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Unformatted text preview: Session 9: The Single Index Model A.
The single index model (SIM) equation is defined as:
(rit − rf ) = αi + βi (rmt − rf ) + ε it
where: rit is the return on stock i observed in period t (rather than expected at the
beginning of period t),
rmt is the return on a market index observed in period t,
rf is the return on the risk-free asset in the same period,
αi and βi are the stock’s alpha and beta, or the parameters of the SIM
εit is the residual return of stock i in period t.
The characteristic line is expressed as:
Rit = α i + βi (rmt − rf )
where: R is the excess return on stock i in period t that is predicted by the model
it (rather than observed in period t).
While the SIM equation is used (i) to explain the sources of excess stock return
observed in a period and (ii) for estimating the parameters of the model (i.e.,
alpha and beta) by running an OLS regression; the characteristic line is used (i) to
summarise OLS regression results by reporting the linear relationship between
the stock excess return and the market index excess return and (ii) for computing
the stock excess return predicted by the model.
CBA Other Statistics
Residual Variance, σ2ε i
Excess Return Variance, σ2i
Historical Average Excess Return on All Ords C. (r CBA − r f ) = − 0.0010 + 0.60(rAll Ords − r f NAB All Ords -0.0010
(4.10) Coefficients of SIM
t-stat of alpha
t-stat of beta 0.0012
(6.48) - 0.42
0.27% ) D.
Firm specific risk refers to the deviation of actual return from that expected as a
direct result of the arrival of unforeseen firm-specific news such as a better than
expected earnings announcement. BHP has the largest firm-specific risk given its
largest residual variance. Since the residual return captures the difference 1 between the actual return and that predicted by the single index model, the
appropriate measure is the standard deviation or variance of residual return. E.
Market risk refers to the deviation of actual return from that expected as a direct
result of the arrival of unforeseen macro-economic news such as a higher than
expected inflation figure. BHP has the largest market risk given its largest beta.
Since beta measures the relative sensitivity of stock returns to market movements,
it is the appropriate measure.
NAB given its largest R2. As the dependent and independent variables in the SIM
regression are the excess return of the stock and the excess return of the market
index respectively, R2 is the appropriate measure since it reflects the proportion of
variation in the dependent variable being explained by the variation in the
independent variable. G.
NAB given the positive alpha. However, since the t-statistic of alpha is less than
two, alpha is insignificantly different from zero. The CAPM implies that alpha is
zero if the stock is correctly priced. H.
The variance of CBA = 0.0009. This is made up of
(i) systematic risk
= beta2 x variance of returns on market index
(ii) firm-specific risk = residual variance
= 0.000052 (from the answers to Question B) I.
Assuming that the market index excess return over the next period is the same as
the historical average of 0.27%, then according to the SIM,
E (R BHP ) = (rBHP − r f ) = − 0.0015 + 1.70(rAll Ords − r f ) = − 0.0015 + 1.70 × 0.0027 = 0.0031
E (RCBA ) = (rCBA − r f ) = − 0.0010 + 0.60(rAll Ords − r f ) = − 0.0010 + 0.60 × 0.0027 = 0.0007
E (R NAB ) = (rNAB − r f ) = 0.0012 + 1.21(rAll Ords − r f ) = 0.0012 + 1.21× 0.0027 = 0.0045
Thus E(Rp) = wBHP E(RBHP) + wCBA E(RCBA) + wNAB E(RNAB)
= (0.3x0.31% + 0.5x0.07% + 0.2x0.45%)
βp = wBHP βBHP + wCBA βCBA + wNAB βNAB
= (0.3x1.70 + 0.5x0.60 + 0.2x1.21) = 0.2145%
= 1.05 2 σ ε2 P 2
= wBHPσ ε2BHP + wCBAσ ε2CBA + wNABσ ε2NAB = (0.32x0.000259 + 0.52x0.000052 + 0.32x0.000085)
σp ( 22
= β Pσ m + σ ε2P ) 1 = 0.000040 2 =( 1.052 x 0.000106 + 0.000040)1/2 = 1.2512% J.
The inputs include (i) the stock alpha, (ii) the stock beta, and (iii) a forecast on the
excess market index return over the forthcoming period. The stock alpha and beta
are estimated by running an OLS regression on the single index model using a set
of historical data. The three inputs are then applied to the characteristic line to
obtain the expected return as follows:
E (Ri ) = (ri − r f ) = α i + β i (rm − r f ) 3 ...
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This note was uploaded on 10/29/2010 for the course FINS 2624 taught by Professor Hneryyip during the Three '10 term at University of New South Wales.
- Three '10