Yield curve bootstrapping: direct market rates vs discount factors interpolation

Question

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 1

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).