but ultimately every bettor is more interested in the actual profit the bettor will make.

A 5% yield from 1,000 bets will give a 50-unit profit from one-unit stakes. In contrast, a 20% yield from 50 bets will give just a 10-unit profit.

Of course, any advocate of Kelly staking will point out that bets with higher excepted value should attract higher stake sizes. In this case, for the same odds, with four times the value we could justify four times the stake size. Increasing this then increases the expected profit to 40 units, much closer to the 50 units profit for the first bettor.

Value pulls a goal back by means of a contentious penalty.

The final score in our competition between volume and value is a narrow 3-2 victory for volume. Whilst this exercise was just a bit of fun, it does help illustrate the competing influences of reducing variance (by increasing the number of bets) and increasing expected value (by being more selective).

It would seem that in a soccer betting market at least, there isn’t a sufficient availability of higher expected value opportunities to fully justify this more selective strategy relative to one that seeks to maximise volume and reduce variance.

Nevertheless, it was pretty close; different markets may yield different relative opportunities that might shift the balance the other way.

Reducing variance is always a great goal for any bettor to have, but this thought experiment should help show that it’s not the only factor one needs to consider.

High-volume bettors will probably continue to mock high-value niche bettors and vice versa, but in fact, a better understanding of the relative influence of volume and value will help any aspiring bettor maximise the outcomes of their goals.

Be sure to read more insightful articles from Joseph Buchdahl at Pinnacle Betting Resources and inform your sports betting predictions.

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