ABSTRACT
We can further illustrate the principles espoused here by supposing that an empirical researcher wants to determine an asset pricing formula in which a firm’s liquidity is viewed as a significant determinant of the return that accrues to its equity holders. Liquidity is measured here by the (natural) logarithm of the firm’s current ratio (i.e. the logarithm of current assets divided by current liabilities), and the researcher has specified that the coefficient associated with the liquidity variable in a multivariate average return/risk measures regression equation is to be as close to 5 as possible. The analysis in this chapter shows that the researcher will be able to determine an inefficient portfolio that implies betas that when taken in conjunction with the asset liquidity measures will have a perfectly linear relationship with the average return earned by N − 2 = 3 of the N = 5 assets on which the analysis is based. We can illustrate the computational procedures by supposing that the empirical researcher determines the logarithm of the current ratio for the third, fourth and fifth firms and summarizes them in the following vector:
c˜=
⎛⎜⎜⎜⎜⎝ c1 c2 c3 c4 c5
⎞⎟⎟⎟⎟⎠= 126,300 ⎛⎜⎜⎜⎜⎝
c1 c2
−53 125 −37
⎞⎟⎟⎟⎟⎠ Thus, the logarithms of the current ratio for the third, fourth and fifth firms are specified to be c3 = − 5326,300 ,c4 = 12526,300 and c5 = − 3726,300 , respectively. Now, it will be recalled that the researcher wants a coefficient of 5 to be associated with the logarithm of the current ratio in an empirically determined asset pricing formula that relates betas and the logarithm of the current ratio to asset average returns. Given this, the researcher will need to determine the inefficient portfolio that leads to the following error vector:
ε˜ = 5c˜= 1
5,260
⎛⎜⎜⎜⎜⎝ ε1 ε2
−53 125 −37
⎞⎟⎟⎟⎟⎠ We can substitute this vector into the expression for
μα ε˜ =
μQ ε˜ = β˜ − b˜
given earlier (in §3-7) and thereby determine the five unknowns, namely,
ψ1 = 1 35
, ψ2 = − 1 35
, ψ3 = 1 35
, ε1 = − 154 5,260
, ε2 = 279 5,260
that will lead to betas that return an error vector with the desired components. Substituting the computed values for ψ1 = 135 , ψ2 = − 135 and ψ3 = 135 into the expression for α˜ (as given in §3-7) shows that the inefficient portfolio that will lead to betas that are compatible with
the desired error vector ε˜ will be
α˜ = 1
⎛⎜⎜⎜⎜⎝ 1 4 7
10 13
⎞⎟⎟⎟⎟⎠+ 135 ⎛⎜⎜⎜⎜⎝
1 −2
1 0 0
⎞⎟⎟⎟⎟⎠− 135 ⎛⎜⎜⎜⎜⎝
2 −3
0 1 0
⎞⎟⎟⎟⎟⎠+ 135 ⎛⎜⎜⎜⎜⎝
3 −4
0 0 1
⎞⎟⎟⎟⎟⎠= 135 ⎛⎜⎜⎜⎜⎝
3 1 8 9
⎞⎟⎟⎟⎟⎠ This in turnmeans the inefficient portfolio is composed of anα1 = 335 proportionate investment in the first asset, an α2 = 135 proportionate investment in the second asset, an α3 = 835 proportionate investment in the third asset, and so on. This will also mean that the betas for this inefficient portfolio will be
b˜ = α˜
α˜Tα˜ = 1
⎛⎜⎜⎜⎜⎝ 280 210 455 490 665
⎞⎟⎟⎟⎟⎠ Hence, the beta relative to the inefficient portfolio for the first asset is b1 = 280526 ≈ 0.5323, the beta for the second asset is b2 = 210526 ≈ 0.3992, the beta for the third asset is b3 = 455526 ≈ 0.8650, and so on.
