Forward Rates Notation

[kausthub]

Hi David,

I have a basic doubt with regard to the way forward rates are denoted. In various spreadsheets, when you write for example - the value 2% under Time column 1.0. Does that mean the six month forward rate that matures in one year is 2% ? As in, is it the six month rate from time period 0.5 to time period 1 ?

Also does F(0.5,1.5) mean the six month forward rate that expires in 1.5 years? which is the six month rate from time period 1.0 to time period 1.5 .

Please clarify as I am having difficulty with regard to valuing bonds under the realised forward carry roll down scenario.

 

[David Harper CFA FRM]

Hi @kausthub It's a good question, I have the benefit (curse? ) of having much opportunity to think about this ... first, we might distinguish between spreadsheet implementation, which of can vary, and textual notation, which (in my opinion) should aspire to consistency, especially for exam candidates, but unfortunately has two versions.

In terms of textual notation, we use and I prefer F(start, end) rather than F(start, tenor) or start_F_tenor such that for example, F(1.5, 2.0) refers to the six-month forward rate beginning in 1.5 years; or F(2.0, 3.0) refers to the one-year forward rate starting in two years. Admittedly we do this because it follows the conventional approach to fixed rate agreements (FRA) where (eg) a 3.0 × 3.25 FRA would refer to Hull's Example 4.3 where we "suppose that a company enters into an FRA that is designed to ensure it will receive a fixed rate of 4% on a principal of $100 million for a 3-month period starting in 3 years." ... that's my notation applied to his example. (Some would assume that 3 × 4 FRA refers to a 30-day rate starting in three months, but I'm not a fan of switching to months unless explicit. If we aren't told otherwise, inputs should always be in per annum terms).

Now, while we use F(1.5, 2.0) to refer to a six month rate starting in eighteen months (1.5 years), it is absolutely true that many authors instead would refer to this as F(1.5, 0.5) so that the second value refers to the tenor of the loan/rate which, in fact, the rate. Related, it's event to refer to this as 1.5y0.5y. There is just no way around the fact that we can be referring either to the tenor of the forward rate (six months) or the end of the forward rate from the perspective of today, time zero, which is how I notate a forward.

With respect to the spreadsheet, it also can be tricky but I *always* have the forward rate in the column that marks the end of the loan/rate period. So if we are painting a term structure with six-month periods, the first column contains the six-month spot rate, S(0.5), which can also be said to be the six month forward rate starting immediately; i.e. F(0, 0.5). The next column contains the six-month forward rate starting in six months; i.e., F(0.5, 1.0), and so if that value in the cell is 2.0% then header timeline will be 1.0 years; in this way, on the XLS, the timeline of 1.0 years corresponds to the end of the rate period. This has a key advantage of "matching" my notation; i.e., the F(X.0, Y.0) rate should be aligned under Time = Y.0 years. I hope that's helpful!

 

[kausthub]

Hi @kausthub It's a good question, I have the benefit (curse? ) of having much opportunity to think about this ... first, we might distinguish between spreadsheet implementation, which of can vary, and textual notation, which (in my opinion) should aspire to consistency, especially for exam candidates, but unfortunately has two versions.

In terms of textual notation, we use and I prefer F(start, end) rather than F(start, tenor) or start_F_tenor such that for example, F(1.5, 2.0) refers to the six-month forward rate beginning in 1.5 years; or F(2.0, 3.0) refers to the one-year forward rate starting in two years. Admittedly we do this because it follows the conventional approach to fixed rate agreements (FRA) where (eg) a 3.0 × 3.25 FRA would refer to Hull's Example 4.3 where we "suppose that a company enters into an FRA that is designed to ensure it will receive a fixed rate of 4% on a principal of $100 million for a 3-month period starting in 3 years." ... that's my notation applied to his example. (Some would assume that 3 × 4 FRA refers to a 30-day rate starting in three months, but I'm not a fan of switching to months unless explicit. If we aren't told otherwise, inputs should always be in per annum terms).

Now, while we use F(1.5, 2.0) to refer to a six month rate starting in eighteen months (1.5 years), it is absolutely true that many authors instead would refer to this as F(1.5, 0.5) so that the second value refers to the tenor of the loan/rate which, in fact, the rate. Related, it's event to refer to this as 1.5y0.5y. There is just no way around the fact that we can be referring either to the tenor of the forward rate (six months) or the end of the forward rate from the perspective of today, time zero, which is how I notate a forward.

With respect to the spreadsheet, it also can be tricky but I *always* have the forward rate in the column that marks the end of the loan/rate period. So if we are painting a term structure with six-month periods, the first column contains the six-month spot rate, S(0.5), which can also be said to be the six month forward rate starting immediately; i.e. F(0, 0.5). The next column contains the six-month forward rate starting in six months; i.e., F(0.5, 1.0), and so if that value in the cell is 2.0% then header timeline will be 1.0 years; in this way, on the XLS, the timeline of 1.0 years corresponds to the end of the rate period. This has a key advantage of "matching" my notation; i.e., the F(X.0, Y.0) rate should be aligned under Time = Y.0 years. I hope that's helpful!

Understood David. I follow now.
Thanks a lot!