P1.T3 Reading 12, notes p.50

[AnZu]

Hi, I was wondering if anyone could explain the formula/calculation at the bottom of p. 50. It says "As another example, what is the 6month semi-annual forward rate starting in 1.5 years?

I think it is an example of forward with discrete compounding, but I can't find the formula in GARP or in the BT notes.

I didn't understand the formula, so I also tried the formula to convert a continuous to a discrete rate (m(e^Rc/m-1)), but I came up with 3.277%.

 

[David Harper CFA FRM]

Hi AnZu,

We need to improve that AIM: the concept is more important than the examples. The forward rate is implied by an equality, in the abstract (i.e., before compound frequency):

• At time zero(ex ante), neglecting liquidity and risk differences: Investing at the 2-year spot rate should have the same expected return as:
• Investing at 1.5 year spot and "rolling over" into the 0.5 year forward rate

With semi-annual compound frequency:
[1+s(2)/2]^(2*2) = [1+s(1.5)/2]^(1.5*2) * [1+f(1.5,2.0)/2]^(0.5*2); <-- this is the equality that determines the implied forward rate; it is the forward rate the equalizes the expected 2-year return for both approaches. The formula on page 50 then solves for the forward rate

The formula above it uses the exact same idea, just with continuous:
exp[s(2)*2] = exp[s(1.5)*1.5]*exp[f(1.5,2)*0.5] = exp[s(2)*2] = exp[s(1.5)*1.5 + f(1.5,2)*0.5] -->
s(2)*2 = s(1.5)*1.5 + f(1.5,2)*0.5; solve for forward.

I have tagged this AIM for revision: need to explain the no-arbitrage concept first.

Thanks, David

 

[AnZu]

Thank you very much for your reply. I am a bit confused by the notation but that is very helpful. Is the 2 at the end (as well as dividing 2.5% and 2.25%) because it is semi-annual?

Also, you mentioned tagging this for revision--does this mean you will post new materials for revision? (And if so, will this show up on my study planner?)

 

[David Harper CFA FRM]

Hi Anzu,

s(2) or r(2) or z(2) typically refers to the 2.0 year spot or zero rate; f(1.5, 2) above refers to the six-month forward rate beginning in 1.5 years, although there are a few variations, but I think this is the most intuitive.

Re the (2): yes the "2" is due to 2 periods per year (i.e., semi-annual). Say you want to compound $1.00 forward over the next three years at a rate of, say, 6.0% per annum

• You could compound at 6% over three years with annual frequency: $1.00*(1+6%)^3.
• Or, alternatively, you could compound at 6.0% over three years with semi-annual frequency: $1.00*(1+6%/2)^(3*2).
• Generally, you could compound at 6.0% over three years with frequency of (k) periods per year: $1.00*(1+6%/k)^(3*k).
• Until you get to the "infinite" variety, which is continuous compounding: $1.00*exp(6%*3)

The general forms of discrete compounding/discounting are given in Hull 4. They are essential, you should have these not only memorized but you should be proficient in their use (and the continuous equivalent):
\(FV=PV\bullet {{\left( 1+\frac{r}{m} \right)}^{mn}}\to PV=FV/{{\left( 1+\frac{r}{m} \right)}^{mn}}\ \)

Re: revision: yes, but given it's not an error in the notes (which would would do ASAP) but rather an internal note to add/elaborate, it is not an internally urgent revision, so the revised PDF appears when it gets done in our workflow. Thanks,