Yield Curve Modelling with PCA for Market Risk assessment

Andrea Ranzato 2020-04-06

Abstract

This dissertation illustrates how principal components computed on a time series of US yield curves identify the most common kind of movements occurring in interest rates of different maturities. In addition, it shows that the significant PCs can be employed to build a risk factor model for an interest rate sensitive portfolio comprised of US Government bonds which might be used to achive interest risk immunization. Conversely to traditional price sensitivity measures of bond securities, such as duration, these models take into account non parallel shifts of the yield curve, usually known as tilt and curvature. Furthermore, we investigate the empirical distribution of the eigenvalues and eigenvectors resulting from the singular value decomposition of the centered and scaled absolute interest rate changes using the bootstrap.

Objectives

1. Understanding the most common movements of the U.S. yield curve from a multivariate time series of interest rates of different maturities.
2. Building a linear risk factor model using the most significant principal components for a portfolio of U.S. Government bonds.
3. Obtaining a “probabilistic” view of the obtained results using the bootstrap.

Seminal paper: Common Factors Affecting Bond Returns (1991)

• Bond portfolios are affected by yield curve risk.
• Yield curve risk: movements of interest rates of different maturities.
• How to manage multidimensional interest rate risk?
• Traditional duration analysis identifies only parallel shifts of the yield curve.
• Statistical approach can identify not only parallel shifts, but also tilts and curvature movements.

PCA in Market Risk Analysis

The Yield Curve

A spot interest rate is the interest rate having validity from now, time zero, until time t in the future.

The concept of the spot yield curve is introduced since the assumption of a flat rate used to discount cash flows at different maturity is not realistic. Conversely, one can observe patterns in the required yields as a function of the maturity of the bond. Usually, the higher the maturity of a bond, the higher is the expected return reflected by the yield as it is possible to see in the figure below. However, in certain circumstances this might not be case.

For the purpose of the current analysis, it will be relevant not much the particular shape assumed by the yield curve on a certain day, instead, it will be of interest understanding its absolute change between two consecutive points in time. The notion of yield curve change is discussed extensively by Jones (1991) who discusses how the knowledge of the most common kind of movements having occurred historically proves useful not only for interest risk analysis, but also to figurate out portfolio allocations that might capitalize on such movements.

Spot Yield Curve on March 10, 2020

PCA on the Interest Rates Term Structure

Fixed-income portfolios are exposed to interest rate risk. In particular, financial institutions are willing to quantify the exposure of bond portfolios to unequal fluctuations in interest rates of different maturities. However, standard measures of bond price volatility[1], such as duration and convexity, do not provide a good estimate of such changes. In fact, their explanation power is limited to:

1. variations of the required yield, disregarding the existence of interest rates of different maturities.
2. parallel shifts of the yield.

In other words, these measures rely on the simplifying assumption of a flat yield curve (Fabozzi 2013).

For instance, “consider the situation of a U.S. government bond trader. The trader’s portfolio is likely to consist of many bonds with different maturities. [Hence], there is an exposure to movements in the one-year rate, the two-year rate, the three-year rate, and so on. […] He or she must be concerned with all the different ways in which the U.S. Treasury yield curve changes its shape through time” (Hull 2018, 185). As a result, bonds portfolio are sensitive to several sources of risk, potentially as many interest rates impact its value. For this reason, there is the need to reduce the number comprising the set of risk factors to a manageable one, usually three or four.

Hence, a statistical approach[2] requires to collect historical observations (see fig. below) of interest rates at different maturities to infer what the future changes might be on the basis of those observed in the past[3]. This would imply to estimate a model which we require to achieve the following objectives:

• Identifying the fundamental yield curve movements.
• Being able to replicate as faithfully as possible the covariation of the overall system of interest rates in a compact way.

The first task is accomplished by principal component analysis, whereas the second by a linear factor model[4], whose factors are selected among the PCs with higher explanatory power. Hence, those principal components represent a reduced set of “basis” that combined linearly with the eigenvectors are able to replicate with accuracy the past behaviour of the interest rates in a compact manner, and more importantly to identify those common movements.

