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# Chapter 10 Financial Forwards and Futures -- and GARP's practice exams

#### NEichi

##### New Member
Hi @David Harper CFA FRM ,

I have some troubles with chapter 10. I absolutely appreciate, that the studynotes of bionic turtle are way more detailed and in depth than the GARP books. But with this chapter I have some problems to match the book contents to the studynotes. For example in the no income case GARP calculates the forward price as F= S(1+R)^T but in the studynotes using cost of carry model it is calculated as F=Se^rT .
I am a little bit unsure now on what to focus on. Is the GARP version a simplification? Maybe I am missing some important points here.
Sorry to be so vague with my question but I am kinda completly lost on this chapter.

Thanks for your help and greetings

Nina

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @NEichi Yes, thank you, greetings to you, too! This has been extensively discussed in the forum. The difference is merely compound frequency. GARP's new chapter 10 happens to assume annual compound frequency, but we tend to default to continuous compound frequency (given that's Hull's approach and Hull is actually the ultimate source here). As has been detailed here in the forum going back almost a decade, I spent years asking GARP to settle on a default compound frequency, for exam purposes, to no avail. Consequently, FRM candidate must be fluent in continuous or discrete compound frequencies, as evidenced by GARP's 2020 P1 Practice Exam which contains both a continuous application of cost of carry (COC) ....
"71. An analyst wants to price a 6-month futures contract on a stock index. The index is currently valued at USD 750 and the continuously compounded risk-free rate is 3.5% per year. If the stocks underlying the index provide a continuously compounded dividend yield of 2.0% per year, what is the price of the 6-month futures contract?"
... and a discrete application as illustrated in the solution to 73.

Although I'm unimpressed with GARP's new chapters (e.g., too shallow/simple in many sections, lack of sufficient numerical examples, too many errors in first version), I do happen to agree with them that it's healthy for candidates to be fluent in compound frequency translations. It's really only one concept cluster and after you get some practice with it, I think you'll see that there is not a big conceptual difference between F=Se^rT and = S(1+R)^T, or the full COC model for that matter. I don't think GARP's new chapter sufficiently frames compound frequencies in this context: it's important that you understand the general discrete form is given by S(1+R/k)^(T*k). My guess is the exam, like the 2020 practice paper hints, will favor continuous compounding because it is more elegant (and slightly less prone to mistakes because it can be a bit of a struggle to infer the discrete compound frequencies for storage and dividends) in COC applications; on the other hand, discrete costs/dividends are clearly more realistic (we receive dividends and pay storage discretely). I do notice that GARP address this issue (for some this will not be enough) with two footnotes in Chapter 10:
"2. It would be more natural to express the three-month interest rate with quarterly compounding rather than annual compounding. We use annual compounding because R is defined with annual compound-ing in Equation (10.1). If the rate were 4% per annum with quarterly compounding, the forward price would be 50 * 1.01 = 50.50 We will discuss compounding frequency issues in Chapter 16.

4. As mentioned in Footnote 2, compounding frequencies will be discussed in Chapter 16. Just as an interest rate can be expressed as 4% with semi-annual compounding or equivalently as 4.04% (= 1.02 * 1.02 - 1) with annual compounding, a yield can be expressed with different compounding frequencies. As the compounding frequency increases, the numerical value of the yield decreases." -- May, B. (2019). 2020 Financial Risk Management Part I: Financial Markets and Products, 10th Edition.

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#### Eustice_Langham

##### Active Member
Hi David, your comments raise an interesting point, do your comments mean that the GARP practice exams indicative of the type of difficulty that can be expected on the exam, in your experience?

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Eustice_Langham Actually, I perceive that GARP's practice exam questions do represent the exam's difficulty. In particular, my observation is that some of their practice questions are quite difficult (a few of them are difficult and time-consuming). They do contain some easy and/or quick questions, but so will the exam (btw, this is why I personally think that, when taking the exam, you should skip really difficult questions rather than risk getting stuck and spending too much time on one question: you want to make sure you answer any of the quick and/or easy questions). I do not think I've ever criticized their practice exams on the dimension of difficulty. All of this is just my opinion, of course.

