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P2.T6.704 Linear discriminant analysis (LDA according to De Laurentis)

Nicole Seaman

Director of FRM Operations
Staff member
Learning objectives: Apply the Merton model to calculate default probability and the distance to default and describe the limitations of using the Merton model. Describe linear discriminant analysis (LDA), define the Z-score and its usage, and apply LDA to classify a sample of firms by credit quality.


704.1. In the Merton approach to credit risk, default probability is given by the this function (note the formula in De Laurentis is incorrect and should be given as follows):

Let us make the following assumptions:
  • The firm's asset value, V(A), is $1.0 billion
  • The expected asset return, µ, is 15.0%
  • The volatility of the assets, σ(A), is 25.0%
  • The face value of debt, F, is $500.0 million
  • The debt matures in four years; i.e., T = 4.0 years
Which is nearest to the default probability (PD) estimated by Merton?

a. 0.5%
b. 1.0%
c. 2.5%
d. 5.0%

704.2. You have proposed a traditional linear discriminant analysis (LDA) for the purpose of predicting default and evaluating credit risk. Members of your firm's Risk Committee politely articulate the following objections to (i.e., arguments against) an LDA approach. Each of the following arguments is a valid drawback or shortcoming to the traditional LDA approach EXCEPT which is incorrect?

a. The model trains on only two outcomes, defaulting or performing, such that gradations of default are not considered
b. The model's weights in the traditional discriminant function (eg, Altman's Z) are constant, but in reality the weights are likely to vary over time
c. A traditional discriminant model will limit the number of variables and therefore is likely to ignore important factors; e.g., qualitative or macroeconomic factors
d. We cannot translate the Z-score into either a credit rating or a default probability; ie, given the Z-score is unitless neither a numerical nor analytical mapping is justified

704.3. The exhibit below (displayed in a format similar to De Laurentis' Table 3.7) shows the Altman's Z-score calculation for a hypothetical company:

(Source for Table 3.7: Giacomo De Laurentis, Renato Maino, and Luca Molteni, Developing, Validating and Using Internal Ratings (West Sussex, United Kingdom: John Wiley & Sons, 2010))

Assume a stress test that downwardly shocks three of the variables as follows:
  • Sales decline of 10%
  • EBITDA decline of 10%
  • Equity market value decline of 20%
No other accounts are affected. Which of the following is nearest to the outcome for the updated Altman's Z-score?

a. Unaffected
b. Z-score drops to 2.95 but the model still predicts safe ("performing"), or at least grey zone
c. Z-score drops to 2.24 and the model predicts distress ("default")
d. Z-score drops to 1.88 and the model predicts distress ("default")

Answers here: