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# P2.T6.211. Spread-based default probability (PD)

#### Suzanne Evans

##### Well-Known Member
Questions:

211.1. The risk-free rate is 2.00% and a corporate bond has a yield of 3.50% per annum. It is estimated that the contribution to the spread by all non-credit factors is 40 basis points; e.g., liquidity risk. The recovery rate estimate is 75.0%. What is an approximation (emphasis on "approximation") for the implied default probability?

a. 1.90%
b. 2.50%
c. 4.40%
d. 6.00%

211.2. The risk-neutral default probability of a one-year corporate BB-rated bond is 5.0% with an estimated loss given default (LGD) of 65.0% while the risk-free rate is 2.0%. If we assume an annual compound frequency, which is nearest to the yield of the corporate bond?

a. 3.57%
b. 4.29%
c. 5.43%
d. 6.60%

211.3. The following three corporate bonds trade
• Bond A has a price of $97.00, pays no coupon, estimated loss given default (LGD) of 50.0%, and matures in one year • Bond B has a price of$95.00, a coupon rate of 1.0% (payable semi-annually), estimated LGD of 45.0%, and matures in 1.5 years
• Bond C has price of $96.00, a coupon rate of 2.0% (payable semi-annually), estimated LGD of 40.0%, and matures in 2.0 years The risk-free rate curve includes: 1.0% at one year; 1.5% at 1.5 years; and 2.0% at two years. If the only contribution to spreads is credit risk (unrealistically, non-credit factors do not contribute to the spread), which bond has the highest market-implied probability of default? a. Bond A b. Bond B c. Bond C d. Cannot determine Answers: #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber FYI, this set contributes to the global topic drill, in which we are currently writing questions for P1.T4 (valuation and risk models) and P2.T6 (credit risk; above is obviously this P2 credit risk). Unlike most of the other questions I write (i.e., AIM by AIM), these global drills are not AIM-centric. Instead, I draw on past questions and sample questions. The three questions above reflect the two approaches GARP tends to employ for spread-based PD (as opposed to structural/Merton/KMV based PD): • The no-arbitrage risk-neutral approach, which has scary-looking formulas, but I feel you are just better to master the no-arbitrage idea (this is often the case). This may just reflect my style: i actually don't have the formula memorized, I just write down the no-arbitrage idea every time and solve for what i need. In this case, for single one-period, the idea is: expected risk-free return should equal = expected risky return; and where Rf = riskfree rate, y = risky yield, PD = default prob, and r = recovery, it should be intuitive that: 1+Rf = (1+y)*(1-PD) + (1+y)*PD*r = (1+y)*[(1-PD) + PD*r] = (1+y)*[(1-PD) + PD*r], and we can solve for y = (1+Rf)/[(1-PD) + PD*r] - 1 = (1+Rf)/[1 - PD*LGD] - 1 • The much simpler approximation given by PD ~= spread/LGD. This is one of those rare cases where the word "approximation" is a hint to take the shortcut! #### Mark W ##### Active Member Hi David, Hate to ask at this late stage but the above doesn't seem to tie to Malz? You say: "1+Rf = (1+y)*(1-PD) + (1+y)*PD*r ...., and we can solve for y = (1+Rf)/[1 - PD*LGD] - 1" Malz has: 1 + r = (1 - π) * (1 + y) + πR giving: y = (1 + r - πR) / (1 - π) - 1 the two do not reconcile...? Apologies if I've missed something basic (we both have different notation - FYI). Thanks - Mark #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi Mark, yes, that is observant. I've uploaded this among submissions to GARP, going back several years, as part of a broader request to standardize certain formulas including this one. It has been surfaced every year for the last 5 or 6 years; it actually arises often. Malz happens to be new reading, but he repeats a version that has occurred previously; whereas the above inherits from Sanders Chapter 11 (primarily). So, basically there are two ways to do it: • assume the recovery term = π * R, or • recovery term = π*R*(1+y) which amounts to, if the exposure is$100 loan with 6% yield, for example, and recovery is 40%, is the equality based on immediate default (recovery 40% of \$100) or default at the end of the period. As a practical matter, I've suggested it is not consequential due to the inherent imprecision of the recovery/LGD; i.e., 40% is less precise anyway than a factor of (1+y) which is likely less than 1.1X.

also, under typical numbers, it doesn't tend to have a dramatic impact; typical use case, the difference in methods is about 10 to 20 bps if solving for PD and (10 to 20 bps)*LGD% if solving for yield. I know GARP is aware of the difference, but frankly I doubt they've absorbed the latest instance of Malz. Sorry, I wish we could just settle on one, the difference doesn't trouble me except it adds to candidate stress. In the meantime, this is definitely an inherent formula difference they should know about such that we should be able to assume specificity in the question. Thanks,

#### Mark W

##### Active Member
Great - thanks David. It's good to know that you know about the discrepancy! I got a difference in the magnitude of ~15 bps when I tried to reconcile between the formulae. Whilst it is fun to note these things in general, two days before an exam, it does slow you down as you are constantly questioning little things that are probably inconsequential. I guess that's the way GARP rolls....

