Aug 22

Conditional versus unconditional expectation and variance

by David Harper, CFA, FRM, CIPM

FRM |

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Learning objective

  • Define and distinguish between conditional and unconditional expectation, and conditional and unconditional expectation variance

Reality tends to be conditional

  • An unconditional expectation is the expected value of the variable without any restrictions (or lacking any prior information).
  • A conditional expectation is an expected value for the variable conditional on prior information or some restriction (e.g., the value of a correlated variable). The conditional expectation of Y, conditional on X = x, is given by: E(Y|X=x)

Linear regression is conditional

Gujarati’s two-variable regression is a important conditional expectation. In this case, we say the expected Y is conditional on X:

prf

Bayes’ Theorem is conditional

In this case, the probability is updated conditional on prior information:

bayes_short

Tips for FRM Candidates:

  • The exponentially weighted moving average (EWMA) and GARCH(1,1) approaches to estimating volatility are applications of condition variance: the variance estimate is conditional on the previous variance
  • Loss given default (LGD) is a conditional mean: the expected loss (1 – recovery) conditional on a default.

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