Showing equation for credit spread

Question

Question: Show that in general, we have s ≈ q × LGD. Where s= credit spread, q= one year risk neutral probability, and LGD is the loss given default.

Answer 1

1 year default probability $= q$

credit spread $= s$

LGD $= l$

t year cumulative default probability $$= 1-e^{-\frac{ts}{l}} = q$$ $1-q = e^{-\frac{ts}{l}}$

Taking natural logarithm on both sides and t = 1, we get

$ln(1-q) = -\frac{s}{l}$

$s = -ln(1-q)\times l$

$s\approx q\times l $= Default probability $\times$ Loss Given Default

Then the question is how do you get the first formula it comes from survival probability.