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Suppose that you interpolate your zero curve, i.e. your discount factors at time $k$, $v_k$, using a log-quadratic approach: $$\ln(v_i) = \alpha + \beta D_i + \gamma D_i^2$$ where $v_i$ is a discount factor between two known discount factors, $v_k$ and $v_{k+1}$, and $D_i$ is the day-count-fraction between the dates associated with $v_k$ and $v_i$. Note ...