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# P1.T3.505. Bond interest payments and zero-coupon bonds

#### Nicole Seaman

Staff member
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Learning objectives: Describe the main types of interest payment classifications. Describe zero-coupon bonds and explain the relationship between original-issue discount and reinvestment risk.

Questions:

505.1. A US corporate bond that matures on October 1st, 2017 with a par value of $100.00 pays a semi-annual coupon with a coupon rate of 9.0% per annum. It pays coupons on April and October 1st and it offers a yield to maturity (yield) of 4.0% per annum. If it settles on September 1st 2015, which is nearest to the bond's flat (aka, quoted or clean) price? a.$109. 89
b. $111.78 c.$113.64
d. $115.53 505.2. Three months ago, a US corporation issued a floating-rate note (FRN) that pays its first coupon in three months and matures in five years. The index (aka, reference rate; eg, six-month LIBOR) was 1.20% at the time of issuance but has dropped to its current level of 0.50%. The index is quoted per annum with semiannual compounding. The quoted margin on the note is 200 basis points, such that the first coupon pays 3.20% = 1.20% reference + 2.00% margin. Assume three months equals 0.25 years and assume the quoted margin equals the required margin; i.e., the margin is appropriate compensation for credit risk. Which is nearest to the note's current value? a.$97.35
b. $99.13 c.$100.00
d. $100.97 505.3. Six months ago Brian Smith purchased a zero-coupon bond with a face value of$100.00 and a remaining term to maturity of seven (7.0) years. When he purchased the bond, the yield curve was flat at 3.0% per annum with semi-annual compounding. While today the yield curve remains flat, it has shifted up by 40 basis points. If Brian sells the bond today, what is his per annum return with semi-annual compounding and approximately how much of the return is due to reinvestment risk?

a. -6.70% with about 30% due reinvestment risk
b. -4.54% with about 50% due reinvestment risk
c. -2.13% with no reinvestment risk
d. +0.40% with no reinvestment risk

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#### Kam

##### New Member
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Hi, what chapter would I find this topic and formulae in the study notes?

#### David Harper CFA FRM

##### David Harper CFA FRM
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Hi @Kam The questions, which I am currently writing, are associated with Fabozzi's Chapter 12 Corporate Bonds (Reading 23); you can see I am using those AIMs. However, because we've previously already written practice questions for this reading (quite detailed), this is a "second pass" and further the actual reading is a bit stale. So what I do is I try to triangulate on other topics/readings such that much of this borrows from Tuckman Chapters in Topic 4. It actually takes more time but I think it's more realistic (and helpful). Fabozzi's Corporate bonds chapter doesn't have any formulas, actually. But I don't think it's useful to query fixed versus floating rate coupons superficially (the FRM will assume you know the superficial difference); so instead I push a little deeper with mechanical problems. I hope that explains, thanks!

#### Yuchao

##### New Member
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Hi,

Question 505.1 is bit misleading without stating the settlement year.

#### Dr. Jayanthi Sankaran

##### Well-Known Member
Hi @Yuchao and @Deepak Chitnis,

You are right in that David has forgotten to give the coupon years. However, I did go to his Answer in the student forum and found that the coupons of $4.5 were paid on April 1st 2015 and October 1st 2015. Using the Bond worksheet in the TI BA II Plus Financial calculator, and plugging in the following values: Settlement date = 9-01-2015, coupon = 9, Redemption date = 10-01-2017, Redemption value = 100, Day count convention = 360*, semiannul compounding = 2/Y, Yield = 4, I get the price of$109.89.

Note: The day-count convention for corporate bonds is 30/360

An alternative method using the TVM function is as follows:

N = 5, I/Y = 2, PMT = 4.5, FV = 100, PV = 111.78
The dirty price of the bond on April 1st, 2015 = $111.78 In order to compute the dirty price of the bond on September 1st, 2015 (the settlement date): we assume reinvestment of the dirty price of the bond on April 1st 2015 such that: Dirty price of the bond on September 1st, 2015 =$111.78*(1 + 4%/2)^150/180 = $113.63 Coupon owed by buyer to the seller = (150/180)*$4.5 = $3.75 Hence quoted price of the bond on September 1st, 2015: Dirty price of the bond on September 1st, 2015 - coupon =$113.63 - $3.75 =$109.88

An important thing to check the veracity of the price of the bond is that it should be a premium bond $109.68 >$100 because Coupon rate 9% > YTM 4% Thanks!

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#### David Harper CFA FRM

##### David Harper CFA FRM
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Hi @Yuchao Good catch. My mistake, the question does omit the settlement year. I have corrected the source to read "If it settles on September 1st 2015, which is nearest to the bond's flat (aka, quoted or clean) price?"

@Jayanthi Sankaran the question already says "If it settles on September 1st 2015, which is nearest to the bond's flat (aka, quoted or clean) price?" such that I think the only missing information was the settlement year ... but totally thank you for showing the solution with the Bond worksheet. I am learning this from you .

#### Dr. Jayanthi Sankaran

##### Well-Known Member
Thanks a lot, David - appreciate it - glad to be of help Jayanthi

#### Taunk

##### Member
Hi, The extract of the answer 505.2 is pasted here:

Therefore, the current value equals $101.160*(1+0.250 /2)^-(0.25*2) =$100.97089.

$101.160*(1+0.250 % /2)^-(0.25*2) =$100.9708.... I believe the % sign is missing in the original answer after 0.25

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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Thank you @taunk , good catch!

#### Nicole Seaman

Staff member
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@David Harper CFA FRM CIPM thank you for letting me know. The member that you mentioned is not a paid member. I've sent them a message to let them know that in order to access the lower sections of the forum they need to be a paid member. Thanks!

