Промышленный лизинг Промышленный лизинг  Методички 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 [ 41 ] 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103

6. Mullifticlor Pricing Models

Chamberlain and Rothschild (1983) show that consistent estimates of the factor loading matrix В can be obtained from the eigenvectors associated with the largest eigenvalues of Т П, where T is any arbitrary positive

idefinite matrix with eigenvalues bounded away from zero and infinity. Both standard factor analysis and principal components fit into this category, for factor analysis T = D and for principal components T = I. 1 lowever, jibe finite-sample applicability of the result is unclear since it is required that I both the number of assets N and the number of lime periods 7go to infinity. J The Connor and Korajczyk principal components approach is also consistent as N increases. It has the further potential advantage that it only requires T > К and does not require T to increase to infinity. However, whether in finite samples il dominates factor analysis or standard principal components is an open question.

6.4.2 Number of Factors

The underlying theory of the multifactor models does not specify the number of factors that are required, thai is, the value of K. While, for the theory to be useful, К should be reasonably small, the researcher still has significant latitude in the choice. In empirical work this lack of specification has Ibcen handled in several ways. One approach is to repeat the estimation* land testing of the model for a variety of values of A and observe if the tests arc sensitive lo increasing the number of factors. For example Lehmann and Modest (1988) present empirical results for five, ten, and fifteen factors. Their results display minimal sensitivity when the number of factors increases from five to ten to fifteen. Similarly Connor and Korajczyk (1988) consider five and ten factors with little sensitivity lo the additional live factors. These results suggest that live factors arc adequate.

A second approach is to test explicitly for the adequacy of A factors. An asymptotic likelihood ratio test of the adequacy of A factors can be constructed using -2 times the difference of the value of the log-likelihood function of the covariance matrix evaluated at the constrained and unconstrained estimators. Morrison (1990, p. 302) presents this test. The likelihood ratio test statistic is

Jr, - -7- l-±(2/V + 5)-A [logn-logBB + D], (0.4.10)

where П is the maximum likelihood estimator of fi and В and D arc the maximum likelihood estimators of В and D, respectively. The leading term is an adjustment to improve the convergence of the finite-sample null distribution to the large-sample distribution. Under the null hypothesis that A factors arc adequate, Jr, will be asymptotically distributed (( -► oo) as a chi-square variatc with 11 (N - A) - N - A ] degrees of freedom. Roll and

6.4. Selection of Factors

Ross (1980) use this approach and conclude that three or four factors are adequate.

A potential drawback of using the test from maximum likelihood factor analysis is that the constrained model assumes a strict factor .structure- an assumption which is not theoretically necessary. Connor and Korajczyk (1993) develop an asymptotic test {N -► oo) for die adequacy of A factors under the assumption of an approximate factor structure. Their test uses the result that with an approximate factor structure the average cross-sectional variation explained by the A+lst factor approaches zero as JV increases,

where the dependence of bA + t on N is implicit. This implies that in a large cross section generated by a A-faclor model, the average residual variance in a linear factor model estimated with A+l factors should converge to the average residual variance with A factors. This is the implication Connor and Korajczyk test. Examining returns from stocks listed on the New York Stock Exchange and the American Stock Exchange they com hide that there are up to six pervasive factors.

6.4.3 Theoretical Approaches

Theoretically based approaches for selecting factors fall into two main categories. One approach is lo specify macroeconomic and financial market variables that are thought lo capture the systematic risks of the economy. A second approach is to specify characteristics of linns which are likely to explain differential sensitivity to the systematic risks and then form portfolios of stocks based on the characteristics.

Chen, Roll, and Ross (1980) is a good example of the first approach. The authors argue that in selecting factors we should consider forces which will explain changes in the discount rate used to discount future expected cash flows and forces which influence expected cash flows themselves. Based on intuitive analysis and empirical investigation a five-factor model is proposed. The factors include the yield spread between long and short interest rales for US government bonds (maturity premium), expected nidation, unexpected inflation, industrial production growth, and the yield spread between corporate high-and low-grade bonds (default premium). Aggregate consumption growth and oil prices arc found not to have incremental effects beyond the five factors.5

Ли allei-unlive implcnicillalion of the liisl a>iioaeh is (jiven by < :aniibell (ItliMia) and is di.scu.sMd in Chapter H.



