Day count convention and compounding example

FRM
#1

Hi,

I have problems with day count conventions.

Example from Hull: price for 90-day T-bill = 10. What is the annualized continuously compounded return (assuming act/365) investor earns in 90 days?

I would solve this as:

  • 10 = 360 / 90 * (100 - cash price) => cash price = 97.5, interest = 2.5
  • annualized return with quarterly comp. and act/360: 2.5 / 97.5 * 360 / 90 = 10.2564%
  • convert this to act/365: 10.2564% * 365 / 360 = 10.3989%
  • convert from quarterly to continuous comp.: 4 * ln(1 + 0.103989 / 4) = 10.2660%

Hull does: 365 / 90 * ln(1 + 2.5 / 97.5) = 10.2678%

I can’t fully understand, why is there a difference (and what exactly his calculation does - it is obviously faster).

I would appreciate any explanation.

#2

The continuous rate associated with a holding period is found by taking the natural log of 1 + holding-period return.

So the first part of his equation is converting to the 365 day convention, and the second part is calculating the continuously compounded return.

#3

With the help of your answer, I found out what I’m doing wrong: converting to act/365 in third step but using 4 for multiplying and dividing in fourth step - which is still convention act/360.

Thank you!