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Financial Mathematics Old FM Forum

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  #1  
Old 10-09-2018, 12:29 AM
Handynasty Handynasty is offline
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Default Spot Rate converted for semiannual payments

Suppose spot rate for payments due in 1 year is 3%, why is the semiannual return 1.5% instead of (1.03)^(1/2) - 1.

Does that mean that spot rates are nominal?
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Old 10-09-2018, 12:36 AM
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Gandalf Gandalf is offline
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What is your source for saying it is 1.5%? One possibility is that 1.5% is just an approximation.
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Old 10-09-2018, 10:56 PM
Handynasty Handynasty is offline
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Hi Gandalf,

It's from section 6.6 of the Financial Mathematics for Actuaries book by Chan, Wai-Sum, and Tse, Yiu-Kuen.

It's not approximation because a formula is given for semiannual coupon bond pricing using spot rates. And it seems to be the convention that spot rate is divided by 2 to get the semiannual spot rate.

An example it gives is spot rate for payment due in three years is 4%, so coupon at 2.5 year is discounted by 1/(1.02^5).

I'm wondering if that's always the convention. And on the exam, would the question specify the way to use spot rates for semiannual payments.
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Old 10-09-2018, 11:15 PM
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The convention in financial markets is to quote bond equivalent yield, i.e. nominal semi-annual rate. Dividing the annual rate by 2 matches up with that.
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Old 10-09-2018, 11:49 PM
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Quote:
Originally Posted by Handynasty View Post
Hi Gandalf,

It's from section 6.6 of the Financial Mathematics for Actuaries book by Chan, Wai-Sum, and Tse, Yiu-Kuen.

It's not approximation because a formula is given for semiannual coupon bond pricing using spot rates. And it seems to be the convention that spot rate is divided by 2 to get the semiannual spot rate.

An example it gives is spot rate for payment due in three years is 4%, so coupon at 2.5 year is discounted by 1/(1.02^5).

I'm wondering if that's always the convention. And on the exam, would the question specify the way to use spot rates for semiannual payments.
I’m very surprised the book says that. What about the coupon due in 3 years? Is it discounted by 1/(1.02^6)?

To my way of thinking, the entire premise of spot rates is that you can value a payment due in k years using the price of a zero-coupon bond maturing in k years. To value a payment due in k-.5 years, you would look at the price of a zero-coupon bond maturing in k-.5 years, not one maturing in k years. (If the only information available was the price of one maturing in k years, you might base the value on that, but it would be an estimate / approximation.)

As to what you would see on the exam, I don´t know. I would hope they wouldn’t throw such a curve ball at you. If they do, hope that the answers aren’t so close together that it matters whether you use 1.02^5 or 1.04^2.5. Good luck.
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Old 10-10-2018, 12:01 AM
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If you have semi-annual payments, there would generally be a separate spot rate for each half year quoted on a nominal annual basis. The payment six months from now would have a discount factor (1 + s1/2) while the payment one year from now would be discounted at (1+ s2/2)^2. .
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Old 10-10-2018, 12:16 AM
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If you have semi-annual payments, there would generally be a separate spot rate for each half year quoted on a nominal annual basis. The payment six months from now would have a discount factor (1 + s1/2) while the payment one year from now would be discounted at (1+ s2/2)^2. .
Yes, that is in effect what I was trying to say. You should (IMO) only be calculating a value where you know the spot rate for that specific period of time. Whether the rate for the 2.5 year payment is expressed as a nominal semi-annual rate or an effective annual rate, I don’t know, but you shouldn’t have to base it on the rate for a payment after 3 years.

Looking at the FM sample questions, specifically 33 and 34, you are given annual spot rates and you treat them as annual effective rates, not annual rates convertible semiannually.
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Old 10-12-2018, 01:05 AM
Handynasty Handynasty is offline
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Thank you Gandolf and Academic Actuary!
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