Fitting Svensson Curves to Coupon-Bond Yields
Summary
The document explains how to fit a Svensson yield curve using prices or quoted yields for coupon-paying bonds, even though the model directly describes zero-coupon rates. The fitted zero rates imply discount factors at each cash-flow date. Applying those factors to a bond’s coupons and principal produces a model price, which can then be converted to a model yield using the usual price-yield relationship.
The parameters can be estimated by minimizing errors between observed and model-implied bond yields. The response notes that practical fitting often instead minimizes weighted price errors, which can make computation faster. This gives the essential route from coupon-bond observations to a smoothed zero curve, and hence to derived par and forward curves. The document provides a modeling procedure rather than empirical evidence or implementation details; it does not discuss choices such as weighting schemes, constraints, or how to handle differences among bonds.
Key ideas
- The Svensson model specifies zero-coupon rates, which can be converted into discount factors for bond cash flows.
- Coupon and principal cash flows discounted by those factors give a model price for each bond.
- Model prices can be translated into model yields and compared with quoted yields.
- Curve parameters can be estimated by minimizing yield errors or, commonly in practice, weighted price errors.
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Full text
# Can you use the Svensson model to fit a smoothed curve for yields on coupon paying bonds rather than spot rates?
# Can you use the Svensson model to fit a smoothed curve for yields on coupon paying bonds rather than spot rates?
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 by Helin (score 2)
https://quant.stackexchange.com/a/63468
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.)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.