Download CHAPTER 10

CHAPTER 9
MULTIFACTOR MODELS OF RISK AND RETURN
Answers to Questions
1.
Both the Capital Asset Pricing Model and the Arbitrage Pricing Model rest on the
assumption that investors are reward with non-zero return for undertaking two activities:
(1) committing capital (non-zero investment); and (2) taking risk. If an investor could
earn a positive return for no investment and no risk, then it should be possible for all
investors to do the same. This would eliminate the source of the “something for nothing”
return.
In either model, superior performance relative to a benchmark would be found by positive
excess returns as measured by a statistically significant positive constant term, or alpha.
This would be the return not explained by the variables in the model.
2.
CFA Examination III (1989)
2a.
The Capital Asset pricing Model (CAPM) is an equilibrium asset pricing theory showing
that equilibrium rates of expected return on all risky assets are a function of their
covariance with the market portfolio. The CAPM is a single-index model that defines
systematic risk in relation to a broad-based market portfolio (i.e., the market index). This
single factor (“beta”) is unchanging:
Rj = Rf + Bj(Rm – Rf)
where
Rj
Rf
Rm
Bj
= expected return on an asset or portfolio
= risk-free rate of return
= expected return on the market
= volatility of the asset or portfolio to that of the market m.
Arbitrage Pricing Theory (APT) is an equilibrium asset pricing theory derived from a
factor model by using diversification and arbitrage. The APT shows that the expected
return on any risky asset is a linear combination of various factors. That is, the APT
asserts that an asset’s riskiness and, hence, its average long-term return, is directly related
to its sensitivities to certain factors. Thus, the APT is a multi-factor model which allows
for as many factors as are important in the pricing of assets. However, the model itself
does not define these variables. Unlike the CAPM, which recognizes only one
unchanging factor, the key factors in APT can change over time.
9-1
Rj = Rf + Bj1(RF1 – Rf) + … + Bjk(RFk – Rf)
where
Rj
Rf
Bj
RFk
j
k
= return on an asset
= risk-free rate of return
= sensitivity of an asset to a particular factor
= expected return on a portfolio with an average (1.0) sensitivity to a factor k
= an asset
= a factor
Research suggests that several macroeconomic factors may be significant in explaining
expected stock returns (i.e., these factors are systematically priced):
(1)
(2)
(3)
(4)
Inflation;
Industrial production;
Risk premia as measured by the spread between low and high grade bonds;
Yield curve, (i.e., slope of the term structure of interest rates.
Other researchers have identified additional factors which may influence an asset’s return.
(5)
(6)
(7)
(8)
Real GNP growth;
Rate of growth of real oil prices (i.e., an energy factor);
Real defense spending;
Market index.
2b.
Because of APT’s more general formulation, it is more robust and intuitively appealing
than the CAPM. Many factors, not just the market portfolio, may explain asset returns.
This permits the stock selection process to take into account as many economic variables
as are believed to be significant in a valuation context – not only as to individual issues,
but also as to groups, sectors or even the market as a whole. For example, given a forecast
of a sudden spurt in the inflation rate, and of the resulting effect on interest rates, the
analyst or portfolio manager can, via APT, arrive at an estimate of valuation changes
across his/her selection universe and adjust portfolio exposures accordingly.
Alternatively, stocks could be selected and portfolios formed based on specific factorsensitivity criteria set up in advance.
3.
The small firm effect refers to the tendency of small capitalization stocks to outperform
large capitalization stocks. In and of itself, such evidence would not necessarily constitute
evidence of market inefficiency, because most tests of such “anomalies” are in fact joint
tests of two elements: (1) an asset pricing model (e.g., the CAPM or APT) which reflects
the risk-return tradeoff; and (2) market efficiency, in the sense of prices reflecting some
set of information. If a test fails to account for such anomalies, it could be due to either
(1) a mis-specified asset pricing model; or (2) market inefficiency, or both.
9-2
4.
Studies of the efficient markets hypothesis suggest that additional factors affecting
estimates of expected returns include firm size, the price-earnings ratio, and financial
leverage. These variables have been shown to have predictive ability with respect to
security returns.
