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Yield Curve denotes the relationship between the yields of a zero coupon bonds and their respective time to maturity. Different estimation methods and mathematical models are used to formulate a yield curve. Parametric models such as Nelson-Siegel model and Nelson-Siegel-Svensson model are used widely to estimate a yield curve.
Nelson and Siegel in 1987 formulated a model to derive the yield at a particular point in time. They used a parametric function considering four parameters in order to estimate the forward rates.
Y(τ) = β0 + β1 * (1-exp(-m/τ))/(m/τ) + β2 * {(1-exp(-m/τ))/(m/τ) – exp(-m/τ)}
In the above function,
Y(τ) denotes the yield for a particular time.
In this model the yield is a sum of different components and four different parameters needs to be evaluated and understood for calculating the yield. These four parameters are β0, β1, β2 and τ.
In order to calculate the values of Beta, Nelson and Siegel suggested to keep τ as fixed and compute Beta using Least Square method. The four parameters that are used in this model can be described as follows:
- β0 - β0 is independent of time and hence denotes the long run yield level and remains constant for all values of τ
- β1 - β1 is considered to be a function of time to maturity. It becomes unity when time to maturity becomes 0 and decreases exponentially as time to maturity increases. The effect of β1 is clearly visible at the beginning of the curve.
- β2 - β2 is also a function of time to maturity and becomes equal to zero when time to maturity becomes zero. It decreases as time increases and then again becomes zero at higher time to maturity. This behaviour of β2 adds a hump to the yield curve.
- τ – τ is used to determine the position of the hump in the yield curve because it effects the weight part of β1 and β2.
A few constraints are present in the Nelson and Siegel method of generating yield curves.
- β0 has to be greater than 0
- Summation of β1 and β2 is greater than 0
- τ is greater than 0
Svensonn worked a bit on the flexibility and fit of the Nelson and Siegel model and added a parameter which could form a second hump in the yield curve.
According to this model,
Y(τ) = β0 + β1 * (1-exp(-m/τ1))/(m/τ1) + β2 * {(1-exp(-m/τ1))/(m/τ1) – exp(-m/τ1)} + β3 * {(1-exp(-m/τ2))/(m/τ2) – exp(-m/τ2)}
In this models six parameters were used - β0, β1, β2, β3,τ1 and τ2.
- β3 - β3 is also a function of time to maturity and is zero when m is zero. When m increases, it decreases and again becomes zero for higher m values. Because of this, a second hump gets added.
Nelson and Siegel yield curve is a continuous curve with a hump. This hump is present in the entire tenor of the curve. As one moves along the curve, the shape of the curve does not changes and is determined mostly by the shape of the preceding tenors. Any change in the slope and curvature is not reflected in this model. NSS curve on the other hand can take 2 humps in its curve. Any change in the slope and curvature can be better reflected in NSS curve. In other words, we can also say that goodness of fit for a NSS curve is better than NS curve.
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