Figure: US Treasury Interest Rates 2006-2020

DATE	MAT1MO	MAT3MO	MAT6MO	MAT1YR	MAT2YR	MAT3YR	MAT5YR	MAT7YR	MAT10YR	MAT20YR	MAT30YR
2006-02-09	4,32	4,52	4,67	4,66	4,66	4,62	4,55	4,55	4,54	4,72	4,51
2006-02-10	4,36	4,53	4,70	4,70	4,69	4,67	4,59	4,59	4,59	4,76	4,55
2006-02-13	4,38	4,55	4,71	4,70	4,68	4,66	4,58	4,58	4,58	4,76	4,56
2006-02-14	4,42	4,55	4,72	4,71	4,69	4,68	4,61	4,61	4,62	4,80	4,60
2006-02-15	4,39	4,55	4,70	4,70	4,71	4,68	4,60	4,60	4,61	4,78	4,58
2006-02-16	4,38	4,55	4,69	4,69	4,69	4,67	4,59	4,59	4,59	4,77	4,57

Table: Sample of the data set of the US Term Structure.

Singular Value Decomposition of the term structure

To illustrate how PCA can be used as a dimension reduction tool for interest risk analysis, we consider the multivariate time series of daily U.S. spot rates for a period of 3521 trading days, from Feb 9, 2006 to Mar 10, 2020.

We observe that the term structure is highly correlated, in particular, among rates of close maturity. The figures in the table below confirm this fact, since the correlation between yields is higher around the main diagonal, meaning that interest rates of similar maturity move closely. Conversely, the correlation is weaker between interest rates of different maturity.

	MAT1MO	MAT3MO	MAT6MO	MAT1YR	MAT2YR	MAT3YR	MAT5YR	MAT7YR	MAT10YR	MAT20YR	MAT30YR
MAT1MO	1,000	0,998	0,994	0,988	0,972	0,950	0,885	0,806	0,709	0,541	0,481
MAT3MO	0,998	1,000	0,998	0,994	0,980	0,959	0,894	0,816	0,718	0,549	0,487
MAT6MO	0,994	0,998	1,000	0,998	0,987	0,968	0,906	0,828	0,731	0,561	0,496
MAT1YR	0,988	0,994	0,998	1,000	0,994	0,977	0,918	0,841	0,743	0,571	0,505
MAT2YR	0,972	0,980	0,987	0,994	1,000	0,994	0,950	0,882	0,785	0,615	0,547
MAT3YR	0,950	0,959	0,968	0,977	0,994	1,000	0,976	0,923	0,835	0,673	0,606
MAT5YR	0,885	0,894	0,906	0,918	0,950	0,976	1,000	0,983	0,927	0,799	0,743
MAT7YR	0,806	0,816	0,828	0,841	0,882	0,923	0,983	1,000	0,977	0,889	0,845
MAT10YR	0,709	0,718	0,731	0,743	0,785	0,835	0,927	0,977	1,000	0,964	0,935
MAT20YR	0,541	0,549	0,561	0,571	0,615	0,673	0,799	0,889	0,964	1,000	0,989
MAT30YR	0,481	0,487	0,496	0,505	0,547	0,606	0,743	0,845	0,935	0,989	1,000

Table: Correlation Matrix of the US Term Structure.

As a result, figures in the table above demonstrate that “treasury yields do not move around in a completely uncorrelated fashion. If they did, it would be impossible to analyze the interest rates risk of a bond portfolio in any meaningful way; even the notion of portfolio duration would be meaningless” (Fabozzi 2012, 797).

Hence, to assess the yield curve risk affecting a fixed-income portfolio, it would be useful to understand, at least historically, what are the most common types of shifts that have occurred in the yield curve.

As stated previously, principal component analysis performed on the interest rates changes is capable of detecting them, in the form of principal components. Usually[5], the first principal component records an almost parallel shift of the yield curve, the second one a change in the slope (tilt), and the third one a change located in the middle of the term structure (curvature or convexity). The first degree of intuition for this representation is provided by the figure below which illustrates the first three eigenvectors resulting from the singular value decomposition of the interest rates changes. In red, the first eigenvector is approximately a parallel line since it takes similar values across the entire spectrum of maturities. For this reason, it captures parallel movements of the yield curve. Subsequently, the eigenvector in green is almost increasing, hence it explains movements which are downward in nature on early maturities and upward on later ones. Ultimately, the third eigenvector in blue, is decreasing at the beginning and increasing at the end. Therefore, it describes inverted “bumps” of the yield curve (Alexander 2008a)[6].