6. MultifactorPricingModels

The second approach of creating lactor portfolios based on firm characteristics has heen used in a number of studies. These characteristics have mostly surfaced from the literature ol CAIM violations discussed in Chapter Г). Characteristics which have been found lo be empirically important include market value of equity, price-lo-carnings ratio, and ratio of book value of equity to market value of equity. The general finding is that factor models which include a broad based market poi tlolio (such as an equal-weighted index) and fat tor portfolios created using these characteristics do a good job in explaining the cross section of returns. 1 lowever, because the important characteristics have been identified largely through empirical analysis, their importance may be overstated because of data-snooping biases. We will discuss this issue in Section 6.6.

6.5 Empirical Results

Many empirical studies of multifactor models exist. We will review four of the studies which nicely illustrate the estimation and testing methodology we have discussed. Two comprehensive studies using statistical approaches to select the factors are 1 .ehmann and Modest (1988) and Connor and Ko-rajczyk (1988). I.eluiiaun and Modest I.M] use lactor analysis and Connor and Korajc/vk CK use (7x 7) principal components. Two studies using the theoretical approach lo factor identification are Fama and French (1993) and Chen, Roll, and Ross (1980). Fama and French [FF] use firm characteristics to form factor portfolios and Chen, Roll, and Ross [CRR] soecify inacrocronoiuie variables as factors. The lirel three studies include tests of the implications of exact factor pricing, while Chen, Roll, and Ross focus on whether or not the factors are priced. The evidence supporting exact factor pricing is mixed, fable 6.1 summarizes the main results from l.M, CK, and FF.

A number of general points emerge from this table. The strongest evidence against exact lac lor pricing comes from tests using dependent portfolios based on market value of equity and book-lo-markel ratios. F.vcn multifactor models have difficulty explaining the size effect and book to market effect. Portfolios which are formed based on dividend yield and based on own variance provide little evidence against exact factor pricing. The CK results for January and non-anuary months suggest that (he evidence against exact lac tor pricing does not arise from the January effect.

Using the statistical approaches, CK and l.M find little sensitivity to increasing the number of factors beyond live. On the other hand FF find some improvement going from Iwo lac tors lo live factors. In results not included, FF find dial with stocks only three factors are necessary and that when bond portfolios are included then live factors are needed. These

Table 6.1. Summary of results for tests of exact factor pricing using vero-intertept F-tesl.

Study

Time period

Portfolio characteristic

p-value

64:01-88:12

market value of equity

0.002

0.002

0.236

0.171

0.011

0.019

63:01-82:12

market value of equity

0.11

0.14

0.42

63:01-82:12 dividend yield 5 5 0.17

5 10 0.18

5 15 0.17

20 5 0.94

20 10 0.97 ,

20 15 0.98 !

li:t:01-82:12 own variance 5 5 0.29 j

5 10 0.57

5 15 0.55

20 5 0.83

20 10 0.97

20 15 0.98

t>3:07-91:12

stocks and bonds

0.010

0.039

0.025

I.M 1.M I.M 1.M 1.M I.M

l.M l.M I.M l.M 1.M l.M

**l.cssihan 0.001.

CK refers to Connor and Korajczyk (1988), LM refers to Lehmann and Modest (1988), and FF refers to Fama and French (1993). The CK factors are derived using (Tx7 ) principal components, the 1 At factors are derived using maximum likelihood factor analysis, and the FF factors are prespecified factor portfolios. For the FF two-factor case the factors are the return cjm a portfolio of low market value of equity Krms minus a portfolio of high market value of equity firms and the return on a portfolio of high book-to-market value firms minus a portfolio of low book-to-market value firms. For the three-facior case the factors are ihose in the two-factor case pins the return on the CRSP value-weighted stock index. For the five-factor case the returhs on a term structure factor and a default risk factor are added. CK include tests separating the intercept for January from the intercept for other months. CiU are results of tests of the hypothesis that thejanuary intercept is zero and CK are results of tests of the hypothesis that the non:)amiary intercept is zero. CK and FF work with a monthly sampling interval. LM use a daily interval to estimate the factors and a weekly interval for testing. The lest results from CK and l.M are based on tests from four five-year periods aggregated together. The portfolio characteristic represents the firm characteristic used to allocate slocks into the dependent pnrttblins. FF use 25 stock portfolios and 7 Ixmd portfolios. The slock portfolios are created using a two way sort based on market value of equity and Ixiok-valtte-to-market-value ratios, file bond portfolios include five US government bond portfolios and two corporate bond portfolios. The government bond portfolios are created based on maturity and the corporate bond portfolios are created based on the level of default risk. iv is the number of dependent portfolios and К is i he number of factors. The /rvalues are reported for (tie zero-intercept /-test.



results arc generally consistent with direct tests lor the number of factors liscusscd in Section 6.4.2.

