continuously compounded zero rate

[Trevor19001]

six month zero rate = 9% per annum with quarterly compounding. The price of a one year bond with a face value of $100 that provides a coupon of 12% per annum semi annually is $96. Calculate the one year continuously compounded zero rate.

please help i cant figure it out !!!

 

[ShaktiRathore]

hi
for quarterly compounding to convert from discrete rate to continuous use formula,
Rc=4*ln(1+R/4) visit https://forum.bionicturtle.com/threads/important-concepts-for-the-frm-exam.7034/#post-24685
after 6 months the $1 amount fetch me,
(1+9%/4)^2=e^(r)T where r is continuously compounded rate per annum
T=1/2yr =>(1+9%/4)^2=e^r/2
take log on both sides,
ln(1+9%/4)^2=lne^r/2
2ln(1+9%/4)=r/2
r=4*ln(1+9%/4)=4*ln(1.0225)=4*.02225=.089 or~ 8.9% is the continuously compounded rate

thanks

 

[Trevor19001]

Thanks for the reply, so 8.90% is the final answer ?

hi
for quarterly compounding to convert from discrete rate to continuous use formula,
Rc=4*ln(1+R/4) visit https://forum.bionicturtle.com/threads/important-concepts-for-the-frm-exam.7034/#post-24685
after 6 months the $1 amount fetch me,
(1+9%/4)^2=e^(r)T where r is continuously compounded rate per annum
T=1/2yr =>(1+9%/4)^2=e^r/2
take log on both sides,
ln(1+9%/4)^2=lne^r/2
2ln(1+9%/4)=r/2
r=4*ln(1+9%/4)=4*ln(1.0225)=4*.02225=.089 or~ 8.9% is the continuously compounded rate

thanks

 

[David Harper CFA FRM]

It's a good bootstrap question, wants to solve for z: 96 = 6/(1+9%/4)^2 + 106*exp(-z)

 

[ShaktiRathore]

oops, I have calculated the continuously compound rate for 6 month only, and not 1 yr continuously compound rate.
Lets assume the 1 yr continuously compound rate be z then,
96 = 6*exp(-.089*.5)+ 106*exp(-z)
96 = 6*.9564+ 106*exp(-z)
96 = 5.738+ 106*exp(-z)
90.2616=106*exp(-z)
exp(-z)=90.2616/106=.8515
=>-z=ln(.8515)=>z=ln(1/.8515)=.1607 or 16.07% please check if this is the final answer

thanks

 

[David Harper CFA FRM]

Yes, checking is the most important step IMO and indeed:
6/(1+9%/4)^2 + 106*exp(-16.07%) ~= 96.00 !

 

[cdbsmith]

Hi David,

Several examples used in the "Products" videos (on Hull chapters 4 & 6) were illustrated using continuously compounded rates. My question is should we also know how to do the same problems with discrete rates as well?

Thanks,

Charles

 

[David Harper CFA FRM]

Hi Charles,

The short answer is: yes.

Almost since we started helping candidates prepare for the FRM (5+ years), we've requested (in our annual feedback to GARP) that the FRM utilize a primary compound frequency (e.g., continuous, annual) and vary only with exceptions (e.g., semi-annual coupon pay bonds), such that trainers/candidates do not have to deal with frequency switches (my argument, to me this is obvious ... , compound frequency is important but can be its own topic such that its variation does not need to "infect" so much P1 with additional cognitive load. It would be more efficient to designate a primary approach).

But likely due to the "anthology approach" of the FRM (i.e., various authors employed), GARP has not (yet) chosen this approach. Instead, GARP reserves the right to vary the compound frequency but the question will always make it clear; e.g., previously they did use "annual compounding" as the default compound frequency, but there is no assurance that will repeat, as i understand.

For the reason, proficiency with compound frequency is important for the exam.

(the reason the videos use continuous for Hull, of course, is that Hull assumes continuous, in most cases)

Here is the official from GARP: http://forum.bionicturtle.com/threa...ntinuous-and-rounding-issues.4743/#post-12512

i.e.,

"You and your forum participant are correct in noting that we sometimes use discrete compounding and sometimes continuous, depending on the application. We realize that it is important to the test taker to be informed of which method is expected of him/her in order to answer a question. To that end we will make note of your observations and re-double our efforts at clarifying which method is expected on exams and practice exams. As always, thanks for your feedback."