#### wooju7533

Subscriber
Hi
Q.505.1
The dirty price of the bond on April 1st, 2015 = $111.78 In order to compute the dirty price of the bond on September 1st, 2015 (the settlement date): we assume reinvestment of the dirty price of the bond on April 1st 2015 such that: Dirty price of the bond on September 1st, 2015 =$111.78*(1 + 4%/2)^150/180 = $113.63 Coupon owed by buyer to the seller = (150/180)*$4.5 = $3.75 why do you use 150/180 to calculate? Difference between April 1st 2015 and September 1st 2015 is 5 month(=150/360). #### Sixcarbs ##### Active Member Subscriber Hi Q.505.1 The dirty price of the bond on April 1st, 2015 =$111.78
In order to compute the dirty price of the bond on September 1st, 2015 (the settlement date):
we assume reinvestment of the dirty price of the bond on April 1st 2015 such that:
Dirty price of the bond on September 1st, 2015 = $111.78*(1 + 4%/2)^150/180 =$113.63
Coupon owed by buyer to the seller = (150/180)*$4.5 =$3.75

why do you use 150/180 to calculate? Difference between April 1st 2015 and September 1st 2015 is 5 month(=150/360).
I think the answer is because the coupon is paid semi-annually.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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HI @wooju7533 In 505.1, there are 180 days between coupons (ie, between 4/1 and 10/1), the coupon rate is 9.0% per annum payable semi-annually (this is typical! So that's 4.5% or $4.50 on 4/1 and then another$4.50 on 10/1). On the settlement date of 9/1, we need to subtract the accrued interest (AI) that has accrued since 4/1, so under the corporate 30/360 day count, that is 5 months * 30 days/month = 150 days since the last coupon. So the AI (to be subtracted from the full/dirty price to retrieve the flat/clean price) is 150/180 * ($100*9.0%/2) =$3.75, or if we can use 150/360 * ($100*9.0%) =$3.75. I hope that's helpful,

#### wooju7533

##### New Member
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Hi
I'm confused with question 505.2

( Three months ago, a US corporation issued a floating-rate note (FRN) that pays its first coupon in three months and matures in five years. The index (aka, reference rate; eg, six-month LIBOR) was 1.20% at the time of issuance but has dropped to its current level of 0.50%. The index is quoted per annum with semiannual compounding. The quoted margin on the note is 200 basis points, such that the first coupon pays 3.20% = 1.20% reference + 2.00% margin. Assume three months equals 0.25 years and assume the quoted margin equals the required margin; i.e., the margin is appropriate compensation for credit risk. Which is nearest to the note's current value?)

i need to solve note's current value and i thought current value is on its first coupon date.
But when i saw answer current value=101.60*(1+0.250/2)^(-0.25*2)=100.97089
Isn't this value three month ago? i tried to read answer several times but i don't understand why i have to multiply this (1+0.250/2)^(-0.25*2)

thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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Hi @wooju7533 In 505.2., you are correct about the 101.60 but that is the value three months (+0.25 years) in the future; it is a future value (FV), not a present value (PV). Please note the assumption (emphasis mine): "Three months ago, a US corporation issued a floating-rate note (FRN) that pays its first coupon in three months ...." Further, semi-annual coupons can be expected (unless otherwise specified) and consistent with the compound frequency: "The index is quoted per annum with semiannual compounding." Given 101.60 is the value in +0.25 years, we need to discount it to today by multiplying by (1+0.250/2)^(-0.25*2).

As @Mkaim wrote in the solution thread at https://www.bionicturtle.com/forum/...and-zero-coupon-bonds-fabozzi.8837/post-38650
"So I just took a look over this again and think David's solution of 100.97 is correct. Here are the mechanics. The 101.16 seems to be a typo (101.6 = correct future value).
• It's a floating rate bond that's only par on coupon dates. We are valuing it off coupon date.
• Next payment was already determined three months ago which is how floaters work. The payment is 3 month out and will be (1.2% Index + 2% Margin = 3.2%). Considering semiannual compounding, it will be (.032 * .5) * 100 = 1.6 ==> Plus par it will be 101.6.
• Now we just need to discount that payment back three months based on the prevailing index rate + Margin adjusted for day count ==> 101.6 / (1.025^.25) = 100.97.
• Note, the credit spread remained the same at 2% (200 bps)."

#### cpsamdavid

##### New Member
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For the question 505.2, Can you please explain why we're discounting for 2 periods ^(0.25 x 2).

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @cpsamdavid In 505.2, we're discounting by one-half a period (i.e., 0.5 periods) where each period is six months (aka, semiannual frequency) because we're discounting 3 months (0.25 years) and there are 2 periods per year.

The general discrete form is PV = FV/(1+r/k)^(T*k) where T is the number of years and k is the number of periods per year. In the case of semi-annual compounding, k = 2, we discount to PV with PV = FV/(1+r/2)^(T*2). To better understand this, which can be confusing, consider the following sequence:
• If we are discounting the FV $101.60 for one year at 2.50% per annum, then PV =$101.60/(1+0.0250/2)^(1.0*2) = $101.60/(1+0.0250/2)^2.0 =$99.10684
• If we are discounting the FV $101.60 for six months (i.e., 0.5 years) at 2.50% per annum, then PV =$101.60/(1+0.0250/2)^(0.5*2) = $101.60/(1+0.0250/2)^1.0 =$100.34568
• If we are discounting the FV $101.60 for three months (i.e., 0.25 years) at 2.50% per annum, then PV =$101.60/(1+0.0250/2)^(0.25*2) = $101.60/(1+0.0250/2)^0.5 =$100.97089. I hope that's helpful,

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