# Interest Rate Convexity - Fundamental Question

I have a very basic question around convexity adjustments in swap valuations. I am comfortable with the mathematical derivation of the convexity adjustment.

My question relates to when and why a convexity adjustment is deemed necessary in some cases and not others.

The general rule seems to be that:

1. For a vanilla IRS, or any other variation in which the floating rate applied to a given period $(T_i, T_{i+1})$ is observed at $T_i$, the "correct" approach is to assume that the expected future spot rates are equal to forward rates, and thus a convexity adjustment is not needed.

2. If on the other hand, the floating rate is set at the end of the period, as in a Libor-in-arrears swap, the "correct" approach is to apply a convexity adjustment to the forward rate to arrive at the expected future swap rate.

Why is the "forward rates will be realized" assumption valid for a simple FRA-based IRS swap, and not for a Libor-in-arrears swap?

Everything I have read seems to simply state this as fact without providing some sort of explanation of why this convention is used?

Vanilla Swap: We observe the LIBOR $L(T_i, T_{i+1})$ at time $T_i$ and payment occurs at $T_{i+1}$. Therefore the correct measure is the classic "forward-risk-neutral" measure with respect to a ZCB expiring at time $T_{i+1}$. In this measure, $L(T_i, T_{i+1})$ is a martingale and therefore $L(t, T_i, T_{i+1})$ = $\mathbb{E}^Q[L(T_i, T_i, T_{i+1})]$ - in other words in this case the forward LIBOR in question is equal to its expected future spot value (in the aforementioned measure) and hence no convexity adjustment is needed.
LIBOR-In-Arrears Swap: We observe the same LIBOR, again at $T_i$, but this time payment also occurs at $T_i$. Therefore the appropriate measure is one which is forward risk neutral with respect to a ZCB expiring at time $T_{i}$. However, in this measure, the LIBOR in question $L(T_i, T_{i+1})$ is not a martingale. Hence the forward LIBOR is not equal to the expected future spot rate, hence the need for the convexity adjustment.