P1 Fromula Sheet Page 136-137 error?

[Meredius]

Hello all,
I'm being very specific as to the formula to Define and Interpret the Forward rate given spot rates.

I understand the formula and the equality of it but I don't understand the powers raised.

So for the formula and example of calculating the 6 month forward we have the one year spot rate raised to the power of 2 (because we have 2 6 month periods to contend with if I understand correctly?)

OK but then on the following page, why are the powers raised to 4 and 3 respectively? Is it because the "unit of time" being measured is in 6 month increments? If this were yearly, would it be the power of 2 for the two year spot rate divided by the power of 1 for the one year spot-rate, assuming calculating for the one year forward one year from now?

Thanks!

 

[Floris]

Hi Meredius,

Perhaps I could give you a hand.

At the top of page 137 it is stated that for the given example it is assumed that:

• 2-year or 24-month spot rate = 6% per year (.060/1)
• 18-month spot rate = 5% per year (.050/1)
• Forward rate compounding frequency is 6 months (2 per year).

Thus:

• 24-month spot rate of 6% per year over 2 (six-month) periods equals 3.0% per 6 months (.060/2 = .030)
• 18-month spot rate of 5% per year over 2 (six-month) periods equals 2.5% per 6 months (.050/2 = .025)
• 24-month period contains 4 periods of 6 months, thus raise 6-month rate to fourth power (1+.030)^4
• 18-month period contains 3 periods of 6 months, thus 6-month rate is cubed (1+.025)^3

Regards

 

[Meredius]

Thank you for responding Floris; although the only thing bugging me is:
18-month spot rate = 5% per year (.050/1) - is the spot rate always given [assumed] per annum then?

The rest is clear.

 

[Floris]

Hi once again,

You are correct: the spot rate is expressed in terms of annual rates normally.

The idea is that one can consider a spot rate like a product that is priced in terms of monetary value (i.e. price) per period, analogous to this is that in a gas station the fuel price is quoted in terms of price per litre (or per gallon in the US). From the GARP FRM study materials I quote Tuckman chapter 2 (page 130 of book Valuation and Risk models):

..Interest rates are more intuitive than prices and, expressed as annual rates, normalize for the investment horizon as well.

I can advise you to take a deeper look into pages 130-135 of the GARP book Valuation and Risk models if you have access to it, then it should be clearer.

 

[Meredius]

I do have the materials. Thank you!

 

[David Harper CFA FRM]

Thank you @Floris , just awesome!
Although I do want to admit there is an interim typo at the top of p 137, which may be the cause of @Meredius confusion:
The question reads "For example, assume the two-year spot-rate is 6% and the eighteen-month spot-rate is 5%. What is the six-month forward rate, f(1.5,2.0)? We can solve for the by re-arranging:"
The relationship which solves for the forward, under semi-annual compounding, is then:

• 1.025^(1.5*2)*(1+f/2)^(0.5*2)=1.03^(2*2), or
• 1.025^3*(1+f)=1.03^4, such that:
• (1+f)=1.03^4/1.025^3 = 1.045, where then f(1.5,2.0) ~= 9.0%

So, while the final calc is wrong, the prior exponents (2 in num, 1 in denom) are incorrect.
Sorry, I think we fixed the notes, but have not updated this corresponding formula sheet (yet)

Two other thoughts:

• You might find helpful my notebook entry attempting to summarize this key idea @ https://forum.bionicturtle.com/threads/shortcut-to-forward-rates-if-you-have-bond-prices.4927/
• I wholeheartedly agree more with "the spot rate is expressed in terms of annual rates normally." Unless explicitly specified otherwise, it's the assumption (and avoids headaches) if all key inputs are expressed on a per annum basis

 

[Meredius]

Thank you David and Floris.
I guess this means I must be learning something.