Eigenvectors

The first table below shows the actual values of the eigenvectors, also known as loadings, whereas the second one illustrates the corresponding eigenvalues in decreasing order of explanatory power. Since the correlation matrix is positive definite, the eigenvalues are all positive. In addition, we know that each principal component contributes to explain an amount of variance corresponding to its associated eigenvalue. For instance, the first principal component PC1 explains 62.97% of the total covariation between changes in the US interest rates. Instead, the first three PCs, considered jointly, capture 91.05% of the total covariation in the system.

Maturity	w1	w2	w3	w4	w5
MAT1MO	-0,123	-0,454	0,537	0,625	0,291
MAT3MO	-0,164	-0,514	0,239	-0,216	-0,722
MAT6MO	-0,221	-0,451	-0,126	-0,495	0,217
MAT1YR	-0,283	-0,336	-0,302	-0,144	0,481
MAT2YR	-0,335	-0,045	-0,412	0,311	-0,078
MAT3YR	-0,354	0,026	-0,306	0,256	-0,149

Table: Slice of the first five Eigenvectors with corresponding loadings

Eigenvalue Id.	Value	%	Cumulative
1	2,632	0,630	0,630
2	1,534	0,214	0,844
3	0,858	0,067	0,911
4	0,673	0,041	0,952
5	0,479	0,021	0,973

Table: First five eigenvalues.

Once the eigenvalues and eigenvectors are found using the singular value decomposition, the principal components are computed using equation , in which:

• contains time series of interest rates changes measured in basis points at the 11 different maturities
• is the orthogonal matrix of eigenvectors
• collects the resulting variables of the transformation performed by matrix on the original set of variables in

In other words, each k-th principal component in , is produced by “weighting” all the columns in with the coefficients in the k-th column vector of . For this reason, each of them is a linear combination of the original variables in .

The role of principal components analysis consists in finding those weights (the columns of ), that lead the derived variables in to have desirable properties:

• highest possible variance
• zero correlation between each others

The figure below shows a small slice of the first three PCs which are the result of the weighted sum of the interest rates changes with the first three eigenvectors , and as weights. Therefore, we might say informally that PC1 “incorporates” the “information” of the first eigenvector which captures a parallel shift of the yield, and so on.

Furthermore, it is useful to recall that the PCs are orthogonal, thus their correlation is zero. This property is particularly useful in market risk models because it allows to handle uncorrelated risk factors.

In conclusion: “the first principal component captures a common trend […] [in] interest rates [changes]. That is, if the first principal component changes at a time when the other components are fixed, then [the interest rates] all move by roughly the same amount. For this reason we often called the first component the trend component. […] Then the second principal component usually captures a change in slope of the term structure. […] For this reason we often called the third component the curvature or convexity component” (Alexander 2008a).

Linear Factor Model

Factor models are conceptually independent from principal component analysis and they constitute an independent field of research in statistics. Nevertheless, PCA provides a feasible estimation strategy for their “factors” . This should not be a surprise, given that “analysis of principal components are more of a means to an end rather than an end in themselves, because they frequently serve as intermediate steps in much larger investigations” (Johnson and Wichern 2014).

We require from a linear factor model[7] to approximate with the highest possible accuracy the covariation of the observed interest rates changes, using just a small set of risk factors.

In particular, we approximate the random vector of interest rates changes

In particular, we approximate the actual interest rates changes by means of three risk factors represented by the first three principal components, and factor weights the corresponding eigenvectors. We call principal component representation at time t the daily interest rates changes provided by the following linear factor model:

We shall see the above matrix multiplication as follows: each m-th column of is computed by “weighting” all the three PCs in , with the coefficients of the m-th column of .

The linear factor model estimated on our data is able to explain about 91.05% of the total covariation of the interest rates changes given that we have used only the first three components. A better approximation can always be achieved by adding more PCs at the cost of increasing the dimensionality of the model. Still, the model demonstrates the power of principal component analysis, since we are capable of explaining almost the entirety of the variance in the system exploiting only three components, instead of the original entire set of variables. This model will be employed later to describe the profit and loss of a portfolio composed of U.S. Government bonds.

The following two plots show the actual interest rates changes and the principal component approximation. On the top, it is shown the linear factor model estimated by means of principal component which makes a pretty good job in replicating the overall covariation of the actual system depicted on the bottom.

