**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)