Yield Curve for India
Month
## [1] Dec-21Nelson-Siegel Model
The model is based on Nelson and Siegel (1987).
\[\begin{equation} y\left(m\right)=\beta _{0}+\beta _{1}{\frac {{\left[{1-e^{{-m \lambda }}}\right]}}{m\lambda }}+\beta _{2}{\left({\frac {{\left[{1- e^{{-m\lambda }}}\right]}}{m\lambda }}-e^{{-m\lambda }}\right)} \end{equation}\]Where \(y(m)\) is yield for maturity \(m\).
\(\beta_0\) is the coefficient of long-term yield component.
\(\beta_1\) is coefficient of the short-term component that goes to 1 for \(m \to 0\) and 0 for \(m \to \infty\) .
\(\beta_2\) is the coefficient of medium-term component that initially decreases and after a certain value it starts increasing. This helps in readjustment of the yield curve for different scenarios.
\(\lambda\) is the decaying parameter that affects the slope of the yield curve along with other parameters.
| Maturity | Actual | Fitted |
|---|---|---|
| 1 Month | 3.33 | 3.365798 |
| 3 Months | 3.55 | 3.566262 |
| 6 Months | 3.87 | 3.840028 |
| 9 Months | 4.04 | 4.084924 |
| 1 Year | 4.34 | 4.304374 |
| 2 Years | 5.15 | 4.982199 |
| 5 Years | 5.77 | 5.968436 |
| 7 Years | 6.19 | 6.252358 |
| 10 Years | 6.41 | 6.481041 |
| 12 Years | 6.68 | 6.571762 |
| 15 Years | 6.75 | 6.662818 |
## [1] "RMSE= 0.0960668181558641"
| Variable | Value |
|---|---|
| beta_0 | 7.03 |
| beta_1 | -3.77 |
| beta_2 | -0.029 |
| lambda | 0.69 |
Svensson Model
This model is based on Svensson (1994).
\[\begin{equation} y\left(m\right)=\beta _{0}+\beta _{1}{\frac {{\left[{1-e^{\left({-m/\tau_1 }\right)}}\right]}}{m/\tau_1 }}+\beta _{2}{\left({\frac {{\left[{1- e^{\left({-m/\tau_1 }\right)}}\right]}}{m/\tau_1 }}-e^{\left({-m/\tau_1 }\right)}\right)}+\beta _{3}{\left({\frac {{\left[{1-e^{\left({-m/\tau _{2}}\right)}}\right]}}{m/\tau _{2}}}-e^{\left({-m/\tau _{2}}\right)}\right)} \end{equation}\]The additional coefficient \(\beta_3\) has similar interpretation as \(\beta_2\) and \(\tau_2\) is decaying parameter associated with it.
| Maturity | Actual | Fitted |
|---|---|---|
| 1 Month | 3.33 | 3.329649 |
| 3 Months | 3.55 | 3.537761 |
| 6 Months | 3.87 | 3.839009 |
| 9 Months | 4.04 | 4.114921 |
| 1 Year | 4.34 | 4.359926 |
| 2 Years | 5.15 | 5.053103 |
| 5 Years | 5.77 | 5.863528 |
| 7 Years | 6.19 | 6.137891 |
| 10 Years | 6.41 | 6.447368 |
| 12 Years | 6.68 | 6.606363 |
| 15 Years | 6.75 | 6.790482 |
## [1] "RMSE= 0.0575484390827892"
| Variable | Value |
|---|---|
| beta_0 | 7.66 |
| beta_1 | -4.43 |
| beta_2 | -2.22 |
| beta_3 | -3.32 |
| tau1 | 0.60 |
| tau2 | 2.79 |
References
Nelson, Charles R, and Andrew F Siegel. 1987. “Parsimonious Modeling of Yield Curves.” Journal of Business, 473–89.
Svensson, Lars EO. 1994. “Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994.” NBER Working Paper, no. w4871.