Bootstrap: Eigenvalues and Eigenvectors

In this section, we apply the bootstrap to approximate the theoretical distribution of the eigenvalues and eigenvector loadings. In other words, we would like to have a probabilistic representation of the results obtained in previous sections. However, an important remark is needed from the very first. The following figures should be interpreted taking in consideration that each bootstrap sample is truly a random sample[8], whereas the figures obtained in the previous section are affected by either the autocorrelation existing within a single interest rate and the cross-correlation subsisting among them. In particular, the proportion of variance explained by the first three components obtained with the bootstrap will substantially outperform those obtained considering consecutive time frames. In fact, in the latter case, the estimated eigenvalues and eigenvectors are inevitably affected by the temporal dependence that underlies two consecutive observations of interest rates changes. For this reason, this application highlights the benefits of handling samples comprised of independent and identically distributed observations, compared with time series samples which might not be stationary in nature.

A potential future analysis of the bootstrap to this particular application should incorporate the estimated autocorrelation and cross-correlation of the interest rates changes, in order to obtain more representative “artificial samples” . Consequently, the estimated eigenvectors and eigenvalues on each sample will be more accurate.

To conclude the premise, we might reasonably state that the results deriving from the non-parametric bootstrap performed in this particular case should be considered significant either in case of stationary interest rates or asymptotically, meaning that these results would be attained only if we had an infinite amount of past observations such that any temporal dependency is zeroed.

After this needed specification, we briefly introduce the bootstrap.

The bootstrap[9] is a computer intensive resampling technique based on the simple but powerful idea of repeated sampling from a collection of available observations, with the objective of evaluating the uncertainty surrounding a parameter of interest. In our case, the original data set is made up of 3520 daily observation across eleven interest rates. Then, the procedure is conducted as follows. We ask R to compose 10.000 distinct cross-sectional samples made of 587 observations each, by drawing randomly[10] from the actual data set. Subsequently, the eigenvectors and eigenvalues are computed on each of the ten thousand samples. As a result, we are capable of obtaining an estimate of their associated variability, under the i.i.d. assumption.

The results of such process are illustrated in the following figures and tables.

The two figures above illustrate the empirical distributions and box plots of the first three eigenvalues resulting from 10.000 bootstrap samples. As we expected, the first eigenvalue consistently attains higher values compared to the second and third one, meaning that, under stationary conditions of interest rates changes, the first component contributes significantly to explain most of the covariation in the system, confirming, with the specifications mentioned above, the figures obtained earlier. Furthermore, we notice a similar degree of variability associated to and which is significantly higher than the one attained by .

Thus, we can conclude, with reasonable confidence, that the third component contributes to a much lesser extent in explaining the overall covariation of the interest rates changes. This is true, because the estimated density is highly concentrated around the measures of central tendencies, whilst the other two exhibit more spread.

After having considered the eigenvalues singularly, we evaluate the variability of the cumulative variance explained by the first three components. The results shown in the figure and in the table below demonstrate that the first three components are capable of explaining an amount of variance equal to 0.916 on average.

Min.	1st Quantile	Median	Mean	3rd Quantile	Max.
0,876	0,909	0,916	0,916	0,923	0,952

Table: Descriptive statistics resulting from 10.000 bootstrap samples of the cumulative variance explained by the first three components

For inferential purposes, it is useful to consider the 95% estimated confidence interval for the four statistics obtained from 10.000 bootstrap samples. Therefore, if we were to repeat the estimation process of the statistics above one hundred times, we are confident that they would be included in the provided intervals 95 times out of 100.

Finally, the last plot represents the “probabilistic” counterpart of the typical structure embodied by the three eigenvectors typical structure which identifies a parallel shift, tilt and curvature of the yield curve, respectively. The figure should be read from the top to the bottom. Each quadrant illustrates the values taken by the corresponding loading resulting from 10.000 bootstrap samples. Even if the typical pattern of the first three eigenvectors can be recognized, we should notice that the symmetry existing along the three eigenvectors tends to hide it partially. This kind of symmetry is due to the centering transformation performed on the interest rates. In fact, if we were to obtaining the eigenvectors from the spectral decomposition of the correlation matrix rather than singualar value decomposition, the typical pattern would be even more manifested. Nevertheless, the first eigenvector depicted in red, assumes with higher probability similar values across the eleven maturities, confirming the idea that it captures a parallel shift of the yield curve.