5.
CFA Examination II (1998)
5(a).
APT vs. the CAPM
Arbitrage pricing theory does not include any of the following three assumptions incorporated in the capital asset
pricing model (CAPM): noted as parts i, ii, and iii:
i.
Investor utility function or quadratic utility function. Capital market theory assumes investors want to
maximize utility in terms of risk and return preferences; maximum return per unit of risk or minimum risk per unit of
return. From the Markowitz model forward, relevant risk has been measured by variance of returns or standard
deviation.
APT makes no assumptions regarding investor preferences; the multifactor model commonly used in APT does
not include any exponents higher than 1.
ii. Normally distributed returns. The probability distribution of expected returns of an
investment and the associated dispersion or variability of those returns form the basis
of Markowitz portfolio theory and the CAPM. Normal or symmetrically distributed
security returns enable estimation of a variance term. In the CAPM, all investors have
identical estimates for the probability distributions of future returns (homogeneous
expectations).
APT does not describe or specify or require an assumption about security return
distributions of any kind.
iii. The market portfolio. The CAPM assumes that pricing, valuation, risk, and return are
solely functions of an asset’s relationship to a market portfolio of all risky assets. In
practice, the market portfolio is difficult to specify, so a mistakenly specified market
portfolio might result. Roll called this misspecification “benchmark error.”
APT does not consider or include an assumption of a market portfolio. APT is
predicated on a common set of several (macroeconomic) factors.
5(b).
Conceptual Difference between APT and the CAPM
Conceptually, in APT, return is a function of a set of common factors. In the CAPM,
return is a function of a market portfolio of all risky assets. Thus, one difference between
APT and the CAPM can be described by the fact that APT is a multifactor model that
attempts to capture several non-market influences that cause securities or assets to change
in price whereas the CAPM is a single-index model that assumes securities or assets
change in price because of a common co-movement with one market portfolio of all risky
assets.
Another conceptual difference between APT and the CAPM is that, in application of the theory, the market portfolio
(or “factor”) required by the CAPM is specified. In APT, the common factors are not identified, but the common
factors in APT are often described or accepted as including inflation or unanticipated deflation; default risk,
9-3
government corporate security spread, risk premiums or interest rate spreads; changes in the term structure of interest
rates; changes in real final sales, GDP growth or a similar proxy for long-run profits on an economy-wide basis;
major political upheavals; and exchange rates.
A third difference between APT and the CAPM lies in the incorporation of sensitivity
coefficients to measure or describe the risk of assets or securities. APT incorporates a
number of sensitivity coefficients. These coefficients determine how each independent
variable or macroeconomic factor affects each asset. Different assets are affected to
different degrees or extents by the common factors. In the CAPM, the only sensitivity
factor is beta, an asset’s sensitivity to the changes in the market portfolio (often called an
asset’s “systematic risk”).
6.
A market factor of 1.2 means the mutual fund is 1.2 times as sensitive as the market
portfolio, all other factors held equal. The SMB (“small minus big”) factor is the return of
a portfolio of small capitalization stocks minus the return to a portfolio of large
capitalization stocks. A value of –0.3 indicates the fund tends to react negatively to small
cap factors; thus the fund is primarily invested in large cap stocks. The HML (“high
minus low”) factor is the return to a portfolio of stocks with high ratios of book-to-market
value less the return to a portfolio of low book-to-market returns. A value of 1.4 indicates
the fund reacts positively to this factor; thus the fund is weighted toward high value
stocks.
7.
CFA Examination III (1990)
The value of stock and bonds can be viewed as the present value of expected future cash
flows discounted at some discount rate reflecting risk. Anticipated economic conditions
are already incorporated in returns. Unanticipated economic conditions affect returns.
Industrial production. Industrial production is related to cash flows in the traditional
discounted cash flow formula. The relative performance of a portfolio sensitive to
unanticipated changes in industrial production should move in the same direction as the
change in this factor. When industrial production turns up or down, so too shares in the
return on the portfolio. Portfolios sensitive to unanticipated changes in industrial
production should be compensated for the exposure to this economic factor.
