Bond pricing question

[The Great Khan]

Hi All,

The following is the question:

203.2. The cash prices of six-month and one-year Treasury bills are $99.00 and $98.00, respectively. An eighteen month (1.5 year) bond that will pay $1.00 coupon every six months (i.e., coupon rate of 2.0% per annum payable semi-annually) currently sells for $97.00. Which is nearest to the 1.5 year zero rate with continuous compounding? (variation on Hull 4.23)

a) 2.84% b) 3.75% c) 4.06% d) 5.39%

Could you help me understand the flaw in my logic? In order to replicate a 1.5 Y zero coupon bond, you could buy the 18M bonds, as well as sell 1% of the two treasuries. This results in zero cash flow at each of the treasury maturities (pay out 1 for the treasury maturing, receive 1 for the coupon) and a final cashflow of $100.

The cost of this set up is $97 - $.99 - $.98 = $95.03

Therefore 95.03e^(1.5*r)=100
e = 3.4%

Which is clearly wrong, but I can't quite figure out what I did wrong. Is this method incorrect, or did I make a mistake along the way?

 

[Deepak Chitnis]

Hi @The Great Khan, I think you need to derive first discount factors. Like six month df=0.99 and 1 year df=0.98, because they are zero coupon then calculate the df of 1.5 year bond like, 97=$1*(df0.5 or 0.99)+1*(0.98)+101*(d1.5). d(1.5)=0.94089. Then you can derive the rate=[(1/0.94089)^1/(2*1.5)]-1*2=0.04103 or 4.103% but it is semi annual we need continuous, now simply convert it to continuous=LN(1+0.04103/2)*2=0.04062 or 4.062%. Hope that helps. Thank you!

 

[The Great Khan]

Ah I see, my error is that I forgot the final cashflow is actually $101 in my set up, not $100.

So formula would be 95.03e^(1.5*r)=101
e = 4.062%

Thanks!

 

[gargi.adhikari]

Hi All,
The Forward Rate Rf for 'Continuos Compounding' can be derived to be as : Rf = ( R2T2-R1T1)/ ( T2-T1). For 'Discrete Compounding' can somebody confirm the Forward rate formula to be : Rf ={ [ ( 1+ R2/ m ) ^ mT2 ] / [ ( 1+ R1/ m ) ^ mT1 ] -1 } * m , where m= the interval. If this formula for 'Discrete Compounding' is correct can someone help me deduce/derive it...? I am missing a (T2-T1) FACTOR in the above formula....

 

[ShaktiRathore]

Hi
please see https://forum.bionicturtle.com/threads/forward-rate-calculation.8419/#post-34269
Rf is forward rate and spot rates for t=T1 is R1 and R2 for time period t= T2 where R1 and R2 are rates compounded m times(T2>T1). Then forward rate Rf(T1,T2) is the rate between time period T1 and T2 over a period of T2-T1, so if i have $1 today then depositing it at R2 compounded m times for T2 period would result in my money growing to $1*[ ( 1+ R2/ m )] ^ mT2 we compounded $1 mT2 times with rate R2/ m ,simultaneously i deposit $1 today at R1 compounded m times for T1 period would result in my money growing to $1*[ ( 1+ R1/ m )] ^ mT1 ,so after time t=T1<T2 i shall have $1*[ ( 1+ R1/ m )] ^ mT1,now i can deposit this amount got at t=T1 at the forward rate Rf(T1,T2) for the time between T1 and T2 (T2-T1) so that at t=T2 i have my money grows to $1*[ ( 1+ R1/ m )] ^ mT1*[ ( 1+ Rf(T1,T2)/ m )] ^ m(T2-T1),i also deposited at t=0 $ 1 at a different rate R2 compounded m times for time period t= T2 therefore my money grows to $1*[ ( 1+ R2/ m )] ^ mT2 after T2,to prevent arbitrage opportunities the amounts that i have got from my different investments of $ 1 for time period T2 shall earn me the same amount so that
$1*[ ( 1+ R1/ m )] ^ mT1*[ ( 1+ Rf(T1,T2)/ m )] ^ m(T2-T1)=$1*[ ( 1+ R2/ m )] ^ mT2
=>[ ( 1+ Rf(T1,T2)/ m )] ^ m(T2-T1)=[ ( 1+ R2/ m )] ^ mT2/[ ( 1+ R1/ m )] ^ mT1
Let P(R1)=1/[ ( 1+ R1/ m )] ^ mT1 and P(R2)=1/[ ( 1+ R2/ m )] ^ mT2
Thus, [ ( 1+ Rf(T1,T2)/ m )] ^ m(T2-T1)=P(R1)/P(R2)
1+ Rf(T1,T2)/ m = (P(R1)/P(R2))^(1/m(T2-T1))
Rf(T1,T2)/ m =(P(R1)/P(R2))^(1/m(T2-T1)) - 1
Rf(T1,T2)= m*{(P(R1)/P(R2))^(1/m(T2-T1)) - 1} (See how these functions P(R1) and P(R2) are used here by David: https://forum.bionicturtle.com/thre...ates-if-you-have-bond-prices.4927/#post-25511)
You can also write the formula in a complicated way as Rf(T1,T2)= m*{([ ( 1+ R2/ m )] ^ mT2/[ ( 1+ R1/ m )]^mT1) ^ (1/m(T2-T1)) - 1}
Thanks

 

[David Harper CFA FRM]

Thanks @ShaktiRathore! Also, source is here at https://forum.bionicturtle.com/threads/p1-t3-203-hulls-interest-rates-ii.6010/#post-24111

 

[gargi.adhikari]

Thank You ! Thank You ! Thank You!
Rf(T1,T2)= m*{([ ( 1+ R2/ m )] ^ mT2/[ ( 1+ R1/ m )]^mT1) ^ (1/m(T2-T1)) - 1} is the formula I was looking for....
I was missing seeing the 1/m(T2-T1) power factor below and was thinking i might be deducing the discrete formula wrong somehow...but now i realize the m(T2-T1) cancelled out ...2 * ( .5) ...so thats why i didn't see it here....heartfelt gratitude for clearing this out for me