Hazard Rates and probability of survival

[Kashif Khalid]

Hello

In one of GARP's practice exam questions it asks:

An Analyst estimates that the harzard rate for a company is 0.1 per year. What is the probability of survival in the first year followed by a default in the second close to?

The answer is the marginal probability (or unconditional probably) of default between year 1 and year 2. My question is why is it not the the unconditional probably for year 2 divided by the probably of survival in year 1. i.e. a conditional probability?

Thanks

Kashif

 

[Mkaim]

Hi @Kashkha,

By definition, Marginal Default Probability = Probability of survival in the first year followed by a default in the second year. Where the probability of default in the second year (only second year without any consideration of what happened in the prior year) = Unconditional Probability. Hence, Marginal Default Probability = Unconditional probability of default in year 2 divided by the probability of survival in year 1. By the way, Marginal Probability of Default = Conditional Probability of default; If GARP wrote it as "Marginal Probability (or Unconditional Probability..." then that's inconsistent with that I know and could just be a typo... they've (GARP) been known to make mistakes on their sample exams.

 

[Chris22]

Their explanation is :
"...the conditional one year default probability given that the firm survived the first year is the difference between the two year cumulative probability of default and the one year probability : 0.18127 - 0.09516 = 0.08611 "

I have the same doubt as Kashif...

 

[David Harper CFA FRM]

Hi @Kashif Khalid , @Mkaim and @Chris22 We just discussed this question at https://forum.bionicturtle.com/thre...swers-is-that-true-or-what-garp16-p2-33.9592/

 

[Stuti]

Hi, I am getting confused between hazard rate and probability of default. Does it mean to say if the hazard rate = 0.1 i.e default occurs once in a decade then the probability of default in 2 years is 0.2? Also what if ask what is the probability of default in 20 years?

 

[Stuti]

What is the difference is saying the probability of survival over a time interval = 1 - lambda*(dt) vs e^(-lambda*t)?

 

[QuantMan2318]

Hi there @Stuti

Those are excellent questions, however, they involve some mathematical properties and I shall endeavor to explain based on some mathematical knowledge that I had to delve into, though not as deep as needed by a STEM person, for an FRM. More advanced practitioners over here will be able to explain in greater detail.

Lambda or Hazard rate is the parameter which determines how intense the default can occur or how high the spreads will be. In short, it is a parameter that influences the probability of Default and thus the default distribution. I don't know how Malz got the lambda, perhaps he was assuming a Poisson process?
In any case, the PD for a very small time interval, dt, conditional on no default before, is lambda * dt. This is opposed to a cumulative PD over a time interval t which is 1-exp(-lambda*t)

You may derive it as follows:

P(t*<t+tau | t*>t) = F(tau) (conditional PD post t conditional on survival till t equals unconditional PD at time tau); (if you look at why this is so, please refer some earlier threads pointed by @David Harper CFA FRM as well as this where you would see a post by one @Delo https://forum.bionicturtle.com/threads/p2-t6-307-hazard-rate-malz-section-7-2.6932/#post-42375, in case you want to know how this is so, perhaps I will explain what Malz states in another post.)

so, for very small time interval tau, where tau tends to zero, he took the limit of the above and assumed that equals

(Marginal PD wrt time t*tau)/Probability of Survival

=F'(t)*tau/(1-F(t)) where I think he assumes the joint probability of survival at t and default at t+tau to equal the numerator.
expanding the above, we get,

F(t) = 1-exp(-lambda*t)
F'(t) = lambda*exp(-lambda*t)
(1-F(t)) = exp(-lambda*t)

so F'(t)*tau/(1-F(t)) equals [lambda*exp(-lambda*t)*tau]/exp(-lambda*t), we get the conditional probability as lambda*tau which is the same as lambda *dt for small intervals of time

Therefore, for extremely small intervals of time, the PD is lambda*dt or lambda*tau, while the cumulative PD is 1-exp(-lambda*t)
So, lambda is the parameter that influences the speed or magnitude of the default, therefore it is called a hazard rate.

You will see that it also influences the spreads as well, with increase in lambda increasing the spread on the Bonds.

