Hull 4.16

[gargi.adhikari]

@David Harper CFA FRM
Hull Chapter 4 Practice Question : Hull 4.16 :-
We are given N =the Time Duration, the PV =$90 and FV= $100 and PMT= 8% Coupon for Bond 1
We are given N =the Time Duration, the PV =$80 and FV= $100 and PMT= 4% Coupon for Bond 2
We are asked to find the 10 yr- 0-Rate
Why could we not just plug in values of N, PV, FV & PMT info and get the value of 1/Y in the calculator..? Is it because the Compounding Information was not given..?

Also, please let me know if I should refrain from posting screenshots in this section...Much Gratitude

View attachment 1321

 

[gargi.adhikari]

Or why could we not simply do it as below:-
FV= PV* EXP(rT)
100= 90 * EXP (r*10 ) => 100/90 = EXP (r*10 ) => r = .03566

intuitively would have followed the above approach first- of using the formula FV= PV * EXP( rT) and then going on to calc r with FV & PV & T being given for the 1st Bond, we get the correct ans for r= > .0357 But , if we used the PV & FV Values for the 2nd Bond, namely PV = $80 ; FV= $100 and plug them into the formula FV= PV * EXP( rT), we get a different value for r....

why would the 1st more simplistic approach not work in this case...something's different about this problem statement...that I might be missing..coz in the exam.. the 1st approach I would take would be to use the FV-PV relationship.. :-( but it would give the wrong result if the 2nd Bond is used in the FV-PV formula... :-(

Would be very grateful for any insights on this...

 

[David Harper CFA FRM]

Hi @gargi.adhikari Hull's solution ultimately does use your simpler approach but please note it is based on net -70 today returns (in the 2-bond portfolio) + 100 in 10 years such that 70*exp(rT) = 100, and r = ln(100/70)*1/T = 3.57% continuously compounded. Why can't we go straightaway to the calculator here with either bond? We can, but it doesn't get us the solution:

• N = 10, PV = -90, PMT = 8, FV = 100 and CPT I/Y = 9.60% yield to maturity
• N = 10, PV = -80, PMT = 4, FV = 100 and CPT I/Y = 6.80% yield to maturity

Yield to maturity (YTM) is a (complex) weighted average of the spot (zero) rates. If we only knew about (solved for) the first bond's 9.6% yield, there are an infinite number of 10-year zero rates (various spot rate curves) that are consistent; e.g., just one of them is the flat zero rate curve at 9.6% but that one does't comport with the second bond, so it's not that one. (Actually, I don't want to get dragged too deeply into Hull's assumptions but technicians will note there is an underlying, unstated simplifying assumption to this problem: it's assuming there are no confounding risk factors if you hold this long/short portfolio. This is a way of leaning on the Law of One Price but the Law assumes an absence of confounding factors w.r.t. identical cash flows so is easiest applied to the riskfree rate curve but these spot rate curves are not risk free, they contain credit spreads etc).

So the key problem with solving for either bond yield (yield to maturity) is that yield by itself doesn't get you the 10-year spot (zero) rate. Hence the elegant solution of setting up a portfolio that replicates a zero-coupon bond and then can be priced by discounting. I hope that's helpful!

 

[gargi.adhikari]

@Thanks so much @David Harper CFA FRM for patiently answering this one

 

[Nicole Seaman]

@Thanks so much @David Harper CFA FRM for patiently answering this one

@gargi.adhikari

It is showing that your original questions have been deleted. David's explanation can't help others if the questions are deleted so we ask that you don't delete questions, especially if they are the original question in the thread or if David has answered them.

Thank you,

Nicole

 

[David Harper CFA FRM]

@gargi.adhikari I agree with Nicole, both of our effort (yours and mind) is sort of lost if you delete the original question. There are no dumb questions and, further, you especially have a knack for asking great questions (or at a minimum, you never ask less than good questions, I mean that sincerely) which benefit future visitors. Thank you!

 

[gargi.adhikari]

@Nicole Seaman @David Harper CFA FRM Yea..that was my bad .. :-( These I inadvertently deleted... :-(

 

[David Harper CFA FRM]

@gargi.adhikari no worries I used the forum's History function to restore them (cc @Nicole Seaman ). As a rule, we don't like to alter/edit anybody's entries, we want the forum to be generally uncensored (i.e., open) except for extreme cases. I hope that's okay. Thanks,

 

[gargi.adhikari]

Thanks so much @David Harper CFA FRM and @Nicole Seaman for using your Magic Wand ...again so sorry for the trouble though....good to see them back ...