# A Refresher on Overround and Vig

/If you pootle around the MoS website - and, by pootle, I mean search - you'll find a number of posts using the terms 'overround' and 'vig' (or 'vigorish'). Today I want to provide a quick refresher on what those terms mean and why they should be important to you if you ever think about placing wagers of your own.

### OVERROUND

Let's start with the term overround, which is defined (for a market where there is only one winner) as the sum of the inverse of all the prices in that market, minus 1.

So, for example, in a head-to-head contest where draws are not possible, if the prices were $2.20 and $1.68, the overround in that market would be calculated by:

Overround = 1/2.2 + 1/1.68 - 1 = 4.97%.

The overround is a measure of how profitable a market is to the bookmaker so, from a bettor's perspective, a market with lower overround should be preferred to one with higher overround.

What we really care about as a bettor, however, is the overround embedded only in the prices for the wagers we plan to make. In the absence of any knowledge - actual or inferred - about a bookmaker's raw probability estimates, we normally assume that the overround embedded in each price is the same as the overround in the market as a whole (though the case for other assumptions about the spread of overround across prices, which usually imply that the prices of favourites carry less overround than those of underdogs, can be made and empirically tested).

If we did know what a bookmaker's assessment of the true probability of an outcome was, we could calculate the overround embedded in any single price using the following formulae:

- Price Offered = 1 / (Estimated True Probability * (1 + Overround))

which gives us

- Overround = 1/(Price Offered * Estimated True Probability) - 1

Now the fair price for any event is the price at which a wager on it would be expected, in the long-run, to break-even, and this can be shown to be equal to the inverse of that event's probability. So, for example, the fair price for a $1 wager on a toss of an unbiased coin would be 1/0.5, or $2.

With that in mind, the first equation tells us that overround can be thought of as deflating the price offered away from the fair price by multiplying that fair price by 1/(1+Overround).

We can see how overround affects the expected return from a bet by using the table below, which considers a range of event probabilities.

Let's walk through the first row.

It relates to an event that the bookmaker assesses as having a 5% chance of occurring. Given that, his (let me assume it's a he for simplicity of exposition) fair price is 1/0.05 or $20 for a $1 bet. If he chooses to embed 1% overround in the final price, that price will instead be $19.80.

Now, if the bookmaker's assessment of the true probability of the event is accurate then the bettor's expected return on every dollar wagered on this outcome is:

5% x ($19.80 - $1) - 95% x $1 = -$0.01 [the $1 terms reflect the initial wager, which is retained by the bookmaker whatever the outcome]

In other words, the bettor should expect to lose about 1c in the dollar on wagers like this.

If, instead, the bookmaker had embedded an overround of 6% by setting a price of $18.87, the bettor's expected return would be a loss of 5.7%.

Roughly speaking, the overround embedded in a price is equal to the bettor's expected loss on that wager, assuming the bookmaker's assessment of the true probability is accurate. As we'll see later, however, vig is a more accurate - indeed, exact - measure of this.

In the columns on the far right of the table we consider a situation where the bookmaker underestimates the probability of events by exactly 5% points. So, for example, for the fourth row where he's assessed an event as having a 35% probability, the true probability is actually 40%.

Here, even for an overround as high as 6%, a bettor would have a positive expectation from a wager on the event. In the 6% overround case, we calculate that expected return as

40% x ($2.70 - $1) - 60% x $1 = +$0.08

Notice that a given sized underestimation on the part of the bookmaker is more disadvantageous to him if it's made about an underdog (eg a team with probability of 25%) than it is about a favourite (eg a team with a probability of 75%). This might in part explain why bookmakers seem to embed more overround in the prices of underdogs than in the prices of favourites (the so-called 'favourite-longshot bias').

### VIG

Vig (or vigorish) is related to overround but is more directly related to the expected return on a wager. In fact, that's exactly what vig is, by definition: the size of the expected loss per dollar from a wager.

We can derive an equation for vig using a generalised form of the expected return calculations we've already been using earlier.

So, we have

Expected Return = (Price Offered - 1) x Estimated Probability - (1 - Estimated Probability)

We define this as the vig of a wager at the given price and probability estimate, and simplify to get

- Vig = Estimated True Probability x Price Offered - 1

A bit of maths yields the following identities with which we can derive vig from overround, and vice versa:

- Vig = Overround / (1 + Overround)
- Overround = Vig / (1-Vig)

We can create a similar table for the effects of vig on expected returns as we did for overround.

Notice that, in this table, the expected returns shown in the middle section are exactly equal to the amount of vig assumed.

### PRACTICAL IMPLICATIONS

So, what does this all mean?

The major implication is that, if you're looking to wager, you should prefer bookmakers who you assess as having embedded less overround or vig in their markets in general (better yet, if you can, in the individual prices for the wagers you're interested in, but that requires some empirical analysis to estimate because you'll almost never know a bookmaker's raw probability estimates).

Here on MoS my major aim is to pit my predictive models against prices offered by widely-available and reputable Australian bookmakers. I don't have the time or inclination to open accounts with a wide range of bookmakers - and I pay a price for that in terms of overround.

Consider, for example, the over/under markets typically posted in 2017 at the opening by the TAB and Centrebet, the prices for which are $1.87 and $1.88 respectively.

In the table at right I've calculated the overround and vig in these markets (assuming that the bookmakers assess the probability of either outcome as 50%).

So, at a $1.87, I'm suffering a 6.5% vig, and at $1.88, a 6% vig. Compared to a bookmaker offering $1.93 that means I'm giving up between 2.5c and 3c on every dollar wagered.

Those calculations all assume that the bookmaker's 50% probability is accurate (in which case I'll still lose money with the bookmaker offering $1.93 prices, but I'll do so less slowly than with a bookmaker offering $1.88). If the bookmaker is miscalibrated, as we saw in the tables above, and if we're able to detect this reliably, then the true overround or vig can be far less than the estimates shown here. In fact, they might even be negative, signifying that a wager has a positive expectation for us.

That aside, however, if you are ever motivated to place wagers based on what you see on MoS, you should be mindful of the vig that you're paying with your bookmaker of choice, and strive to find one with the smallest vig possible - certainly smaller than the apparent vig that MoS Funds are paying if you possibly can.