The Back Half Pays 20.227 Percent, and the Fifth Root Cancels
Because the horizons are ten and five, both sides of the no-arbitrage equation are fifth powers and the root disappears, leaving 1 + f = 1.15 squared over 1.10 = 529/440, so f = 89/440 exactly. Reflecting 10 percent around 15 to get 20 is low by exactly (b - a) squared over (1 + a), a square over a positive number, which is why the reflection can never overshoot for any pair of rates. Under continuous compounding the same problem is linear and 20 percent is exactly right, so the instinct is correct machinery pointed at the wrong convention.