Which of the following is false?

Question

BlackTom AI · Accepted Answer

Cumulative default probability is computed from survival probabilities, so the correct formula uses (1 − marginal probability) terms.

1. Relevant knowledge points and concepts
- Adjustable-rate mortgages (ARMs): interest rates reset periodically based on an index — a standard definition.
- Marginal default probability: probability of default in a specific period conditional on having survived to the start of that period.
- Survival probability through period t: product of (1 − marginal default probability) for each period up to t.
- Cumulative default probability to time T: 1 minus the survival probability through T.
- Loan commitment: contract specifying available amount and time window to draw the loan.
- Discriminant models (e.g., Altman Z): provide weighted linear combinations of financial ratios; an often-cited advantage is interpretability and relative stability of weights compared with more complex models.

2. Step-by-step reasoning
- Let p1, p2, ..., pT be marginal default probabilities in periods 1..T.
- Survival through period T (no default to T) = ∏_{i=1..T} (1 − p_i).
- Cumulative default probability by T = 1 − survival = 1 − ∏_{i=1..T} (1 − p_i).

3. Calculations / intermediate steps (numerical example)
- Suppose p1 = 0.10 and p2 = 0.20.
- Survival through 2 periods = (1 − 0.10) × (1 − 0.20) = 0.90 × 0.80 = 0.72.
- Cumulative default by period 2 = 1 − 0.72 = 0.28 (28%).
- If one used the formula in option c (1 − product of the marginal probabilities): 1 − (0.10 × 0.20) = 1 − 0.02 = 0.98 (98%), which is clearly incorrect.

4. Why the correct answer is correct
- Option c is false because it omits the necessary (1 − p_i) survival factors; it incorrectly uses the product of p_i instead of the product of (1 − p_i).

5. Why the other options are incorrect/true
- a: True — ARMs reset based on an index.
- b: True — that is the definition of marginal default probability.
- d: True — a loan commitment specifies amount and time available to draw.
- e: Considered true here — discriminant models’ coefficients are relatively stable and interpretable compared with some alternative methods, which is cited as an advantage.

6. Conclusion
Option c is the false statement because the correct cumulative-default formula is 1 − ∏(1 − p_i). This confirms the provided answer: c.

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