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# Enter our weekly multiple choice Trivia Contest and Win!!!! (Fixed Income Basics)

#### Nicole Seaman

##### Director of FRM Operations
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.) The price of a two-year zero-coupon government bond is$96.68. The price of a similar three-year bond is $93.13. If rates are given with an annual compound frequency, which is nearest to the one-year implied forward rate from year two to year three? a .2.75% b. 3.81% c. 4.60% d. 5.35% Assume the following spot rate curve (rate are given with annual compound frequency): Which is nearest to the yield-to-maturity (YTM) of a three-year bond that pays an annual coupon with a coupon rate of 4.0%? a. 1.50% b. 1.83% c. 2.94% d. 3.76% Assume the following spot rate curve (rates given with annual compound frequency): Consider two bonds : • Bond A is a three-year bond that pays an annual coupon with coupon rate of 2.0% • Bond B is a three-year bond that pays an annual coupon with coupon rate of 6.0% Over the next year, as both bonds decrease in maturity and become two-year bonds, what happens to their prices? a. Both prices decrease b. Bond A increases in price, while Bond B decreases in price c. Bond A decreases in price, while Bond B increases in price d. Both prices increase Assume the following spot rate curve (rates given with annual compound frequency): Each of the following is true about this downward-sloping spot rate curve EXCEPT which is false? a. A three-year bond the pays an annual coupon of 2.0% has a price of about$102.31
b. A three-year bond the pays an annual coupon of 2.0% has yield (YTM) of about 1.210%
c. As the yield (YTM) is an average of spot rates, the yield is not variant to (does not depend on) the bond's coupon rate
d. If a three-year bond prices to par, over the next year (as its maturity decreases to two years), if the spot rate curve is static, the bond's price will decrease

Consider the following a three-year bond that pays a 10.0% annual coupon while the spot rate curve is upward sloping:

The bond's price (which is the sum of its discounted cash flows) is $122.94. Which is nearest to the bond's Macaulay duration? a. 1.84 years b. 2.76 years c. 3.00 years d. 3.15 years #### abhinav0131 ##### New Member Hi Nicole Please let me know if I am right. 1.b 2.c 3.b 4.c 5.b I have another question if you don't mind, how much time should ideally be enough to answer the above questions. I took like 20-25 mins. Best Abhinav #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi @abhinav0131 Is it okay if we wait until Friday to share the trivia answers (we don't want to "spoil" the fun ). If you really can't wait, here is the underlying XLS that I prepared (for backup), which contains the correct answers in ORANGE cells https://www.dropbox.com/s/hioi2gbuqwcdgrq/0421_trivia_fixed_income.xlsx Re: how much time should ideally be enough to answer the above questions? I wrote the above in response to @Ryan S request here on Friday for help with bond "relationship" concepts, but I realized it would be more helpful to set those up with prior, testable fundamentals. So the above, in my opinion, are both basic and, on average and approximately, somewhat "representative" of test difficulty; e.g., question 1 is an altered copy of a somewhat popular question "as is." However, these questions are deliberate such that three (or four depending on how you look at it) of the above can be eyeballed without calculations; as each one isolated on a basic concept. Therefore, if the P1 exam gives an average of 2.4 min per question (240/100), then I think a good target time for these five questions is 12 to 15 minutes (i.e., 2.4*5 = 12 as lower bound then add 3 minutes for upper bound due to slightly above average difficulty implied by questions 3 and 4 which are above average only because they "compound" multiple questions). I hope that helps, thanks! #### abhinav0131 ##### New Member Hi David I understand, thanks vm for the prompt response. Abhinav #### mshah6490 ##### New Member Subscriber 1. B 2. C 3. B 1. C 5. B #### Alex_1 ##### Active Member Hi, 1 - b 2 - c 3 - b 4 - c 5 - b #### Ryan S ##### Member Subscriber These definitely took me longer than 12-15, and as suspected, its because I'm not yet comfortable with some of the basic concepts. I want the "eyeball" intuition you speak of. Looking forward to more discussion Friday. I got the same answers as Abhinav. #### Nicole Seaman ##### Director of FRM Operations Staff member Subscriber @Ryan S, Just make sure that you go to our Facebook to enter your answers or list them on here. At the end of the week I have to go through each post individually to get the answers for users who posted on the forum. If you answer here, all you need to do is list them 1-5 like the others did above. Thanks! Nicole #### PaulHugan ##### New Member Hi. 1 - B 2 - C 3 - B 4 - D 5 - B #### edmondclchou ##### New Member Subscriber 1-B 2-C 3-B 4-D 5-B #### FRMCAND ##### Member 1. B 2. C 3. B 4. D 5. B thanks, #### shardasb ##### New Member 1.B 2.C 3.B 4.C 5.B thanks #### Nicole Seaman ##### Director of FRM Operations 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: @Ryan S and Kahlid Noor. 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!! Nicole #### Alex_1 ##### Active Member Hi all, I am slightly confused (had a pretty bad learning day with lots of calculation errors and stuff like this), is the correct answer for question 4 c) or d)? Thanks! #### Nicole Seaman ##### Director of FRM Operations Staff member Subscriber Here are the answers to last week's trivia. I apologize for the delay! 1. B. 3.81%; i.e.,$96.68/$93.13-1 = 3.81%. 2. C. 2.94%. Price = 4/1.01+4/1.02+104/1.03 =$102.98. With TI BA II+ calculator, solving for I/Y: 3 N, -102.98 PV, 4 PMT, 100 FV and CPT I/Y = 2.9475
3. B. Bond A increases in price, while Bond B decreases in price
4. C. False. Yield is variant to coupon rate; the yield is an average of spot rate but "weighted" by cash flows. In regard to (A), (B), and (C), each is TRUE.
5. B. 2.76 years