@David Harper CFA FRM , I had to interpolate much to get this interpretation, in the absence of more mathematical knowledge, I am not sure if my answer is correct.

 

[Linghan]

@David Harper CFA FRM I am very confused with which method to use in terms of getting cumulative default, given credit spread and recovery rate-
1, exponential (1-e^(-PD*t))
2. 1-(1-PD)^t
Credit/LGD =PD ?? I assume.
Thanks!!

Ling

 

[QuantMan2318]

Dear @Linghan

I don't have the material with me at the moment but as you want the cumulative PD, we have the following formula:

P(t*<t+tau|t*>t) = F(tau) where F (tau) is the cumulative PD at tau which is 1 - exp(-lambda*tau) and where lambda can be approximated as z/(1-R) where z is the credit spread and 1-R is the LGD

Does this answer your question?

Thanks
Mani

 

[Linghan]

Dear @Linghan

I don't have the material with me at the moment but as you want the cumulative PD, we have the following formula:

P(t*<t+tau|t*>t) = F(tau) where F (tau) is the cumulative PD at tau which is 1 - exp(-lambda*tau) and where lambda can be approximated as z/(1-R) where z is the credit spread and 1-R is the LGD

Does this answer your question?

Thanks
Mani

@Quant

Dear @Linghan

I don't have the material with me at the moment but as you want the cumulative PD, we have the following formula:

P(t*<t+tau|t*>t) = F(tau) where F (tau) is the cumulative PD at tau which is 1 - exp(-lambda*tau) and where lambda can be approximated as z/(1-R) where z is the credit spread and 1-R is the LGD

Does this answer your question?

Thanks
Mani

But isn't that Credit spread also equal PD*LGD???? Is that mean lamda is = PD??!! (unconditional first year).

 

[David Harper CFA FRM]

@Linghan The hazard rate (aka, default intensity), λ, is the instantaneous conditional default probability, so it's the continuous version of the discrete (conditional) PD. For example, we might assume a conditional PD of 1.0%; i.e., conditional on prior survival, the bond has a default probability of 1.0% during the n-th year. Let's assume we want a 5-year cumulative PD:

• For (continuous) λ = 1.0%, 5-year cumulative PD = 1 - exp(-λ*T) = 1 - exp(-0.010*5) = 4.88%;
• For discrete conditional PD = 1.0%, 5-year cumulative PD = 1 - 5-year cumulative survival probability = 1 - (1-PD)^T = 1 - (1-0.010)^5 = 4.90%, which is pretty close as expected.

Because (spread) ~= λ*LGD, λ ~ = spread/LDG approximates the (instantaneous) conditional default probability.

@QuantMan2318 I think maybe your P(t*<t+ τ |t*>t) = F(τ) is (Malz's representation of) the unconditional PD, where he distinguished unconditional F(tau) from cumulative F(t). Unconditional PD is the product of cumulative survival (t*>t) and conditional PD. Thanks!

 

[Linghan]

@Linghan The hazard rate (aka, default intensity), λ, is the instantaneous conditional default probability, so it's the continuous version of the discrete (conditional) PD. For example, we might assume a conditional PD of 1.0%; i.e., conditional on prior survival, the bond has a default probability of 1.0% during the n-th year. Let's assume we want a 5-year cumulative PD:

• For (continuous) λ = 1.0%, 5-year cumulative PD = 1 - exp(-λ*T) = 1 - exp(-0.010*5) = 4.88%;
• For discrete conditional PD = 1.0%, 5-year cumulative PD = 1 - 5-year cumulative survival probability = 1 - (1-PD)^T = 1 - (1-0.010)^5 = 4.90%, which is pretty close as expected.

Because (spread) ~= λ*LGD, λ ~ = spread/LDG approximates the (instantaneous) conditional default probability.

@QuantMan2318 I think maybe your P(t*<t+ τ |t*>t) = F(τ) is (Malz's representation of) the unconditional PD, where he distinguished unconditional F(tau) from cumulative F(t). Unconditional PD is the product of cumulative survival (t*>t) and conditional PD. Thanks!

Thanks!!