P1.T3.Ch10 - Hull’s example 5.3 - Doubt in Effective rate compounded continuously

[yatin_93]

@David Harper CFA FRM Hulls example 5.3 - isn’t the formula for converting nominal rate to effective rate compounded continuously = e^r-1.

can you explain how was n*ln(1+r/n) was arrived at?

 

[David Harper CFA FRM]

HI @yatin_93 the general form for converting a discrete rate into a continuous rate, when k is the the number of periods per year (e.g., with semi-annual compounding k = 2), is given by R_c = k*ln(1 + R_k/k) where discrete rate R_k is translated into its equivalent continuous rate R_c. Consequently, as Hull's Ex 5.3 involves income provided at a rate of 4.0% per annum with semiannual compounding (i.e., k = 2 because its 2.0% every six months), the equivalent continuously compounded dividend rate is 2*ln(1 + 4.0%/2) = 3.96%.

If, instead and alternatively, the assumption were that the asset provides income at a rate of 4.0% every year, then k = 1, and the we have R_c = 1.0*ln(1 + R_m/1.0) = R_c = ln(1 + R_m). Conversely, if we wanted to translate a continuous rate into its equivalent discrete rate with annual compound frequency (k = 1), then we'd use an instance of R_k = k* [e^(R_c/k) - 1] where k = 1 such that R_k = 1.0* [e^(R_c/1.0) - 1] = e^R_c - 1. So, it's true that R_k = e^R_c - 1 translates a continuous (aka, continuously compounded) rate, R_c, into its equivalent discrete rate, R_c, with annual compound frequency (aka, effective annual rate because k = 1).

A note about terminology, each of the R_c or R_m rates is a nominal (aka, stated) rate, so it's not sufficiently specific to say that we are converting from a nominal rate. We need to specify a nominal rate with a compound frequency; e.g., "4.0% per annum with semi-annual compound frequency" indicates 4.0% as the nominal rate. Also, "effective rate compounded continuously" is too many words. The effective annual rate (EAR; or even effective rate is possible here but not best) is a term which refers to the discrete rate with annual compound frequency. I hope that's helpful!

 

[yatin_93]

@David Harper CFA FRM thank you for clarification.

 

[yatin_93]

@David Harper CFA FRM i was thinking - does this mean that a Nominal Rate can be discrete or visa versa?

 

[David Harper CFA FRM]

Hi @yatin_93 Yes, because a nominal (aka, stated) interest rate--which btw is the default format for both input and output (we never say the bond's yield is 3.5% per six month period; we always say the bond's yield is 7.0% per annum)--is just the annualized expression of an interest rate regardless of whether it compounds continuously or discretely. For example, we might say the return on an investment is "8.0% per annum" or that a bank deposit account pays "3.0% per annum" but these nominal/stated interest rates are insufficiently defined until we specify the compound frequency. The "8.0 per annum" can be either continuous (i.e. ,"with continuously compounding") or discrete (e.g., "with quarterly compounding", "with semi-annual compounding). Thanks,

 

[yatin_93]

@David Harper CFA FRM got it. Thank you