# Forward Rate

Assume the following spot rates (as BEYs). Yrs to Maturity***Spot Rates 0.5************4% 1.0************4.4% 1.5************5% 2.0************5.4% What’s 0.5f1 and 1f1? Any quick, clean solution to this? Thx.

what exactly do you mean with 0.5f1? Is it the forward rate from time 0.5 to time 1.5, or something different?

Hyang, For period 0.5 you should know the spot rate, remember this will be your z1.

martin.heidegger Wrote: ------------------------------------------------------- > what exactly do you mean with 0.5f1? Is it the > forward rate from time 0.5 to time 1.5, or > something different? Sorry for the confusion. All numbers around f should be tagged - I mean “subscript”. 0.5 means half year and 1.5 mean one and half years.

The questions are:

1. What’s the 6-month forward rate one year from now? - 0.5f1
2. What’s the 1-year forward rate one year from now? - 1f1

I have a bit trouble to think this through and calculate it fast & accurately. In general, when a spot rate is given, is it semi-annual based or annual based?

I think that for 1) What’s the 6-month forward rate one year from now? it would be (1.05)^(1.5) = (1.044)^(1.0) * (1+x)^(0.5) solve for x. just guessing! what does the answer key say? 2) (1.054)^(2) = (1.044)^(1.0) * (1+x)^(1.0) solve for x??

hyang Wrote: > The questions are: > 1) What’s the 6-month forward rate one year from > now? - 0.5f1 > 2) What’s the 1-year forward rate one year from > now? - 1f1 > > I have a bit trouble to think this through and > calculate it fast & accurately. In general, when a > spot rate is given, is it semi-annual based or > annual based? In the real world or in the CFA texts? Think of it as this: Compound from 0 until t # Compound from t to T -> Compound from 0 to T With B(t) being a discount factor: (1/B(t)) * (1+ forward rate …)^… = (1(B(T)) ergo: (1+ forward rate …) = B(t)/B(T) where B are Zero-Bond prices. Easy: the smaller one is B(t), a sthe whole thing needs to be > 1. Now choose your convention(s). If you’re calculating forward Libor, convention is eg simple compounding with ACT/360, ie 1 + L x days/360 = B(t)/B(T) or you have (semi-annual compounding): (1 + m*f/2)^m = B(t)/B(T) where m is the number of half-years. The right hand side doesn’t change (although you have to know what the convention for spot rates is, to calculate the bond prices)

Honestly I really hate this type of calculation - not the fun part of the study for me. Ok. Spot rates for a maturity of N periods are the geometric mean of forward rates over the N periods. I think we are expected to know how to calculate S rates given f rates and vice versa. Since the spot rates are given in every six months in this case, so I assume it’s semi-a based. The answer sheet shows the following: 1) (1 + S1.5/2)^3 = (1 + S1.0)^2 x (1 + 0.5f1.0/2) --> 0.5f1.0? 2) (1 + S2/2)^4 = (1 + S1/2)^2 x (1 + 1f1/2)^2 --> 1f1? Can anyone help to decipher the above based on the geometric mean forumula??? Thx.