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
1. This is highly testable concept: to extract forward rates from prices or spot rates. The question happens to use annual compounding, but it's possible to be asked the same question in semi-annual or even continuous compounding. This question happens to give the prices as assumptions, but of course, you could alternatively be given the spot rates. In this case, the prices are based on an upward-sloping spot rate curve of 1.0% @ 1.0 year, 1.7% @ 2.0 year, and 2.4% @ 3 years. In this way spot rates are "interchangeable" with the prices of zero-coupon risk-free bonds.

2. Extremely common question. Why can the answer be eyeballed? Because yield is an average of spot weights, but averaged by the cash flows. The cash flows of a vanilla bond are weighted heavily toward the final cash flow (which includes the principal). Consider the extreme: if this were a zero-coupon bond, then the yield of the bond would be 3.0% (equal to the 3-year spot rate). Even if we increase the coupon to, say, 10%, the yield only goes down to 2.88%.

3. Bond A has a coupon (2%) that must be less than it's yield (as the curve only starts at 2%), so the 3-year price of Bond A must be less than par. As it's term shortens to two years, it's price increases per a pull to par. Similarly, Bond B has a coupon (6%) that must be greater than its yield (as the curve only goes up to 2.8%), so the 3-year price of Bond B must be greater than par. As it shortens, it also pulls to par, but that is a decrease for Bond B.

4. A harder question, as @Alex_1 shows, between (c) and (d). In regard to (C), again we can keep in mind that yield is a (weighted by cash flows) average of the spot rates; in this way, yield is function of both the spot rate curve and the instrument's particulars. This curve slopes down: 2% @ 1.0 year, 1.6% @ 2.0 years, and 1.2% @ 3.0 years. Here is how I think about yield, I start with the zero coupon bond. In this case, a 3-year zero coupon bond will have a yield equal to the 3-year spot rate or 1.2%. As the coupon increases, the two other cash flows adds weight, such that the yield slowly "creeps up" this curve, from 1.2% and toward (the higher) 1.6%, but it is very gradual because most of the weight will still be on the final cash flow.

In regard to (d), if the 3-year bond prices to par, what do we immediately know? The coupon rate equals the yield. So, if we get really comfortable with the concepts, we don't have to do any calculations to declare: the coupon rate is a little higher than 1.2% for a 3-year bond that prices to par. What happens if the bond shortens to 2 years yet the spot curve remains static? The coupon rate of >1.2% is going to be below the yield because the shorter spot rate curve is [1.6%, 2.0%] such that, the 2-year bond is going to price below par (i.e., price decreasing).

5. This is to illustrate, as simply as possible, that the Macaulay duration is just the weighted average maturity of the bond, where the weights are the cash flows as a percentage of price. So, Mac duration = (1.0 years * 8.05%) + (2.0 years * 7.88%) + (3.0 years * 84.07%) = 2.76 weighted average years is the Mac duration.
Any of these ideas can be further explored in the associated spreadsheet, each question has a tab (sorry the formatting isn't greatly organized as this is just a one-time use): https://www.dropbox.com/s/hioi2gbuqwcdgrq/0421_trivia_fixed_income.xlsx

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#### Tatase

##### New Member
Would there be a difference in the analysis regarding answer 4d if the bond was a floating rate?