KMV's sigma - a note


According to the KMV1 model, the standard deviation for the default of the \(i\)th borrower \(\sigma_{Di}\) is given by \(\sigma_{Di}=\sqrt{(EDF)(1-EDF)}\) where \(EDF\) is the expected default frequency.


Specifically, let \(p\) be the probability of complete repayment of the loan, and \((1-p)\) be the probability of default, or the expected default frequency (EDF)


Probability of repayment


A Bernoulli trial2, repayment \(p\) returns 1 and default \((1-p)\) returns 0


Expected outcome:

$$ E(\mbox{payoff}) = 1 \times p + 0 \times (1-p)=p$$

Standard deviation:

\begin{align*}\sigma & =\sqrt{(1-p)^2 \times p + (0-p)^2 \times (1-p)} \\ & =\sqrt{(1-2p + p^2) \times p + p^2 \times (1-p)} \\ & =\sqrt{(p-2p^2 + p^3) + (p^2 -p^3)} \\ & =\sqrt{p-p^2} \\ & =\sqrt{p(1-p)} \end{align*}

Probability of default


A Bernoulli trial3, repayment (EDF) returns 0 and default (1-EDF) returns 1


Expected outcome:

$$ E(\mbox{payoff}) = 0 \times EDF + 1 \times (1 - EDF)= 1 - EDF $$

Standard deviation:

\begin{align*}\sigma & =\sqrt{(0 - (1 - EDF)^2 \times EDF + (1 - (1 - EDF))^2 \times (1 - EDF)} \\ & =\sqrt{(-1 + EDF)^2 \times EDF + EDF^2 \times (1-EDF)} \\ & =\sqrt{(1-2EDF + EDF^2) \times EDF + EDF^2 - EDF^3} \\ & =\sqrt{EDF - 2EDF^2 + EDF^3 + EDF^2 - EDF^3} \\ & =\sqrt{EDF - EDF^2} \\ & =\sqrt{EDF(1-EDF)} \end{align*}
Notes
  1. The KMV model is named after Stephen Kealhofer, John McQuown and Oldrich Vasicek. The model was acquired by Moody's Analytics in 2002
  2. Saunders and Cornett (2010, p330)
  3. KMV

References
  1. Saunders A and M Cornett, (2010), Financial institutions management: a risk management approach, 7 ed., McGraw-Hill