The two observations that I've made about their practice exams are (i) breadth and (ii) quality. Breadth refers to the fact that due to significant recycling of their PQs and the math (i.e., 80 + 100 questions against at least 1,500 Learning Objectives, by definition, cannot nearly sample every concept), the current set of PQs can only ever cover a small fraction of the testable universe. Quality refers to the (related) fact that 20 to 35% of the practice exam questions have required subsequent error correction such that some historical errors have been counterproductive to learners; the 2018 PQ set itself actually represents about four years worth of cumulative error corrections (most submitted by us to GARP) such that there is little new information to be gained by going back earlier than 2018. Some of these details are in my memo https://www.bionicturtle.com/forum/...-committee-and-garps-board-of-trustees.22758/ I have not been shy to assert to GARP that I continue to believe candidates and EPPs--given the time, money and opportunity cost invested--have every right to expect that GAPR's PQs (and the exam questions) meet a quality threshold. It should goes without saying ...

That said, I have not yet had time to go through the newest 2020 PQ set. Mathematically, it can't overcome the breadth problem, but I'm *hopeful* that it meets the quality standard. I hope that's helpful,

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#### Eustice_Langham

##### Active Member
Thanks David, that is very useful to know. I have copies of the GARP practice questions so will focus on those until the exam time. This brings me to a nagging question, unfortunately this pandemic isn't going away any time soon and I have a fear that the scheduled exams due in October may well be delayed again.

#### NehaKey

##### New Member
Thanks David, that is very useful and answered few of my questions. I am working on GARP practice questions as part of my prep for the FRM exam. Since, I am really focusing on practice exam, I want to know how close or similar are the questions in terms of difficulty to the actual exam administered.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @NehaKey Re difficulty, I think I answered that already above when I wrote "Actually, I perceive that GARP's practice exam questions do represent the exam's difficulty. In particular, my observation is that some of their practice questions are quite difficult (a few of them are difficult and time-consuming). They do contain some easy and/or quick questions, but so will the exam (btw, this is why I personally think that, when taking the exam, you should skip really difficult questions rather than risk getting stuck and spending too much time on one question: you want to make sure you answer any of the quick and/or easy questions). I do not think I've ever criticized their practice exams on the dimension of difficulty. All of this is just my opinion, of course." I hope that's helpful,

#### Eustice_Langham

##### Active Member
David, I have nagging question concerning this topic. I am still at a loss to understand what is the difference between the expected spot price and the forward price, or is there no distinction, ie its just a matter of semantics.
I am comfortable with the premise of the LO in as much as futures markets generate arbitrage opportunities when there is a disequilibrium, but again and no doubt this is due to my misunderstanding, this disequilibrium is due to lack of a convergence between the futures markets and the spot markets. Again, however and perhaps I am rambling here, but the lack of convergence is expected or anticipated. I would appreciate your thoughts. Thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Eustice_Langham The forward price, as hinted by its extended notation that includes the zero, F(0,T), is observable today. If F(0,1) = $10.367, that's a predetermined price at which the long/short (makes a promise today to) buy/sell one year in the future. That's why I like to say "you can see the forward price today, but nobody can today see the expected future spot price." The expected future spot price, E[S(t)], is a theoretically expectant price of the spot price. But as spot prices are immediate, by definition, it's a debatable and unobservable value. E[S(t)] is a prediction. Notice the "E[.]" notation is meaningful and represents a difference between an expected future value versus the forward price that is observable today. It's meaningful to say (eg) "the spot price is$10.00" which, being immediate, we can represent as S(0) = $10.00. But it's at least confusing to write S(0, 1.0) or S(1.0) =$10.00; what is a spot price in one year? What we probably mean is, "we expect or predict the spot price in one year, when we get there, will be $10.XX;" i.e., E[S(1.0)]. This thread https://www.bionicturtle.com/forum/threads/contango-backwardation-and-trading-cheap.10654/ is recommended and contains a illustration, an numerical example (e.g., the$10.367) due to a very inquisitive previous customer. The numerical example (and XLS) includes S(0) = $10.00; F(1.0) = F(0, 1.0) =$10.367; and E[S(1.0)] = $10.747. It is the case that F(1.0) = F(0, 1.0) mean the same thing: today's price of a forward/future contract that matures in one year; not to get too complicated, but you could imagine F(t, 1.0). So the XLS exhibit in the thread contains the scenario F(0,1) < E[S(1)]; specifically, F(0,1) =$10.376 < E[S(1)] = \$10.747. Under the theory of normal backwardation, this is thought to be the "natural state" because, imagine you are the "speculator" who takes the long futures position: this difference represents your expected future profit on the long futures contract, your compensation for the risk. Under the theory, maybe I am the offsetting SHORT position who is the "hedger." But this inequality implies that I am am expecting a corresponding loss?! Yes, it's true (!): I am hedging and the expected loss is the price (aka, cost) of my hedge ("insurance has negative NPV," a saying goes, as an instance of a broader presupposition that hedges have a cost). There is more detail than you probably ever want in the thread, I hope that's helpful!

