# ANALYSING AND INTERPRETING THE YIELD CURVE PDF

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6 Theories of the yield curve. 7 -- Expectations hypothesis -- Liquidity preference Formal relationship: spot and forward rates. 10 Interpreting the. Analyzing and Interpreting the Yield Curve. MOORAD CHOUDHRY, PhD. Head of Treasury, KBC Financial Products, London. Shapes Observed for the Yield. Considerable effort is expended by bond analysts and economists in analyzing and interpreting the shape of the yield curve. This is because.

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PDF | Anyone with an involvement in the bond markets must become keenly interested in the B. Analysing and interpreting the yield curve. The yield curve is the defining indicator of the global debtcapital markets, and an understanding of it is vital to the smoothrunning of the economy as a whole. The U.S. Treasury bond yield curve is usually considered inverted In the past, inverted yield curves and subsequent recessions have been .. agree with this analysis; today's forward spread implies higher interpretation of the AQR findings could be that the U.S. is uniquely vulnerable to Fed actions that.

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This belief is an implied one, in that the shape of the curve today implies what interest rates and inflation should be in the future. At any one time, the yield curve reflects market expectations based on all known information up to that point. As new information is received and analyzed, the shape and level of the yield curve changes to reflect this latest information. As such, a yield curve is a static snapshot of a dynamic situation.

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Tools Request permission Export citation Add to favorites Track citation. The tests described in this section were developed by the statisticians, Lawley, James, and Anderson. We rely on the exposition by Muirhead Although PCA does not assume the data are normal, like most parametric statistics, the tests depend on the assumption of normality.

## Yield curve

This property gives rise to a natural test for the dimension of an interest rate model. In a sense, this is a test for randomness. For example, in the extreme case in which hypothesis H0 were true, the distribution of X would be spherical, with no direction of interest rate movements being any more likely than another.

That is, there would be no coherence whatsoever among movements at the different maturities.

On the other hand, H K can also be related to the dimension of the space needed to model yield curve shifts. Then a single component effectively describes the coherent movement of the yield curve. The p value 9 See Table 3 for the results of this test.

Similar conclusions are reached using five year data and using coarser layout of nodes than given in Table 1. It appears that there is no natural dimension for interest rate fluctuations. Even though all the eigenvalues were statistically significant, perhaps as a practical matter the contribution of the last K components is small. For this approach to work, we need to specify the proportion h of total variation that is considered small. Suppose we are satisfied with a K component model so long as the fraction of variance explained by the remaining principal components is less than or equal to h.

These computations provide a practical measure of the dimension of the interest rate fluctuation space. A six component model gives a nearly perfect fit with a confidence interval ranging from J Econ Finan — Table 4 Confidence intervals and actual observed values of proportions of variation captured by the first six principal components 1 p.

Figure 3 allows us to calculate the confidence interval for different levels of significance.

## Yield curve

The confidence intervals provide a basis for comparison with other measurements of proportional variation. For matrices V having large condition numbers, as ours do, very small projections along U K for K large can result in large contributions to the statistic W. The 0.

The smallest value of 2 W obtained was Note that W is dominated by W1 , the part generated from the lengths of the projections of the test vector along the principal components divided by the eigenvalues. Nonetheless, these results do not seem to be numerical artifacts since the same conclusion is reached both with coarser meshes and with the smallest eigenvalues eliminated from consideration.

Table 5 Values of the statistic W for linear first principal components with slopes as given in the first column Slope W1 W2 W 0. The second column shows how it is the projection of the linear vector along the principal components with small eigenvalue that overwhelm W J Econ Finan — 4.

Visually, this seems plausible, as illustrated in Fig. Let the specified vector U 10 equal the first eigenvector computed using the first five years of data. Therefore, if the last 42 principal components are assumed to be noise, we fail to reject the null hypothesis that the shape of the first eigenvector is the same in the two time periods. However, the hypothesis that the last 42 components are not significant was tested in Section 4.

So we are reluctant to use this test to support a claim that the shape of the first principal persists over time. The principal components corresponding to the largest covariance matrix eigenvalues are shown. A statistical test for equality of the principal components corresponding to the largest variances is negative unless data corresponding to the 42 smallest variances is neglected J Econ Finan — 5 Conclusion Principal components analysis indicates strong support for a multifactor model.

On the other hand, the hypothesis that the remaining components are not significant cannot be rejected. In particular, the hypotheses that the q remaining eigenvalues are equal cannot be rejected for q ranging from all eigenvalues to the last two.