In the same way we can observe that the second eigenvector most of the times represent a tilt of the yield curve, meaning that yields on shorter maturities witness a positive change, whereas the yields on longer ones change negatively. We might notice also that, with lower probability, the second eigenvector captures an inverted behaviour in which the longer yields maturities have positive changes.

Finally, the third eigenvector almost always shows the typical curvature component.

Bond Portfolio

This section demonstrates how to build a principal component factor model for interest rate sensitive portfolios.

The portfolio chosen for this application comprises 26 U.S. Government bonds expiring in a range of time that goes from a few months to 30 years (see Table below for all the details). The market prices of the securities refers to March 10, 2020, as provided by Business Insider. Bonds data were retrieved by means of a Python script which have semi-automated the process of data collection. If run on a terminal, the program keeps asking for the bond data which can be copied and pasted from the site. Afterwards, the figures are stored in a dictionary which is then converted into a .csv file. Subsequently, that file has been read using R in order to perform the analysis.

For the sake of simplicity, we assume to have acquired from the market one unit of each security at the market price listed on March 10, 2020 which we set to be our reference time point t.

NAME	MRKT_PRICE	CP_RATE	POSTED_YTM	ISN	ISSUE_PRICE	ISSUE_DATE	FACE_VAL	MATURITY_DATE	COUPON_PYMT_DATE	NUMB_PAYMENTS	STARTCOUPON_DATE	FINALCOUPON_DATE	PURCHASE_DATE	DAYS_TO_COUPON	YTM2_POSTED	SEMI_COUPON_AMOUNT	N_CFs	SEMI_ANN_YTM	ANNUAL_YTM	DELTA_YTM	GROUP
US TREASURY 2047	137,04	2,750	2,22	US912810RZ30	98,97	2017-11-15	100	2047-11-15	2020-05-15	2	2018-05-15	2047-11-14	2020-03-10	66 days	0,794	1,375	56	0,012	0,024	0,175	II
US TREASURY 2020	99,30	1,500	1,76	US9128282Q23	99,94	2017-08-15	100	2020-08-15	2020-08-15	2	2018-02-15	2020-08-14	2020-03-10	158 days	0,661	0,750	1	0,029	0,059	4,166	VI
US TREASURY 2043	131,44	2,875	2,20	US912810RB61	97,93	2013-05-15	100	2043-05-15	2020-05-15	2	2013-11-15	2043-05-14	2020-03-10	66 days	0,789	1,438	47	0,013	0,026	0,448	II
US TREASURY 2036	155,39	4,500	1,91	US912810FT08	99,51	2006-02-15	100	2036-02-15	2020-08-15	2	2006-08-15	2036-02-14	2020-03-10	158 days	0,706	2,250	32	0,008	0,016	-0,296	VI
US TREASURY 2026 15.02	127,43	6,000	1,51	US912810EW46	98,37	1996-02-15	100	2026-02-15	2020-08-15	2	1996-08-15	2026-02-14	2020-03-10	158 days	0,584	3,000	12	0,012	0,025	0,989	VI
US TREASURY 2028	105,56	0,521	0,15	US9128283R96	99,54	2018-01-15	100	2028-01-15	2020-07-15	2	2018-07-15	2028-01-14	2020-03-10	127 days	0,072	0,261	16	-0,002	-0,003	-0,487	IV

Table: A sample of the Bond Portfolio as of March 10, 2020

The portfolio cash flow along the term structure is illustrated in the figure below wherein the gray lines represent the eleven vertices of the risk factors for which the historical observations are available. We recall that the value of portfolio is sensitive to movements in those key rates. However, since most of the projected cash flows at time t occur between those vertices, we employ Svensson’s model[11] in order to obtain an interpolated value for the spot rates at the maturities dictated by each of the portfolio cash flows. This is possible since, each cash flow can be regarded as a zero-coupon bond whose corresponding yield is the one approximated by means of Svensson’s model.

Figure below shows the interpolation performed by Svensson’s model[12] on the yield curve observed on March 10, 2020 at the requested cash flow maturities measured in years. In other words, since the portfolio cash flow is a vector having dimension (1, 170), Svensson’s model adds 159 points to the yield curve at the constant desired time points indicated by the cash flow maturities.

