According to the KMV^{1} 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 trial^{2}, **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 trial^{3}, **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**

- The KMV model is named after Stephen Kealhofer, John McQuown and Oldrich Vasicek. The model was acquired by Moody’s Analytics in 2002
- Saunders and Cornett (2010, p330)
- KMV

**References**

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