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Weekly trivia 4/28/14 (Duration, DV01 and convexity)

Nicole Seaman

Chief Admin Officer
Staff member
Subscriber
Head on over to our Facebook page to enter our Trivia Contest! You will be entered to win a $15 gift card of your choice from Starbucks, Amazon or iTunes (iTunes is US only)!

If you do not have Facebook, you can enter right here in our forum. Just answer the following questions:


(Note: All participants will be entered into our random drawing regardless of correct or incorrect answers. There will be two winners drawn at random.)

Question 1
The prices of five bonds, with maturities from six months to 2.5 years, that all pay 4.0% semi-annual coupons, allow us to infer the spot rate curve:



A 2.5-year 2.0% semi-annual coupon bond has a price of $98.18. Assume the spot rate curve is static (unchanged). As the bond approaches maturity, when does its price decrease?

a. From 2.5 to 2.0 years
b. From 2.0 to 1.5 years
c. From 1.5 to 1.0 years
d. From 1.0 to 0.5 years

Question 2
Consider a 5.0% semi-annual coupon bond with a price of $98.14 based on a yield (YTM) of 6.0%:



What happens to the bond's duration if the yield decreases to 2.0%?

a. Decreases
b. Unchanged
c. Increases
d. Unclear

Question 3

Consider a 6.0% semi-annual coupon bond with a price of $94.62 based on a yield (YTM) of 6.0%:



Which is nearest to the bond's DV01?

a. $0.0173
b. $0.0181
c. $0.0240
d. $0.0329

Question 4

Consider a zero-coupon bond with a yield of 5.0% at various maturities?


At which displayed maturity is the bond's DV01 the highest? (bonus: under continuous compounding, find the formula for maturity with peak DV01.)

a. One (1) year
b. Ten (10) years
c. Twenty (20) years
d. Forty (40) years

Question 5

Consider a 5.0% annual coupon bond with a price of $84.44 based on a yield (YTM) of 9.0%:


Which is nearest to the bond's convexity?

a. 4.5 years^2
b. 17.3 years^2
c. 21.9 years^2
d. 26.0 years^2
 

Thierry S

New Member
1.C
2.C
3.A
4.C
5.C

Bonus 4: T_DV01_max = 1/delta_y * ln [(y + delta_y) / y]
where y is the yield 5% and delta_y the increment 1bp
 

Nicole Seaman

Chief Admin Officer
Staff member
Subscriber
Congratulations to our winners for this week's trivia contest! Winners have the choice of receiving a gift card from Amazon, iTunes (iTunes) and Starbucks.

Our winners are: @shardasb and @Alex_1!! Please email me at [email protected] or post here on the forum to let us know if you would like to claim your prize now or if you would like it to accrue.

Thank you to everyone who participated this week!! The answers are below :)

Nicole
Answers to this week's trivia

  1. C. From 1.5 to 1.0 years
  2. C. Increases
  3. A. $0.0173
  4. C. Twenty (20) years
  5. C. 21.9 years^2

    (David will expand on these over the weekend)
 

Nicole Seaman

Chief Admin Officer
Staff member
Subscriber
@shardasb,

Just let me know if you want me to use the email address that we have on file, and I will get that sent to you :)

Thanks!

Nicole
 

Alex_1

Active Member
Congratulations to our winners for this week's trivia contest! Winners have the choice of receiving a gift card from Amazon, iTunes (iTunes) and Starbucks.

Our winners are: @shardasb and @Alex_1!! Please email me at [email protected] or post here on the forum to let us know if you would like to claim your prize now or if you would like it to accrue.

Thank you to everyone who participated this week!! The answers are below :)

Nicole
Answers to this week's trivia

  1. C. From 1.5 to 1.0 years
  2. C. Increases
  3. A. $0.0173
  4. C. Twenty (20) years
  5. C. 21.9 years^2

    (David will expand on these over the weekend)

Hi @David Harper CFA FRM CIPM , this will probably annoy you, but would you have the more detailed answers for the trivia questions? Sorry for the reminder and many thanks in advance! :)
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @Alex_1 You could *never* annoy me :) Thanks for the reminder! Here are my comments:

Question 1:
This illustrates a relationship between maturity and bond price. The FRM AIM is "Assess the impact of maturity on the price of a bond..." and Tuckman's related example starts with the question "If the term structure of rates remains completely unchanged over a six-month period, will the price of a bond or the present value of the fixed side of a swap increase or decrease over the period?"

