Feb 24

FRM 2008 Early Bird Episode #7 (duration)

by David Harper, CFA, FRM, CIPM

FRM |

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Contents

  • Update and a question for you
  • This week's learning outcomes: Duration
  • Video tutorial (Episode #5)
  • Practice questions (four)
  • Previous Early Bird emails

Update and a question for you (do you want webinars?)

Hello! As the 2008 FRM registration is fast approaching (March 3rd), GARP will soon publish the 2008 FRM Study Guide. I am excited because our new site will launch then too. And downloads should be easier (which have given some people trouble lately). Due to popular demand, this year we are adding an iPod format for all 2008 screencast tutorials. In the meantime, please bookmark our forum. It continues to be a popular, free, resource-rich area for risk-related inquiry!

Also, you might find helpful a couple my recent blog posts:

webinars

Finally, I am thinking about conducting a few live webinars this year. Not to replace the recorded screencast (movie) tutorials, but rather to give me a chance to field queries in real-time. My thinking is that the first webinar could be: "your study strategy for the year." We'd take the new curriculum and break it down, so that you might get a better sense of where you should focus.

If you like the idea of live webinar(s), please reply/comment/write me back and tell me you approve. Basically, if there is enough demand, I'll do it! I can be reached at .

Learning Objectives for Episode #7 (Bond Duration)

This week's Early Bird screencast episode is a 30-minute introduction to bond pricing and duration.

TI BA II+ Keys

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For exam candidates, you'll need to get comfortable with the calculator keys.

Bond math is the essence of finance. I'll show you how the generic time value of money (TVM) keystrokes on the Texas Instrument BA II+ (TI BA II+) map to their bond equivalents. As in, [PMT] = coupon payments and [I/Y] = yield to maturity (YTM). (If you'd like a refresher on yield to maturity, I recorded a nine-minute tutorial last week here. This explains, too, why the realized yield is likely to vary from the YTM.)

Duration

Duration has several definitions and a few interpretations. We are most interested in following Fabozzi with an interpretation based on sensitivity (see modified duration below):

  • For any given yield, we can solve for a line which is tangent to the price/yield curve.
  • The slope of the tangent line is dollar duration. Dollar duration = duration multiplied by price.
  • Modified duration is the sensitivity measure: what is the approximate percentage price change given a 100 basis point yield change.
  • Macaulay duration is the weighted average time to receipt of cash flows. This is the duration that can be expressed in terms of years. For example, a 30-year zero has a Macaulay duration of 30 because we wait 30 years for the single cash flow.
  • The dollar value of an '01 (DV01) is closely related to the modified duration. It is just the dollar change given a one basis point yield change (1 basis point = .01% = .0001). Therefore, we can solve for the DV01 this way: DV01 = [(Price)(modified duration)]/10,000

Convexity

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Duration is a primary but partial risk measure. It is accurate only "locally" for small yield changes. The convexity adjustment improves on the merely linear duration approximation.

Unlike duration, which at the portfolio level is simply a weighted average, convexity is a function of the square of maturity. Consequently, as illustrated, given similar durations, a barbell strategy implies greater convexity than a bullet strategy.

 

Video Tutorial

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The 30-minute video tutorial that introduces duration is located here.

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 wrote four compound questions to stretch your understanding of duration. Did you notice several have working, browser-based spreadsheet (EditGrid) solutions provided also? We do this a lot a bionic turtle: mastery is when you can follow the model...

Question #1 (yield to maturity, YTM)

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Assume a $1,000 par 6% semi-annual coupon bond with 10 years to maturity. The bond's price is $750.76.

(i) What is bond's yield to maturity (YTM) on a bond-equivalent basis?
(ii) Under what conditions will this YTM match the realized yield?
(iii) Given the yield (YTM) we just solved for, what is the bond's corresponding effective annual yield (EAY), also know as the effective annual rate (EAR)?

The next two parts do not really involve the bond; rather, they are just conversion practice.

(iv) Convert the EAY in step (iii) into a continuously compounded rate
(v) Convert the continuously compounded rate in step (iii) into a monthly compounded rate

Question #2 (bond price)

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A newly issued 10-year bond with a face value of $1,000 pays a 4% coupon and yields 4.5% (i.e., time to maturity = 10.0 years and yield to maturity = 4.5%).

(i) If the bond pays an annual coupon (annual compounding), what is the bond's price?

(ii) If the bond pays a semi-annual coupon (semi-annual compounding), what is the bond's price?

(iii) If the bond continues to pay a semi-annual coupon, but yield changes to 4.0%, what is the new price?

Note: You do not need to re-key all of the time value of money (TVM) keys if they do not all change. Try solving (iii) by only changing the required input.

(iv) For this bond, could we use the Taylor series expansion to estimate sensitivity of the bond price to changes in yield? Why or why not?

(v) In three months, if nothing else changes (except the maturity is 7.75 instead of 10.0), will the bond's duration be higher or lower than the bond's current duration?

(vi) In three months, if nothing else changes, will the bond's convexity be higher or lower than the bond's current duration?

Question #3 (duration)

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Assume a $100 par 10-year bond with a 4% semiannual coupon (the annual coupon is $40 but $20 is paid every six months) that yields 6%.

(i) Without doing calculations (just by looking), is the bond's price above/below its $100 par?

(ii) What is the bond's price?

(iii) What is the bond's modified duration?

(iv) What is the bond's Macaulay duration?

(v) Can you use the result in (iii) or (iv) to estimate the bond's dollar value of an '01 (DV01)?

(vi) How can we interpret (i.e., express into a sentence) both (a) the modified duration and (b) the Macaulay duration?

Question #4 (duration grab bag)

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(i) Given a yield of 6% for a 15-year zero coupon bond, what is the bond's Macaulay duration and its approximate modified duration?

(ii) Assume a simple bullet portfolio consisting of two equally-weighted bonds (i.e., their market values are equivalent). Both bonds have a duration of 10.0 years. What is the portfolio's duration?

(iii) Assume a barbell portfolio consisting of two equally-weighted bonds (i.e., their market values are equivalent). The first bond has a duration of 2.0 years and a the second bond 18.0 years. What is the portfolio's duration?

(iv) Can we say anything about the convexity of the bullet portfolio in part (ii) compared to the convexity of the barbell portfolio in part (iii)?

(v) What is the general impact on bond duration of the following: (a) an increase in maturity, (b) an increase in yield, and (c) a credit rating upgrade?

Answers to practice questions

Previous Early Bird Episodes

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

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. Webinar Consultant 25 Feb 2008

      Hi David, If your interested, I could offer the webinar service for free, as a trial run.  This could give you an opportunity to test with your audience.

      The technology we’re using is powered by Live Meeting 2007, and Epop Web Conferencing.  Both web conferencing platforms have features such as share Word & PowerPoint files, desktop & application share, web-browse, video & audio media.  Also, you could record sessions, for anyone that might have missed.

      All the best,

      David Corcoran
      Webinar Consultant
      http://www.batipi.com

    2. Dipranil 03 Mar 2008

      Hi David,

      Please let me know where can I find the ipod version of this video tutorial (Early Bird week#7).

      Thanks,
      Dipranil

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