Afterwards, the process shown for the daily yield curve of March 10 is repeated for each of the past curves, starting from February 09, 2006 to March 10, 2020, in order to reconstruct the cross-sectional term structure. The final result is shown in bottom figure. In practical terms, applying Svensson’s model to the historical yield curves grows the dataset of interest rates from an original dimension of to a size of . In other words, we use Svensson’s model to reconstruct the past term structure as if we were capable of observing each day the yield curve at the maturities dictated by the portfolio cash flows.

The functional form for the estimated spot yield curve by Svensson’s model, implemented in the YieldCurve R package can be seen in its documentation.

The approximated yield curve on March 10, 2020 is then employed to compute the vector at time t of present value of basis point move associated to each of the portfolio cash flow. As a result we obtain an exact representation of what would happen at time t to the present value of each if the yield curve was to move downward by 0.01% at each of the n maturities.

Maturity	C($)	R10032020	PV($)	R10032020-0.01%	PV($)	PV01
0,05833	1,3125	0,00570	1,31206	0,00560	1,31207	0,00001
0,18333	17,4375	0,00501	17,42152	0,00491	17,42184	0,00032
0,22778	1,0625	0,00482	1,06134	0,00472	1,06136	0,00002
0,35278	0,6175	0,00443	0,61654	0,00433	0,61656	0,00002
0,39722	0,9375	0,00433	0,93589	0,00423	0,93593	0,00004
0,43889	123,0000	0,00425	122,77138	0,00415	122,77675	0,00537
0,48333	1,3125	0,00418	1,30986	0,00408	1,30992	0,00006
0,56667	1,3125	0,00409	1,30947	0,00399	1,30954	0,00007
0,69444	17,4375	0,00403	17,38887	0,00393	17,39007	0,00120
0,73611	1,0625	0,00403	1,05936	0,00393	1,05944	0,00008
0,86389	0,6175	0,00406	0,61534	0,00396	0,61540	0,00005
0,90833	0,9375	0,00408	0,93404	0,00398	0,93413	0,00008
0,95000	22,2500	0,00410	22,16365	0,00400	22,16575	0,00210
0,98611	1,3125	0,00413	1,30718	0,00403	1,30731	0,00013
1,07222	1,3125	0,00419	1,30663	0,00409	1,30677	0,00014
1,19722	17,4375	0,00431	17,34804	0,00421	17,35011	0,00207
1,24167	1,0625	0,00435	1,05679	0,00425	1,05692	0,00013
1,36667	100,6175	0,00448	100,00500	0,00438	100,01861	0,01361
1,41111	0,9375	0,00452	0,93155	0,00442	0,93168	0,00013
1,45278	22,2500	0,00457	22,10312	0,00447	22,10632	0,00320

Table: Risk factors sensitivites

Prior to proceed the analysis, we introduce the profit and loss of the portfolio which is defined as the change in value between two consecutive days.

Then, “the [profit and loss] on the portfolio is approximated as a weighted sum of the changes in the interest rate risk factors with weights given by the present values of a basis point at the maturity corresponding to the interest rate” (Alexander 2008a). Therefore, we shall represent the profit and loss at time t in the following way:

where the vector contains the changes between two consecutive yield curves estimated using Svensson. The minus sign is due to the convention of representing the losses as positive quantities. “The […] vector [] is held fixed at its current value so that we are measuring the interest rate risk of the current portfolio” (Alexander 2008a).

At this point, if principal component were not known, we would be constrained to model the joint behaviour of the entire set of 159 interest rates changes. This would imply to consider 159 variances and 25281 covariances, for a total number of 14535 parameters which is clearly unfeasible. Conversely, using the principal component approximation derived previously, we are able to represent the interest rates changes using just three principal components, and at the same time being still capable of explaining most of the covariation in the system.

Then, the principal component representation using the first three components is obtained from the singular value decomposition of the Svensson interest rates changes.

Using matrix notation the PCA approximation for the entire set of observations would be as follows:

Then, the j-th principal component risk factor sensitivity of the portfolio is computed as:

or, equivalently,

which measures the change in the portfolio value when the principal component risk factor changes, keeping all the other principal component risk factors constant (Alexander 2008b, 1:32).

