yield curve for the short maturities and above it for the long maturities, illus- trating that the second principal component can be interpreted as correspond- ing to a change in the slope of the yield curve. Due to the pattern of negative and positive factor loadings on the third principal component, the curve corre- sponding to the third principal component is below the initial yield curve for the very shortest maturities, above it for intermediate maturities, and then again below it for the long maturities. This pattern is often called a change in the curvature of the yield curve.
One caveat worth mentioning is that Figure 8.1 overstates the relative importance of the second and third components, because the shocks to the principal components p1, p2, and p3 were each set equal to the same value of 100 basis points. Table 8.4 shows that, on a monthly basis, the volatility (stan- dard deviation) of the first principal component p1 is 88.93 basis points, so a change of 100 basis points is only standard deviations and is not unusual. However, the monthly volatilities of the second and third princi- pal components are 25.22 and 12.36 basis points, respectively, so changes of 100 basis points correspond to 100 25.22 3.97 and 8.09 standard deviations, respectively. Thus, the relative importances of the second and third factors are slightly more than one-fourth and one-eighth as large as the differences between the curves in Figure 8.1 suggest.
USING THE PRINCIPAL COMPONENTS TO COMPUTE
to) the partial derivatives with respect to the principal components. Letting denote the change in the price of the kth bond due to a one basis point change in the ith principal component, the exposure of the kth bond to the ith principal component is
For example, if then a 10 basis point change in the ith principal component results in a change in the bond price of basis
points or –0.159% of its value.
Table 8.5 shows the exposures of the three bonds to the first three prin- cipal components. The exposure to the first principal component is negative for all three bonds, consistent with the interpretation of the first principal component as corresponding to a shift in the level of interest rates (and the choice of the sign of the factor loadings). The exposure of the five-year bond to the second principal component is approximately zero because a large part of the value of this bond is due to the return of the principal at the end of the years, and the factor loading five-year zero-coupon rate on the second principal component is only 0.0282 (see Table 8.3). The effect of this factor loading on the bond price is then offset by the negative factor loadings for the shorter maturities. The 10- and 30-year bonds have negative exposures to the second principal component because the long-term zero-coupon rates have positive factor loadings on the second principal component. Similarly, the bonds’ exposures to the third principal component depend upon the zero-coupon rates that determine the bond prices and the factor loadings of these rates on the third principal component. For example, the 30-year bond has a positive exposure because the long-term interest rates have negative factor loadings on the third principal component.
TABLE 8.5 Exposures of the bonds to the first three principal components Principal Components
Bond Value Weight 1 2 3
1 (5-year) 100 0.2 −1.59 0.00 −0.11
2 (10-year) 200 0.4 −2.55 −1.08 0.35
3 (30-year) 200 0.4 −3.72 −2.91 2.47
Portfolio: 500 −2.83 −1.60 1.10
Volatility:
(basis points) 88.93 25.22 12.36
⌬Pki
ki
⌬Pki⁄0.0001
( )
Pki ---.
=
ki = –1.59,
1.59×10 –
1.59
– ×10×10–4= –0.00159,
=
Given these exposures, the first-order approximation to the portfolio variance used in the delta-normal approach is
where, as above, i is the variance of the ith principal component, we have used the exposures from Table 8.5 and the variances (eigenvalues) from Table 8.4, and the factor 10–8 adjusts for the fact that the volatilities in Table 8.1 are expressed in basis points (1 basis point 10–4). This formula for the portfolio variance does not include any covariance terms because the principal components are uncorre- lated. Also, because the covariance matrix in Table 8.2 is expressed in annual terms, the is and this portfolio variance also are in annual terms. The monthly portfolio variance and volatility are then 0.007797 12 6.4973×10−4 and
0.02549, or 2.549% of the value of the portfolio.
The formula for the delta-normal value-at-risk is
where as usual the constant k is determined by the confidence level of the value- at-risk. One remaining detail is that we need the expected changes in the bond prices, which in turn depend on the expected changes in the principal compo- nents. The expected changes in the principal components can be computed using equation (8.15) along with a model giving the expected changes in interest rates x, and the expected rate of return on the portfolio is given by E[return]
p1E[p1] p2E[p2] p3E[p3]. Rather than go through these steps, for simplic- ity we assume that the expected rate of return on the portfolio is 1/2% per month.
Using this assumption, the monthly volatility computed above, and a horizon of one month, the 95% confidence delta-normal value-at-risk is
or 3.71% of the value of the portfolio.
The benchmark-relative VaR can be computed by adjusting the portfolio to reflect a short position in the benchmark. Using the 10-year bond as a benchmark, the adjusted portfolio exposures are −2.83− (−2.55) (−2.26)
portfolio variance = p12 1+p22 2+p32 3
2.832
(– )
[ (94 900, )+(–1.602)(7636)
=
1.102
( )(1834) ]
+ ×10–8
0.007797,
=
=
⁄ =
0.007797 12⁄ =
VaR = –k E( [return]–portfolio volatility),
=
+ +
VaR = –k E( [return]–portfolio volatility) 1.645 0.005( –0.0255)
–
=
0.0371,
=
p1 relative
= =
−1.60− (−1.08) −0.52, and 1.10− (−0.35) 0.75, and the monthly volatility is 0.0029, or 0.29% per month. Assuming that the expected relative return is zero, the benchmark-relative VaR is
or 0.48% per year.