344 RISK BUDGETING measure of the linear relationship between excess returns and the return on the market portfolio (over the

investment funds

risk-free rate). Although simple, the market model may not offer the practitioner a useful way to measure and explain risk. A manager may mistakenly select the wrong market portfolio in the analysis or may simply be interested in a richer model to help explain sources of risk and return. Also, Fama and French (1996) have shown that the market portfolio does a rather poor job at explaining movements in individual stock returns. The market return is not the only factor that may explain movements in excess stock returns, and therefore more factors are needed. Macroeconomic Factors It is natural to think that stock returns reflect the state of the economy so that various measures of macroeconomic conditions serve as a basis for a set of additional factors. Chen, Roll, and Ross (1986) have investigated whether macroeconomic factors can explain stock returns. Examples of macroeconomic factors include: (1) the growth rate in monthly industrial production; (2) a measure of default premium (discussed earlier), measured as the difference between the monthly return on a high-yield bond index and the return on long-term government bonds; (3) the real interest rate; (4) the maturity premium, measured as the difference between return on the long-term government bond and the one-month Treasury bill return; and (5) the change in monthly expected inflation. We incorporate macroeconomic factors into the market model as follows. Assume that, in addition to the market factor, there are K - 1 other factors that impact the nth security's excess return at time t. These additional factors enter into the market model through the residual or error, which for each security reflects the extent to which a stock's return is out of alignment with the expected relationship to the market portfolio return. Residual returns for common stocks arise in part from common factors that extend across many stocks, and in part from specific returns, which are unique to an individual company. Taking these issues into consideration, the market model now takes the following form: rjt) - At) = ajt) + $n(t)[f"(t) - rf{t)} + ejt) (20.9) en{t) = ln^)fl{t) + lnM2(t) + --- + ln,K-l(t)fK-l(t) + Un(t) (20.10) where fk(t) = Return on the kxh macroeconomic factor at time t jn k{t) = Loading (exposure) of the kxh factor on the nxh asset ujt) = nxh security's idiosyncratic return Note that in (20.10) we no longer assume that the residual error term, ejt), has a zero mean. In fact, its expected value will depend on the macroeconomic factor returns and factor loadings. Combining (20.9) and (20.10), we get the standard form of the so-called market model: rjt) - At) = ajt) + Vn(t)[t"(t) - At)] + Jjmt) + yjt)f2(t) + ... + yniKJt)fKJt) + ujt) {20A1) Time series regression methods can be used to estimate (20.11).

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