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# General question

#### PJAYAKUMAR

##### Member
Hello David,

Can you please explain the math portion on this question

A five-year bond provides a coupon of 5% per annum payable semiannually. Its price is 104.

What is the bond's yield? You may find Excel's Solver useful.

The answer (with continuous compounding) is 4.07%

I performed

2.5e^-1y + 2.5e^-2y+2.5e^-3y+2.5e^-4y + 106.5e^-5y = 100

I did take ln on both sides

ln 2.5 (-1y) +ln 2.5 (-2y)+ln 2.5 (-3y) + ln2.5 (-4y) + ln 106.5 (-5y) = ln 100

I am getting answer of 0.850%.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @PJAYAKUMAR when you take the natural log of both sides, on the left you are taking the natural log of a summation but LN(X + Y + Z) <> LN(X + Y + Z). Rather, you are justified in distributing the LN if it were a series of products. That is, LN(X * Y * Z) = LN(X) + LN(Y) + LN*(Z); but you don't have products on the left, you have a series of discounted cash flows that are added.

You can also just use the calculator: 10 N, -104 PV, 2.5 PMT, 100 FV and CPT I/Y = 2.0534% is the s.a. rate, then ÷ 100 + 1 = [display 1.0205] LN [display 0.0203 .. mine only goes to 4 decimals] * 2 = [display 0.0407 which is the continuous such that I agree with the answer]. I hope that's helpful!

#### Flashback

##### Active Member
One silly question but it drives me insane a bit.

When to use conditional probability of default (or death) in risk calculation:

(PD in Y2 - PD in Y1) / (PS in Y1)

or just:

(PS in Y1) * (PD in Y2)

to get probability of default (or death) in Y2.

Thus, what is the prob. of death in Y2 given the fact that he survived a previous year? Also, when to use each of two formulas above?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Flashback We recently discussed this here @ https://www.bionicturtle.com/forum/...ts-and-mortality-tables-hull.10259/post-70531 in reference to Hull's (RM & FI Chapter 3) Mortality Tables, see below.
• The default probability analog is: The Joint (aka, Unconditional) Prob [Default during Year T and Survives thru end of Year T-1] = Cumulative Prob[Survive thru end of Year T-1] * Conditional Prob [Default during Year T | Survive thru end of Year T-1] = Cumulative Prob[Default, end of Year T] - Cumulative Prob[Default, end of Year T];
• In Hull's Mortality Table (below), the "Death w/n one year" is a Conditional Death Probability while the "Survival" column is a Cumulative Survival Probability. Consequently, the Conditional prob of death during the first year after the 80th birthday, Conditional Pr(death over the next year | Survival up to 80th birthday) = Unconditional (aka, joint) Prob (death during 80th year) / Cumulative Pr(survival up to 80th birthday) = (0.5062880 - 0.4762130)/(0.5062880) = 0.059403.
For me, the key relationship is the simple Conditional Pr(A | B) = Joint Pr(A, B) / Cumulative Pr(B); which is the same as more familiar Conditional Pr(A | B) = Joint Pr(A, B) / Unconditional Pr(B) because the Cumulative Pr(survive up to end of year T-1) is also an Unconditional probability. So the potentially confusing part is ...
• The cumulative probability of survival up to (eg) the end of the 80th year is an unconditional probability because it is from the perspective of time zero (age zero),
• But the Joint Prob (survive to 80th birthday and die during 80th year) is also an unconditional probability because it is also from the perspective of time zero (age zero).

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#### Flashback

##### Active Member
Still am not smarter than before. The likelihood of appearing of those calculations on both P1 and P2 are about 99.9 % at least in one question. Yet, I'm using trial and error method with success of above 80 % in each Mock and TTs. I hope same would happen on real testing.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Flashback I took some of my time to try and explain the concept. We can't control the exam. If you don't want help, I would appreciate if you just wouldn't ask and I'll save myself some time. I'm certainly not smarter due to your comment.

If I just copy the formulas you "need to use" then, well for most people, they'll just get trapped up when the question/application is slightly different. We try to understand things around here.

#### Flashback

##### Active Member
I appreciate your effort and your time but I gave a feedback and that's how I feel. I'm sorry if it's offended you. This was not my intention.
I can stop posting here if I cannot give my opinion or feedback. I don't want to offend anyone.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
You are free to give your opinion. And I am too, and my opinion is that your reply to my effort was not constructive (aka, lame). If you have constructive feedback, me and @Nicole Seaman (I think it's been shown) want to hear it. The unconditional/conditional/joint PD issue is challenging, and as you point out, it's almost certain to appear on the exam. My mere preference is that instead of "Still am not smarter than before," etc etc blah blah blah ... you might contribute something, and maybe we can generate a more efficient summary; or if you don't want to contribute, but if you just want to express negative opinions, then yes, my preference is that you just not bother ... because it will save me the time of an effort to write a careful reply.

#### Flashback

##### Active Member
Look, David. First I am not a native English speaker and such response sounds normal talking in my language. I can agree only that I used an incorrect expression which sounds misappropriated in your language. I'm sorry if this is the case.

Furthermore, as I told, I appreciate your effort and I'v given the positive feedbacks on your helpful postings many times thus far.

I can contribute only if I masterized some concept or at least feel like I did.
I cannot contribute if I also have some doubts or queries.

I will not bother again. Enjoy your time. I'm sorry, 'bro.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
No worries, at all. I appreciate your aren't a native English speaker. I am sorry if I misinterpreted your original postings, I always try to read carefully so maybe i did ....

And by the way, among the error corrections we've submitted to GARP over the last decade, no concept cluster has required more corrections (of GARP's own materials) than the conditional/unconditional/joint/cumulative default probability cluster. If you go back over all the prior sample exam questions, they've required many revisions/corrections. The cumulative (pun intended) appearance on the modern FRM owes much to us (our constructive members and myself) providing input, to help GARP utilize the terms (and their math) consistently and accurately. I relay this because:
• We don't know of easy shortcuts to these PD type problems. The shortcuts that have been utilized have often required subsequent fixes, and engendered confusion
• If GARP's own shortcuts led to mistakes, that's probably a good hint that shortcuts don't exist or aren't advisable (at least w.r.t this PD cluster)
• The PD cluster does take time to master, but this is risk (after all) and it makes sense that PD concept fluency is fundamental, so it's not a terrible spend of time. I hope that's helpful!

#### Flashback

##### Active Member
Thank You. These days I will pay a special attention to this key concept and will search the forum in depth. This is over the top on my To-Do list.

PS
'Bro was a joke.