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# FRM Part 2. Tuckman´s chapters

#### Juan B.

##### Member
Hi David / BT Team,

I wanted to ask you for your insight on the Tuckman Readings for Part 2 and what would you consider a low / high priority from the exam planning perspective.

I have retrieved the below info from your summary Focus from 2012. I get the impression that some chapters (the important ones) were moved to part 1 which leaves not many high testable matters from the dense 4 chapters. Do you have any different feedback after 2012 & 2013 exams for this year´s exam?

Thanks a lot!

Juan

2012 FOCUS REVIEW (related to Tuckman)

• T5.b.2. Tuckman Durations (because single-factor sensitivities are highly testable in P2, like they were in P1). If you can comprehend all of this (relatively small) 5.b.2. spreadsheet, I would venture to say that you know most of what you need from these Tuckman chapters.

Bond sensitivities (single, multi-)
Here's the reality of the three assigned Tuckman chapters, from an exam standpoint only: they are progressively more difficult and less testable. The final (Science of Term Structure) is quite dense but, historically at least, has been tested only superficially. It's an important topic in bond pricing generally (outside the exam), but you don't want to skip other FRM topics because either Chapter 7 (key rates) or Chapter 9 (Term structure) are slowing you down.

• Single-factor sensitivities: get very comfortable with durations, DV01, and convexity, much of which overlaps with P1. The most important difference here in P2 is that you want to be sufficiently comfortable with DV01 and dollar duration such that you can perform hedging calculations.
• Key rates: it's probably enough to have a superficial understanding. Why is it? (it's multi-factor, so we can overcome the unrealistic assumption of parallel yield curve shift) How does it work basically? The one additional thing here, that i would understand, is how KR01 and key rate durations are analogous to DV01 and duration.
• Term structure: understand the logic of the binomial tree, which is similar to Hull's binomial tree for option pricing (except here the underlying risk factor is a mean reverting interest rate rather than a stock price). The FRM historically has mostly, if not almost exclusively, tested one idea here: the calculation/meaning of the risk-neutral probability (p). To my knowledge, much of the depth in Chapter 9 has yet to be tested (such that I requested it be removed last year); this whole chapter historically has been borderline optional w.r.t the exam.

#### Juan B.

##### Member
Hi David / BT Team,

Just wondering if you could shed some light on the tuckman´s chapters for Part 2 as per my previous post.

Thank you.

Juan

#### fjquinon1

##### New Member
I find the definitions of risk neutral and real world probability to be extremely unintuitive. I would assume in a risk neutral world, the weights assigned to an up and down movement to be equal (50% up, 50% down) and a real world to weight the movements more realisitically (we'll say for arguments' sake 80% up, 20% down). However, the readings seem to suggest the exact opposite - a real world or true probability is .5 assigned to each movement with risk neutral assigning varied weights. How does an interest rate have an equal chance of moving up and moving down in the real world?

#### hamu4ok

##### Active Member
I find the definitions of risk neutral and real world probability to be extremely unintuitive. I would assume in a risk neutral world, the weights assigned to an up and down movement to be equal (50% up, 50% down) and a real world to weight the movements more realisitically (we'll say for arguments' sake 80% up, 20% down). However, the readings seem to suggest the exact opposite - a real world or true probability is .5 assigned to each movement with risk neutral assigning varied weights. How does an interest rate have an equal chance of moving up and moving down in the real world?

It was for me as well. Tuckman's reading was tough one for me. I reread those chapters on interest rate models several times until I could be assured that my brain has some snapshot of so called bird's eye picture of all the stuff.
Regarding real world and risk neutral probabilities, it is unintuitive. For real world one, I suggest to look at it as a predicting side of coin before flipping it, if there is no other factors affecting the result, the probability would be 50/50. In binomial world of fixed income it must be also 50/50: up or down state of the world.
The risk-neutral probabilities, is a tough one to picture. After reading for example, Malz on credit risk, Merton Model, Gregory on CP Risk, and other overlapping reading in FRM, you realize that real-world probabilities do not give you the price you have on the market now. The reason for that is numerous: credit risk premium, liquidity premium, time value of money, irrational behaviour, etc. All these (except for the last one) when added would drive the price down or up from the our "perfect idealized world" model of 50/50 chance of being of either up or down. So you need to adjust something to match the modeled value with market price. You could do it by different ways, changing the discount rate (adding spread to it, Z-spread, OAS), or changing probabilities (to risk-neutral one) or discount using risk-free rate and then deduct risk premium (CVA) to come up with risky real value. Helps you to memorize also that risk-neutral often means market implied factors (such as using implied volatilities in BSM).