#### Eustice_Langham

##### Active Member
David, Thanks, I havent had the chance to digest that thread but will do so. If I can ask for some more insight, I have copied below an extract from another educational providers notes on this chapter:

"Backwardation refers to a situation where the futures price is below the spot price. It occurs when the benefits of holding the asset outweigh the opportunity cost of holding the asset as well as any additional holding costs. A backwardation commodity market occurs when the lease rate is greater than the risk-free rate"

"Contango refers to a situation where the futures price is above the spot price. It is likely to occur when there are no benefits associated with holding the asset, i.e., zero dividends, zero coupons, or zero convenience yield. A contango commodity market occurs when the lease rate is less than the risk-free rate."

My question concerns the second sentence in the first paragraph, can you provide some insight as to why this is the case, if this is answered in the thread that you have referred me to, please advise. Thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Sure @Eustice_Langham Those statements are correct but they do not involve the expected future spot price ("normal backwardation" is not the same as mere "backwardation," alas). Contango is simply an upward-sloping forward curve (or segment): S(0) < F(0,T) < F(0, T+Δt). Backwardation is a downward-sloping forward curve: S(0) > F(0,T) > F(0, T+Δt). The cost of carry model describes the expected relationship between the current spot price (observed, by definition) and a forward price, in this case: F(0) = S(0)*exp[(r-L)*T] where L = y - u; here L is lease rate, y is convenience yield and u is storage cost. You can see by relationship, F(0) = S(0)*exp[(r-L)*T], that if L > r then forward curve is downward sloping (aka, backwardation) and if L < r then the forward curve is upward-sloping (aka, contango). So far, this does not involve E[S(t)]. I am using the continuous version while GARP's notes contain the discrete analog (see https://www.bionicturtle.com/forum/threads/lease-rate-formula-in-chapter-11-book-3-frm-part-1.23323/ ). Again, a forward curve is a vector of observed prices (ie., spot plus series of forward prices) but the expected future spot price is unobservable and not an element in the forward curve (in a direct sense).

I have a video focused on the lease rate as L = y - u, in which case it can be called the "net convenience yield." See here (hope this helps ...): https://www.bionicturtle.com/forum/threads/t3-18-commodity-eg-gold-lease-rate.22431/

#### Eustice_Langham

##### Active Member
Thanks David, that is very useful and answered few of my questions. I am working on GARP practice questions as part of my prep for the FRM exam. Since, I am really focusing on practice exam, I want to know how close or similar are the questions in terms of difficulty to the actual exam administered.
The only issue is that the GARP practice exams are the same from year to year, so if you were anticipating completing say three, ie 2020, 2019 and 2018...only bother with one. Its disappointing as I would have thought that GARP would have plenty of old questions to choose from.

