# Risk Neutral Probability of Default

Assume a 1-year, zero coupon bond trading at \$95. One year benchmark rate is 3%.

Recovery rate = 60%

So,

(60p+100(1-p))/1.03 = 95

(100-40p)/1.03 = 95

p = 5.38%, this is the risk neutral probability of default

Can someone please explain why they use the risk-free rate here? I understand to calculate the risk neutral probability, you would use a rate without any risk premiums which is the risk free rate, but why do we even calculate a “risk-neutral” probability? What is the point of it here? Thanks!

Although they use the term “risk-neutral probability”, it’s nothing more nor less than a weight.

Specifically, it’s the weight that leads to an arbitrage-free price.

Thanks, but why is the risk free rate used to discount instead of the YTM?

Because arbitrage is risk-free.

It’s the same reason that you use the risk-free rate to determine the forward price for, say, GOOG. You don’t honestly think that GOOG is going to increase in value at the risk-free rate, but the price in the forward contract uses the risk-free rate to avoid the possibility of arbitrage.

Makes sense, thank you!

My pleasure.