Inflation. Inflation is reflected in the discount rate, which generally is as follows: K = real
return + hedge for inflation + risk premium. Unexpected inflation will quickly be factored
into k by the market as investors attempt to hedge the loss of purchasing power. Thus, the
discount rate would move in the same direction as the change in inflation.
High inflation could also affect cash flows in the DCF formula, but the effect will
probably be more than offset by higher Ks nominal cash flow growth rates do not always
match expected inflation rates. If the growth in nominal cash flows lags the inflation rate,
the relative performance of a portfolio sensitive to rising inflation should decline over
time. Investments in bonds are subject to significant adverse inflation effects. Hence,
higher unanticipated inflation will negative affect portfolio values.
9-4
Risk premia or quality spreads. Risk premia affect the magnitude of the DCF discount
rate. The risk premium measure represents investor attitudes toward risk-bearing and
perceptions about the general level of uncertainty. When the return in low versus high
quality bonds widens, there is likely to be a negative impact on the values of stock and
bonds, particularly for lower quality companies. Riskier cash flows require higher
discount rates, and widening quality spreads often signal greater uncertainty about profit
levels and debt service requirement.
Yield curve shifts. Yield curve shifts affect the discount rate. If a parallel upward shift
occurs, it would mean investors are requiring higher return to hold all assets. Hence, with
high discount rates, portfolio values would fall. If the curve becomes steeper, longer
duration assets, such as long-term bonds and growth stocks, would be negatively affected
more than shorter-term assets.
8.
The macroeconomic approach to identifying the factors in a multi-factor asset pricing
model tries to find variables that explain, the underlying reasons for variations in the cash
flows and investment returns over time. The microeconomic approach concentrates on the
characteristics of the securities themselves.
Conceptually, it is possible for the two approaches to lead to the same estimate for
expected returns. Practically speaking, it is not likely that they will. The two sets of
factors, while probably significantly correlated, are not likely to be perfectly correlated,
leading to differences in estimates.
9.
The three factors used by Fama and French in heir microeconomic model are the excess
market return; SMB (“small minus big”) – he difference between the returns to a portfolio
of small capitalization stocks and a portfolio of large capitalization stocks; and HML
(“high minus low”) - the difference in returns to a portfolio of high book-to-market value
stocks and low book-to-market stocks. This type of approach focuses on the various
characteristics of the underlying securities. The macroeconomic approach, as its name
implies, concentrates on more general, economy-wide variables that cause variation in the
cash flows and investment returns.
10.
Multifactor model typically include a market portfolio return as one of the factors. Thus
the coefficients on the other factors in the model indicate the marginal sensitivity of
returns to those specific factors. Higher values indicate greater sensitivity of returns to
that factor.
In a macroeconomic-based model such as the one developed by Chen, Roll, and Ross,
returns on a particular security might be much more sensitive to the credit spread than to
one of the other macro variables. The investor could then assess whether that security fit
his particular investment profile.
9-5
Likewise, a microeconomic-based model would identify sensitivity to certain
characteristics, such as small vs. large firm, etc. Again, the investor could then select
securities based on the sensitivities to the various characteristics.
11.
11(a). One study that does not support the APT is that of Reinganum on the small-firm effect.
Reinganum hypothesized that if the APT were a superior theory to CAPM, the APT
would explain the small-firm effect where the CAPM could not. However, his results did
not support that hypothesis. The small-firm portfolio still had significant positive excess
returns, while the large firm portfolio had significant negative excess returns.
A study that supports the APT is that of Cho, Elton, and Gruber. Their study tended to
support the basic contention of the APT that factors other than the market portfolio affect
security returns.
11(b). Shanken’s contention that the APT is not testable is in essence that the APT is not
falsifiable. If a test finds that returns are not explained by a particular set of factors, that is
not taken as evidence that the APT is incorrect, but that the particular set of factors is
wrong. On the other hand, if a study finds that some other set of factors does explain
returns, that is taken as evidence if favor of the APT. The APT gives no guidance as to
the proper set of factors, so in that sense it cannot be falsified.
