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June 01, 2023, 04:37:40 pm

Author Topic: Coupon Bonds with Changing Interest Rate  (Read 8833 times)  Share 

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Maths Forever

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Coupon Bonds with Changing Interest Rate
« on: October 05, 2015, 06:16:42 pm »
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Does anyone know how to break down this question?

You want to purchase a 10-year coupon bond with semi-annual coupons and an annual coupon rate of 6%. The face value of the bond is $10,000. You observe a yield curve with spot rates that increase 25 basis points for every six month increase in the term of a loan. These rates are nominal, annual quotes. Given that the present value of a $2,000 face value zero-coupon 6-month bond is $1,941.75, what is the price of the 10-year bond? Assume the risk on both bonds is the same, and interest is compounded on a semi-annual basis. Provide calculations out to four decimal places for partial credit.

Thanks in advance!  :)
Currently studying at the University of Melbourne.

pi

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Re: Coupon Bonds with Changing Interest Rate
« Reply #1 on: October 05, 2015, 08:32:59 pm »
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Despite my doing a different degree at a different uni, this still sounds like a Finance 1 assignment question. We, ATAR Notes, endorse students doing their own work if it's for assessment.

Just a heads up to anyone replying, keep this in mind and post tips only! :)

stolenclay

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Re: Coupon Bonds with Changing Interest Rate
« Reply #2 on: October 06, 2015, 01:24:39 am »
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In case anyone got scared, I would be very very surprised if Finance 1 at UoM got close to this hard.

The prices of the zero-coupon bond and the bond we are pricing is the sum of the respective constituent cash flows discounted at the appropriate rate. The appropriate discount factors for a given point in time will be the same for both securities, since both bonds have the same risk, so we can work out the implied discount factors by inspecting the zero-coupon bond, which has been priced for us already.

You'll want to use the fact that
the present value of a $2,000 face value zero-coupon 6-month bond is $1,941.75
and that you
observe a yield curve with spot rates that increase 25 basis points for every six month increase in the term of a loan
to set up an equation and solve for the spot rates corresponding to a term of 6 months and every 6 months after that (until the maturity of the bond we are pricing, which is 10 years in this case).

After that it's only a matter of discounting the cash flows of the bond that we are pricing at the calculated rates. Note that the spot rates given by the yield curve are nominal annual rates, so they will require quick conversion into effective rates before you use them to discount anything.
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Maths Forever

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Re: Coupon Bonds with Changing Interest Rate
« Reply #3 on: October 06, 2015, 07:40:11 pm »
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Thanks very much stolenclay!

This question is actually from Finance 1. I thought it seemed a bit advanced too at first!

I think the problem was that I was trying to derive a formula for the present value of all the coupon payments and the bond, where it is best to use an excel spread sheet.

I found that r=0.060 = 6.0% p.a., since it is a 6 month security (the zero coupon bond), and F = P (1 + r/2).

Hence, for the first 6 months, the return on the coupon bond would be 6.0% /2 = 3.0%, with the same risk.

Following this logic, does it make sense that for the second 6 months the return would be (6.0% + 0.25%)/2 = 3.125%, third 6 months it would be 3.25% and so on, when using these values to discount from the face value of the bond and each of the coupons?

Thanks again.  :)
Currently studying at the University of Melbourne.

stolenclay

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Re: Coupon Bonds with Changing Interest Rate
« Reply #4 on: October 08, 2015, 01:11:11 am »
+1
This question is actually from Finance 1.
Oh dearie me. I seriously didn't think they threw you around this badly in Finance 1... Guess the moans of impossible Finance 1 exams are true then.

I think the problem was that I was trying to derive a formula for the present value of all the coupon payments and the bond, where it is best to use an excel spread sheet.
Yep, definitely use an Excel spreadsheet (or some other computational tool) for this one. If the term of the bond was a lot shorter (2 or 3 years) you could be expected to do it with a scientific calculator I suppose. But good to hear you didn't actually have a problem in the first place!

Following this logic, does it make sense that for the second 6 months the return would be (6.0% + 0.25%)/2 = 3.125%, third 6 months it would be 3.25% and so on, when using these values to discount from the face value of the bond and each of the coupons?
Yep, that looks good to me, i.e. the price is
Spoiler
where is the discount factor for a payment occurring at the end of the th half-year.
« Last Edit: October 08, 2015, 01:12:42 am by stolenclay »
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Maths Forever

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Re: Coupon Bonds with Changing Interest Rate
« Reply #5 on: October 08, 2015, 08:21:10 pm »
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Thanks Stolenclay,

I didn't use that exact formula you've listed. But I found that P (bond) = $4,404.4889, P (all coupons) = $4,176.6139, and so Price = $4,176.6139 + $4,404.4889 = $8,581.1029. Does that seem to match with your formula?

Thanks so much for your reply, and all your help!  :)
« Last Edit: October 08, 2015, 08:23:46 pm by Maths Forever »
Currently studying at the University of Melbourne.

stolenclay

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Re: Coupon Bonds with Changing Interest Rate
« Reply #6 on: October 08, 2015, 11:28:23 pm »
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Price = $4,176.6139 + $4,404.4889 = $8,581.1029. Does that seem to match with your formula?
Yep, matches with what I got.

Thanks so much for your reply, and all your help!  :)
Haha, I didn't help all that much anyway, but no problem!
Thoughts on my journey through university
20142016 BCom (Actl), DipMathSc @ UoM
20172018 Master of Science (Mathematics and Statistics) @ UoM