Converting Bootstrapped Discount Factors to Zero Rates
Summary
The document explains how to convert a bootstrapped Ibor discount-factor curve into zero rates. It contrasts simple interest conversion, where the discount factor uses a linear accrual term, with annual compounding, where the rate is exponentiated by the year fraction. The choice describes the rate convention, not the instruments used to build each segment of the curve.
The answers emphasize that rates require a compounding convention and a day-count convention. They also give a continuously compounded conversion using the negative logarithm of the discount factor divided by the accrual period. Zero rates are not directly observable and often serve as interpolation inputs, so consistency matters; comparisons with another source require matching its conventions. One answer recommends keeping conventions aligned with the underlying Libor curve, while another notes that market and zero-rate conversion day counts can differ. The document offers guidance rather than a worked numerical example, and does not prescribe one universal convention for every market or application.
Key ideas
- Simple and annual-compounded formulas encode different rate conventions.
- A discount factor alone does not determine a quoted zero rate without a compounding convention.
- The year fraction depends on the selected day-count convention.
- Continuous zero rates can be derived from the negative logarithm of the discount factor divided by the accrual period.
- Use consistent conventions when interpolating or comparing zero-rate curves.
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Full text
# Discount Factors to Zero Rates
# 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).
Any comments are welcome! Thank you very much in advance.
## Answer by dm63 (score 2)
https://quant.stackexchange.com/a/41450
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.
## Answer by Ismael Martínez Santiago (score 1)
https://quant.stackexchange.com/a/83969
In most CCP’s I’ve seen, zero rates are always the logarithm of the negated accrual factor divided by the accrual period, as in the last answer. There is a preferred day count convention for the accrual period, but it’s not always the market’s recommended one. For instance, €STR prefers Act/360, but for ZR/DF conversion, Act/365 is used.
Now let me show my two cents: as long as you are coherent, the divisor in the accrual period formula has no bigger consequences. As a matter of fact, zero rates are not observable. They are just a convenient artifact for interpolation. But if you want to compare your zero rates with zero rates from other sources, then you have to use the exact conversion they are using.
## Answer by Ivan (score 0)
https://quant.stackexchange.com/a/41451
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 $\frac{Act}{365}$.
## Answer by Rahul Gupta (score 0)
https://quant.stackexchange.com/a/66207
following is the formula to get the Zero Rates from bootstrapped DF-
= -LN (DF) / Alpha
Where Alpha is the Accrual periodShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.