Dybvig and Ross counter this argument by stating that the PT is testable as an equality,
rather than the “empirical APT” of Shanken.
12.
CFA Exam III (1986)
12(a). The basic Capital Asset Pricing Model (CAPM) assumes that investors care only about
portfolio risk and expected return; i.e., they are risk averse. From this assumption comes
the conclusion that a portfolio's expected return will be related to only one attribute - its
beta (sensitivity) relative to the broadly based market portfolio.
Arbitrage Price Theory (APT) takes a different approach: it is not much concerned about
investor preferences, and it assumes that returns are generated by a multi-factor model.
APT reflects the fact that several major (systematic) economic factors may affect a given
asset in varying degrees. Further, unlike the CAPM, whose single factor is unchanging,
APT recognizes that these key factors can change over time (as can investor preferences).
Summarizing, APT 1) identifies several key systematic macroeconomic factors as part of
the process that generates security returns vs. only one factor recognized by the CAPM, 2)
recognizes that these key factors can change over time, whereas the CAPM’s single factor
is unchanging, 3) makes fewer assumptions about investor preferences than the CAPM,
and 4) recognizes that these preferences can change over time.
9-6
12(b). The four systematic factors identified by Roll and Ross are unanticipated changes in: 1)
inflation, 2) industrial production, 3) risk premiums, and 4) the slope of the term structure
of interest rates.
The APT asserts that an asset’s riskiness and, hence, the average long-term return, is
directly related to its sensitivities in unanticipated changes in these four economic
variables. This relationship may be expressed in equation form as follows:
Ri = R0 + bi1 F1 + bi2 F2 + . . . + bin Fn
This means the return (Ri) that a certain asset (i) will produce is a combination of some
“base” return (R0) plus returns occasioned by the influences or sensitivities (bn) of some
systematic external factors (Fn).
13.
CFA Examination II (2001)
Argument
a. Both the
CAPM and APT
require a meanvariance efficient
market portfolio.
b. Neither the
CAPM nor APT
assumes normally
distributed
returns.
c. The CAPM
assumes that one
specific factor
explains security
returns, but APT
does not.
State whether
Argument is Correct or
Incorrect
Indicate, for each incorrect
argument, why the argument is
incorrect
Incorrect
CAPM requires the mean-variance
portfolio but APT does not.
Incorrect
Correct
9-7
CAPM assumes normally
distributed security returns, but
APT does not.
CHAPTER 9
Answers to Problems
1.
1(a).
In general for the APT, E(Rq) = 0 + 1bq1 + 2bq2
For security J:
E(RJ) = 0.05 + 0.02x0.80 + 0.04x1.40
= 0.05 + 0.016 + 0.056
= .1220 or 12.20%
For Security L:
E(RL) = 0.05 +0.02x1.60 + 0.04x2.25
= 0.05 + 0.032 + 0.09
= .172 or 17.20%
1(b).
Total return = dividend yield + capital gain yield
For security J, the dividend yield is $0.75/$22.50 = 0.033 or 3.33%
For security L, the dividend yield is $0.75/$15.00 = 0.05 or 5%
For security J, the expected capital gain is therefore 12.20%– 3.33% = 8.87%
For security L the expected capital gain is therefore 17.20% - 5.00% = 12.20%
Therefore:
The expected price for security J is $22.50x(1.0887) = $25.50
The expected price for security L is $15.00x(1.1220) = $16.83
2.
2(a).
For 1981-2000:
RINTC
= 0.615x9.09 - 0.64x(-1.1) – 1.476x4.48
= 5.59 + 0.704 – 6.61
= -0.32%
RJPM
= 1.366x9.09 – 0.387x(-1.1) + 0.577x4.48
= 12.41 + 0.43 + 2.59
= 15.43%
RWFMI
= 1.928x9.09 +0.817x(-1.1) +1.684x4.48
= 17.53 - 0.90 + 7.55
= 24.18%
9-8
For 1928-2000:
RINTC
= 0.615x7.02 – 0.64x3.09 –1.476x4.39
= 4.32 – 1.98 – 6.48
= -4.14%
RJPM
= 1.366x7.02 –0.387x3.09 +0.577x4.39
= 9.59 – 1.20 + 2.53
= 10.92%
RWFMI
= 1.928x7.02 + 0.817x3.09 + 1.684x4.39
= 13.53 + 2.52 + 18.66
= 34.71%
2(b).
