Discount Factors to Zero Rates

I have obtained a Ibor-6Months curve using bootstrapping techniques. For the short-term of the curve I used spot, for the middle-term FRAs and for the long-term IRS.

The curve that I have obtained is given in discount factors...(using the configuration detailed above). The question is, how can I now obtain the zero rate curve once the discount factors are known?

Shall I use equation (1):

$DF(t;T)=\frac{1}{1+r(t;t,T)\cdot\alpha\left(t;t,T\right)}$

Or shall I use equation (2):

$DF(t;T)=\frac{1}{\left(1+r\left(t;t,T\right)\right)^{\alpha(t;t,T)}}$

where $\alpha$ refers to the year fraction and $r$ is the zero rate, $t$ is the actual time and $T$ is the maturity time.

Is the equation the same for any tenor (taking into account that the instruments involved are different)? I would say IRS tenors follow the equation (2) while spots or FRA tenors follow the equation (1).

• Personally I have used both depending upon the context and to be consistent with the market terminology for different products. In all honesty I have never found the zero rate to be useful for anything really, certainly not analysis wise. For what purpose are you interested in its calculation?
– Attack68
Aug 25, 2018 at 21:52
• I think i got it...for a fix-float Euribor-6M IRS, we have the floating leg following semi-annually coupons while in the fixed leg we have annually coupons. This annual coupons does not appear in FRA instruments (due to the fact that both legs are semi-annually). This makes short and middle-term following Equation (1), while the long term follows Equation (2), which is annually compounded due the annual coupons on IRS Aug 26, 2018 at 8:02

Equation 2 gives the annual zero rate for all tenors. In practice, people sometimes quote rates f less than one year using Equation 1, but in general , equation 2 is used.

You can use either but a rate and a curve are only well defined if given alongside calculation conventions.

The convention in Equation 1 is that the rate is linear, in Equation 2 it is (annually) compounded.

Moreover you need a daycount convention to calculate the year fraction between two dates, for example $\frac{Act}{365}$.

My suggestion is to stick to the convention of the Libor you’ve used i.e. likely linear $\frac{Act}{365}$.

following is the formula to get the Zero Rates from bootstrapped DF-

= -LN (DF) / Alpha

Where Alpha is the Accrual period