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# Question about marginal default probability

#### maoxindi

##### New Member
I have a question on Credit Risk Study Note Page 90.

Marginal default probability = lambda*exp(-lambda*t).

Under the formula, it says that "This is always a positive number, since default risk "accumulates", i.e. the probability of default increases for longer horizons. If lambda is small, it will increase at a very slow pace."

I don't understand "This is always a positive number, since default risk "accumulates", i.e. the probability of default increases for longer horizons. If lambda is small, it will increase at a very slow pace."

What I understand: marginal default probability is a decreasing function of t, is it right? If lambda is small, marginal default probability will decrease at a slow pace?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi maoxindi,

Here is a snapshot from the T6 XLS, this plots the cumulative PD for different hazard rates (default intensities).
Here is the XLS b/c it also contains the marginal PDs: https://www.dropbox.com/s/i7m9iocwenerebe/malz_7.2.xlsx

The point Malz is making (the text is straight out of Malz 7.2) is:
• Cumulative PD, given by 1 - exp(-λt) and plotted below; i.e., "the [cumulative] probability of default increases for longer horizons"
• The marginal PD, not plotted and given by λ*exp(-λt), which is the derivative of cumulative PD, if lamda is small "will increase at a very slow pace;" e.g., compare the rate of change (slope) difference between hazard = 0.10% and 10.0%. Meanwhile, this marginal PD is itself decreasing with maturity. I hope that helps, #### Sujit9960

##### New Member
Subscriber
Hi David,

How does the marginal default distribution change for investment grade bonds? This is because the cumulative default distribution of investment grade bonds tend to increase strongly over time over time unlike non-investment grade bonds that increase much less strongly over time.

I believe the below diagram is applicable for non-investment grade bond only. I am trying to understand how marginal default probability distribution would change for investment grade bond having below cumulative probability distribution assuming constant hazard rate: Thanks,
Sujit