Let me also share my bird's-eye picture of Tuckman fixed income and market risk in bullet points.
• the goal of everyone is to price the asset, which is simply a NPV of its cash flows.
• For fixed income products, there is one big issue, at what rate to discount those cash flows as cash flows are more or less predictable (there is a reason why they are called fixed one).
• The issue of discount rates is twofold, one is that you need to use risk-free rate (Libor vs OIS ) and also that interest rates are pretty unpredictable.
• Regarding Libor vs OIS, Mr Hull already proved that OIS is clearly more risk-free rate than Libor, so no issue here (2007-08 credit crisis and Libor scams proved it).
• Regarding unpredictability of interest rates, there are all sorts of interest rates models that help you to guess the future interest rates, which you will use to discount the CFs.
• Interest rate models that use only one factor, such as short-term rate (e.g. 6m rate is proved to be helpful enough to deduct from it the whole term structure), are most simpliest and testable for FRM exam - called short-rate interest rate models.
• The general formula of interest rate movements (they call it dynamics), may be called Brownian motion or ITO process (I may be wrong though), but it goes like this dr= mean/drift(t) * dt + sigma/volatility(r)* dw
• In general you can divide all short-rate interest rate models into two broad categories: one with CONSTANT VOLATILITY (+ varying drift) and another with VARYING VOLATILITY (+ varying drift). This is helpfull to categorize and memorize all those numerous models, Ho-Lee, Vasicek, Lognormal, Model 1,2,3,4, ect.
• dt is time step, for bonds like products, it is every 6 months (t=0.5), for some other products you may want to use monthly (dt=1/12) or weekly (dt=1/52)
• The memorizing the central node of each models, it is helpfull to note that E[dw] = 0, so only the first part - drift is needed.
• For upper/lower nodes of binomial trees of interest rates, you need to use St.Dev [dw] = SQRT (dt), so you add/deduct sigma * SQRT (dt) to the drift part.
• mean-reverting drift is kind of varying drift, but instead of varying to time, it varies with some long-time value of interest rate.
• volatility tends to vary more with the level of interest rates (r) than time (t).
• BINOMIAL TREES is simply a graphical tool to picture the possible interest rates across time periods which are predicted by one of your models.
• You apply BACKWARD INDUCTION on these Binomial Trees to value derivative products (such as options).
• All these crazy actions you conduct to come up with discount rates, so that you can value those nasty derivative products.
• For ABS, MBS and other structured products, other factors need to be considered, such as prepayment risk (PSA model) and refinancing burnout/path-dependency issue (you cannot use binomial trees for that purpose, instead Monte-Carlo simulations need to be done)
That's my bird's eye view on fixed income material and interest rate risk.
The only thing left for me is to tackle VAR, ES, Extreme Value, Copulas in the same way or another. So many VARs to memorize, and not only you need to memorize the formulas, but also know why you need a specific VAR and what you do with it.

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#### hamu4ok

##### Active Member
the main ideas in one slides:

#### HALCOLM1

##### New Member
Subscriber
Hi David,

In the Tuckman study notes Under the AIM
"Construct and apply an arbitrage argument to price a call option on a zero-coupon security using replicating portfolios"

Could you explain:
how we know the payoff can be replicated by going long a 1y bond and short a 6m bond?
In the study notes why the 1y and 6m have the same price of $612.50? How$629.34 less $612.50 equals$0.58?

