Hull Problem 4.23

[Adelaide]

Hi,
can someone please explain why:

6-month rate is: 2ln(1+6/94)
12-month rate is: ln(1+11/89)

Thanks.

 

[Alex_1]

Hi, I am actually also confused, I would have computed for the 12-month rate as follows: 1*ln(1+12/89) - as per the formula m * ln (1 + R_m/m) for continuous compounding... Maybe there is something I am missing. @David would you happen to know the logic behind this computation?

 

[David Harper CFA FRM]

I find easier than Hull's explain:

• 94*exp(r*0.5) = 100; i.e., 94 today receives 100 in 0.5 years at continuous (r). So: 100/94 = exp(r*0.5) --> ln(100/94) = r*0.5 --> r = 2*ln(100/94)
• 89*exp(r) = 100; i.e., 89 today receives 100 in 1.0 year at continuous (r). So: 100/89 = exp(r) --> ln(100/89) = r

@Alex_1 your conversion would be okay, but 12 is not R(m). If m = 1, then your input is the annual rate, which is 11/89; i.e., pay 89 today and receive 100 in one year, is a return of 100-89 = 11 on 89 in annual discrete terms, so R(m) = 11/89, and for m = 1, the continuously compounded equivalent return = 1 * ln (1 + [11/89]/1), just like Hull has. I hope that helps,

 

[Alex_1]

Hi @David Harper CFA FRM CIPM , thanks a lot, your answer explained this topic very, very well and I discovered I have forgotten some basic things which I went through a couple of weeks ago.

 

[Adelaide]

Helps a lot indeed!