Sovereign curve return

Sovereign curve return is generated by changes in the AAA sovereign curve (reference zero-coupon yield curve). FIA offers the following options:

No effect calculated (NONE)

Any returns due to changes in the sovereign curve is assigned to residual. This approach is appropriate for portfolios that are purely credit driven and have no exposure to changes in the sovereign curve.

Aggregate effect calculated (AGGREGATED)

Using the yield curve at the start and end of each calculation interval, each security is repriced using both curves and the return is generated. No sub-effects are calculated and a single return figure is generated. This type of decomposition is suitable for the simplest possible attribution.

Duration attribution (DURATION)

Calculates return due to (i) parallel shifts in yield curve, (ii) non-parallel shifts

For simple duration attribution, FIA calculates three yield curves:

The curve at the start of the interval
The starting curve, plus the parallel change in the curve;
The starting curve, plus parallel changes, plus non-parallel changes (equivalent to curve at end of interval)

Each security is priced on each curve to generate three prices $p0, p1, p2$ .

Return due to parallel shift is given by

r_{parallel}=\frac{p_1-p_0}{p_0}

Return due to non-parallel shift is given by

r_{non-parallel}=\frac{p_2-p_1}{p_0}

The sum of the two terms is

r_{total}=\frac{p_2-p_0}{p_0}

which is the overall return of the security.

Shift, twist, curvature return (STB)

FIA calculates four yield curves:

The curve at the start of the interval
The starting curve, plus the parallel change in the curve;
The starting curve, plus parallel changes, plus twist changes
The starting curve, plus parallel changes, plus twist changes, plus other higher-order changes (equivalent to curve at end of interval)

Each security is priced on each curve to generate four prices $p0, p1, p2, p3$ . Return due to parallel shift is given by

r_{parallel}=\frac{p_1-p_0}{p_0}

Return due to twist shift is given by

r_{twist}=\frac{p_2-p_1}{p_0}

Return due to higher-order shifts is given by

r_{curvature}=\frac{p_3-p_2}{p_0}

The sum of the three terms $r_{total}$ is

r_{total}=\frac{p_3-p_0}{p_0}

which is the overall return of the security.

Key rate duration return (KRD)

A key rate duration analysis isolates the effects of changes at particular maturities along the yield curve, rather than measuring the effect of different types of movements.

Key rate duration analysis may be appropriate when running attribution on portfolios of securities that have cash flows spread across a range of maturities, rather than having the bulk of their yield curve exposure concentrated at maturity. Securities in the former category include mortgage-backed bonds and other amortizing securities, and related securitized securities.

In order to run a KRD analysis on a given security, FIA uses a zero coupon yield curve at the start and end of an interval, and a set of reference maturities.

The security is first priced off the start curve.
The start curve is modified so that its level at the first reference maturity is changed to the corresponding level at the end curve. Yields that lie at or beyond neighbouring reference maturities are left unchanged, while yields that lie in the interval adjoining the current reference maturity are linearly scaled. The security is then priced off this intermediate curve.
The start curve is then successively modified so that its value at the nth reference maturity is changed to the value from the end curve, as described above. At each change, the security is priced using the new curve.
At the end of the process, the pricing curve is identical to the end curve.

The return due to the changes in the prices is now calculated. The sum of the returns will equal the overall return for the security over the interval, and the individual sub-returns are generated by changes at the given reference maturities.

The sensitivity of a security's price to changes at a particular maturity is measured by the key rate duration, just as the sensitivity to parallel curve shifts of a security's overall price is measured by the modified duration. FIA does not currently export key rate durations, but this feature may be introduced in future releases.

CCB attribution (CCB)

CCB attribution uses the Colin-Cubilie-Bardoux algorithm to calculate the twist and curvature movements of a yield curve. This algorithm uses a conventional approach to calculating the parallel shift of a yield curve, but performs a least-squares fit of a first-order polynomial to calculate the twist of the curve. This removes many of the inherent problems involved when fixed twist points are defined.

PCA attribution (PCA)

Principal component analysis (PCA) uses a suitably large number of historical yield curve changes to determin a small set of basis functions that can be linearly combined to represent these curve movements in the most economical way.

This is accomplished by forming the variance-covariance matrix V from the sample of spot rate changes at the N maturities selected. If we then calculate the N orthogonal eigenvectors of V and rank by order of eigenvalue size, the highest ranked eigenvector forms a basis function that explains as much as possible of the observed curve motion in terms of a single vector. By using a combination of this vector and lower ranked eigenvectors, the underlying data can be approximated to any degree of accuracy required.

The variances of the principal components are given by the magnitudes of the eigenvalues, so that the eigenvector with the highest value has the most explanatory power on the underlying data. If the values of the majority of eigenvalues are low, then this indicates that the underlying data can be closely modelled by a small number of functions, which represent some underlying structure in the data. PCA is therefore a useful technique for reducing the dimensionality of a modelling problem. In particular, PCA has been found to work well on yield curve changes (Phoa, 19981; Barber, Copper, 19962), since in practice practically all yield curve changes can be closely approximated using linear combinations of the first three eigenfunctions from a PCA.

PCA on historical yield curve data shows that curve movements fall into a number of fairly clearly defined types. Typically, the first eigenfunction is close to a flat line, the second rises monotonically (but is seldom a straight line), and the third imposes some curvature motion. These functions are usually interpreted as shift twist, and curvature.

However, these movements are typically slightly different from more conventional interpretations of these terms. The shift movement from a PCA is usually close, but not identical to, a parallel curve shift, and the twist movement is not uniform across all maturities. For these reasons, a PCA may not directly represent investment outcomes in terms of the decisions that were taken by the trader.

References

1 Phoa, Wesley, Advanced Fixed Income Analytics, Frank J. Fabozzi Associates, 1998

2 Barber, Joel R., Copper, Mark L., Immunization Using Principal Component Analysis, Journal of Portfolio Management, Fall 1996