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Unformatted text preview: UNIVERSITY OF NORTH CAROLINA
KENAN-FLAGLER BUSINESS SCHOOL
BUSI 580: INVESTMENTS
PART I: PORTFOLIO THEORY AND ASSET PRICING
Prof. Günter Strobl Spring 2010 Solution to Problem Set 3
A. The Arbitrage Pricing Theory
1. (a) In a two-factor economy, the expected return of any well-diversified portfolio is
E (rp ) = rf + λ1 b p ,1 + λ 2 b p , 2 . Thus, in order to find the factor risk premia, we have to solve the following
system of two equations in two unknowns: 13.1% = 5% + 1.2 × λ1 + 3.4 × λ 2
15.4% = 5% + 2.6 × λ1 + 2.6 × λ 2
The solution to this set of equations is λ1 = 2.5% (risk premium for factor F1) and
λ2 = 1.5% (risk premium for factor F2).
(b) If the risk premium for F1 increases from 2.5% to 2.75%, then the expected
returns for portfolios X and Y are:
E(rX ) = 5% + 1.2 × 2.75% + 3.4 × 1.5% = 13.4% E(rY ) = 5% + 2.6 × 2.75% + 2.6 × 1.5% = 16.05% 1 B. Testing Multifactor Models
2. Please see the Excel spreadsheet PS3_solution.xls.
(a) Of the five stocks in our sample, only IBM and Eastman Kodak do not show an
intercept coefficient that is significantly different from zero at the 5% level.
Merck and Coca Cola have significantly positive intercepts, so these stocks are
underpriced relative to the three-factor model. General Motors has a significantly
negative intercept. Given that the three-factor model is rejected for three of the
five stocks in our sample, we can probably reject the three-factor model (although
we would need to run a joint test of zero intercepts, to be certain).
The R2’s for the individual stocks range from a low of 27.7% for Eastman
Kodak to a high of 36.9% for IBM, so the three-factor model is able to account
for about a third of the monthly variation in (excess) returns for the individual
Based on the slope coefficients for SMB, all of the stocks except for General
Motors behave like large-cap stocks. Based on the slope coefficients for HML,
General Motors behaves like a value stock, whereas the other four stocks behave
like growth stocks.
(b) Of the five stocks in our sample, three (IBM, Merck, and Coca Cola) have an
intercept coefficient that is significantly different from zero. In all three cases, the
intercept coefficient is positive, which indicates that the stocks are underpriced
relative to the four-factor model. Based on this evidence, a joint test of zero
intercepts is very likely to reject the four-factor model.
Overall, the four-factor model does not do a better job of explaining the
variation in returns than the three-factor model. Except for IBM, the R2’s of the
two models are virtually the same.
(c) In a factor-pricing model, alpha represents the pricing error, while the factor
sensitivities represent exposure to one or more sources of systematic risk. The one
risk factor common to the one-, three-, and four-factor models is market risk.
Across all three models, the pricing errors are relatively constant. In particular,
Merck and Coca Cola are significantly underpriced relative to all three models.
The apparent failure of the three- and four-factor models to improve on the
one-factor model is accounted for by the relative constancy of the factor
sensitivities across stocks. In the three-factor model, the additional risk factors
represent size and value risk. However, the stocks in our sample are primarily
large-cap stocks with only slight variation in terms of value risk. In the fourfactor model, the additional risk factor represents momentum risk, but only two of
the stocks in our sample exhibit significant exposure to this source of systematic
risk. 2 ...
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- Fall '09