I am confussed as to how to think through when to take the square root or 1/2 degree of the formula? I assume it is because we are getting the 2 year rate and need to account for compounding? Something isn’t clicking for me with this problem and I was hopoing someone could walk me through the logic? I greatly appreciate any help!

The following interest rate information is observed:

Spot Rates 1 year 10% 2 years 11% 3 years 12%

Based on this data, the 2-year forward rate one year from now is closest to:

Your thinking is correct: the square root accounts for the fact that money is invested for 2 full years at time 1. The 2 year forward rate 1 year from now should be such that if I invest money for 1 year at 10%, then for 2 years at the calculated forward rate, I should end up with exactly the same future value as if I invested for 3 full years at 12% at time 0. A little algebraic shuffling gets you to the solution as presented.