Deriving Survival Probability from Credit Spreads

# Relation between Survival probability and Credit spread


# Relation between Survival probability and Credit spread












Let ${CS}_t$ is the term structure of the Credit spread obtained from market.

There is a standard relation between this and and term structure of Survival probability as follows,

${SV}_t = \exp[\left(-CS_t \times t \right) / LGD]$

My question is what is the basis of this formula i.e. how this relationship can be derived?

Thanks for your time.

## Answer by achirikhin (score 4, accepted)

https://quant.stackexchange.com/a/79540

This formula only holds exactly if both discount rate and intensity are constant and credit spread is paid in continuous time. Otherwise, it is only an approximation.

To derive it, you write the pricing equation for a CDS:

$(1-R)\int_0^T \lambda exp(-(r+\lambda)u)du = CS\int_0^T exp(-(r+\lambda)u)du$

and integrate to obtain

$(1-R)\lambda = CS$

The result is sometimes referred to as "credit triangle".

## Answer by Mild_Thornberry (score 1)

https://quant.stackexchange.com/a/79537

If you define $CS_t$ as your spot default spread, then it comes from an actuarial understanding of what your credit spread is. $CS_t / LGD$ is the risk-neutral probability of default over each year in t years. Default is like a company "dying", so it can be interpreted as a type of mortality rate. Your $CS_t$ needs to be adjusted by the reciprocal of your recovery rate (LGD), because recovery rates are baked into CDS spreads. See the answer below if you want to know why recovery rates are baked into spreads:

How to compute the implied probability of default from a CDS spread?

Also check out "Force of Mortality" on Wikipedia for intuition on your survival probability:

The simplest example is when the force of mortality is constant:

$\mu(y) = \lambda$

Then survival is:

$S_x(t) = e^{-\int_{x}^{x+t} \lambda \,dy}=e^{-\lambda t}$

Think of $CS_t / LGD = \lambda$.