question from FRM II 2013 credit risk

[southeuro]

hi all,

here's a question from FRM level 2 2013.

1-year zero-coupon bond with face value 1mil, and 0% recovery rate issued by company A. Bond is currently trading at 80% of its face value. Assuming excess spread only captures credit risk and that the risk-free rate is 5% per annum, the risk neutral 1-year probability of default on company A is closest to which?

a. 2
b. 14
c. 16
d. 20

I put the formula spread = - [ -(1/t) * (ln(current debt value/face value))] - risk-free rate and get 17.3%

but the answer is given as c thru a strange formula namely, 1+r = (1 - PD) * (1 + y) - (1 - PD) * (Face value / market value)

Thanks

 

[hamu4ok]

1 step. Find risky rate 0.8=1/(1+y) => y=25%
2 step. insert in formula (1+Rf)=(1-pd)*(1+y) + pd* RR
1.05=(1-pd)*1.25 + pd*0
(1-pd)=1.05/1.25=0.84 => pd=1-0.84=0.16

(1+Rf)<=(1-pd)*(1+y) + pd* RR is important formula, concept that risky asset should provide return equal or greater than risk-free rate after adjustment for default risk of the survived portion and recovery of defaulted portion.

another formula would give just an approximation pd= spread / LGD = (0.25-0.05) / 1 = 0.20 which is close to a more correct pd, but turns out to be incorrect answer for this question.

To reconfirm the answers:
- rechecking with pd=0.20
1.05=0.8*1.25+0*0.2 ==> 1.05=1.00 INCORRECT

- rechecking with pd=0.16
1.05=0.84*1.25+0*0.16 ==> 1.05=1.05 CORRECT

 

[southeuro]

thanks hamu4ok! wonderfully explained!