The excess return on Intel (INTC) in the 1981-2000 period seems reasonable, in that it is
very close to zero. INTC’s excess return in the 1928-2000 period is unusual in that it is
negative. The excess returns of J. P. Morgan (JPM) and Whole Foods (WFMI) appear to
be extremely large. This is partly due to the fact that we are using out-of-sample numbers
to do the estimating. That is, the regressions were estimated using 1996-2000 data, but
the estimates were made using data from much longer periods.
2(c).
No, we wouldn’t expect the factor betas to remain constant over time. The sensitivity of a
particular company’s return to a specific factor will change as the character of the firm
changes. For instance, growth companies don’t remain growth companies forever, but
tend to mature. Thus their factor betas would change to reflect the slowdown in growth
but increased stability of earnings.
3(a).
RQRS
= 4.5 +7.5x1.24
= 4.5 + 6.825
= 13.8%
RTUV = 4.5 + 7.5x0.91
= 4.5 + 6.825
= 11.325%
RWXY = 4.5 + 7.5x1.03
= 4.5 +7.725
= 12.225%
3(b).
RQRS = 4.5 + 7.5x1.24 + (-0.3)x(-0.42) + 0.6x0.00
= 4.5 + 9.30 + 0.126 +0.00
= 13.926%
RTUV
= 4.5 + 7.5x0.91 + (0.3)x(0.54) + 0.6x0.23
= 4.5 + 6.825 – 0.162 + 0.138
= 11.301%
9-9
RWXY = 4.5 + 7.5x1.03 + (-0.3)x(-0.09) + ).6x0.00
= 4.5 + 7.725 + 0.027 + 0.00
= 12.252%
3(c).
Assuming that the factor loadings are significant the three factor model should be more
useful to the extent that the non-market factors pick up movements in returns not
captured by the market return.
3(d).
Because the factor loadings on MACRO2 are zero for two of the stocks, it appears that
MACRO2 is not a systematic factor, i.e., one that generally affects all stocks. It may
represent industry- or firm-specific factors.
4(a).
E(RD) = 5.0 + 1.2x1 + 3.4x2 = 13.1%
E(RE) = 5.0 + 2.x + 2.x
Solving the second equation for 1 in terms of 2, we get:
1 = [(10.4 – 2.6)/2.6]2
Substituting that into the first equation:
1.2[(10.4 – 2.6)/2.6]x2 + 3.4x2 = 8.1
Solving for 2, we find 2 = 1.5. Using this value we can determine from either equation
that 1 is equal to 2.5.
4(b).
Because neither stock pays a dividend, the total return is all due to price appreciation.
Therefore for stock D:
P0x(1.131) = $55
P0 = $55/1.131
= $48.63
And for stock E:
P0x(1.154) = $36
P0 = $36/1.154
= $31.20
4(c).
From part (a), the risk premium for factor 1 was 2.5%. The new risk factor is thus 2.5% +
0.25%, or 2.75%. The new expected returns are:
9 - 10
E(RD) = 5.0 + (1.2x2.75) +(3.4x1.5)
= 5.0 + 3.3 + 5.1
= 13.4%
E(RE) = 5.0 + (2.6x2.75) + (2.6x1.5)
= 5.0 + 7.15 + 3.9
= 16.05%
4(d).
5(a).
D:
PD0(1 + 0.134) = $55
PD0 = $55/1.134
PD0 = $48.50
E:
PE0(1 + .1605) = $36
PE0 = $36/(1.1605)
PE0 = $31.02
Because no stock pays a dividend, all return is due to price appreciation.
