Hull - Chapter 6 Int Rate Futures - Question 6.13

[notjusttp]

Hi David,

Hull end of chapter questions Q 6.13 I am unable to understand how do we get

1) 9 Month Libor rate is 8% P.A and 6 Month Libor rate is 7.5%

Answer states like this

The fwd int rate for time periods between 6 months and 9 months is 9% becos 9% p.a for 3 months when combined with 7.5% p.a for 6 months gives an avg interest rate of 8% p.a for the 9 month period.

Can you pls kindly clarify how do we calculate the above statement.( i.e 9%). I would appreciate if you can show how to calculate this manually without using any of your excel sheets so that the same can be replicated by me in case this happens to be an exam question.

Thanks & best Rgds
Amit

 

[David Harper CFA FRM]

Hi Amit:

We analyzed Hull 06.13 here: http://forum.bionicturtle.com/viewthread/1503/

This is an *important* analytic. Yes, there is a formula but more useful to be able to apply the no-arbitrage idea
pleast try Hull 04.14 for good practice @ http://www.bionicturtle.com/learn/article/hulls_forward_rates_practice_market/

in this case, imagine two investment scenarios that, as of today, should have the same expected final value such that you are indifferent to which investment:

1. invest at 9 month spot, or
2. invest at 6 month spot and "roll over" into a 3 month invest (best estimate = today's forward rate F(0,6,3))

unless there is an arbitrage, under continous:

exp(9/12*8%) = exp(6/12*7.5%)*exp(3/12*Forward)
This formula (above) is is the key, or at least this is how i deal with it. (I know the formula but i prefer to use this b/c i don't 100% trust my formula recollection. See how left hand expresses a single 9 month spot invest and the right hand represents the "roll over" over alternative, and how as of today, you should be sort of indifferent (no arbitrage)?

exp(9/12*8%) = exp(6/12*7.5% + 3/12*Forward)
and
9/12*8% = 6/12*7.5% + 3/12*Forward, so that
Forward = (9/12*8% - 6/12*7.5%) / (3/12)

which is same as Hull's formula 4.5, so there isn't any special calculator, rather trick is getting the formula.

David

 

[notjusttp]

thats a brilliant explanation..once i got it i cud crack it..thks a ton..cheers..Amit