Feb 04

FRM 2008 Early Bird Episode #4

by David Harper, CFA, FRM, CIPM

FRM |

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Contents

  • The importance of quality practice questions (and thanks for feedback)
  • Where to find the previous Early Birds
  • This week's learning outcomes: Functions and Factor Models
  • Video tutorial (Episode #4)
  • Practice questions (3)

The importance of quality practice questions

Thanks for the positive feedback on the Early Bird. I am thrilled some of you find it helpful. I had a request for more practice questions. Although this week only includes three practice questions, I will try to add a couple more questions to next week's episode (week #5: an introduction to partial derivatives) and going forward. Of course, there are plenty of practice questions available on the member page.

Rather than pull questions from our database, I write the Early Bird questions de novo with the goal of covering key ideas. Last year, we added the forum and we learned something: not all practice questions are created equally. Customers posted questions from various sources. And, to be helpful, we worked out many of them. We found the quality and relevance of practice questions can vary widely. Some questions frankly are not good; some questions are good but not timely. Sometimes, from an exam standpoint, the topic is obsolete. For example, hedge accounting (FASB No. 133) was in the 2006 curriculum but not really in the 2007 curriculum. Will in be in the 2008 FRM? I don't know yet. But you want to practice questions that are both good and relevant (timely).

Consider, for example, questions about the Basel Accords (the original and Basel II). Due to the swirling, global, incredibly complex dynamics around the implementation of the Basel framework, Basel II test questions are necessarily asked at a "principles" level (versus at a detail or, especially, an implementation level). Principle-level questions tend to be stable as the core Basel II framework hasn't really changed in two years (i.e., the July master Framework is essentially unchanged from November 2005). On the other hand, GARP does a good job keeping abreast of changes. Last year they added three strong Basel readings: a dense review of credit risk concentration, a helpful note on the IRB function, and a strong paper on outstanding issues (maybe the best place to start!). So, you can see, even good Basel questions need to be updated.

Previous Early Bird Episodes

In case you missed them, the previous early bird emails (with the practice questions) are found here:

Last week, I introduced linear regression. We saw that an OLS method produces a line that minimizes the sum of squared errors (SSE) and is given by: y = mx + b + e (error). We broke a regression into two pieces: SSR (sum of squared regression) + SSE (sum of squared errors); such that the coefficient of determination (R-squared) = SSR/(SSR+SEE). In the forum, somebody noticed that I did not provide the equation for the slope coefficient even as one of the questions asked for it. Sorry! It is worth knowing: Slope = Covariance (independent, dependent) / Variance (independent). Please note this slope is the same as beta under a particular definition of the independent variable (i.e., market portfolio return) and the dependent variable (i.e., security return). In this case, the regression line is the characteristic line and the slope is beta.

Learning Objectives (Functions and Factor Models)

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This week, I segue to multifactor models with a regression line: the characteristic line. As mentioned, the slope of this line is beta (a.k.a., equity beta). Beta is the quantity of risk in the capital asset pricing model (CAPM). And the CAPM is just a special case of multifactor models.

The theme of this week's 30 minute episode is functions and multifactor models. To illustrate, I show the simplest function for each of three asset classes: equity, bonds, and forwards (commodities). Here are the key Early Bird ideas:

  • The characteristic line regresses a security's return against the market's return. The slope of this line is the security's beta
  • Beta is "quantity of risk" in the capital asset pricing model (CAPM).
  • CAPM says the security's expected return = riskless rate + (beta)(equity risk premium)
  • Beta = COVARIANCE (security return, market return) / VARIANCE (market return). It is a standardized measure of systematic risk.
  • Here is the key point. Under CAPM, the security's excess return (i.e., expected return - riskfree rate) is the product of a factor (the ERP) and a sensitivity to the factor (beta).
  • If you understand that, you should see why CAPM is just a special case of a multifactor model
  • We have several ways to classify multifactor models (e.g., APT versus empirical). If you are seeing multifactor models for the first time, do not be intimidated. On a superficial level, they are all the same. All express the security's excess return as a function of: (1st factor)(1st sensitivity) + (2nd factor)(2nd sensitivity) + ... + (nth factor)(nth sensitivity)
  • To introduce bond pricing, I start with the simplest example: pricing a zero-coupon bond. Note how elegant this is if you assume continuous compounding. The Price = Face * EXP[(-rate)(time)]
  • If the price of a bond is a function of its yield [P = f(yield)], then the change in price is a function of change in yield. This sensitivity is duration. Generally, this is a core idea in risk measurement: if our (multifactor) model expresses the price of an asset as a function of several factors, risk is the sensitivity (a derivative) of the asset price to changes in the factor(s).
  • Finally, we look at the cost of carry model which links the forward price to the spot price.
  • Note the difference between the price of a forward and the value of a forward
  • The value of a forward on a stock is (f) = Stock (S0) - (Delivery)EXP[(-rate)(time)]. In English: stock price minus discounted delivery price. I am fond of this elegant formula because it is a great learning "building block."
  • It is the same formula, after all, as the minimum value (lower bound) of a stock option: Minimum value (option) = Stock (S0) - (Strike)EXP[(-rate)(time)]. We can go right from this minimum value to put-call parity. Then wrap it with a couple of functions to arrive at the Black-Scholes option pricing model. Which, of course, we'll use in the Merton (structural) model for credit risk. That's what I call a reusable building block!

Video Tutorial

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The 30-minute video tutorial is located here (with a table of contents).

If you are a paid member, you can also access this in the member section (where you will also find the downloadable slides, if you would like to view those. As well as an ipod format file.)

Practice Questions

This week I've got three practice questions for the theme Functions and Factor Models. One question per pricing model for each of equity, bond, and forward.

Question #1 (equity function)

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As of Jan. 30th, Google's beta was 1.26 (see snapshot at left, as reported by Yahoo finance). Further, the one-year Treasury rate is 3.26% (assume Treasuries are a good proxy for the riskfree/riskless rate).
Finally, assume the equity risk premium (ERP) is 4%.

(i) Under the capital asset pricing model (CAPM), what is the expected return on Google's stock?
(ii) If CAPM is a simple case of a (fundamental) multi-factor model, which variable (or term) is the FACTOR and which is the SENSITIVITY (a.k.a., factor sensitivity, factor loading)?
(iii) How is beta defined?

Question #2 (bond function)

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Assume a 5-year zero-coupon bond that yields 8% with a face value of $100.

(i) What is the current price, under continuous compounding?
(ii) What is the current price, under semi-annual compounding?
(iii) What is the bond's Macaulay duration?
(iv) What is the bond modified duration?
(v) Bonus: The duration formula is dP/P = (-Duration)(dy). In English, the relative change in bond price is a linear function (where -Duration is the constant) of the yield change. If we "reprice" the bond by using only this duration equation (i.e., only "first-order" effects), by shocking the yield up or down, will our new price estimate be accurate? If not, what is the bias?
(vi) Turtle Tough Bonus: Using continuous compounding, solve for the bond duration by taking the first derivative of this equation: Bond Price = (Face)exp[(-rate)(years)]

Question #3 (forward function)

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Assume a long forward contract on a non-dividend-paying stock has one year to maturity. The current price of the stock is $10 and the delivery price is $10.

If the riskless rate is 5%, what is the value of the forward contract (assume continuous compounding under the generic cost of carry model)?

Answers to practice questions

That's all for this week. Good luck and see you next week!

David-Harper_100w

David Harper, CFA, FRM, CIPM
Founder
www.bionicturtle.com

Comments

  1. Gopal Krishnamurthy 04 Feb 2008

    Dear David,

    your lessons are excellent. Do you have a programme suited for PRM. If so i would like to join immediately.

  2. David Harper, CFA, FRM, CIPM 05 Feb 2008

    Gopal,

    Thank you, I really appreciate that kind feedback! Sorry, I do not currently have plans for PRM. (But I would always be interested in talking to a potential instructor to help with that....). David

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