The entire set of risk factor sensitivities are computed as follows:

Eventually, the principal component factor model representation of the portfolio profit and loss is:

where and denote the principal component risk factor at time t, and their (constant) risk factor sensitivities.

Using PCA we reduced the number of risk factor from 159 to .

The estimated risk factor sensitivities are shown in the following table:

w1	w2	w3
-0,355	0,205	0,052

Table: Risk factors sensitivites

Therefore, the resulting PCA factor model for this example is:

which can be used to immunize the portfolio against the most frequent movements of the yield curve.

Conclusions

This text examined the US spot term structure observed between 2006 and 2020. In particular, the empirical analysis was conducted on a multivariate time series comprised of eleven key interest rates provided by the US Treasury which showed high correlation, in particular between yields of closer maturity. As a consequence, we were able to approximate the dynamics of the interest rates changes with an accuracy of almost 91% using only the first three principal components computed on those interest rates. In addition, we confirmed the existence of the traditional structure for the eigenvectors of the sample correlation matrix identifying a parallel shift a tilt and curvature as main yield curve movements occurring between consecutive time instances. However, as regard to stability, this was not entirely true when the analysis was performed on shorter successive time windows, even if some regularities were still evident, in particular as far as the first eigenvector was concerned. This might suggests that on shorter time windows principal component analysis is affected by short term volatilities of the interest rates.

Further, we showed the approximated empirical distribution associated to the first three eigenvalues and eigenvectors resulting from 10.000 bootstrap samples. We were immediately able to conclude that even if the first two eigenvalues attained consistently higher values compared to the third one, there were a greater amount of uncertainty underlying their distributions. Moreover, we were able to show that the first three components were able to explain about 92% of the variation in the interest rates changes within a 95% confidence interval, under the i.i.d. assumption which is not necessarily true in the context of financial time series.

In the last section we included in the analysis a portfolio made of bonds expiring at different maturities. We then succeeded to approximate the historical yield curves using Svensson’s model to interpolate the historical daily yield curves, in order to obtain a series of risk factors associated to portfolio cash flows projected from March 10, 2020. Finally we employed the principal component representation to approximate the profit and loss of the portfolio which is of interest in risk management applications.

To conclude, it seems useful to provide some ideas for potential future analysis:

• Coding a bootstrap function that might take into consideration the autocorrelation of the interest rates.
• Investigating more deeply the differences arising from applying spectral decomposition on the correlation matrix instead of the covariance.
• Applying the technique of cash flow mapping and investigating the possible differences with the results obtained with Svensson’s model.
• Making further steps in the understanding of the uses of the profit and loss obtained by means of PCA.

Bibliography

Alexander, Carol. 2008a. Practical Financial Econometrics. Vol. 2. Market Risk Analysis. Wiley.

———. 2008b. Quantitative Methods in Finance. Vol. 1. Market Risk Analysis. Wiley.

Choudhry, Moorad. 2004. Analysing and Interpreting the Yield Curve. Sixth. Wiley.

Connor, Gregory. 1995. “The Three Types of Factor Models: A Comparison of Their Explanatory Power.” Financial Analysts Journal 51 (3): 41–43.

Embrechts, Paul, Alexander J. McNeil, and Rudiger Frey. 2015. Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press.

Fabozzi, Frank J. 2012. The Handbook of Fixed Income Securities. Eighth. McGraw-Hill.

———. 2013. Bond Markets, Analysis, and Strategies. Eight. Pearson.

Hull, John C. 2018. Risk Management and Financial Institutions. Fifth. Wiley.

Johnson, Richard A., and Dean W. Wichern. 2014. Applied Multivariate Statistical Analysis. Sixth. Pearson.

Jolliffe, I. T. 2002. Principal Component Analysis. Second. Springer.

Jones, Frank J. 1991. “Yield Curve Strategies.” The Journal of Fixed Income 1 (2): 43–48.

Rao, Radhakrishna. 1964. “The Use and Interpretation of Principal Component Analysis in Applied Research.” The Indian Journal of Statistics 26 (4): 329–58.

Redfern, David, and Douglas McLean. 2004. “Principal Component Analysis for Yield Curve Modelling.” Moody’s Analytics Research.

Sironi, Andrea, and Andrea Resti. 2007. Risk Management and Shareholders’ Value in Banking. Wiley.

