Discounting - Discount Factor

The discount factor, DF(T), is the factor by which a future cash flow must be multiplied in order to obtain the present value. For a zero-rate (also called spot rate), taken from a yield curve, and a time to cashflow (in years), the discount factor is:

In the case where the only discount rate you have is not a zero-rate (neither taken from a zero-coupon bond nor converted from a swap rate to a zero-rate through bootstrapping) but an annually-compounded rate (for example if your benchmark is a US Treasury bond with annual coupons and you only have its yield to maturity, you would use an annually-compounded discount factor:

However, when operating in a bank, where the amount the bank can lend (and therefore get interest) is linked to the value of its assets (including accrued interest), traders usually use daily compounding to discount cashflows. Indeed, even if the interest of the bonds it holds (for example) is paid semi-annually, the value of its book of bond will increase daily, thanks to accrued interest being accounted for, and therefore the bank will be able to re-invest these daily accrued interest (by lending additional money or buying more financial products). In that case, the discount factor is then (if the usual money market day count convention for the currency is ACT/360, in case of currencies such as USD, EUR, JPY), with the zero-rate and the time to cashflow in years:

or, in case the market convention for the currency being discounted is ACT/365 (AUD, CAD, GBP):

Sometimes, for manual calculation, the continuously-compounded hypothesis is a close-enough approximation of the daily-compounding hypothesis, and makes calculation easier (even though it does not have any real application as no financial instrument is continuously compounded). In that case, the discount factor is: