[Interest rate] Forward rate of LIBOR

 Interest rate mentioned in this post means short rate such as LIBOR or EURIBOR. Forward rate is the expected interest rate in future, derived from financial market. The forward rate in USD typically going up along with tenor.

 The question is how the forward rate is reliable while the interest rate has always stayed low for a last several years despite forward rate expected higher level all the time.

 Ultimately and generally, no one knows interest rate in the future, but mathematical model has challenged to predict the trend of interest rate in stochastic differential formulae which non-mathematician do not have to know details as it may annoy you.

 Even though the mathematical methodology is hard for the most of people to understand, it is important to know two approaches to predict the future rate in the modelling. One estimates the parameters by observing financial instruments in the market, which can be interest rate swap. Another estimates the parameters from historical data of the interest rate.
 When bankers or traders talk about forward rate, it typically implies the first approach which estimates higher rate. On the other hand, despite the gap between the forward rate and reality, people has not mentioned about the forward rate derived from the historical data, so now time to see how different between those two approaches in the forward rate.

 The chart describes historical USD 3M LIBOR in last 10 years, the forward rate derived from historical data in last 30 years and the forward rate derived from current financial market.
The point is the huge gap between the two different forward curves. As the lowest rate in a last few years, the market consensus seems to expect the lower rate in the future. However, the market still expect relatively higher interest rate compared with the rate estimated from historical data. Historical period covers the period when the interest rate had been certainly high level.
 Forward rate by market has been always shape like that in the figure, but the rate has stayed low. The forward rate will betray again or it is actually time interest rate going up?