Value at Risk - Ch.8: Multivariate Models


Reading Source: Textbook - Value at Risk

Topics Covered in this Reading:

  • Why Simplify the Covariance Matrix
  • Factor Structures
    • Simplifications
    • Diagonal Model
    • Multifactor Models
    • Application to Bonds
    • Comparison of Methods
  • Copulas
    • What is a Copula?
    • Marginals and Copulas
    • Applications
  • Conclusions
  • Appendix - Principal Component Analysis


For the principle component analysis, this chapter gives us an example with 3 factors.

-The first factor has even coefficients across all maturities, but why it can be defined as yield level factor?:roll_eyes:
-Also for the second factor and third factor, I am so confused why they describe the change in slope of yield curve and short-term yield respectively?:worried:

Any further explanation will be greatly appreciated!:grimacing:


Are you referring to Table 8-6?

The first factor has similar values across all maturities. This indicates that interest rates are changing similarly across all maturities; in other words, a level shift. To be a perfect level shift, all values would be exactly the same.

The second factor has an increase to the shorter-term rates and a decrease to the longer-term rates: this is a change in the slope of the yield curve.

The third factor has its largest value in time 1, so I guess it can be considered to represent mostly a short term yield change (although there are still impacts to the other maturities).


Very appreciated for your replying.

May I ask a further question? I still can not understand why the second factor refers to the change in the slope of the yield curve?:grimacing:


Sure, let’s say the yield curve currently looks like this:

1 year: 1.5%
2 year: 1.75%
5 year: 2%
10 year: 2.5%
20 year: 3.15%

we can see the slope of the yield curve is currently positive (as it starts low, and the yields increase with time).

If we apply the following factor values:
1 year: 0.52%
2 year: 0.34%
5 year: 0.13%
10 year: -0.13%
20 year: -0.41%

We get the resulting interest rate curve:
1 year: 1.5%+0.52%=2.02%
2 year: 1.75%+0.34%=2.09%
5 year: 2%+0.13%=2.13%
10 year: 2.5%-0.13%=2.37%
20 year: 3.15%-0.41%=2.74%

The slope is still positive, but it is much more flat now as the rates only slightly increase with time. So the slope has changed due to this factor


Thanks for your patience,

I start to getting confused about PCA approach, does it mean that the variance of yield change across all maturities can be explained by some common factors? Such as yield level factor, slope of term structure, etc.

Additionally, the Principle component is z1 = β11R1 + β21R2 +…+ βN1*RN + …, where βi1 is eigenvector for i maturity and z1 is 1-th principle component, Ri is defined as yield change for maturity i. I can not image why we can say z1 is yield level shift if β11 = β21 =…= βN1. And same one the 2nd and 3rd principle component.:tired_face::tired_face:

Any help will be appreciated!