How Discount Factors Are Calculated

Discount factors for interest rate swaps are calculated using the following formula:

$$d_t=1/(1+r_t)^t$$

where d is the discount factor for a cash flow at time t. r is the zero rate for a deposit paying at time t and t is the number of years between now and the date of the cashflow.

For example, if we are valuing a swap on the 25th May 2022 and there is a cashflow on 31st December 2022, the discount factor for that specific cashflow is calculated as follows:

Time t

t is calculated using the following (using an Act/365 day count convention).

The number of days between the 25th May 2022 and 31st December 2022 is 220.

$$t = 220/365 = 0.60273973$$

Zero rate r

r is the zero rate between now and time t. It is the rate I would get if I deposited in a EONIA paying account between 25th May 2022 and 31st December 2031.

The zero rate r is calculated using our bootstrapping models, so not straightforward to imply calculations for, but is essentially calculated as

$$(1+r_t)^{t} = (1+r_{0,1})^{1/365}(1+r_{1,2})^{1/365}...(1+r_{t-1,t})^{1/365}$$

where each interest rate on the right hand side of the equation is an EONIA forward rate between day x and x-1.

If the discount factor for 31st December 2022 is 1.000235461 then we can imply that r is -0.0391%.

$$d = 1/(1-0.0391\%)^{0.60273973}= 1.000235461$$