View Full Version : Effective Duration and Convexity

CaffeineJunky

04-26-2010, 03:19 PM

I just finished Hoffman exam #1, and there is a question on there (#11) that has be a bit confused about the proper use of the effective duration formula. The formula for effective duration is [P(down) - P(up)]/[2P* delta y]. In practice, I used the bond function on the calculator to move the yield up and down by .1 to get the P(up) and P(down). So in the case above my delta y would be .001. Is this correct? The yield on the calculator is expressed as a semiannual (or annual) rate, is the delta y expressed as a semiannual rate as well, or should this be based on continuous compounding?

EG: annual coupons, 7% continuous yield, and I want to use .001 as delta y. In the calculator, the yield is converted to annual compounding ( e^(.07) = 1.0725), so enter a yield of 7.25 in the calculator. Move the yield up to 7.35 to get P(up) and move the yield down to 7.15 to get P(down). Or should I have used e^(.071) = 1.0736 (7.36%) and e^(.069) = 1.0714 (7.14%) to get P(up) and P(down)?

Hope some of this makes sense.

triplea

04-26-2010, 03:25 PM

Your delta would be .002 (+.001 - (-.001))

triplea

04-26-2010, 03:33 PM

And if that does not solve the problem, I think you have to add .001 to the continuous yield, and then convert to annual compounding.

CaffeineJunky

04-26-2010, 03:34 PM

Your delta would be .002 (+.001 - (-.001))

I did multiply by 2, I think the issue is whether the yield used in the bond calc (semi or annual) is consitent with the yield used in the delta y component (continuous?).

CaffeineJunky

04-26-2010, 03:35 PM

And if that does not solve the problem, I think you have to add .001 to the continuous yield, and then convert to annual compounding.

I think that is the issue. So is that always the case that the delta y is expresses as a continuous rate?

triplea

04-26-2010, 03:38 PM

Wait a minute, isn't #11 asking for duration, not effective duration?

triplea

04-26-2010, 03:45 PM

Effective duration can be used when the bond has embedded options. If it doesn't, I would stick with the traditional duration formula. I would think we would use effective duration in a Fabozzi context (embedded options with a binomial interest rate lattice).

CaffeineJunky

04-26-2010, 04:01 PM

Wait a minute, isn't #11 asking for duration, not effective duration?

Shouldn't we get the same answer (for small changes in y)? Eff duration is easier to calculate quickly in exam situations (or so I thought).

CaffeineJunky

04-26-2010, 04:09 PM

On a side note, what did you think of the 1st Hoffman exam? The only ones that stumped me were 14, 17 (forgot the P&G details), 20 (forgot to add .0035 to each node, just assumed the bond would be called in one year :duh:), 23 (is this exam 6?), 25 (par yield?), 28 was messy, 32 (not even going there), and 35 (thrown off by the "create them synthetically"). You?

triplea

04-26-2010, 04:10 PM

I would agree - it should be close. I (just) got 2.8307 using the approximation.

triplea

04-26-2010, 04:11 PM

On a side note, what did you think of the 1st Hoffman exam? The only ones that stumped me were 14, 17 (forgot the P&G details), 20 (forgot to add .0035 to each node, just assumed the bond would be called in one year :duh:), 23 (is this exam 6?), 25 (par yield?), 28 was messy, 32 (not even going there), and 35 (thrown off by the "create them synthetically"). You?

I plan on doing it tonight (I am way behind - I took the CAS 2003 today which has been slowing me down).

triplea

04-26-2010, 04:12 PM

I would agree - it should be close. I (just) got 2.8307 using the approximation.

(97.00663 - 96.459) / (.002 * 96.73)

I added 0.1% and subtracted 0.1% from 7% and calculated the price using continuous compounding.

CaffeineJunky

04-26-2010, 04:43 PM

Duration works but convexity is way off. He gets 8.26, and the effective convexity approximation is around 20. I must be doing something wrong.

Effective convexity is [P(up) + P(down) -2P]/[P * (delta y)^2] right? Because if the denominator was [2*P * (delta y)^2] I would be a lot closer.

:exams::exams::exams:!!!

triplea

04-27-2010, 10:09 AM

On a side note, what did you think of the 1st Hoffman exam? The only ones that stumped me were 14, 17 (forgot the P&G details), 20 (forgot to add .0035 to each node, just assumed the bond would be called in one year :duh:), 23 (is this exam 6?), 25 (par yield?), 28 was messy, 32 (not even going there), and 35 (thrown off by the "create them synthetically"). You?

#14 - my answer was "?"

#17 - definitely have to remember these

My answer to #20 - No :)

#23 - :shake:

#25 - definitely par yield is something that could catch candidates off guard, so look out for that one

#35 = :swear: definitely my weak point

I hope #30 shows up instead of memorizing all the different strategies

#32 - how to memorize the differential equation:

Dr S and T and half of Gooss equal rf

"delta" * rS + "theta" + 1/2*"gamma"*sigma^2S^2 = rf

:shrug:

triplea

04-27-2010, 10:10 AM

#36 is also a good question to ask on VaR. Definitely on the syllabus and easily within reach.

CaffeineJunky

04-27-2010, 11:04 AM

#32 - how to memorize the differential equation:

Dr S and T and half of Gooss equal rf

"delta" * rS + "theta" + 1/2*"gamma"*sigma^2S^2 = rf

:shrug:

Thanks for that one, I'm using it!

triplea

04-27-2010, 12:50 PM

6 assumptions of CAPM:

WHAT, ME?

Wealth

Holding period

Assets

Transaction costs & taxes

Markowitz

Expectations are homogenous

dumples

04-27-2010, 02:18 PM

Duration works but convexity is way off. He gets 8.26, and the effective convexity approximation is around 20. I must be doing something wrong.

Effective convexity is [P(up) + P(down) -2P]/[P * (delta y)^2] right? Because if the denominator was [2*P * (delta y)^2] I would be a lot closer.

:exams::exams::exams:!!!

Fabozzi's effective convexity is 1/2 the convexity of the formulas in Hull and BKM. His formula for dP/P doesn't have the 1/2 in the convexity term (because it's already built into the number)

Goldfarb talks about this on page 301 of his manual, but the moral of the story is that if you want to approximate convexity for use in the formulas in Hull and BKM you should say convexity = [P(up) + P(down) -2P]/[P * (delta y)^2]

triplea

04-27-2010, 08:33 PM

4 types of informational biases

SCOF

Sample size neglect

Conservatism

Overconfidence

Forgot :(

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