Two bonds illustrate linear EL and non-linear UL (StdDev)

[David Harper CFA FRM]

A forum thread yesterday had me thinking how instructive (to a few FRM ideas) is the simple two-bond portfolio. Take a two-bond portfolio with only a few assumptions:

• $100 Bond A with PD (aka, EDF) = 10%,
• $100 Bond B with PD = 20%
• Start with correlation = 0

Here is the spreadsheet, pretty basic (inputs in yellow)

• http://sheet.zoho.com/public/btzoho/twobonds-1117-1

This is an exercise in two Bernoulli random variables; e.g.,

• The standard deviation of Bond A = SQRT(10% * 90%) = 30% or $30 <<-- but you already knew that
• What's the variance of the two-bond portfolio? The XLS details the variance calculation but we know it's = SQRT[100^2*10%*90% + 100^2*20%*80% + 100*100*30%*40%*0 correlation] = $50.0 <<-- and you already know that Variance(100*PD_A) = 100^2*Variance (PD_A) = 100^2 * 10% * 90%.

What I find interesting and not intuitive is that the E[portfolio loss] = E[Bond A loss] + E[Bond B loss] regardless of correlation. The EL[portfolio] is totally indifferent to the correlation input:

• E[two-bond portfolio loss] = $30, regardless of the correlation assumption

For example, if we increase the correlation assumption to 0.40, what happens is:

• The standard deviation increases from $50.0 to $58.8. The XLS illustrates this. The two-bond portfolio has three outcomes: neither default, one bond defaults, or both bonds default. The impact of increasing correlation (e..g, from 0.0 to 0.4) is to increase the probability of both tail outcomes. In other words, the distribution gets more disperse.
  This is analogous to the credit risk idea that unexpected loss is a non-linear function of PD/EDF and impacted by correlation
• However, the mean of the distribution is totally unaffected by changes to the correlation input: it remains $30 regardless!!
  This is the idea the portfolio expected loss (EL) is a linear function of PD/EDF and unaffected by correlation

And notice the difference between a sum and product of random variables:

• E[sum of portfolio losses] = E[loss Bond A] + E[loss Bond B], always regardless of correlation.
  Because E(a + b) = E(a) + E(b)
• Prob[Both bonds default] = E[PD_A * PD_B] = PD_A * PD_B + Covariance(A,B)
  Because E(a*b) = E(a) * E(b) + covariance(a,b)
  This is worth memorizing because it has two other versions:
  Covariance(a,b) = E(a*b) - E(a) * E(b); and what if a=b? Then:
  Variance(a) = E(a^2) = [E(a)]^2