#### Eustice_Langham

##### Active Member
Sure @Eustice_Langham Those statements are correct but they do not involve the expected future spot price ("normal backwardation" is not the same as mere "backwardation," alas). Contango is simply an upward-sloping forward curve (or segment): S(0) < F(0,T) < F(0, T+Δt). Backwardation is a downward-sloping forward curve: S(0) > F(0,T) > F(0, T+Δt). The cost of carry model describes the expected relationship between the current spot price (observed, by definition) and a forward price, in this case: F(0) = S(0)*exp[(r-L)*T] where L = y - u; here L is lease rate, y is convenience yield and u is storage cost. You can see by relationship, F(0) = S(0)*exp[(r-L)*T], that if L > r then forward curve is downward sloping (aka, backwardation) and if L < r then the forward curve is upward-sloping (aka, contango). So far, this does not involve E[S(t)]. I am using the continuous version while GARP's notes contain the discrete analog (see https://www.bionicturtle.com/forum/threads/lease-rate-formula-in-chapter-11-book-3-frm-part-1.23323/ ). Again, a forward curve is a vector of observed prices (ie., spot plus series of forward prices) but the expected future spot price is unobservable and not an element in the forward curve (in a direct sense).

I have a video focused on the lease rate as L = y - u, in which case it can be called the "net convenience yield." See here (hope this helps ...): https://www.bionicturtle.com/forum/threads/t3-18-commodity-eg-gold-lease-rate.22431/
David, Thanks again, for my clarification though, you make the comment that..."normal backwardation" is not the same as mere "backwardation,", you also go on to say in the "trading rich/cheap thread", you make the following comment.."Also note, per the four light E[S(t)] that "normal backwardation | contango" is independent of contango or backwardation. We can have four different permutations, although theory expects "normal backwardation" and "contango."..I have searched and I cannot locate where there is a distinction applied between backwardation versus normal backwardatin. Can you elaborate please. Thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Eustice_Langham Those distinctions are in the CFA (or at least they were; the original source is CFA author Don Chance). Please see below. Note there are four possible combinations; the natural state is "contango" [i.e., an observably upward sloping forward curve per F(0,T) > S(0)] simultaneous with "normal backwardation" [i.e., F(0,T) < E[S(T)] per the long position's expectation to realize a future profit]. Contango (observable upward sloping) is the default because, for consumption commodities, it's arguably natural to presume that L < Rf, or equivalently, that y < (Rf + u); in the diagram c = cost of carry and for most consumption commodities it should be the case that c = Rf + u where (u) is storage cost, hence contango is illustrated as c>y but that's equivalent to L<Rf. Frankly, I don't much anymore utilize "normal backwardation" and "normal contango" terminology except which asked because I worry they are confusing on top of the observable contango/backwardation.

The concept of "trading rich/cheap" is very important and is represented by the difference between dark F(0,T) versus light F(0,T). The forward contract trades rich (cheap) when it's actual price is higher (lower) than its theoretical price per the cost of carry model (or another pricing model). Trading rich/cheap is not isolated to futures: it's universally important anywhere there is a pricing model ... I hope that's helpful,

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#### Eustice_Langham

##### Active Member
Thanks for the clarification David, appreciate your assistance.

#### jihan w

##### New Member
Dear all,
During my study I noted that you still use the old formula for pricing forward/future. Example for forward without any cah flows in Garp material they use S×(1+r)^t.

Thank you

#### David Harper CFA FRM

Staff member
Subscriber

##### New Member
Hi David

I sat jan 2021 exam (postponed from Nov 2020) and unfortunately didnt make it. I see that GARP are emphasizing on discrete compounding in forward and future prices. However, in Jan exam GARP stated “always assume continuos compounding”. Is this the case for July 2021 exams? I find continuous compounding more efficient and time saving

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nadaalshahabi Well, for years we asked GARP to genuinely prioritize continuous/discrete, if only for the sake of candidate/EPP convivence, but they just never seemed to care about a strong declaration. Of course you are correct that the practice exam(s) lately advise to “always assume continuous compounding” yet many of the actual source text examples employ discrete (e.g, annual) compounding. I agree with you about the elegance of CC, but our guidance must continue to be that candidates be facile in either and facile in translation back and forth. Of course, some instruments basically imply their frequency: it makes sense that s.a. coupon bonds would discount with semi-annual frequency as the default (while Hull would nevertheless discount s.a. bond continuously). Finally, obviously the question needs to be super clear about the compound frequency. Personally, I believe the practice paper's up-front guidance to “always assume continuous compounding” is insufficient by itself: each question itself should be explicit (ie., super clear) about which frequency is wanted. Thanks,

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