Given the assumption of an UNCHANGED term structure, the price pattern of the bond is:
  • $98.178 @ 2.5 years (shown in question)
  • $98.912 @ 2.0 years
  • $100.062 @ 1.5 years
  • $99.733 @ 1.0 year, and
  • $99.911 @ 0.5 years
The only price decrease is from a maturity of 1.5 years to 1.0 year. The reason is that the bond's coupon rate is 2.0% and the only forward rate less than this coupon rate is the F(1.0, 1.5) of 1.323%.; the other forward rates are greater than the 2.0% coupon rate. Tuckman explain this in the 3rd edition but here is his (simpler!) explanation from the 2nd edition: "More generally, price increases with maturity whenever the coupon rate exceeds the forward rate over the period of maturity extension. Price decreases as maturity increases whenever the coupon rate is less than the relevant forward rate."

Question 2:
Here is what happens to duration as the yield decreases, in this case from 6.00% (left) to 2.00% (right). Notice that the weight of the final, largest cash flow increases slightly, from 92.79% to 93.05%:



Here is Tuckman: "As it turns out, increasing yield also lowers duration. The intuition behind this fact is that increasing yield lowers the present value of all payments but lowers the present value of the longer payments most. This implies that the value of the longer payments falls relative to the value of the whole bond. But since the duration of these longer payments is greatest, lowering their corresponding weights in the duration equation must lower the duration of the whole bond. Conversely, decreasing yield increases DV01 and duration"

Question 3:
This question just combines to ideas: the relationship between Mac and modified duration, and the relationship between DV01 and dollar duration. The Macaulay duration is 1.912 years, as it is the weighted average maturity of the bond. We need to substitute the modified duration = Mac duration/(1+yield/k) into a key relationship: DV01 = dollar duration/10,000 = (mod duration*price)/10,000. In this case, then, DV01 = (1.912/1.03)*$94.62/10,000 = $0.01756; without rounding the bond price and Mac duration, the exact DV01 is $0.01731

Question 4:
This is sort of a trick question, because it requires only that we realize DV01 = dollar duration/10,000; i.e., they differ only by their units: DV01 is "useful" because duration is a linear approximation in the first place, and dollar duration assumes a huge (one unit or 100%) change in yield, so it's totally "unusable" by itself. DV01 is "useful" because the linear approximation is only good for very small changes in yield. In any case, given their direct relationship, DV01 peaks at the same maturity as dollar duration, which is 20 years.

In regard to the bonus, all we need to do is take the derivative of the price, P = F*exp(-rT), with respect to maturity, T, and set it equal to zero, to identify the local maximum take the derivative of DV01 (or dollar duration) with respect to maturity and set this function equal to zero, on order to find the local maximum. Here is the answer but don't peek until you've tried! :).

Question 5:
The point of this is really just to remind, conceptually, that convexity is denoted in units of years^2 and that convexity will be in the neighborhood of the bond's maturity squared. Exam-wise, it's possible that's all you need to know: a 5 year bond's convexity will be somewhere in the rough neighborhood of 5^2 = 25 years^2 (true, the modified convexity is variant to coupon, yield and compound frequency, but in most scenarios you still end up near T^2 or less, so realistically, a 5-year bond will tend to have a convexity of something a little less than 5^2).

The above is a Macaulay convexity (analogous to a Macaulay duration) such that it needs a similar adjustment to retrieve the sensitivity-type metric which we call convexity, but could be called modified convexity = 26.04/(1+9%)^2 = 21.918 years^2. If this were instead a zero-coupon 5-year bond, the (modified) convexity, under annual discounting, would be 5*6/1.09^2 = 25.25.
 
Last edited:

mshah6490

New Member
Subscriber
@David, in bonus part of Q4, if we take the derivative of the price, P w.r.t. T, won't that give us the time, T when we get the maximum price. In my opinion, we would probably have to take the derivative of $Dur with respect to time to get the local maximum. Please correct me if I am wrong.
Thanks
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @mshah6490 Yes, that was sloppy of me :oops: You are totally correct, it is the derivative with respect to time (not maturity)! We are looking for the maximum DV01, so we want the d[DV01]/dT. But your expression is even better perfect (because DV01 is a just a scaled dollar duration, by constant 1/10,000, and your statement is exactly true: "We would probably have to take the derivative of $Dur [i.e., dollar duration] with respect to time to get the local maximum. The answer is worked out here, but try it yourself before peeking! https://www.bionicturtle.com/forum/threads/frm-fun-1.5950/
 
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