Problem bank

Problem 285 of 333MediumLinear AlgebraP285

Level, slope and curvature coordinates

  1. A rates desk describes daily moves in the 2y, 5y and 10y yields (in basis points) with three shape vectors:

    ℓ=(111),s=(−101),c=(1−21)\ell = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \qquad s = \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}, \qquad c = \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}

    (a) Show these form a basis of R3\mathbb{R}^3, and write today's move of +8+8bp, +1+1bp, 00bp as a combination of them. Is the answer unique?

    (b) A colleague replaces cc with k=(0,1,2)Tk = (0, 1, 2)^T and calls it a bear steepener. Can ℓ,s,k\ell, s, k describe every move? Can they describe today's move?