###### 9.4.5 Example: US Treasury Securities

Kenneth Garbade (1986) discusses value-at-risk techniques developed by the Bankers Trust Cross Markets Research Group10 for US Treasury securities. His paper is historically important because it details sophisticated value-at-risk research performed during the mid-1980s. It also represents the first published use of principal components in a value-at-risk measure. The following example loosely parallels a principal-component remapping described in that paper.

Measure value-at-risk as 1-week 99% USDvalue-at-risk. Today is May 27, 1986. A portfolio has holdings **ω** in various US Treasury bills, notes and bonds. A primary mapping is constructed, and a linear remapping is applied:

[9.74]

Key vector ^{1}** R** represents basis point changes in constant-maturity Treasury yields for various maturities:

[9.75]

It has conditional mean vector ^{1|0}**μ** = **0** and conditional covariance matrix11

[9.76]

Eigenvectors *ν** _{i}* of

^{1|0}

**Σ**comprise columns of the matrix

[9.77]

These are graphed in Exhibit 9.28.

*ν**calculated from covariance matrix [9.76] are ordered according to their respective eigenvalues λ*

_{i}*. Eigenvalues equal the variances of corresponding principal components.*

_{i}Recall from Section 3.7 that we likened eigenvectors *ν** _{i}* to “modes of fluctuation.” The first three eigenvectors loosely correspond to parallel shifts, tilts, and bends in the yield curve. The mathematics has produced “modes of fluctuation” that our intuition might expect for a yield curve. The remaining eigenvectors reflect more complicated fluctuations that contribute modestly to actual fluctuations in the yield curve. We have

[9.78]

where components ^{1}*D _{i}* are respective principal components. By construction, the

^{1}

*D*are uncorrelated random variables. Each has variance equal to the eigenvalue λ

_{i}*of the corresponding eigenvector. The covariance matrix of*

_{i}^{1}

**is**

*D*[9.79]

The variance for ^{1}*D*_{1} is 3623, which is far larger than the other variances. It contributes more to the variability in ^{1}** R** than do any of the other principal components.12 If we consider the variances of

^{1}

*D*

_{1},

^{1}

*D*

_{2}, and

^{1}

*D*

_{3}, together they account for almost all the variances of the components of

^{1}

**. The variances of the remaining**

*R*^{1}

*D*are all less than or equal to 33, which is less than 1% of the variance of

_{i}^{1}

*D*

_{1}. Intuitively, this means that parallel shifts, tilts, and bends account for most of the variability in the Treasury yield curve.

For our principal-component remapping, we discard the contributions ^{1}*D _{i}* of the last seven principal components. We define a three-dimensional key vector comprising the first three principal components:

[9.80]

Let be the 10 × 3 matrix with columns equal to the first 3 eigenvectors **ν*** _{i}* of

^{1|0}

**Σ**:

[9.81]

We obtain

[9.82]

Our portfolio remapping has form

[9.83]

The first row of the schematic represents the primary mapping. The second represents the linear remapping = (^{1}** R**). The third represents the principal-component remapping , which is the focus of this example.

To assess the quality of the principal component approximation, we consider a diversified portfolio of US Treasury bonds. We generate 1,000 pseudorandom13 realizations ^{1}*d*^{ [k]} of ^{1}** D**, and calculate corresponding realizations

^{1}

*r*^{[k]}for

^{1}

**and for . From these, we calculate realizations and for and . Pairs (, ) are plotted as a scatter diagram of Exhibit 9.29.**

*R*The approximation is good, but not as good as those obtained in some earlier examples. Compare Exhibit 9.29 with Exhibits 9.10 and 9.15. The approximation could be improved by discarding fewer principal components.

###### Exercises

In Section 8.7, we constructed a primary mapping for a coffee portfolio. In Section 9.4.1, we applied a holdings remapping, replacing a 30-dimensional vector of coffee spreads ^{1}*R** ^{Spreads}* with an 8-dimensional vector . Now let’s apply a principal-component remapping to , replacing it with a 6-dimensional14 vector .

Measure spreads in units of .01 USD per pound.[6] Assume has conditional mean vector

[9.84]

and conditional covariance matrix

[9.85]

- Calculate the determinant of the correlation matrix corresponding to .
- Is conditionally singular, multicollinear, or neither of these?
- Based upon its first six principal components, construct a principal-component remapping for . Denote your new key vector .
- What are the conditional mean vector and conditional covariance matrix of ?
- Redraw schematic [9.54], adding your principal-component remapping.

Given a principal-component remapping [9.72], if ^{1}** R** is non-singular and joint-normal, is necessarily joint-normal?

Solution