Thus, the inherent dimension of the data is quite large, with the components associated with the smallest eigenvalues making statistically significant contribution to the sample covariance matrix. We can establish a practical limit on the dimension of the interest rate model if we are willing to designate a threshold error variance. Principal component analysis does not support the hypothesis that the first principal component is a level shift.

Nevertheless, as shown in Fig. The direction associated with the first principal component is estimated independently for the two periods. Then the null hypothesis that two direction vectors are equal is tested. If all but the first six components are excluded or filtered out as noise , we accept the null hypothesis.

However, if all the components are included, the null hypothesis is rejected. The tests described in this section were developed by the statisticians, Lawley, James, and Anderson. We rely on the exposition by Muirhead Although PCA does not assume the data are normal, like most parametric statistics, the tests depend on the assumption of normality. This property gives rise to a natural test for the dimension of an interest rate model.

In a sense, this is a test for randomness. For example, in the extreme case in which hypothesis H0 were true, the distribution of X would be spherical, with no direction of interest rate movements being any more likely than another. That is, there would be no coherence whatsoever among movements at the different maturities.

On the other hand, H K can also be related to the dimension of the space needed to model yield curve shifts. Then a single component effectively describes the coherent movement of the yield curve. The p value 9 See Table 3 for the results of this test.

Similar conclusions are reached using five year data and using coarser layout of nodes than given in Table 1. It appears that there is no natural dimension for interest rate fluctuations.

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Even though all the eigenvalues were statistically significant, perhaps as a practical matter the contribution of the last K components is small. For this approach to work, we need to specify the proportion h of total variation that is considered small.

Suppose we are satisfied with a K component model so long as the fraction of variance explained by the remaining principal components is less than or equal to h.

These computations provide a practical measure of the dimension of the interest rate fluctuation space. A six component model gives a nearly perfect fit with a confidence interval ranging from J Econ Finan — Table 4 Confidence intervals and actual observed values of proportions of variation captured by the first six principal components 1 p. Figure 3 allows us to calculate the confidence interval for different levels of significance.

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The confidence intervals provide a basis for comparison with other measurements of proportional variation. For matrices V having large condition numbers, as ours do, very small projections along U K for K large can result in large contributions to the statistic W. The 0. The smallest value of 2 W obtained was Note that W is dominated by W1 , the part generated from the lengths of the projections of the test vector along the principal components divided by the eigenvalues.

Nonetheless, these results do not seem to be numerical artifacts since the same conclusion is reached both with coarser meshes and with the smallest eigenvalues eliminated from consideration. Table 5 Values of the statistic W for linear first principal components with slopes as given in the first column Slope W1 W2 W 0. The second column shows how it is the projection of the linear vector along the principal components with small eigenvalue that overwhelm W J Econ Finan — 4.

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Visually, this seems plausible, as illustrated in Fig. Let the specified vector U 10 equal the first eigenvector computed using the first five years of data. Therefore, if the last 42 principal components are assumed to be noise, we fail to reject the null hypothesis that the shape of the first eigenvector is the same in the two time periods. However, the hypothesis that the last 42 components are not significant was tested in Section 4.

So we are reluctant to use this test to support a claim that the shape of the first principal persists over time. The principal components corresponding to the largest covariance matrix eigenvalues are shown.

A statistical test for equality of the principal components corresponding to the largest variances is negative unless data corresponding to the 42 smallest variances is neglected J Econ Finan — 5 Conclusion Principal components analysis indicates strong support for a multifactor model.

On the other hand, the hypothesis that the remaining components are not significant cannot be rejected. In particular, the hypotheses that the q remaining eigenvalues are equal cannot be rejected for q ranging from all eigenvalues to the last two.Yield curves are usually upward sloping asymptotically : the longer the maturity, the higher the yield, with diminishing marginal increases that is, as one moves to the right, the curve flattens out.

Like many empirical models, some formal predictive models that forecast output growth based on the term spread seem to have a structural break around As such, a yield curve is a static snapshot of a dynamic situation. For this approach to work, we need to specify the proportion h of total variation that is considered small. In particular, the hypotheses that the q remaining eigenvalues are equal cannot be rejected for q ranging from all eigenvalues to the last two.

How do binary models that predict recessions compare with models that forecast continuous dependent variables e. Your password has been changed.

Steep yield curve[ edit ] Historically, the year Treasury bond yield has averaged approximately two percentage points above that of three-month Treasury bills.

What maturity combinations work best? Learn more.

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