Overround when the two prices are lopsided

Question

A bookmaker prices a two-outcome match at decimal odds of $1.50$ on the favorite and $2.50$ on the underdog. (Decimal odds $d$ mean a $\$ 1 $winning stake returns$ $d$ in total.)

What is the overround, what fraction of stakes does the bookmaker keep on a balanced book, and what fair probabilities is the book implying for each side?

Accepted Answer

Implied probabilities: \frac{1}{1.50} = 0.6667, \qquad \frac{1}{2.50} = 0.4000. Overround is the excess of their sum over 1: S = 0.6667 + 0.4000 = 1.0667, \qquad 	ext{overround} = S - 1 = 6.67\%. Hold on a balanced book. If money splits so the payout is equal either way, the winning side is paid 1/S per \1 of stakes, so the bookmaker keeps 1 - \frac{1}{S} = 1 - \frac{1}{1.0667} = 1 - 0.9375 = 6.25\%, matching the identity 	ext{hold} = 	ext{overround}/(1+	ext{overround}) = 0.0667/1.0667. De-vig to fair probabilities. Divide each implied probability by S: \frac{0.6667}{1.0667} = 0.625 \ (	…

Overround when the two prices are lopsided

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