GARP Practice Exam 2013 P2 Question 8

[prebhan27]

Hi everyone,
In this question I do not get to the term (1+r) = (1-PD)*(1+y) - (1-PD)*(FV/MV)
Could you explain why you use the term behind the minus and how you get to the term?
I would have solved it with the following term:
(1+y)*(1-PD)+PD*RR = 1+r , where PD*RR is canceled out as RR is zero.
So you have y and PD as unknown but you get to y with the following formula regarding the bond: FV*exp(-y)=MV --> 100*exp(-y)=80
But then you get about 23% for y and, if you plug that in the formula above you get around 14% for PD, which is wrong.
Could someone help me with the problem?
Thanks in advance
Peter

 

[afterworkguinness]

Hi Prebhan, can you post the entire question ?

 

[prebhan27]

Here is the question incl. answer and explanation:

8. Consider a 1-year maturity zero-coupon bond with a face value of USD 1,000,000 and a 0% recovery rate
issued by Company A. The bond is currently trading at 80% of face value. Assuming the 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 of the following?
a. 2%
b. 14%
c. 16%
d. 20%
Correct answer: c

Explanation: This can be calculated by using the formula which equates the future value of a risky bond with yield
(y) and default probability (π) to a risk free asset with yield (r):

1 + r = (1 - π) * (1 + y) + πR
π = Probability of default; R = Recovery rate

In the situation where the recovery rate is assumed to be zero, the risk-neutral probability of default can be derived
from the following equation:

1 + r = (1 - π) * (1 + y) - ( 1 - π) * (FV/MV)
where MV = market value and FV = face value.

Inputting the data into this equation yields π = 1 - (800,000*1.05)/1,000,000 = 0.16.


Reference: Allan Malz, Financial Risk Management: Models, History, and Institutions (Hoboken, NJ: John Wiley & Sons,
2011), chapter 6, p. 203.
AIM: Explain the relationship between the yield spread and the probability of default and calculate default probability
of a debt security using the credit spread.
Section: Credit Risk Management and Measurement

 

[Bester]

Hi

I get the answer to be c as well.

To calculate r I use my calculator
FV = 1000000
PV = -800000
N = 1
PMT = 0
Thus r = 25%

Then using the following formula

(1-p)(1 + r) = (1 + rf)

Note recovery rate = 0 thus p x recover = 0

(1-p) = (1+ 0.05)/(1+0.25)

p = 16 thus c is correct

 

[prebhan27]

Ok thanks for the answer. My fault was to take continous discounting to come from Face and yield to the Market value: FV*exp(-y)=MV. If you just use FV/(1+y)=MV you get to the yield of 0.25, which would be the same type of calculation that the financial calculator does.

 

[afterworkguinness]

When I calculate a hazard rate to get the default probability, I am a bit off from the correct answer. Is there something wrong with my approach below?

Bond return = [(Face/Price)^(1/maturity)]-1 = 25%
Spread = Bond Return - Risk Free Rate = 20%
Hazard Rate = (Spread/1-Recovery) = 20%
1 year Probability of Default = 1 - e^(-hazard rate) = 18.13%

Reference Malz, page 246

EDIT:
Also, looking at Malz page 203 as sourced by GARP's answer; I don't see how we get from

1 + r = (1 - π) * (1 + y) + πR
to here:
1 + r = (1 - π) * (1 + y) - ( 1 - π) * (FV/MV)

That can't be correct at all because it results in 1.05 = 0

(1+risk free) = (1-pd)*(1+risky yield)-(1-pd)*(FV/MV)
(1+risk free) = (1-pd)*(1+risky yield)-(1-pd)*(1+ risky yield)
1.05 = (1-x)*(1.25)-(1-x)*(1.25)
1.05 =0


EDIT: 2
My first attempt was off due to the compound type (simple vs exponential). The posted solution still is questionable. The answer couldn't have come from that formula.