Editing Discounting (section)

== 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) ''r'', taken from a [[yield curve]], and a time to cash flow ''T'' (in years), the discount factor is:

: <math> DF(T) = \frac{1}{(1+rT)}. </math>

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

: <math> DF(T) = \frac{1}{(1+r)^T}. </math>

However, when operating in a bank, where the amount the bank can lend (and therefore get interest) is linked to the value of its [[asset]]s (including [[accrued interest]]), traders usually use daily compounding to discount cash flows. 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 [[United States dollar]], [[euro]], [[Japanese yen]]), with ''r'' the zero-rate and ''T'' the time to cash flow in years:

: <math> DF(T) = \frac{1}{( 1 + \frac{r}{360} )^{ 360T } } </math>

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

: <math> DF(T) = \frac{1}{( 1 + \frac{r}{365} )^{ 365T } }. </math>

Sometimes, for manual calculation, the continuously-compounded hypothesis is a close-enough approximation of the daily-compounding hypothesis, and makes calculation easier (even though its application is limited to instruments such as financial derivatives). In that case, the discount factor is:

: <math> DF(T) = e^{-T\ln(1+r)}. \,</math>