A Mortgage Pool was securitized with mortgages of monthly payments from homeowners. With an assumed Default Modeling using an Annual Default Rate ADR of 14%, the corresponding monthly default rate SMDn for period n is closest to:

Question

BlackTom AI · Accepted Answer

Use the discrete survival relationship: (1 − monthly default)^12 = 1 − ADR. Therefore monthly default SMD = 1 − (1 − ADR)^(1/12).

1. Relevant concepts
- ADR = annual default probability (14% → survival over one year = 86%).
- If monthly default probability is constant, annual survival = (monthly survival)^12.
- Convert by taking the 12th root of annual survival.

2. Step-by-step reasoning
- Annual survival = 1 − ADR = 1 − 0.14 = 0.86.
- Monthly survival = 0.86^(1/12).
- Monthly default SMD = 1 − monthly survival = 1 − 0.86^(1/12).

3. Intermediate calculations
- ln(0.86) = −0.150822 (approx).
- (1/12)·ln(0.86) = −0.150822 / 12 = −0.0125685 (approx).
- monthly survival = e^(−0.0125685) ≈ 0.987503.
- monthly default = 1 − 0.987503 = 0.012497 ≈ 0.01250 = 1.2497% ≈ 1.25%.

4. Why this answer is correct
- The formula accounts for compounding over 12 months. Using the exact 12th root of annual survival gives the constant monthly probability that, compounded across 12 months, produces the 14% annual default.

5. Why other options are incorrect
- Simple division ADR/12 = 14%/12 = 1.1667% → 1.17% (listed) is a linear approximation that ignores compounding; it understates the monthly rate.
- Options like 1.00%, 1.05%, 1.08% are too low; 1.33%, 1.42%, 1.55% are too high relative to the correct compounded value.
- “Cannot calculate” choices are incorrect because ADR fully determines the equivalent constant monthly default rate under the stated assumption (no need for PSA or a specific month n).

6. Conclusion
The computed monthly default rate is approximately 1.25%, which matches the provided correct answer: 1.25%.

A Mortgage Pool was securitized with mortgages of monthly payments from homeowners. With an assumed Default Modeling using an Annual Default Rate ADR of 14%, the corresponding monthly default rate SMDn for period n is closest to:单项选择题

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