 

[cdbsmith]

David,

Thanks for the response. I will make sure that I am comfortable working with discrete and continuous rates. That said, I certainly appreciate your reaching out to GARP on our (the test takers') behalf requesting that they clarify what to assume in their questions. The less ambiguity and confusion the better, right?

Thanks,

Charles

 

[cdbsmith]

Hi David,

Thanks again for answering my questions on continuously compounded interest rates.

However, I've been thinking about it further and developed a sort of tug-of-war about in my mind. Let me explain. I am a CFA candidate as well and will be taking the Level 2 exam in 2014. I must say, the material covered for the FRM exam makes me feel as though I will be "overly prepared" for certain topics (e.g., Quant, Fixed Income, Derivatives, etc.) on the CFA exam as the breadth and depth of the FRM material is incredible. Which brings me to what I've been struggling with lately. Specifically, the way the CFA curriculum covers compound interest rates is totally different from the way the FRM curriculum (Hull) does. For instance, regarding derivatives, the CFA curriculum emphasizes the use of compound interest rates when determining the initial price of forwards/futures contracts on equity indexes and currencies. For all other types of derivatives (forward/futures on commodities and bonds, plain vanilla interest rate swaps, currency swaps, etc.), the CFA curriculum uses discretely compounded interest rates. In contrast, the FRM curriculum (i.e., Hull) emphasizes the use of continuously compounded rates for virtually all derivatives price and value computations.

From my perspective, it makes sense to use continuously compounded interests when dealing futures/forwards on equity indexes and currencies since any increases/decreases in values are instantaneous and all gains are continuously reinvested per se. But, regarding futures or swaps with bonds as the underlying, for example, it makes sense (to me) to use discrete rates as the coupon payment intervals are pre-specified (i.e., discretely defined) and such payments by nature are not continuously reinvested. Likewise, in the case of plain vanilla interest rate swaps, and FRAs, where payment intervals are also pre-specified (discretely defined), the CFA curriculum emphasizes the use of discretely compounded interests (which makes sense to me), wheres as Hull emphasizes continuously compounded interest rates (which makes less sense to me). So, therefore, I'm struggling with the reason Hull is using continuously compounded interests to price and value every type of derivative.

Before reading the Hull material, I assumed that I would not have to spend significant time reviewing derivative pricing and valuation since I had already covered derivatives in the CFA program and thought it would be similar in the FRM program. Of course, I've now had to reprogram my mind with so much emphasis being placed on the use of continuously compounded interest rates. I suppose at the end of the day, all of this will make me that much stronger as a practitioner, but it has been bugging me since I continue to automatically assume discrete rates when doing practice questions resulting in a lot of mistakes.

Am I missing something? Is my line of thinking off base? I would love to get your perspective on this, especially sense you have experienced both the CFA and FRM programs.

Thanks,

Charles

 

[David Harper CFA FRM]

Hi Charles,

I empathize, if only because FRM candidates manifest a perennial struggle with continuous/discrete compound frequency conventions; e.g., this week a question that is interesting to me (http://forum.bionicturtle.com/threads/forward-rates.7297/#post-26086) because shall we view it negatively (i.e., the FRM creates more work by showing the same concept in two different ways) or shall we view it positively (to comprehend there is only one concept, not two, at play here is to be on the way to basically mastering compound frequencies)?

Here are my thoughts:

• I mostly agree with your perspective about when it make sense to use either. I think the conventional view is: discrete is realistic/practical since actual cash flows arrive discretely; but continuous is analytically convenient/tractable. In the thread above, @Shaki wrote a sentence that I find telling: "second one [continuous] is approximation to the first one [discrete] to make the calculations a little simpler." To me, that's what's typically happening: real life is discrete, but we're use continuous to approximate
• Re: the CFA: I notice my Level 2 (2012) CFA textbook (Don Chance) includes both discrete and continuous pricing of forwards/futures; So technically, it looks like their curriculum is serving both? Nevertheless, I tend to agree with you with respect to questions, I perceive the CFA asks questions predominantly in the discrete domain.
• Hull's default usage of continuous is wholly consistent with his one foot (or two feet!) in financial engineering; e.g. the Black-Scholes assumes a continuous-time stochastic process. As many of the models (pricing, especially) are developed/manipulated in continuous-time, you tend to see continuous compounding deployed in association, out of convenience, with higher-level models where FEs are very comfortable with (continuous) calculus. That said, most of these authors (IMO) could easily go either way. Hull could easily show corresponding discrete versions for almost everything; instead, he has one section showing translations that can be used throughout.
• About the FRM, please note: Hull does default to continuous, but the FRM does not "default" to continuous (as above). To date, the FRM expects fluency between the two. (In fact, lately, annual compounding appears to be the default. Could this have anything to do with the fact that P. Jorion, who is but one author and not the derivatives author, prefers annual? hmmm ....)
• Why does the FRM do that? To be honest, because the FRM is more of an anthology with less coherence than the CFA. It is a byproduct justified ex post, it is not an ex ante intention. GARP basically must insist on fluency because they employ different authors with different preferences.
• In theory, should discrete/compound fluency be an issue? No. Once mastered, the difference should barely give a candidate pause.
• In practice, is it extra work on candidates? Absolutely positively, it's clearly a headache, or it wouldn't come up so much. So, I agree with you generally, except for one thing: you are not wrong, in practice, to assume discrete: actual exam questions (as of today) are more likely to declare discrete assumptions! (even though Hull employs continuous, sorry)

I hope that's interesting (if you don't mind, I am going to include a link to this post in my upcoming feedback to GARP: we get to see an advance draft of 2014 AIMs/Study Guide and that's when I get a chance to submit my itemized feedback). Thanks,

 

[cdbsmith]

Thanks for your eloquent response, David!

To be clear, I have no issue with applying continuously or discretely compounded interest rates. My only struggle is with how Hull applies the continuous rates (i.e., plain vanilla interest rate swap, equity index future/forward, currency future/forward, etc). As I previously stated, it makes total sense to me in applying continuously compounded interest rates to futures/forwards on equity index and currencies. It does not makes sense to me, however, to apply continuously compounded interest rates to plain vanilla interest rate swaps with discretely defined payment intervals
(i.e., 6 months, 12 months, etc.).

In a nutshell, that really has been my concern. Performing the calculations with either is not a problem for me. Although, prior to entering the FRM program, it was clear to me that I should use discretely compounded rates for pricing and valuing swaps and FRAs, for example, and to use continuously compounded rates when pricing and valuing futures/forwards on equity indexes.

That said, I really do appreciate your explanation as it is good to know that I'm not the only one struggling with Hull's approach. However, I have made the decision to just be prepared for whatever the FRM folks throw at me on the exam. At this point, I don't want to use the precious time I have left on something that certainly won't be resolved before this year's exam. I will now just look for the keywords in derivatives questions before attempting to answer instead of allowing myself to assume discrete rates every time.

And yes, please feel free to pass this thread to GARP. It's really good that you have the ability to communicate with GARP and influence how they administer the exams.

Thanks,

Charles

 

[David Harper CFA FRM]

Hi Charles,

Thank you! Okay I see what you are saying about Hull. I think Hull is (largely) internally consistent, specifically: he employs continuous compound frequency as the default (and, in particular, for the zero/spot/LIBOR rates used for discounting; basically, I *think* all of his market interest rates are expressed with continuous). But he employs discrete when the instrument (and often the instrument's cash flows) basically require it or suggest it, for example:

• swap payment rates are semi-annual (b/c they settle semi-annually)
• bond coupon rates are semi-annual (b/c the coupon pays semi-annually)
• Eurodollar rates are quarterly (b/c of course it's a contract specification, but that's because it's a 90-day rate)

The interest rate swap is a good example because both continuous and discrete are invoked, Hull's Example 7.2:

• The fixed rate payer pays 8.0% per annum (i.e., of the notional) with semi-annual compounding; the floating rate payer (initially) pays 10.2% LIBOR with semi-annual compounding. These are semi-annual rates to match the semi-annual (cash flow) settlements.
• Simultaneously, to price (value) the swap, Hull uses LIBOR rates with continuous compounding.

Similarly, when pricing a bond, he necessarily gives a coupon rate "per annum with semi-annual compounding," but discounts the cash flows with continuous rates (of course, he could discount with semi-annual). I think Hull is internally consistent, but I don't think the FRM necessarily follows Hull because it employs other authors. Thanks,