We had over 80 attendees on Saturday’s webinar (the third of five in our Early Bird series). As you may know, last year I only recorded tutorials, so I am learning myself about the challenges of a live format. I am especially grateful for some well-placed feedback I’ve gotten, including the following:

• To help field questions, a colleague will now accompany me on future webinars (just like Bryce did on Saturday)
• It was noted that I entertained all sorts of questions Saturday, even off the day’s topic of calculus. While I enjoy fielding questions, even after almost 2 1/2 hours, we didn’t get to the Taylor Series exercise. So, going forward, we will do our best to contain the Q&A to the particular webinar topic. You are welcome to ask any question, of course, but some questions I may have to defer until after the webinar. (and I will reply to those typically in the forum)
• To make the Early Bird material easier to find, we created a (category) page for the Early Birds. Just bookmark this page and any Early Bird updates will be posted here!

Are you doing the homework? Please see bottom of post for links to the homework.

To Register for Early Bird Webinar #4

The next Early Bird webinar is March 14th at 9 AM EST US (same time as before).

To register, simply click on this link

Recording of the EB #3 Webinar

Here is the recorded webinar. Please do not download both, these are the same single file, only one is in the smaller zip format:

• Windows media (.wmv)
• Same but a .wmv inside a zip file (.zip) 

Important: you need to install this CODEC in order to view the webinar

PowerPoint Presentation Used

The PowerPoint decks used in each of the Early Bird webinars can be found here:

• Early Bird #1 (financial returns)
• Early Bird #2 (volatility)
• Early Bird #3 (partial first derivatives)

To stay updated with Early Bird registration, presentations, and practice homework, simply bookmark this page.

Key Concepts

I hope the webinar successfully emphasized the following concepts:

• The partial first derivative is very common in risk measurement. It appears in various asset classes and metrics; e.g., option delta, bond duration, risk contribution, marginal value at risk (marginal VaR)
• The first derivative is an instantaneous rate of change; i.e., the limiting ratio illustrated by the convergence of a secant line to a tangent line.
• We looked at some basic differentiation rules (e.g., power rule).
• I highly recommend the excellent, free calculus resources at www.analyzemath.com. My favorite calculus texts are The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guides) and Calculus Know-It-ALL: Beginner to Advanced, and Everything in Between by Stan Gibilisco. I also like the affordable Schaum's Outline of Calculus.
• We looked at a handy idea: The first derivative of the natural log of a function equals the function’s growth rate or relative rate of change: if f(x) = ln g(x), f’(x) = g’(x)/g(x).
• Given the price function of a 30-year zero-coupon bond under continuous discounting, p(y) = 100*EXP(-y*30), we showed that the first derivative is the dollar duration: p’(y) = –3000*EXP(-y*30). Further, while duration has various specific "flavors," all are variations on (functions of) this first derivative dollar duration. For example, by dividing by price [i.e., p(y)], we produce the modified/Macaulay duration of 30.
• It is helpful to be mindful of the axis units. In the case of dollar duration, our first derivative at 5% yield was about –670. What are the units? In this case (i.e., dollar duration), this refers to –670 $/%. Specifically, 670 dollars in price change for a one-unit (100 basis point) change in yield. Still confused about duration units? See the comments to this post.
• The option delta is a first derivative; i.e., the change in call price with respect to a change in stock price. What are the units in this case? Since we have dollars (option price on y axis) divided by dollars (stock price on x axis), they cancel and option delta is unitless.
• We applied the first derivative rule to confirm that (i) futures contract delta is EXP(r*t) and (ii) the Eurodollar futures contract implies a $25 dollar change for each one basis point move.
• Finally, I hope I succeeding in conveying the one big idea underlying this webinar. We may characterize portfolios as responding to (mapped to) a set of underlying risk factors (the call option analogy: the value of the call option reacts to risk factors such as volatility and interest rates). The partial first derivative returns a linear approximation of the portfolio’s sensitivity to an underlying risk factor.
• Phillip made an excellent point about the key weakness of the first derivative linear approximation: is it is only locally accurate; the larger the change in the underlying factor, the less accurate it becomes.

Previous Follow-ups

• Here is follow-up to EB #1 and here is the practice homework for financial returns. Please try and do these exercises: mastery of discrete/continuous compound frequencies will pay dividends, I promise!
• Here is the follow-up to EB #2 and here is the associated practice homework on volatilities.

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