E(RA) = 1.1x0.04 + 0.8x0.02
= 0.044 + 0.016
= 0.06 or 6%
E(Price A) = $30(1.06) = $31.80
E(RB) = 0.7x0.04 + 0.6x0.02
= 0.28 + .012
= 0.04 or 4%
E(Price B) = $30(1.04) = $31.20
E(RC) = 0.3x0.04 + 0.4x0.02
= 0.12 + 0.008
= 0.02 or 2%
E(Price C) = $30(1.02) = $30.60
5(b).
In order to create a riskless arbitrage investment, an investor would short 1 share od A
and one share of C, and buy 2 shares of B. The weights of this portfolio are WA = -0.5,
WB = +1.0, and WC = -0.5. The net investment is:
Short 1 share A = +$30
Buy 2 shares B = - $60
Short 1 share C = +$30
Net investment = $ 0
9 - 11
The risk exposure is:
Risk Exposure
A
B
C
Net Risk Exposure
Factor 1
(-0.5)x1.1
(+1.0)x0.7
(-0.5)x0.3
0
Factor 2
(-0.5)x0.8
(+1.0)x0.6
(-0.5)x0.4
0
At the end of the period the profit is given by:
Profit = ($30 - $31.50) + 2x($35 – 30) + ($30 - $30.50)
= -$1.50 + $10 - $0.50
= $8
6(a).
Using a spreadsheet program we find for monthly returns:
Portfolio A: RA = 0.0187 or 1.87%
Portfolio B: RB = 0.01497 or 1.497%
Factor 1 average = 0.01148 or 1.148%
Factor 2 average = 0.00035 or 0.035%
Factor 3 average = -0.01287 or –1.287%
Standard deviation A = 0.0559 or 5.59%
Standard deviation B = 0.0465 or 4.65%
Standard deviation 1 = 0.053 or 5.3%
Standard deviation 2 = 0.069 or 6.9%
Standard deviation 3 = 0.0497 or 4.97%
At annual rates:
Portfolio A: RA = 22.44%
Portfolio B: RB = 17.96%
Factor 1 average = 13.78%
Factor 2 average = 0.42%
Factor 3 average = -15.44%
Standard deviation A = 19.36%
Standard deviation B = 16.11%
Standard deviation 1 = 18.36%
Standard deviation 2 = 23.90%
Standard deviation 3 = 17.22%
6(b).
It is not clear from the numbers in part a whether one portfolio outperformed the other
because A has higher return and higher risk.
6(c).
r(1,2) = 0.2207
r(1,3) = -0.5505
r(2,3) = -0.7531
These answers were obtained using a spreadsheet program.
6(d).
In theory the correlations should be zero, because we want the factors to be independent
of each other.
6(e).
Factors 1 and two are not highly correlated, but it appears that there is significant
correlation between factors 1 and 3 and factors 2 and 3. Statistically speaking, this leads
to problems of multicollinearity, which would affect regression estimates.
9 - 12
7.
7(a).
Using a basic regression package, we get the following results (t-statistics given in
parentheses):
RA = .0057 + 0.99x(Factor 1) - 0.201x(Factor 2) - 0.133x(Factor 3)
(2.92*) (22.27*)
(-4.61*)
(-1.90)
adj. R2 = .967
RB = .0080 + 0.969x(Factor 1) + 0.059x(Factor 2) + 0.344x(Factor 3)
(2.85*) (14.67*)
(0.91)
(3.29*)
adj. R2 = .900
*Significant at 5% level of significance.
7(b).
The adjusted R2’s in both regressions are very high (.967 and .900, respectively.) This
means the regressions account for over 90% of the variation of each portfolio’s returns
around its respective mean.
7(c).
Factor 1 is the most likely candidate for the market factor, because it has a large,
significant, and positive effect on both portfolios. Factors 2 and 3 have different signs in
the two equations.
7(d).
A positive HML factor loading indicates a value stock or portfolio. Portfolio B has a
positive loading on this factor, and therefore is the more likely candidate for the valueoriented portfolio. A negative HML factor loading indicates a growth stock or portfolio.
Portfolio A has a negative loading on this factor, and therefore is more likely to be a
growth-oriented portfolio.
9 - 13