Can someone please explain me the following example concerned with Odds for and against probability

Question

Example:
Take an example of two financial-services companies, whose publicly traded share prices reflect a certain probability that interest rates will fall. Both firms will receive an equal benefit from lower interest rates. However, an analyst's research reveals that Firm A's shares, at current prices, reflect a 75% likelihood that rates will fall, and Firm B's shares only suggest a 40% chance of lower rates. The analyst has discovered (in probability terms) mutually inconsistent probabilities. In other words, rates can't be (simultaneously) both 75% and 40% likely to fall. If the true probability of lower rates is 75% (i.e. the market has fairly priced this probability into Firm A's shares), then as investors we could profit by buying Firm B's undervalued shares. If the true probability is 40%, we could profit by short selling Firm A's overpriced shares. By taking both actions (in a classic pairs arbitrage trade), we would theoretically profit no matter the actual probability since one stock or the other eventually has to move. Many investment decisions are made based on an analyst's perception of mutually inconsistent probabilities.

Sophie Macon · Answer

Hi @spunkyaditya - have you tried thinking about it if the probabilities for both are the same? Work through the same logic and you'll see why it the arbitrage works (by definition) with different probabilities...

Michael_in_Scotland · Answer

Maybe this will help. Imagine both firms' shares would be fairly priced at $10 if there is NO interest rate reduction, and $20 if there IS a reduction. Think about how our view of the probability of a rate reduction would effect the amount we'd pay for a share in either company. If we thought a rate reduction was 50% likely, then we'd pay $15 for either share; if we thought it was 25% likely, we'd pay $12.50, if we thought it was 100% likely then we'd be willing to pay the whole $20, and so on. (This is a bit simplistic but I think the point should be clear.)

Given all that, it appears from the question that shares in firm A must currently be trading at $17.50, and shares in firm B must be at $14, even though both have the same intrinsic value regardless of whether the rate reduction does or does not happen. So now, imagine if you shorted A at $17.50 and bought B at $14:

- If the rate reduction does happen, both firms will increase to $20, because that is the fair value of both if the rate rise occurs. You'll lose on A (- $2.50/14% ), but you'll gain more on B (+ $6/43%).

- On the other hand, if the rate reduction does not happen, both firms will drop to $10. You'll lose on B (- $4/29%), but you'll gain more on A (+ $7.50/43%).

Either way you make a profit.

You can work out the exact numbers to convince yourself! You should also be able to see that the exact prices don't matter - I just used $10 and $20 for both companies to make the example simpler.