# Lognormal vs Normal for Interest Rate Curves

## Table of Contents

I've been doing some reading for interest rates at work, especially in this new highly volatile environment, so I figured I'd sum up the research here.

An interest rate curve is a curve representing various interest rates at different points in time. Two types of curves exist: swap curves and basis spread curves (also known as S Curves) and bond curves (F and M curves).

## 1 S Curves

S curves and basis spread curves are built out of fixed coupon rates of market quoted interest rate swaps across different maturities in time. Typically, this means that the short end of the swap curve (< 3 months) is using unsecured deposit rates, typically, overnight, one-month, two-month, and three-month deposit rates. 3 months to 2 years is typically constructed via forward rate agreement contracts and interest rate futures, where as the long end of the curve is constructed generally used observed quotes of swap rates out to 10 years or more.^{1} S curves are typically used to price fixed-income instruments (corporate bonds, mortgage securities, etc) or to price cash flows, swaps, and other derivatives.

## 2 F & M Curves

F and M curves are built of maturity indices, with each index representing a hypothetical bonds par yield at certain tranches, similar to mortgage tranching^{2}. The construction of these is typically dependent on the platform the curves are being pulled from.

F curves are divided into either single issuer curves (government/sovereign), or segmented based on different credit grades.

M curves are US municipal bond curves, using the AAA, AA, A, or BB credit grades.

## 3 Lognormal vs Normal

Lognormal vs normal usage is a long standing discussion within the finance industry. The schism is based off of how historical changes to applied: are rates log-normally distributed, so we should apply historical changes multiplicatively, or are rates normally distributed, so we should apply the historical changes in absolute terms. In other words, we can "bump" the interest rate curves by applying changes to interest rates, but the exact metholodogy has yet to be settled.

The main problem is when a low interest rate environment is applied to a high interest rate environment: if the historical interest rates are low compared to current rates, a few basis points movement will correspond to massive relative moves. Usage of "normal" or absolute historical changes will therefore **underestimate** market moves, but usage of "lognormal" or relative will **overestimate** it.

The exact opposite applies from shifting from a high interest rate environment to a low one: normal changes will overestimate market movement, while relative will underestimate it.

The current solution appears to be using the shifted log normal: https://jfi.pm-research.com/content/27/2/37.