Anyway, many thanks again for the response. Mark

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Thanks Mark, it's not a time for me to ask you favors, but after the exam, feel free to correspond to GARP (this thread here happens to be public, feel free to link to anything on our forum) to the extent you want them to get a candidate's perspective . Actual candidate feedback has natural weight and sometimes, i think, it requires a pattern of feedback to get heard (not just me repeatedly asking!). This Malz is another great example (in my opinion) of how rapid syllabus changes can induce additional stress, the anthology of different authors already creates natural discrepancies but when the anthology also rotates rapidly, a consequence is that some defintions/formulas which you might expect could be "standardized," if only for exam purposes, are nevertheless potentially conflicted both across the syllabus (authors) and across time (rotation). thanks,

#### cash king

##### New Member
Hi David,

211.1 C
211.2 C
211.3 B

Not quite sure about the last one, since I took two approximations: 1. treat all three bonds as zero-coupon bonds, 2. approximate PD using PD=z_spread/LGD. My method gives PD_A=4%, PD_B=7.8%, PD_C=5%

I wonder if my approximation is conceptually corect.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @cash king your answers are correct. This question does use PD=z_spread/LGD for PD but for spread it uses, for example, Bond C yield given by: N = 4, PV = -96, PMT = 1, FV = 100 and CPT I/Y = 2.051816 * 2 = 4.1036 and Bond C yield = 4.1% with spread of 4.1% - 2.0% ~= 2.1%. and implied PD = 2.1%/40% = 5.26%. Thanks,

#### cash king

##### New Member
Got it. You simply calculated YTM for each bond, then approximated Z-spread by yield-spread.

Now I realize even if my assumption works (assuming all bonds are zero-coupon), my way of calculating their yields((100-price)/100, ) is incorrect. The correct way should be (100/price) ^(1/n)-1

THANKS!

PS: I find (100-price)/100 is the so called "discount rate" of zero bonds (r), while by convention "100*r" is the quoted price of a zero bond.

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#### southeuro

##### Member
Hi David,

This may be a very basic question but am confused.. I have the following question from Jorion's handbook edt 6 (21.3 FRM question exam 2002, page 506):
A risk analyst seeks to find out the credit linked yield spread on a BB-rated 1-year coupon bond issued by a multinational. If the risk-free rate (RFR) is 3%, the default rate for BB-rated bonds is 7%, and the LGD is 60%, then the yield to maturity of the bond is?

I can use 2 formulae (or so I thought).. Either: Premium = RFR + (PD) (LGD) or 1 + RFR = (1+Premium) [1 - (PD)(LGD)]

when I use the first one I get x = 3 + (.6)(.07) hence 7.2%
when I use the second I get 1.03 = (1 + premium) [1 - (.07)(.6)] and get 7.52%

Can you help? Thanks!

ps. btw, Jorion's book (esp. questions) are strewn with mistakes ... even if this one may not be.

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @southeuro

Yes, there is a history behind this. Neither can be said to be wrong. In the current exam, I do not think there is an assigned reading that would lead us to utilize your second method (the answer given). That approach is based on previously assigned Saunders which (legitimately) equates the expected return on a risk-free loan to the weighted expectation of a risky loan:
(1 + Rf) = p*(1 + yield) + (1-p)*recovery*(1+yield), where p = 1 - pd; which is the same formula as Jorion 21.3 but uses probability of repayment (p) rather than probability of default (pi),
but there was also an acceptable variation: (1 + Rf) = p*(1 + yield) + (1-p)*recovery
... both of which the problems of which we quite did share with GARP and it has long been the case that the simpler approximation (your second) is preferred and assigned.

On the current test, I see no reason why a candidate would use anything other than (via Malz but also unassigned Hull) your first approach, which is the simple approximation:
pd ~= spread/LGD, or spread ~= PD*LGD. In this case, spread = 7.0%*60% = 4.2% such that yield = 4.2%+3.0% = 7.2% (same as you).

This is another reason why I do not recommend taking the Handbook questions too seriously (they are generally too old, is simply the problem. Plus, they absolutely contain too many errors). I actually think Jorion's text is great, the weakness is the included sample questions. It's getting to the point where I'd almost ignore them, given their dated timing. In any case, your instincts here look good, to me. I hope that helps.