Can you use the Svensson model to fit a smoothed curve for yields on coupon paying bonds rather than spot rates?

Question

i've been struggling to find an answer to this question online, I know most applications of the model are used on zero (aka spot) rates. But could you use yields from a sovereign yield curve (i.e coupon paying bonds) to find a smoothed curve?

thanks,

Answer 1

It is in fact more common to fit this kind of model to coupon bonds. After all, the purpose of such curve fitting exercise is typically to obtain smoothed zero coupon curves (and by extension, smoothed par curves and forward curves).

Recall that the zero coupon rates under the Svensson model can be calculated from $$ y(t) = \beta_0 + \beta_1 \frac{1 - \exp(-t/\tau_1)}{t/\tau_1} + \beta_2 \left( \frac{1 - \exp(-t/\tau_1)}{t/\tau_1} - \exp(-t/\tau_1) \right) + \beta_3 \left( \frac{1 - \exp(-t/\tau_2)}{t/\tau_2} - \exp(-t/\tau_2) \right), $$ where $\beta_0$, $\beta_1$, $\beta_2$, $\beta_3$, $\tau_1$, and $\tau_2$ are the model parameters, and $t$ is time to maturity. The discount factor for time $t$ is then $$ d(t) = \exp(-t\cdot y(t)). $$ In other words, given the model parameters, you can calculate discount factor corresponding to any future cash flow.

In a typical curve fitting exercise, you have a large collection of bonds, whose theoretical prices, or prices as determined by the Svensson model, can be obtained as $$ P_i^\text{theoretical} = \sum c_i(t) \cdot d(t), $$ where $c_i(t)$ is the cashflow at time $t$ for bond $i$ (coupon payments and principal payments). From this $P_i^\text{theoretical}$, you can use the standard price-yield formula to calculate the theoretical yield, $y_i^\text{theoretical}$.

Our objective is then to find the model parameters (i.e., the $\beta$'s and $\tau$'s), such that the yield errors across all bonds are minimized (typically in a least squares sense): $$ \min \sum_i \left(y_i^\text{quoted} - y_i^\text{theroetical}\right)^2. $$

(In practice, the objective function above is typically expressed as weighted price errors, instead of yield errors, so as to make computation faster.)