Thanks

#### tosuhn

##### Active Member
hi @David Harper CFA FRM CIPM & @Nicole Manley, two quick questions:
Am studying Reading 29, Tuckman Chapter 7 & 8 at the moment and discovered the below:
Tuckman Chapter 7 notes, pg 14, there is a missing interest rate tree.
Tuckman Chapter 8 notes, pg 28, under Model 1, the figure 4.076% under node [2,0], is it a typo as if I used the formula of r0 - 2sigma*sqrt(dt), I am not able to get this figure.

Hope to hear from you soon.
Thanks.
Regards,
Sun

#### ami44

##### Well-Known Member
Subscriber
Hi David,

In the Tuckman study notes Under the AIM
"Construct and apply an arbitrage argument to price a call option on a zero-coupon security using replicating portfolios"

Could you explain:
how we know the payoff can be replicated by going long a 1y bond and short a 6m bond?
In the study notes why the 1y and 6m have the same price of $612.50? How$629.34 less $612.50 equals$0.58?

Thanks

Maybe you found you're answer already, but I just finished the chapter and feel inclined to test my newly acquired knowledge.

how we know the payoff can be replicated by going long a 1y bond and short a 6m bond?

The equations for calculating the hedge are:
F(0.5) + P(1.0, up) * F(1.0) = 0
F(0.5) + P(1.0, down) * F(1.0) = 3

with P(1.0, up) = 0.97324 the price of the 1y bond after 6 month and under the condition, that we had an up move.
equivalent P(1.0, down) = 0.97800

As I understood it, it's not necessary to use the 1y bonds. The math would work with any security for which the value depends on the 6month rate. If we use a 5y bond instead, of the 1y bond, the prices in the above equations would be different of course, but the resulting option value should be the same. Tuckman just uses the simplest example, with the 6month and 1 year bond.

My feeling tells me, that if we get rid of the 0.5 year bond also and use for example a 1year and 5 year bond, we will end up with linear dependent equations.I'm not sure though.

In the study notes why the 1y and 6m have the same price of $612.50? This is not the price of the bond, but the value of the hedge. The hedge consists of 629.34 long (1y) and 612.50 short (6month). In case of an up move the long position is worth 629.34 * 1/(1+0.055/2) = 612.50 and the short position has a value of -612.50. Which is in total zero, which equals the value of the option in case of an up movement. How$629.34 less $612.50 equals$0.58?
Because these are facevalues, that have to be discounted to get the value of the hedge:
629.34 * 1/(1+0.0515/2)^2 - 612.50 * 1/(1+0.05/2) = 0.58

Hope that helped.

#### ami44

##### Well-Known Member
Subscriber
Am studying Reading 29, Tuckman Chapter 7 & 8 at the moment and discovered the below:
Tuckman Chapter 7 notes, pg 14, there is a missing interest rate tree.
...

The odd thing is, that there is no mention of embedded options in the core readings, but a learning objective "Describe the impact of embedded options on the value of fixed income securities". I guess its some legacy objective from another edition, or was I just to dumb to find it?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @ami44 That is a great observation: it's been there for at least three years and frankly I missed it. The concept is important, but I agree with you: technically it is not covered in Tuckman's The Science of Term Structure Models. I just wrote to my contacts at GARP, to alert them, and linked back to this thread. Thanks!

#### tosuhn

##### Active Member
Hi @David Harper CFA FRM CIPM i think my question below was missed out:
Tuckman Chapter 8 notes, pg 28, under Model 1, the figure 4.076% under node [2,0], is it a typo as if I used the formula of r0 - 2sigma*sqrt(dt), I am not able to get this figure.

Hope to hear from you soon.
Thanks!
Regards,
Sun

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @tosuhn It looks correct to me, either:
• 5.00% - 2*1.60%*sqrt(1/12) = 4.076%; i.e., 0.08333 = 1/12, or
• 4.538% - 1.60%*sqrt(1/12) = 4.076% (keep in mind 4.538% is rounded so this is not as good). I hope that helps!