Fixed Income Trading and Risk Management by Alexander During

Author:Alexander During [During, Alexander]
Language: eng
Format: epub
ISBN: 9781119756354
Published: 2020-12-19T00:00:00+00:00

FIGURE 19.3 Sensitivities of zero rates given by a Nelson‐Siegel spline to 25% bumps in each curve parameter.

19.4.1 The Nelson‐Siegel and Nelson‐Siegel‐Svensson splines

The Nelson‐Siegel spline is a curve model that expresses spot (zero) rates as:

(19.18)

The contributions of the 3 terms prefixed by represent a constant, an exponentially declining short‐term contribution and a hump‐shaped medium‐term component. The time scale of the second and third components is calibrated by the parameter. This easy interpretation means that this type of model is often used by economists, including central banks, to represent bond yield curves in a parsimonious manner.

A useful way to demonstrate the role of each parameter is to fit the curve to a given market so as to obtain a realistic set of parameter values. One can then measure the changes on interest rates induced by changing the value of one parameter at a time. For the Nelson‐Siegel spline Equation (19.18), this is done in Figure 19.3 for the example of zero rates.

The Nelson‐Siegel‐Svensson model is simply the Nelson‐Siegel model Equation (19.18) with a second hump term added:

(19.19)

The additional two parameters ( and ) make the Nelson‐Siegel‐Svensson model somewhat more flexible. At the same time, the separation between the two parameters can be weak, which leads to identification problems between the terms led by and .

A Nelson‐Siegel spline is used in the calculation of the REX index of Germany government bond yields. The REX index is administered by Deutsche Börse AG and the associated REXP performance is used by some German domestic fund managers. Because German government bonds used to exhibit a strong tax‐driven coupon effect, the Nelson‐Siegel form of a polynomial yield curve is augmented by a second‐order polynomial of the coupon for each bond.

The REXP is based on the performance of 30 theoretical bonds with integer maturities of 1, 2, …10 years and coupons of 6%, 7.5%, and 9% to calculate the daily performances. The REXP is not representative of the German debt market because it uses fixed weightings for the 30 theoretical bonds instead of actual market weights for the maturity sectors. More fatally for investors looking to replicate the REXP performance, current German government coupons are lower than the theoretical benchmark coupons and the tax effect is such that the yield of higher coupon bonds is higher than that of lower coupon bonds. The running carry of the REXP benchmark basket is therefore higher than what can be achieved with the actual underlying bonds. Managers benchmarked to the REXP index therefore generally carry spread risk to earn back the basis between the benchmark and the actual underlying market.

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