John Hull, Calculating forward interest rates from a set of spot rates

surojitpalb

Member
I am unable to follow the given answer(highlighted in yellow). Its page 52 of John Hull, Financial market and products. Please breakdown the formula so I can understand what values or parameters have been considered. I am new to this exam.
 

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Matthew Graves

Active Member
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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:

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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.
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Once we have the GF between T+1.5 and T+2 we can use the first formula to back out the corresponding rate:

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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.

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Hopefully that helps (and I haven't made any errors!).
 
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