Excel
02 Dec 2008
Feb
11
FRM 2008 Early Bird Episode #5
by David Harper, CFA, FRM, CIPM
Contents
- How to purchase the 2008 Bionic Turtle program
- Previous Early Bird emails
- This week's learning outcomes: Derivatives
- Video tutorial (Episode #5)
- Practice questions (5)
How to purchase the 2008 Bionic Turtle program
Several people have written to ask, "How do I buy the 2008 program?" I regret the new website is not yet ready; it will be ready on March 3rd when registration for the exam opens at GARP. In the meantime, you can still purchase here. Any current purchase earns a membership through the end of the year and automatically applies to the 2008 FRM.
The Early Bird tutorials are free to anybody. The only difference for paying customers (members) is their access to downloadable PowerPoint slides and the ipod format. You don't need to buy anything yet, the new website will contain details of the 2008 program. I remind you that GARP has not yet published either the new Study Guide or the new Learning Outcomes.
Finally, our forum is openly public (e.g., and feedback from previous customers is unmoderated). Our mission is to share great learning about finance, so you are always welcome. If you have a tough practice question you'd like help with, we'll try to work it out with you.
Previous Early Bird Episodes
In case you missed them, the previous early bird emails (with the practice questions) are found here:
- Episode #1 (notation, compounding, matrix)
- Episode #2 (distributions: binomial, Poisson, normal)
- Episode #3 (linear regression)
- Episode #4 (functions and factor models)
- And the answers to the practice questions are in this forum area
Learning Objectives for Episode #5 (Derivatives)
Last week, we looked at Functions and Factor Models. This week I briefly (20 minutes) introduce a few differentiation rules (i.e., taking the derivative of a function). If you already know how to take a derivative, you can skip this introductory episode. If you do not, please try to watch (or at least flip through the slides) because the first derivative is key building block with significance in the formal curriculum. Why are derivatives so fundamental? Because if we can express assets or portfolios as functions of risk factors, then asset/portfolio risk can be characterized as price sensitivity to factor changes. This price sensitivity can be mathematically approximated by first- and second-order (partial) derivatives.
Specifically:
- A first-order derivative and a partial derivative (i.e., when the function has more than one independent variable, we still take the derivative with respect to one of the independent variables while holding the others constant)
- Constant rule (derivative of a constant is 0), linear rule (derivative of a line is its slope), and power function rule
- Rules for exponential function and natural log (including the nice idea that we can take the derivative of the natural log of a function to obtain the function's growth rate)
- The Taylor series expansion: a function is deconstructed into an infinite series of derivatives. We use the first few terms of the Taylor series to approximate changes produced for nonlinear but smooth functions
- Maybe most importantly, we emphasize that you can look at the first derivative as a single idea across asset classes. For a bond, the first derivative is duration (i.e., a linear approximation of the change in price given a small change in yield). For an option, it is delta (i.e., a linear approximation of the change in call value given a small change in stock price); for portfolio VaR, it is marginal VaR (i.e., a linear approximation of the change in portfolio VaR given a small change in the position)
Video Tutorial
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 five practice questions for the derivatives theme. I tried to involve some concepts you are likely to encounter in the formal curriculum.
Question #1 (gold futures)
Setup:
As of Feb 2008, the spot price of gold is $900. The August 2008 gold futures price is $921 (+ 6 months) and the Feb 2009 gold futures price is $923 (+ 12 months). Finally, the current riskless rate is 3%.
In summary:
Gold spot = $900
Gold futures (+ 6 months) = $921
Gold futures (+ 12 months) = $923
Riskless rate = 3%
Let's assume continuous compounding/discounting. (Please note: all numbers are significantly rounded from actual, for convenience's sake).
Questions:
(i) If storage costs are 6% per annum, what does the cost of carry model predict for the Feb 2009 futures price (+12 months)?
(ii) If we assume gold has no convenience yield (nor financing costs above the riskless rate), what does the six month contract imply is the per annum storage cost of gold?
(iii) If storage costs are $2 per ounce per year, rather than a proportional cost, what does the cost of carry model predict for the Feb 2009 futures price (+12 months)?
(iv) What is the delta of six month futures contract (including the storage cost above)? Please try and show this by taking the first derivative!
(v) Bonus: (i) Is this gold futures curve in contango? (ii) is this gold futures curve in normal contango?
Question #2 (Fama & French)
Setup:
The Fama & French is a three-factor (multifactor) model that adds two additional factors to the capital asset pricing model (CAPM). It says that a security's expected return is given by:
Expected [r] = Rf + [beta3][RMRF] + [beta2][SMB] + [beta1][HML]
where:
r = security's return
Rf = riskfree rate
beta3 = market beta similar to beta in CAPM
RMRF = market risk premium (return on market - riskless rate) similar to ERP in CAPM
beta2 = factor sensitivity to size (i.e., exposure level to size risk)
SMB = size (market capitalization) factor
beta1 = factor sensitivity to value (i.e., exposure level to value risk)
HML = value factor (excess return due to low market-to-book value)
If this is unfamiliar, please note its resemblance to CAPM. Subtract riskfree rate from both sides so that left side is excess expected return. Then excess return [expected return - riskless rate] is simply the sum of three RISK PREMIUMS where each risk premium is the product of a SENSITIVITY (which is specific to the security) and a RISK FACTOR (which is common to all securities). So generically this Fama French is:
(r - Rf) = [sensitivity to...][common market premium factor] + [sensitivity to...][common size factor] + [sensitivity to...][common value factor]
Questions:
(i) If RF = 3%, RMRF = 4%, SMB = 2% and HML = 2%, what is the expected return on a security that has a market beta (beta3) = 1.0, a size beta (beta2) = 0.0 and a value beta (beta1) = 1.0 (this might describe a large cap value stock that covaries tightly with the market)
(ii) What is the partial derivative of the expected return with respect to RFRF? To put another way, what is the rate of change of the expected return with respect to RMRF? We could also write:
if Expected Return (r) = f (RMRF, SMB, HML) = (market beta)[RMRF] + [size beta][SMB] + [value beta][HML], what is dr/dRMRF?
