R12.P1.T3.Hull_v3 - Calculate forward interest rates from a set of spot rates

[nilz]

In this reading i do understand how to compute the Continuous Forward Rate using Spot Rate/Zero Rate.
Formula - RF = (R2T2-R1T1)/(T2-T1)

However in the example on Page 50 the question asked is -
"what is the six-month semi-annual forward rate starting in 1.5 years " But the a different formula is applied.

My initial guess looking at the numbers shown in formula its a discrete Forward Rate.
Can you explain what would be the actual formula to use for such problems?

Thanks
Nilesh

 

[David Harper CFA FRM]

Hi @Nilesh The concept is the same but, as you suggest, the only difference is the compound frequency. I wrote an explanatory post on this, please see https://forum.bionicturtle.com/threads/shortcut-to-forward-rates-if-you-have-bond-prices.4927/ Thanks!

 

[nilz]

Thanks for the reply. I went through the thread and used the following formula -

f = m { (p1/p2) ^ [(1/t2-t1) . m ] - 1 }

However i see the that the numerator and denominator are opposite compared the the example given in study notes. ( Please refer the earlier post in this thread)

ie. P(1) = 2.25 p(2) = 2.50 hence as per the formula i divide p1/p2 which yields a different result. Can you example how and when should i use the formula given above. Your example in Study notes does p(2)/p(1) but the explanation given in url uses p1/p2.

 

[David Harper CFA FRM]

Hi @nilz

The difference, it appears to me, is rates versus prices (I don't see where our notes perform the strictly price-based calculation). Please keep in mind the above price-based calculation assumes you have prices for zero-coupon bonds.

So, in the notes we are given

• 1.5 year spot rate = 2.25%
• 2.0 year spot rate = 2.50%

We can infer the six-month forward, f(1.5, 2.0):

• Continuous f(1.5, 2.0) = (2*2.50% - 1.5*2.25%)/(2.0 - 1.5) = 3.250%
• Semi-annual f(1.5, 2.0) = [(1+2.5%/2)^(2*2)/(1+2.25%/2)^(1.5*2)-1]*2 = 3.252%

The example does not give prices, p(1.5) or p(2.0), but we can infer them (keep in mind: zero-coupon prices):

• zero p(1.5) = 100/(1 + 2.25%/2)^(1.5*2) = 96.6995; i.e., semi -annual prices
  
• zero p(2.0) = 100/(1 + 2.50%/2)^(2.0*2) = 95.1524

Then we could use these prices, this is the long way around now, but just to confirm (from the thread):

• s.a. f(1.5, 2.0) = m * {[P(s1)/P(s2)]^(1/mn) - 1}, where here m = 2 (s.a.) and n = 0.5, where (n) is that formula is 1.0 - 0.5 = 0.5, so that m*n = 1.0:
  f(1.5, 2.0) = m * {[P(s1)/P(s2)]^(1/mn) - 1} = 2 * [(96.6995/95.1524)^(1/1.0) - 1] = 3.252%

I hope that helps,

 

[nilz]

That makes sense but i am still dont understand if there is any formula that we could apply for
Semi-annual f(1.5, 2.0) = [(1+2.5%/2)^(2*2)/(1+2.25%/2)^(1.5*2)-1]*2 = 3.252% {this is what i am trying to figure out... }
Just like we have for Continuous = (R2T2-R1T1)/(T2-T1).

 

[David Harper CFA FRM]

Hi @nilz
The generic form in the linked thread:

(1+s1/m)^(xm) * (1+f/m)^(nm) = (1+s2/m)^[(x+n)m]; where s1 and s2 are spots and f is forward with k periods per year. Then:
(1+f/m)^(nm) = (1+s2/m)^[(x+n)m] / (1+s1/m)^(xm), and since P(s1) = 100/(1+s1)^xm and P(s2) = 100/(1+s2/m)^[(x+n)m]:
(1+f/m)^(nm) = P(s1)/P(s2), and
f/m = [P(s1)/P(s2)]^(1/nm) - 1,
f = m * {[P(s1)/P(s2)]^(1/nm) - 1}

 

[Jo_]

Hi @nilz

• Continuous f(1.5, 2.0) = (2*2.50% - 1.5*2.25%)/(2.50% - 2.25%) = 3.250%
• Semi-annual f(1.5, 2.0) = [(1+2.5%/2)^(2*2)/(1+2.25%/2)^(1.5*2)-1]*2 = 3.252%

@David Harper CFA FRM CIPM

Was wondering the same thing and happy to find this thread. Little type in the first formula though(should be 2.5-1.5). Given that the difference in outcomes is really minor and the continuous method is easier to use, i guess it's safe to use that one on the exam - yes?

 

[David Harper CFA FRM]

Hi @Jo_ Great, glad it helps. Thank you for spotting my typo above, I edited denominator to (2.0 - 1.5) such that continuous forward of 3.250% remains correct, I think.
Re: Given that the difference in outcomes is really minor and the continuous method is easier to use, i guess it's safe to use that one on the exam - yes?
Yes, that is true, in most cases it should not matter (and I think the modern exam is good about giving options that are not clustered such that, related, rounding is not a problem). However, compound frequency is a well-known assumption, from GARP's perspective, such that the desired frequency will be stated (e.g., they may prefer annual in some cases). Thanks,