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

#### Nicole Seaman

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 #### shardasb ##### New Member Hi, I find duration,DV01, Convexity etc little difficult concepts. My answers are: 1.b 2.c 3.a 4.c 5.c #### 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 #### Alex_1 ##### Active Member 1 c 2 b 3 a 4 c 5 c #### 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)

#### Alex_1

##### Active Member
Thanks a lot! I guess I'll let it accrue. Looking forward to the detailed answers! Best regards.

#### shardasb

##### New Member
Thanks a lot! I think,I'll claim Amazon15 gift card.

#### Nicole Seaman

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

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

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:
Thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @mshah6490 Yes, that was sloppy of me 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/

Staff member
Subscriber
@shardasb,