credit linked yield spread (YTM) q from 2010 practise question

[[email protected]]

Hi david,

Here is a question from 2010 nov FRM practise questions issued by garp

Ref is : Arnaud de Servigny and Olivier Renault, Measuring and Managing Credit Risk

Q:A risk analyst seeks to find out credit linked yield spread on a BB rated ,2 yr zero copupn bond issued by a multinational petroleum company. if the prevailing annual risk-free rate is 3%, the default rate for BB-rated bonds is 7%. per year, and the loss given default is 60%, the the yield to maturity of the bond is:

a.2.57%
b. 5.90%
c.7.45%
d. 7.52%

The answer uses some formula like (1+rfr)^t= (1+r*)^t *[(1-pie)^t+ f(1-(1-pie)^t)]
Answer is C: 7.45%

which I couldnt get..


Please can you tell how to solve this problem and which formula to use exactly (could not find it in books/ jorion ??

Thanks a lot
snigdha

 

[David Harper CFA FRM]

snigdha,

can you refer to this thread were we discussed: http://forum.bionicturtle.com/viewthread/2516/
... GARP has an error in referencing de Servigny, the source is really Saunders (assigned 2009 but not 2010). Also, note i have an XLS as well there.

Thanks, David

 

[[email protected]]

David,

I guess , I do not have the rights to view that thread ...
Please,if u can copy the same here.I would be very grateful to you.

Thank you
snigdha

 

[David Harper CFA FRM]

Hi snigdha, copied here:

Here in the spreadsheet version: http://sheet.zoho.com/public/btzoho/l1-27b

This is based on Saunders (not de Servigny):
(1 + riskless_rate)^2 = (1 + risky_rate)^2 * [cumulative prob of repayment + cumulative PD * recovery_rate]

Where:
cumulative prob of repayment (cuml p) = 93%^2, and
cumulative prob of default (cuml PD) = 1-93%^2
recovery_rate = 1 - LGD

Such that:
(1 + riskless_rate)^2 = (1 + risky_rate)^2 * [93%^2 +(1-93%^2)* 40%], and
(1 + risky_rate)^2 = (1 + riskless_rate)^2 / [93%^2 +(1-93%^2)* 40%], and
risky_rate = SQRT[(1 + riskless_rate)^2 / [93%^2 +(1-93%^2)* 40%]] – 1
risky_rate = SQRT[(1.03)^2 / [93%^2 +(1-93%^2)* 40%]] – 1 = 7.45%

It’s his extension of the same no-arbitrage idea: that you should be indifferent between riskless and expected risky investment so that:
risky = riskless, and over 1-year:
(1+riskless) = prob of survive/repay * (1+risky_rate) + prob of default * (1+risky_rate) * recovery

And for a 2-year expected return, the return on riskless = (1+riskless)^2
and this should equal (time 0) expected return on risky bond which equals (where p = prob of survive/repay):
weighted prob of carrying without default + weighted prob of defaulting
= p^2*(1+risky_rate)^2 + (1-p^2)*(1+risky_rate)^2)*recovery
i.e., our expected return is the weighted average of the two return outcomes: survival over the two years (93% ^2) or not survival over the two years (1 - 93%^2) = cumulative PD). So this could also be:
= (1-conditional PD)^2*(1+risky_rate)^2 + 2-year cumulative PD*(1+risky_rate)^2)*recovery

 

[[email protected]]

David,
Thanks For the wonderful reply.
I wish, I would have known about BT and your wisdom a bit sooner

Though I understood the explanation but I wanted to ask that whether the formula


(1 + riskless_rate)^2 = (1 + risky_rate)^2 * [cumulative prob of repayment + cumulative PD * recovery_rate]


Is this concept captured as a part of any specific AIM for L1 FRM. If yes which one??
if no , do I need to memorize it to solve this type of questions.
Is there any related/specific learning you might like to highlight to better grasp the topic?

thanks a ton again,
Snigdha

 

[David Harper CFA FRM]

Hi snigdha,

There is no direct source in either current (2010) L1 or L2 FRM. (Jorion comes close is Chapter 20 of the handbook but not exactly...)
The source is Chapter 11 of Anthony Saunders' Financial Institutions http://www.amazon.com/dp/0077211332/
... this Chapter of Saunders was dropped from the curriculum. I have alerted GARP to this problem.

I would probably memorize at least the one-period version. Historically, GARP likes to test PD ...

Re: Is there any related/specific learning you might like to highlight to better grasp the topic?
Yes, absolutely: the no-arbitrage idea the makes possible the formula. It is the start of many formulas. In this case, I recommend a meditation on the starting idea:
(ex ante riskless investment = risky investment

Thanks for your kind words!

David