Tsay, Ruey S. 2010. Analysis of Financial Time Series. Third. Wiley.

1. Standard measures and models to manage interest rate risk such as the Duration Gap are discussed by (Sironi and Resti 2007). Particular attention is devoted to the asset-liabilities mismatch.

2. The random process generating the behaviour charachterizing the yield curve is also modeled by means of stochastic differential equations. Those models are reviewed in (Choudhry 2004). In contrast, the approach adopted here is statistical in nature. The difference between the two frameworks can be well appreciated in Redfern and McLean (2004)

3. Yet, this approach is subject to another kind of risk known as historical bias. It might not be the case that the past will repeat itself in the future. In trying to reduce the historical bias risk, one might plug into the model subjective information adopting a bayesian approach. Model risk should also be taken into account.

4. Multi-factor models are extensively used in finance. Their aim is to explain the covariance structure of portfolios’ asset returns, as a function of p underlying factors. In the literature, one can identify three types of factors models depending, on the estimation strategy adopted. A comparison between their explanatory power on U.S. equities is provided by (Connor 1995). On the other hand, the estimation process of the three models, - macroeconomic, fundamental and statistical - is discussed in (Embrechts, McNeil, and Frey 2015) and, more extensively, in (Tsay 2010). Generally speaking, both the macroeconomic and fundamental require the researcher to provide the observations on a factor, for example in the form of a economic indicator or a financial index, whereas the latter allow the analyst to estimate directly from the data the main factors that might drive the overall risk of the portfolio.

5. Fabozzi (2012) reports the academic studies which applied PCA on the term structure of different countries. In particular: Robert Litterman and Jose Scheinkman, “Common Factors Affecting Bond Returns”, Journal of Fixed Income (September 1991), pp. 54-61; Alfred Buhler and Heinz Zimmermann, “A Statistical Analysis of the Term Structure of Interest Rates in Switzerland and Germany”, Journal of Fixed Income (December 1996), pp. 55-67; Joel R. Barber and Mark L. Copper, “Immunization Using Principal Component Analysis”, Journal of Portfolio Management (Fall 1996), pp. 99-105; Rita L. D’Ecclesia and Stavros Zenios, “Risk Factor Analysis and Portfolio Immunization in the Italian Bond Market”, Journal of Fixed Income (September 1994), pp. 51-58; Bennett W. Golub and Leo M. Tilman, “Measuring Yield Curve Risk Using Principal Components Analysis, Value at Risk, and Key Rate Durations”, Journal of Portfolio Management (Summer 1997), pp. 72-84; Lionel Martellini and Philippe Priaulet, Fixed-Income Securities: Dynamic Methods for Interest Rate Risk Pricing and Hedging (Chichester, England: Wiley, 2000); Sandrine Lardic, Philippe Priaulet, and Stephane Priaulet, “PCA of Yield Curve Dynamics: Questions of Methodologies”, Journal of Bond Trading and Management (April 2003), pp. 327-349. Fabozzi (2012) reports also the number of factors (PCs) employed and the percentage of variance explained. In all of the cases it is greater than 80%.

6. It is worth noting that, within this case, the eigenvectors, and consequently the principal components, have straightforward interpretation since the system of interest rates is ordered and comprised of variables of the same kind. Differently, the principal components would be the result of linear combinations of variables having independent meaning. Therefore, they would be still valid from a mathematical point of view, yet their interpretation would be more obscure. These issues are discussed by Rao (1964) in a very technical paper, and more accessibly by (Jolliffe 2002) in Chapter 11.

7. The general formulation would be

8. The bootstrap produces approximately samples with i.i.d. observations, notwithstanding technicalities attributable to the process with which a computer generates random numbers.

9. This technique was conceived by renowned statistician Bradley Efron as an improvement over the jackniffe.

10. The random draws should be performed with replacement.

11. An alternative strategy relies on Cash Flow Mapping which consists in assigning each cash flow falling on non key maturities to the vertices by keeping the main financial characteristics of the portfolio invariant. Since this process would have been quite laborious to carry out in R, we have decide to embrace Svensson’s model which is capable of approximating the spot rates at the requested non standard maturities.

12. In R the YieldCurve package implements Nelson-Siegel, Diebold-Li and Svensson models.