Quarterly rate to Semi Annual rate

[bhar]

What is a quarterly rate of 8.00% converted into its bond-equivalent (semi-annual) rate?

Answer: we can take the long way and find the continuous equivalent, which is equal to LN(1.02)*4 = 7.92105%. Then convert that to the semi-annual rate, which is equal to [EXP(7.92105%/2) – 1]*2 = 8.080%

Could you please give a easier and direct method to do this.

Thanks
Bhar

 

[ShaktiRathore]

Bond equivalent yield =2[(1+r/4)^2-1]=8.08%
Let n be the number of months in a period than rate per month=r/12*n
Compounding the rate over 6 months 6/n times,
Bond equivalent yield =2[(1+nr/12)^(6/n)-1] is the general formula
where n:number of months in the period mentioned
e.g. for semiannual yield converted to BEY, n=6
Bond equivalent yield =2[(1+6r/12)^(6/6)-1]
Bond equivalent yield =2[(1+r/2)^(1)-1]=r

thanks

 

[bhar]

In a similar manner, if 5 % is the semi annual rate, what is the Monthly compounding rate. I am a bit confused. Could you please assist. If I use your formula - I am getting 1.67%. Could you please correct me.

To a similar question, the quarterly compounding rate is 14%. What is the the annual compounding? the answer i got was 14.75 = [(1+.14/4)^4]-1. Could you please confirm if this way we can calculate for the above question.

 

[ShaktiRathore]

please dont get confused the BEY with other compounding measures. the formula is for BEY don't use for other purpose. please use simple compunding to arrive at your results.
e.g. if 5% is semiannual rate than and i is monthly rate than,
(1+i)^6-1=5%=>(1+i)^6=1.05=>i=1.05^(1/6)-1
5% is the semi annual rate than BEY is same as twice semi annual rate. Monthly compounding rate is not required as semi annual rate is already given. if monthly rate is given as i the case of MBS than we need to compound over 6 months the monthly rate to get semi annual rate and than double it to get BEY.You mean to say what is monthly rate of 8.00% converted into its bond-equivalent (semi-annual) rate?
we need to convert monthly rates to 6 month equivalent because bond coupons are paid every 6 months with yield BEY/2 we have to first convert the given yield for a period into 6 months yield and double that to arrive at BEY.

thanks

 

[bhar]

A small correction - we need to multiply with 6 or 12 as the case may be - 1.05^(1/6)-1*6.
This would give the rate = 4.88
Please correct me, if I am wrong. I thought i was quite clear until a while. I am thoroughly confused. Please help

 

[David Harper CFA FRM]

bhar ShaktiRathore There is an example of this conversion in learning XLS 3.b.1., it is an instructive exercise (IMO), we just need to start with Hull's example in Chapter 4 (4.2) but use the equivalence between two different discrete rates. Given discrete rates R(m1) and R(m2):
[1+R(m1)/m1]^(m1*n) = [1+R(m2)/m2]^(m2*n) and solve for R(m1) or R(m2)

 

[bhar]

Thanks David. This confirms. I do not have access to the XLS. However, I solved for the qtr compounding based on the above formula and get a rate of 4.969. Could you please correct.

 

[bhar]

Bond equivalent yield =2[(1+r/4)^2-1]=8.08%
Let n be the number of months in a period than rate per month=r/12*n
Compounding the rate over 6 months 6/n times,
Bond equivalent yield =2[(1+nr/12)^(6/n)-1] is the general formula
where n:number of months in the period mentioned
e.g. for semiannual yield converted to BEY, n=6
Bond equivalent yield =2[(1+6r/12)^(6/6)-1]
Bond equivalent yield =2[(1+r/2)^(1)-1]=r

thanks

With David's equating method also, I am able to arrive at the 8.08%. Thanks Shakthi. Very timely help. This clears the air.