YouTube T3-08: Interest rates: compound frequencies

[Nicole Seaman]

The full specification of a rate is something like "discount at 8.0% per annum with annual compounding" or "compound at 8.0% per annum with continuous compounding).

David's XLS is here: https://www.dropbox.com/s/dcxd8qp8xrp7big/0328-irate-compound-freq.xlsx?st=rtruoiqf&dl=0

 

[superturtle]

Hi @David Harper CFA FRM , I see that you used 2 different approaches to get the semi-annual forward rate.

Firstly in P1.T3,Chp16,Pg18: you used the "longer method" ( the last equation in the page) to compute the semi-annual forward rate from C.C. zero rates.

Secondly in P1.T3,Chp20,Pg13: you calculated the semi-annual forward rate by converting from the C.C. forward rate. If I used this method for the case in Chp16, I get a different answer from the method used there.

The confusion is that I don't see the distinction in these two questions for me to decide which approach to use, which gives different answers.

Thanks
Ken

 

[David Harper CFA FRM]

Hi @superturtle There is no important difference except compound frequency. Please See below. I quickly prepped an XLS for you (open it here at https://www.dropbox.com/s/lvs9mt3d0yn18a1/2021-04-29-forward-rates.xlsx?dl=0). As discussed on dozens of threads, the forward rate is solving for an equality:

• exp(z1*t1) * exp[ f(t1,t2)*(t2-t1)] = exp(z2*t2), if the rates are continuous (cc), and
• (1+z1/2)^(t1*2) * (1+ f(t1,t2)/2)^[(t2-t1)*2] = (1+z2/2)^(t2*2), if the rates are semi-annual (s.a.). Those formulas solve for the forward rates.

Below illustrates. The inputs are the yellow s.a. spot rates (conveniently round 1.000%, 2.000%, etc). If we use semi-annual rates (e.g., 3.00% per annum with semi-annual compounding), then the s.a. formula retrieves the corresponding semi-annual rates. The price of the bond ($112.0821) is solved discounting those forward rates. Below that are the CC forward rates, and the bond prices the same discounting with those. (Each year's discount factor must be the same). The only difference is whether (for example) the six-month forward rate beginning in one year, F(1.0, 1.5) is expressed as "5.0149% per annum with s.a. compound frequency" or "4.9530% with CC frequency." I hope that helps.

 

[superturtle]

Thank you @David Harper CFA FRM for the explanation and spreadsheet. You used s.a. spot rates to calculate the s.a. forward in this spreadsheet. However, in P1.T3,Chp16,Pg18: you used c.c. spot rates to calculate s.a. forward rate. I am wondering is the approach on pg18 correct. Shouldn't I convert the c.c. spot into s.a. before using the 2nd formula you highlighted ((1+z1/2)^(t1*2) * (1+ f(t1,t2)/2)^[(t2-t1)*2] = (1+t2/2)^(t2*2)).

Sorry for the duplicate posts elsewhere, was afraid you missed it.

 

[David Harper CFA FRM]

Sure @superturtle you are correct: at the bottom of page 18, we are incorrect to do that. We're using the (same as above) continuous rates (i.e., 2.25% CC at 1.5 years 2.50% CC @ 2 years) to solve in a formula that assumes they are semi-annual (I think we meant that if they were s.a. rates but obviously that is super confusing and, more accurately, wrong). Just as you say, to use that formula, the continuous rates should be translated to their semi-annual equivalents. Then we would get s.a. forward rate of 3.2765% (rather than 3.252%) which should translate into 2*ln(1+3.252%/2 = 3.250%. We can always test it at the end. I'll tag page 18, sorry it's confusing/wrong. Thanks,