(iii) Given the answer to (ii), if the RMRF increases by 1% (i.e., the price of market risk increases from 4% to 5%), what is the change in the security's expected return?
(iv) What is the second-order partial derivative of the expected return with respect to RFRF?
Question #3 (derivatives of functions)
Assume we have fitted a few nonlinear functions to explain a company's annual sales over time (as a function of the year T). In these functions, sales are given as a function of the year (T):
1. S(T) = 2-(T^2)+(T^3)
2. S(T) = (10)EXP[T^0.25]
3. S(T) = LN[6*T^2]
4. Bonus: What is the relative growth rate of function 2: S(T) = (10)EXP[T^0.25]?
For each of the equation, evaluate the first derivative at the fifth year (T=5); i.e., the derivative of S with respect to T.
Question #4 (European call option)
Setup:
Assume a European call option with 11 months to maturity and a strike price of $10. The riskless rate is 3% and the stock's volatility is 41%. The stock pays no dividend (this is just background that you don't actually need to solve the problem).
If the stock price is $9, then the value of the call option (given by Black-Scholes) is $1.12. The delta of a European call option on a non-dividend-paying stock is simply N(d1). N() is the standard normal cumulative distribution function; in Excel, it is =NORMSDIST(). In this case, d1 happens to be zero (0).
Questions:
(i) What is the stock option's delta, given the stock price is $9?
(ii) If the stock price increases by $1 (from $9 to $10), what is the approximate increase in the value of the call option (as implied only by the delta, the linear approximation)?
(iii) Given a $1 increase in the stock price (from $9 to $10), how does the actual change in the price of the call option compare: is it higher, lower, or about the same?
(iv) At this same position, if we are LONG 1,000 options, how can we achieve a delta neutral hedge?
(v) Bonus: If everything else remains the same, what happens to delta as the stock price increases? What happens to delta as the stock price decreases?
(vi) Bonus: delta is the first partial derivative of the call price with respect to the stock price. What is the second partial derivative (with respect to stock price)? Given the answer to (vii), can we say anything about it (e.g., zero, constant, nonzero)?
Question #5 (coupon-paying bond)
Setup:
Assume a $100 face value, five (5) year zero-coupon bond trading at a yield (YTM) of 4%. On a semi-annual basis, therefore, T = 10 and y = 2%.
The price of the bond is given by:
P = ($100)[(1+y)^(-T)]
Questions:
(i) What is the price of the bond?
(ii) Solve for the first derivative of the bond price (P) with respect to the yield (y). What is this duration equation; i.e., what is dP/dy?
(iii) Use this linear approximation (duration) to re-price the bond given a +1% increase in the yield (i.e., shock the yield +1% and use the duration equation above to estimate the new price)
(iv) How is this re-priced bond (using the duration) different than simply re-pricing the bond as direction function of T=10 and y=2.5%? i.e., new price = ($100)[(1+2.5%)^(-10)]
Answers to practice questions
That's all for this week. Good luck and see you next week!
David Harper, CFA, FRM, CIPM
Founder
Comments
David: Really great tutorial ! I bet this section will be a great brush-up for people who are a little bit rusty on math…
I wonder if what you covered in this tutorial is THE most difficult (advanced) math in the exam OR you will have more coverage on calcus later on ?
thanks
Lee
I meant “Calculus”
Lee,
Thanks I really appreciate that.
Re calculus: It depends on the LOs because we don’t like to stray too far from the testable material. The highest math may go a notch higher but that’s about all. The test is broad not mathematically deep, frankly. I hope this year we see a a notch higher (e.g., Black-Scholes PDE; I’d love to see copulas covered; given subprime I would hope we see more math vis a vis credit derivatives) but the FRM is not really a heavy duty quant exam.
Like the Taylor Series Expansion mentioned: it will be covered qualitatively but there is hardly any mathematical application of it (with the slight exception of option portfolio VaR).
(I used to say the toughest math was reflected by the following: you have to price a CDS. But they removed that LO in 2007!? Although i think that still is about right vis vis level of difficulty)
So, there is a lot of math in the exam but it is rather basic math applied to different asset classes (e.g., bond pricing, option pricing, portfolio VaR), discussed in conceptual ("principles") terms or where needed for regulatory (e.g., the Basel IRB function is sort of mathy) rather than advanced math.
David: in your tutorial, you gave example of calculating the first-order deriv of Y = 10 + 3X - X^3, you gave result of Y’ = 3 + 3X^2. I think it should be Y’ = 3 - 3X^2. if not, could you please tell why we have to change the sign before X^3.
thanks
Lee
Lee,
My mistake. The term -X^3 evaluates to -(3)X^2, you are correct. You are right, you don’t change signs; e.g., if f(x) = w(x) - z(x), then f ‘(x) =w ‘(x) - z ‘(x).
David
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