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# Shortcut to forward rates (if you have bond prices)

#### ShaktiRathore

##### Well-Known Member
Subscriber
David your method is perfect but i just rearranged the formula and value of P to arrive at the answer mine formula is also working fine enough.please take a look.
P(x)=100/(1+s1/m)^x[ my P(x) is different then yours this saved me from embedding extra m in power)
P(.5)=100/(1+.02020/2)^.5=99.498 [m=2 for semiannual compounding]
P(1)=100/(1+.030692/2)^1=98.488
f(.5)=m * {[P(s1)/P(s2)]^(1/n) - 1}=2*([99.498/98.488]^(1/.5)-1]=2(1.0206-1)=2*.0206~.0412 which is coming out as correct .this saved me from including extra m as mn.

thanks

#### David Harper CFA FRM

Staff member
Subscriber
Hi Shakti,

• a one-year bond price of P[0,1] = $97.00 And then we can apply the solved-for formula (above) that gives us directly the implied semi-annual six month forward : F[0.5,1.0] = [P(0.5)/P(1.0) - 1] * 2 = (99/97 - 1)*2 = 4.124%. So we can be finished. I just meant that I like to test the result to make sure it works. I can't speak for other, but while the derivation of the formula is intuitive, the formula by itself isn't necessarily intuitive. So, the "test" is simply that we should get the same one-year return in each approach where: 1. We just invest in the one-year bond for a return given by (1+3.069%/2)^2, or 2. We invest in the six-month bond then "roll-over" as the solved-for forward rate as given by (1 + 2.020%/2)*(1 + 4.124%/2). They should be equal, is a "test" of whether the solved-for forward rate of 4.124% is correct. Same idea for annual compounding or any frequency. The archetypal no-arbitrage idea is: invest over longer horizon should equal (=) invest in shorter horizon and "roll over" at the forward rate (if we are not including adjustments for the advantage of liquidity and lower risk of the shorter horizon). I hope that helps. Thanks! #### adamszequi ##### New Member Hi @adamszequi The example assumes we are given two bond prices: • a six month bond price of P[0,5] =$99.00, and
• a one-year bond price of P[0,1] = \$97.00
And then we can apply the solved-for formula (above) that gives us directly the implied semi-annual six month forward : F[0.5,1.0] = [P(0.5)/P(1.0) - 1] * 2 = (99/97 - 1)*2 = 4.124%. So we can be finished. I just meant that I like to test the result to make sure it works. I can't speak for other, but while the derivation of the formula is intuitive, the formula by itself isn't necessarily intuitive. So, the "test" is simply that we should get the same one-year return in each approach where:
1. We just invest in the one-year bond for a return given by (1+3.069%/2)^2, or
2. We invest in the six-month bond then "roll-over" as the solved-for forward rate as given by (1 + 2.020%/2)*(1 + 4.124%/2). They should be equal, is a "test" of whether the solved-for forward rate of 4.124% is correct. Same idea for annual compounding or any frequency. The archetypal no-arbitrage idea is: invest over longer horizon should equal (=) invest in shorter horizon and "roll over" at the forward rate (if we are not including adjustments for the advantage of liquidity and lower risk of the shorter horizon). I hope that helps. Thanks!

Got it!!
Thanks David

##### Active Member
Hi,

I am not able to locate as to where I had found the formula for the "Discrete"-Forward-Rate Formula....
and I had wrongly noted this formula with slight differences in 2 different places - Can someone please help me figure which is the right one ..?

I wanted to confirm which is the correct "Discrete"-Forward-Rate Formula
1 ) Rf = [ { ( ( 1 + R2/m ) ^ mT2 ) / ( ( 1 + R1/m ) ^ mT1 ) } ^ 1/ m * (T2-T1) - 1 ] * m

OR

2) Rf = { [ ( ( 1 + R2/m ) ^ mT2 ) / ( ( 1 + R1/m ) ^ mT1 ) ] - 1 }^ 1/ m * (T2-T1)

#### Nicole Seaman

Staff member
Subscriber
Hi,

I am not able to locate as to where I had found the formula for the "Discrete"-Forward-Rate Formula....
and I had wrongly noted this formula with slight differences in 2 different places - Can someone please help me figure which is the right one ..?

I wanted to confirm which is the correct "Discrete"-Forward-Rate Formula
1 ) Rf = [ { ( ( 1 + R2/m ) ^ mT2 ) / ( ( 1 + R1/m ) ^ mT1 ) } ^ 1/ m * (T2-T1) - 1 ] * m

OR

2) Rf = { [ ( ( 1 + R2/m ) ^ mT2 ) / ( ( 1 + R1/m ) ^ mT1 ) ] - 1 }^ 1/ m * (T2-T1)

I'm sure someone here can provide you with a definite answer, but I did find this thread that you posted previously: https://www.bionicturtle.com/forum/threads/calculation-of-cash-flow-from-fras.9321/#post-40155. It looks similar to what you are asking in this post.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@gargi.adhikari I moved this to my notebook entry here (tag = "forward rate"). I think it's your former (first) not your latter. See above at https://www.bionicturtle.com/forum/...f-you-have-bond-prices.4927/page-2#post-26345. We are starting with the no-arbitrage:
• (1+r1/m)^(t1*m)*(1+f/m)^[(t2-t1)*m] = (1+r2/m)^(t2*m) -->
• (1+f/m)^[(t2-t1)*m] = (1+r2/m)^(t2*m) / (1+r1/m)^(t1*m) -->
• (1+f/m) = [ (1+r2/m)^(t2*m) / (1+r1/m)^(t1*m) ] ^ [1/ ((t2 - t1)*m) ] -->
• f = ( [ (1+r2/m)^(t2*m) / (1+r1/m)^(t1*m) ] ^ [1/ ((t2 - t1)*m) ] - 1 ) * m ... I hope that helps,