# Actual default probability

“Actual default probabilities do not include the default risk premium associated with the uncertainty in the timing of the possible default loss.”

Can anyone explain what this means? I vaguely remember that real-world default probabilities are lower than risk-neutral default probabilities due to higher risk premium the market ascribes to risky debt securities. Does anyone know the source of this higher risk premium?

I am wondering the same because there is question in the curriculum about this and i got it wrong

Actual Default Probabilities are are assumed from historical information.

For instance, One year bond with face value of 100

• Risk free for one year = 2%
• Coupon = 5%
• Recovery rate = 50%
• Default prob = 1% (Based on historical information)
{102.4020} = \frac{{(105\times0.99)}+{(100\times50\%\times0.01)}}{(1+2\%)}

Therefore, {P} = {1\%}

Risk- neutral Probability of Default is solved by using given quoted price, coupons, risk free rate, and recovery rate

For instance, One year bond with face value of 100

• Risk free for one year = 2%
• Coupon = 5%
• Recovery rate = 50%
• Quoted price = 90 (Currently trading in the mkt)
90 = \frac{{(105\times(1-P^{*}))}+{(50(P^{*}))}}{(1+2\%)}

Solve for {P^{*}}= {24\%}

Therefore, {P}<{P^{*}}

The difference between Actual Default Prob and Risk- neutral Probability of Default reflects risk premiums required by investors to take risks. (Actual Default Prob overstates the price of the bond)

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so does this mean that the actual default probability contains the risk premium or not?
My understanding is that risk neutral probability is not considering risk at all when pricing a security, whereas the actual default probability does. But in the book, it says that “Actual default probabilities do not include the default risk premium associated with the uncertainty in the timing of the possible default loss”. This is the sentence that is not making sense to me

ADP do not include the default risk premium ASSOCIATED with the uncertainty in the timing of the possible default loss. However, under the assumption of risk-neutrality, risk premium is purely the expectation of the possible loss computed with risk-neutral probabilities by using the quoted price.

Oversimplification here:
Risk neutral probability is just the mkt implied probability by showing that everything could be bought or sold as there was a single probability for everything. It is just taking the price into the equation and solving for the probability.

In this case, the risk premium is just the amount that needs to be subtracted from the price of the bond (using ADP) in order to match the price of the bond in the mkt.

Because the mkt-implied risk neutral default probability, the price of the bond can be seen as it were purely priced from investors whom are not risk adverse by assuming the default probability to be 24%. Therefore, the book just states that the difference reflects the risk premium for default timing risk.