2012.P1.-Focus-Review-4 | Forward Rates Question..

[atandon]

Hi David,

I am referring to your video - 2012.P1.-Focus-Review-4 where you have framed a question in regard to Forward rates and supposed to calculate 2yr forward rates starting in 3 yr. Could you pls remind me of the formula you used for the answer. I was relating to the discrete and continuous formula but finally gave up.

Answer provided on the video -
[(1.055^5/(1.045^3)^(1/2)-1 = 7.02%]

 

[David Harper CFA FRM]

Hi atandon,

Yes, this is a key relationship for the exam. It starts with a equality, if you imagine yourself today with two investment choices.

1. One allows you to invest for five years at the 5-year spot rate, S5. On this 5-year investment, your expected cumulative, annual compounded, return = (1+S5)^5.
2. Suppose instead you can invest for 3-years and "roll over" at the end of 3-years into an forward rate instrument that will pay the two-year forward rate (starting in three years); i.e., 2F3 or F[3,5]. Your expected cumulative, annual compounded, return on this "two step" strategy = (1+S3)^3*(1+2F3)^2

That is the key equality, you should be able to replicate it for both continuous compounding and semi-annual compounding. Under the above annual compounding, note that the anchor idea here is that you should have the same expectation under either investment choice (this does require that you don't care about the "liquidity" difference between the two, which is unrealistic, so this isn't a fully realistic situation). But if indifference then we have:

(1+S5)^5 = (1+S3)^3*(1+2F3)^2, and the above solves for the implied forward rate, 2F3; i.e., the forward rate implied by the two spot rates that would cause your indifference between a 5-year spot investment compared to a 3-year spot rolled over into a 2-year forward.

I hope that explains, it would be good to practice by deriving the same formula for both continuous and semi-annual compounding. Thanks,

 

[Bryon]

Hi David,
It seems that the formula given here is consistent with CFA level2 curriculum not FRM ( Hull assumes continuous compounding).
Is it out of scope here? Of course, the arbitrage-free derivation for the relationship remains the same.

Thanks

 

[David Harper CFA FRM]

Hi Bryon, it is true that the assignment (Hull) assumes continuous, but the application in annual and semi-annual (i.e., it is also in Tuckman) is clearly in scope. In fact, in my opinion, as in the above example, annual compounding is more likely to appear (e.g., was the default frequency in the last exam). The historical sample of forward rates includes more discrete frequencies than continuous, thanks,

 

[atandon]

Thanks a lot David for explaining the concept in most immaculate way.

 

[suntanv]

Dear David,

I have a question on what happens when you are dealing with money market curves rather than a medium or long term curve? Suppose we are considering a 12 months term in total and we have the rates for 1x2, 2x3, 3x4 (annualized rate act/360)and now how do we calculate for the spot rates?

 

[David Harper CFA FRM]

Hi suntanv,

Setting aside compound frequency (which has no impact on the concept), but using continuous out of convenience, the basic symbolic idea is:

exp(s1*t1)*exp(f1*[t2-t1]) = exp(s2*t2); i.e., rolling over into the forward up thru t2 should equal spot alone thru t2
so that (s1*t1)+(f1*[t2-t1]) = (s2*t2), and
f1 = [(s2*t2) - (s1*t1)]/[t2-1]; i.e., an implied (not observed) forward rate is something that requires two spot rates

this shows that an implied forward requires two spot prices, or:

• if we have only a 12 month spot rate, there are an infinite combination of short term spots, s(<1) and associated forward, or
• if given a short term spot less than 12 months, then we can solve for an implied forward. I don't know if that answers your scenario, feel free to re-ask with values. Thanks,

 

[suntanv]

HI David, Thanks for the feedback. let me give you the values suppose I have the values of (0x1) 0.001, (1x2) 0.003, (2x3) 0.004 and (3x4) 0.005 and the settlement dates are Feb 21, 2013, March 21,2013, April 19,2013 and May 20,2013. In that case If i want to derive the spot rate of 0x3 and then the implied forward rate of (3x1) then how should I go about it. Two things that I find complicated here that it is not simple period and then the rates are quoted in a different way. Thanks in advance.