futures - How is this probability (45%) of Fed raising rates 3 times in 2017 calculated from Fed Funds market?

The probability of the Fed raising rates 3 times in 2017 is above 45%.

What data and formula is used to calculate this probability?

This Financial Times article is published on 17Dec2016.

She portended that three rate rises were in the pipeline for next year, more than had been expected.

Second, the market actually believed her. The sharp rise in bond yields that has followed, with the US 10-year Treasury now yielding more than 2.6 per cent for the first time in more than two years, shows this.

So does the fed funds futures market, where investors place bets on the future path of rates. On election eve, this market put the chance of three rate rises next year at less than 5 per cent. By Wednesday, before Ms Yellen spoke, this had risen to about 30. Now it stands just above 45 per cent.

Answer

Using the following data from 12/18/16: Jan 2017 Fed funds futures =9936, Jan 2018 Fed Funds futures =9877 implies that 99.36-98.77 = 59bp of hikes are built in for 2017. IF you assume the only two possibilities are 2 hikes or 3 hikes (meaning, 50bp or 75bp of hikes, assuming each hike would be 25bp), then by simple linear interpolation the probability of 3 hikes must be (59-50)/(75-50)= 36%

The above is a commonly used method when there are only two realistic possibilities, so the only data you need is the correct fed funds futures contract. If there are more possibilities, you need to utilize the options market in order to obtain more information. For example, if the Fed hikes 3 times the Fed fund futures will be at about 98.61. We look at the options market:

Jan 2018 98.6875 put option = 0.095, Jan 2018 98.625 put option = 0.0775, Therefore Jan 2018 98.6875-98.625 put spread = 0.0175, but the max payoff of this put spread is 0.0625

so the implied probability of the put spread being in the money (ie futures<98.625) is 175/625 = 28%, a slightly different answer. This is equivalent to the probability of at least 3 hikes.

For some reason I'm not getting 45%, but the market may have moved, or the option prices i am using may be stale. Anyway i hope this illustrates the methods typically used.