The LM results display considerable sensitivity to the number of depen-Jent portfolios included. The Rvalues arc considerably lower with fewer portfolios. This is most likely an issue of the power of the test. For these :csts with an unspecified alternative hypothesis, reducing the number of portfolios without eliminating the deviations from the null hypothesis can cad lo substantial increases in power, because fewer restrictions must be estcd.

The CRR paper focuses on the pricing of the factors. They use a cross-sectional regression methodology which is similar lo the approach presented n Section 6.3. As previously noted they find evidence of live priced factors. The factors include the yield spread between long and short interest rates or US government bonds (maturity premium), expected nidation, unexpected inflation, industrial production growth, and the yield spread between corporate high-ami low-grade bonds (default premium).

6.6 Interpreting Deviations from Exact Factor Pricing

We have just reviewed empirical evidence which suggests that, while multi-factor models do a reasonable job of describing the cross section of returns, deviations from the models do exist. Given this, it is important lo consider the possible sources of deviations from exact factor pricing. This issue is important because in a given finite sample it is always possible to find an additional factor that will make the deviations vanish. However the procedure of adding an extra factor implicitly assumes that the source of the deviations is a missing risk factor and does not consider other possible explanations.

In this section we analyze the deviations from exact factor pricing for a given model with the objective of exploring the source of the deviations. For the analysis the potential sources of deviations are categorized into two groups-risk-based and n on risk-based. The objective is lo evaluate the plausibility of the argument that the deviations from the given factor model can be explained by additional risk factors.

The analysis relies on an important distinction between the two categories, namely, a difference in the behavior of the maximum squared Sharpe ratio as the cross section of securities is increased. (Recall that the Sharpe ratio is the ratio of the mean excess return to the standard deviation of the excess return.) For ihe risk-based alternatives the maximum squared Sharpe ratio is bounded and for the nonrisk-based alternatives the maximum squared Sharpe ratio is a less useful construct and can, in principle, Le unbounded.

6.6.1 Exact Factor Pricing Models, Mean-Variance Analysis, and the Optimal Orthogonal Portfolio

For the initial analysis we drop back to the level of the primary assets in the economy. Let N be the number of primary assets. Assume that a riskfree asset exists. Let Z, represent the (N x 1) vector of excess returns for period /. Assume Z, is stationary and ergodic with mean ц and covariance matrix П that is full rank. We also take as given a set of A factor portfolios and analyze the deviations from exact factor pricing. For the factor model, as in (6.2.2), we have

Z, = a + BZA, + £,. (6.6.1)

Here В is the (/VxA) matrix of factor loadings, ZSl is the (Axl) vector of time-/ laclor portfolio excess returns, and a and e, are (Nx I) vectors of asset return intercepts and disturbances, respectively. The variance-covariance matrix of the disturbances is E and the variance-covariance matrix of the factors is ClK, as in (6.2.3)-(6.2.6). The values of a, B, and £ will depend oa the factor portfolios, but this dependence is suppressed for notalional convenience.

If we have exact factor pricing relative to the A factors, all the elements of the vector a will be zero; equivalenlly, a linear combination of the factor portfolios forms the tangency portfolio (the mean-variance efficient portfolio of risky assets given the presence of a riskfree asset). Let /.,jt be the excess return of the (ex ante) tangency portfolio and let wr/ be die (Nx 1) vector of portfolio weights. From mean-variance analysis (sec Chapter 5),

w, = (til V) il V (6.6.2)

In the context of the A-factor model in (6.6.1), wc have exact laclor pricing when the tangency portfolio in (6.0.2) can be formed from a linear combination of the A factor portfolios.

Now consider the case where we do not have exact factor pricing, so the tangency portfolio cannot be formed from a linear combination of the factor portfolios. Our interest is in developing the relation between the deviations from the asset pricing model, a, and the residual covariance matrix, E. To facilitate (his, we define the optimal orthogonal portfolio, which is the unique portfolio that can be combined with the A factor portfolios to form the tangency portfolio and is orthogonal to the factor portfolios.

Definition (optimal orthogonal portfolio), lake as given К factor portfolios which cannot be combined to form the tangency port folio or the global minimum-variance portfolio. A portfolio h will be defined as the optimal orthogonal portfolio with respect lo these К factor portfolios if

шч = W/(u> + u>;,< 1 -.ш) (6.6.3)

Sec Roll (I<ISO) lor general piopci lies (it in lliogonal pen ilolioy



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 [ 41 ] 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103