How Discount Factor Interpolation Affects Forward Curve Smoothness
Summary
The document compares two yield curve construction approaches: interpolating quoted spot rates and deriving discount factors, or interpolating discount factors directly. It asks why the latter may produce a smoother implied forward curve when using linear interpolation.
The answer derives forward-rate expressions for both approaches between two LIBOR maturities. With spot-rate interpolation, the resulting expression includes quadratic terms in the intermediate maturity; with linear discount-factor interpolation, the expression lacks those terms. The answer uses this algebraic difference to suggest that the forward rate changes more rapidly under spot-rate interpolation, while discount-factor interpolation tends to be smoother.
This is a local comparison based on a simplified two-point setup. It does not address alternative interpolation schemes, instrument conventions, or the broader calibration choices involved in constructing a market yield curve.
Key ideas
- Interpolating spot rates and interpolating discount factors are distinct curve construction choices.
- Forward rates can be derived from the discount factors implied by either interpolation method.
- The spot-rate interpolation expression contains quadratic maturity terms in the example.
- The discount-factor interpolation expression is simpler and is presented as producing a smoother forward rate.
- The comparison is limited to a two-maturity illustration and does not establish a universal result for all curve setups.
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# Yield curve bootstrapping: direct market rates vs discount factors interpolation
# Yield curve bootstrapping: direct market rates vs discount factors interpolation
From my understanding, there are (most generally speaking) two approaches for bootstrapping the yield curve (with an exact method). We can either interpolate between the market quotes (interbank deposits, futures, FRAs, swaps, etc.) and then infer the discount factors, this would not require a minimization technique as far as I understand. Alternatively, we could interpolate the discount factors such that we match the market quotes (which would require interpolation and minimization simultaneously). Please correct me if I am wrong here (conceptually).
It is well known that using linear interpolation with both way would result in a irregular looking forward curve. However, a text that I read recently (I cannot link it), claims that when interpolation is performed on the discount factor rather than the direct market quotes, everything else equals (so same interpolation method), the implied forward curve would be smoother in the case of discount factors (as interpolating variable). What is the particular reason for this?
## Answer by Kurt G. (score 2, accepted)
https://quant.stackexchange.com/a/70535
If we interpolate between the two libors $L_1$ and $L_2$ (spot rates) with maturities $T_1$ and $T_2$ the discount factor at $T\in(T_1,T_2)$ is $$ P(T)=\frac{1}{1+TL(T)}=\frac{1}{1+T\frac{(T_2-T)L_1+(T-T_1)L_2}{T_2-T_1}}\,. $$ The forward rate for $[T,T_2]$ then becomes \begin{align}\tag{1} F(T,T_2)&=\frac{1}{T_2-T}\Bigg(\frac{P(T)}{P(T_2)}-1\Bigg)=\frac{1}{T_2-T}\Bigg(\frac{1+T_2L_2}{1+T\frac{(T_2-T)L_1+(T-T_1)L_2}{T_2-T_1}}-1\Bigg)\\ &=\frac{1}{T_2-T}\frac{T_2(T_2-T_1)L_2-T(T_2-T)L_1-T(T-T_1)L_2}{T_2-T_1+T(T_2-T)L_1+T(T-T_1)L_2}\,. \end{align} If instead we interpolate between the discount factors then \begin{align}\tag{2} F(T,T_2)&=\frac{1}{T_2-T}\Bigg(\frac{(T_2-T)P(T_1)+(T-T_1)P(T_2)}{(T_2-T_1)P(T_2)}-1\Bigg)\\ &=\frac{1}{T_2-T}\Bigg(\frac{1+T_2L_2}{T_2-T_1}\Bigg(\frac{T_2-T}{1+T_1L_1}+\frac{T-T_1}{1+T_2L_2}\Bigg)-1\Bigg)\,. \end{align} Since (1) contains $T^2$ terms and (2) does not we can expect that the $T$-derivative of (1) is larger than that of (2).Shown 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.