Forward rates

[vakshay]

Hi all,

This might be a very basic question but it is really confusing for me at the moment.

There are two formulas to calculate the implied forward rates using the spot rate curve.

1. (AIM 37.5) -> (1+f2) = (1+Z2)^2 / (1+z1)^1
2. AIM 26.4 -> R(fwd) = r2t2-r1t1/(t2-t1)

I am trying to understand the difference here and which formula should be applied where?

Really appreciate help on this.

Thanks

 

[ShaktiRathore]

Hi
second one is approximation to the first one to make the calculations a little simpler.
Let f be the forward rate and r2 be rate for period T1 and r1 be rate for period t1 then foraward rate is the rate for period starting after time t1 till time t2 where t2>t1 for a total time of t2-t1,
usually we calculate forward rate as (1+f)^(t2-t1)=(1+r2)^t2/(1+r1)^t1 ; t1=1;t2=2 for formula 1. (AIM 37.5) -> (1+f2) = (1+Z2)^2 / (1+z1)^1
we can write the formula as
(1+f)^(t2-t1)=(1+r2)^t2*(1+r1)^-t1
(1+f*(t2-t1))=(1+r2t2)*(1-r1t1) {approx by binomial theorem (1+x)^n~1+nx}
1+f*(t2-t1)=1+r2t2-r1t1-r2t2*r1t1
f*(t2-t1)=r2t2-r1t1-r2t2*r1t1 (cancelling 1s on both sides)
f*(t2-t1)=r2t2-r1t1 (negnecting r2t2*r1t1 as its of 1/10000 order or small)
f=(r2t2-r1t1)/(t2-t1) {dividing by t2-t1 both the sides) which is our formula for 2. AIM 26.4 -> R(fwd) = r2t2-r1t1/(t2-t1)

thanks

 

[ShaktiRathore]

you can also use continuous compounding to derive our second formula which can come very handy during the exam, suppose we have 1$ at time 0 we invest it for period t1 at rate r1 and at f for period t2-t1 thus our overall return should be same as the return on $1 invested at r2 for period t2 otherwise arbitrage is possible so in the end investor should end up with the same return whether he invest at r2 for period t2=t2-t1+t1 or just invest for period t1 at rate r1 and at forward rate f for period between t1 and t2 i.e. t2-t1 so that ends up with the same return at time t2.
[$1*e^(r1t1)]*e^f(t2-t1)=$1*e^(r2t2)
cancelling 1$ on both sides,
[e^(r1t1)]*e^f(t2-t1)=e^(r2t2)
e^[(r1t1)+f(t2-t1)]=e^(r2t2) adding powers by laws of exponents
(r1t1)+f(t2-t1)=r2t2 (equalising powers of base e)
f(t2-t1)=r2t2 -r1t1
f=r2t2 -r1t1/(t2-t1) this is our formula
hope u understood
thanks

 

[vakshay]

Hi Shakti,

Thanks for the detailed explanation. ( also, sorry for late reply, I was travelling for work )

I understand that both the equations would bring same result but just one last thing to ask on this.

Would there be any specific question where we would choose one formula over the other?

thanks again

 

[David Harper CFA FRM]

Hi vakshay,

Exam-wise IMO ShaktiRathore 's 2nd method is more relevant such that it should really depend on how the rate is quoted. For example, if we are told the 1-year zero rate is 2.00% and the 2-year zero rate is 3.0%, we actually do not have quite enough information (per the approximation; i.e., Shakti's first response). But the the FRM knows to specify a rate, for example, as "2.00% per annum with continuous compounding" or "2.00% per annum with annual compounding"

• If these given rates--i.e., s(1) = 2.0% and s(2) = 3.0%--are continuous, then the implied forward(1,2) is 3%*2-2%*1 = 4.0% with continuous compounding
• if these given rates are annual, the implied f(1,2) = 1.03^2/1.02 - 1 = 4.00xx% with annual compounding

There isn't really a difference except the spot rates (2 and 3%) are actually different inputs; i.e., if we assume the spot rates are continuous, we can translate them into their annual equivalents: 1-year = exp(2.0%) ~= 2.02% and 2-year = exp(3.0%) ~= 3.05%, then if we solved for the implied forward of those (with annual compounding), we'd get the annual equivalent of the implied forward under continuous, that is, we'd get exp(4.0%) - 1.

So, there is only one method, really, it's just that 2.0% per annum with continuous compounding is a slightly different rate (input) than 2.0% per annum with annual compounding. We can remind ourselves of this by compounding $1.0 under each: $1.0*exp(2%) > $1.0*1.02.

 

[vakshay]

thanks David.

This makes complete sense.