1) Taking into consideration that BEY = 2* Semiannual effective yield, why then BEY DOES NOT EQUAL to 4 * quarterly effective yield?

2) Is YTM our “stated rate”? So to compute Effective annual return (EAR) we use this YTM.

EAR = [ (1 + stated annual yield / m)m ] - 1 where m - number of compounding periods THEN we can compute

Effective rate for period = [ (1 + EAR)1/m ] - 1 where m number of compounding periods. So using this formula we can obtain semiannual effective / quarterly effective yield etc Is this is other way to compute effective rate fo period without computing effective annual yield before?

3) We convert what yield (effective or stated annual (YTM)) using this formula?:

(1 + APR/m)m = (1+ APR / n)n

4) Why we need to use BEY rather than EAR (makes more sense for me)

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romainnnnn wrote:

1) Taking into consideration that BEY = 2* Semiannual effective yield, why then BEY DOES NOT EQUAL to 4 * quarterly effective yield?

Because to go from one effective yield (e.g., quarterly) to another effective yield (e.g., semiannual), you compound.

So, if the quarterly effective yield is 1%, then the semiannual effective yield is 2.01% (= 1.012 − 1), the EAY is 4.0604% (= 1.014 − 1), and the BEY is 4.02% (= 2.01% × 2).

romainnnnn wrote:
2) Is YTM our “stated rate”? So to compute Effective annual return (EAR) we use this YTM.

YTM means yield to maturity.  It can be quoted as an EAY, a BEY, a nominal, annual rate compounded monthly, or any other way you want to quote it.

In fixed income it is commonly quoted as BEY because bonds commonly pay coupons (at half the annual coupon rate) twice per year.

romainnnnn wrote:
EAR = [ (1 + stated annual yield / m)m ] - 1 where m - number of compounding periods THEN we can compute

Effective rate for period = [ (1 + EAR)1/m ] - 1 where m number of compounding periods. So using this formula we can obtain semiannual effective / quarterly effective yield etc Is this is other way to compute effective rate fo period without computing effective annual yield before?

Your formula starts with the effective annual rate, so somehow you must have computed it or must have been given it.

romainnnnn wrote:
3) We convert what yield (effective or stated annual (YTM)) using this formula?:

(1 + APR/m)m = (1+ APR / n)n

That will convert from a nominal rate with one compounding frequency (m times per year) to a nominal rate with a different compounding frequency (n times per year), for a given effective annual rate.

romainnnnn wrote:
4) Why we need to use BEY rather than EAR (makes more sense for me)

Because that’s the convention that the bond market has adopted.

And they adopted it because most bonds pay coupons semiannually.

Simplify the complicated side; don't complify the simplicated side.

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Many thanks! But I still have some questions, please clarify:

1) What is clear: semiannual-pay bond (periodicity of two) with an 8% YTM has a yield of 4% every six months and an effective yield of 1.042 − 1 = 8.16%. Question: suppose we want to calculate YTM for for a 5-year, semiannual pay 7% coupon bond. We get first “YTM/2” (=3.253%)  and then double it to get thy YTM expressed as an annual yield (=6.506%). In the notes it is “called” effective. WHY? It must be then (1+0.03253)2 ………

2) So when we calculate BEY as 2*semiannual, it is a kind of “proxy”. Normally it should be compounded as (1+x)2?

3)An Atlas Corporation bond is quoted with a YTM of 4% on a semiannual bond basis. What yields should be used to compare it with a quarterly-pay bond and an annual-pay bond?

So, the effective annual yield compounded quarterly should be 1,014 - 1 = 4.06%

Why the solution in Notes:  For the annual YTM on the quarterly-pay bond, we need to calculate the effective quarterly yield and multiply by four (??? we need to compound, even in your above answer from effective to effective)

The quarterly yield (yield per quarter) that is equivalent to a yield of 2% per six months is 1.021/2 − 1 = 0.995%. The quoted annual rate for the equivalent yield on a quarterly bond basis is 4 × 0.995 = 3.98%.

romainnnnn wrote:
@S2000magician

Many thanks!

You’re quite welcome.

romainnnnn wrote:
But I still have some questions, please clarify:

1) What is clear: semiannual-pay bond (periodicity of two) with an 8% YTM has a yield of 4% every six months and an effective yield of 1.042 − 1 = 8.16%. Question: suppose we want to calculate YTM for for a 5-year, semiannual pay 7% coupon bond. We get first “YTM/2” (=3.253%)  and then double it to get thy YTM expressed as an annual yield (=6.506%). In the notes it is “called” effective. WHY? It must be then (1+0.03253)2 ………

I’ve never seen this called an effective rate, and it shouldn’t be because it isn’t.  It’s a nominal rate compounded twice per year.

Where, exactly, did you see this called an effective rate?

romainnnnn wrote:
2) So when we calculate BEY as 2*semiannual, it is a kind of “proxy”. Normally it should be compounded as (1+x)2?

It’s not so much a proxy as a different convention for quoting rates.  It’s centimeters instead of inches.

romainnnnn wrote:
3)An Atlas Corporation bond is quoted with a YTM of 4% on a semiannual bond basis. What yields should be used to compare it with a quarterly-pay bond and an annual-pay bond?

You should always compare effective yields; that keeps it simple for you.  So, if the YTM is quoted as a BEY (which is what I presume that you mean when you say that it’s 4% on a semiannual basis), then you first compute the semiannual effective yield (= 4% ÷ 2 = 2%), then compute the effective quarterly yield (= 1.021/2 − 1 = 0.9950%) and the effective annual yield (= 1.022 − 1 = 4.04%) and compare those effective yields to the effective yields of some quarterly investment and some annual investment, respectively.

romainnnnn wrote:
So, the effective annual yield compounded quarterly should be 1,014 - 1 = 4.06%

That’s the effective annual yield for an investment that has a quarterly effective yield of 1%.  Such an investment might be quoted as having an annual (nominal) yield of 4%.

romainnnnn wrote:
Why the solution in Notes:  For the annual YTM on the quarterly-pay bond, we need to calculate the effective quarterly yield and multiply by four (??? we need to compound, even in your above answer from effective to effective)

Whose notes?  Can you give me a volume and page number for reference?  Without knowing exactly what they were asking, I cannot explain their answer fully.

romainnnnn wrote:
The quarterly yield (yield per quarter) that is equivalent to a yield of 2% per six months is 1.021/2 − 1 = 0.995%. The quoted annual rate for the equivalent yield on a quarterly bond basis is 4 × 0.995 = 3.98%.

That’s the quoted annual nominal rate.

Simplify the complicated side; don't complify the simplicated side.

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Sorry for my late answer and so many thanks again. I must say your explanations are the best form all I’ve seen.

Concerning where I’ve seen it was called “effective”. It was not clear in notes, now I understood that they meant, however I thing they have a mistake - Curriculum. Fixed income. Volume 5. Reading 52. Paragraph 3.3