John Hull, Calculating forward interest rates from a set of spot rates
- Thread starter surojitpalb
- Start date
I always prefer to think of these formulas in terms of growth factors. What we need to work out is the growth factor between T+1.5y and T+2y which we can then convert into the corresponding rate.
GFs (discrete) are defined as follows:
^{nt})
where n is the compounding frequency (in this case 2 for semi-annual) and t is in years.
For example,
$1 at time T is expected at time T+1.5 to be worth $1 * GF1.5 = $1 * (1+0.0225/2)^(1.5*2) = $1 * 1.034131.
$1 at time T is expected at time T+2 to be worth $1 * GF2 = $1 * (1+0.025/2)^(2*2) = $1 * 1.034131.
In order to get the forward GF from T+1.5 to T+2 you need to divide GF2 by GF1.5 to give the growth between T+1.5 and T+2.
i.e.
Once we have the GF between T+1.5 and T+2 we can use the first formula to back out the corresponding rate:
^{nt}=(1+\frac{r}{2})^{2*(t2-t1)}&space;\therefore&space;r=((GF_{t1,t2})^{\frac{1}{2*(t2-t1)}}-1)*2)
In this case, t2 - t1 = 0.5 so 2*(t2-t1) = 1 which means the above simplifies to the formula you see in the notes substituting GFt1,t2 for GFt2/GFt1.
^{\frac{1}{n(t2-t1)}}=n\left&space;(&space;\frac{(1+r_2/n)^{n*t2}}{(1+r_1/n)^{n*t1}}-1&space;\right&space;)^{\frac{1}{n(t2-t1)}}=2\left&space;(&space;\frac{(1+0.025/2)^{2*2}}{(1+0.0225/2)^{2*1.5}}&space;-1\right&space;))
Hopefully that helps (and I haven't made any errors!).
GFs (discrete) are defined as follows:
For example,
$1 at time T is expected at time T+1.5 to be worth $1 * GF1.5 = $1 * (1+0.0225/2)^(1.5*2) = $1 * 1.034131.
$1 at time T is expected at time T+2 to be worth $1 * GF2 = $1 * (1+0.025/2)^(2*2) = $1 * 1.034131.
In order to get the forward GF from T+1.5 to T+2 you need to divide GF2 by GF1.5 to give the growth between T+1.5 and T+2.
i.e.
Once we have the GF between T+1.5 and T+2 we can use the first formula to back out the corresponding rate:
In this case, t2 - t1 = 0.5 so 2*(t2-t1) = 1 which means the above simplifies to the formula you see in the notes substituting GFt1,t2 for GFt2/GFt1.
Hopefully that helps (and I haven't made any errors!).
surojitpalb
Member
Thanks @Matthew Graves, this will be of much help to me.
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