Conditional Survival Probabilities from Hazard Rates

Summary

The document asks how the survival probability of a default time relates to its hazard rate, and why a conditional survival probability has a shifted hazard integral. It presents the standard unconditional relation as an exponential of the negative cumulative hazard. It then defines the chance of surviving an additional interval given survival to a starting time, and cites a formula that integrates the hazard from that starting time onward.

The questioner’s derivation identifies the key distinction: unconditional survival through the combined horizon is not the same quantity as survival over the next interval conditional on having survived already. The conditional probability divides the survival probability at the later time by survival through the conditioning time. The document contains no answer or worked resolution, so it serves mainly to frame the distinction and the probability identities involved. Its application is credit default modeling, and it does not discuss estimation of hazards or dependence between defaults.

Key ideas

• Unconditional survival through a horizon is the probability that the default time exceeds that horizon.
• Conditional survival over an additional interval divides later survival by survival to the conditioning time.
• The hazard integral for conditional survival begins at the time already survived.
• The document poses this distinction in a credit default modeling context but does not provide a full answer.

Tags

# Survival Probability and Hazard Rate Function


# Survival Probability and Hazard Rate Function












I'm currently reading the article written by David X.Li "On Default Correlation: A copula Function Approach". I'm deepening my interest in subprime mortgage crisis. In the introduction of the paper the author talks about survival probability and hazard rate function. They are linked by the following formula: $$S(t)=e^{-\int_0^th(s)ds},$$ where $S$ denotes the survival probability and $h$ the hazard rate function. The author also defines the following quantity: $$p_{t,x}=\mathbb P\left\{T>t+x|T>x\right\},$$ where "T" is a random variable denoting the default event. Then, he states that: $$p_{t,x}=e^{-\int_0^th(x+s)ds}.$$ According to my calculations: \begin{align} e^{-\int_0^th(s)ds}&=S(t+x)\\ &=\mathbb P\left\{T>t+x\right\}\\ &=\mathbb P\left\{T-x>t\right\}, \end{align} whereas \begin{align} p_{t,x}&=\mathbb P\left\{T>x+t|T>x\right\}\\ &=\frac{\mathbb P\left\{T>x+t,T>x\right\}}{\mathbb P\left\{T>x\right\}}\\ &=\frac{\mathbb P\left\{T>x+t\right\}}{\mathbb P\left\{T>x\right\}}\\ &=\frac{\mathbb P\left\{T-x>t\right\}}{\mathbb P\left\{T>x\right\}}. \end{align} So in the first part of my calculations, I'm missing the denominator since the two expressions must match. Can I kindly ask you where I am wrong?

Thanks in advance.

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