Week 9: Week 9 Single Factor Models, Single Index Models, and CAPM

Question 1

What is the Single Factor Model (SFM) ?

Answer 1

The Single Factor Model (SFM) assumes that all firms in the stock market are influenced by a single common economic factor—a proxy for systematic risk (e.g., interest rates, business cycles).

• Beta (β) measures a stock’s sensitivity to changes in this factor.
• All non-systematic risk (firm-specific variation) is treated as uncorrelated noise—e.g., beef prices for Tesco or coffee bean costs for Costa.
• The SFM simplifies market behavior by summarizing all systematic risk with one macroeconomic indicator.

Question 2

What is the Single Index Model (SIM)?

Answer 2

The way of making the SFM works, is to assert that the return on a broad index of securities is a valid proxy for the common macro factors .i.e. replace the common macro factor (F) with a broad index of securities that can mimic the common macro factor such as the FTSE100 or S&P500 indexes.

Question 3

what are the advanatges of the single index model?

Answer 3

The SIM:

Reduces the number of estimates required, compared to the efficient frontier of Markowitz.

In Markowitz’s approach, analysts cannot specialise by industry.

Question 4

What are the drawbacks of the single index model?

Answer 4

Drawbacks:

Uncertainty is classified into macro vs. firm-specific risk – is this realistic?

What about industry events that affect firms within an industry without affecting the broad economy?

e.g. a new method to store hydrogen safely would affect car and oil industries without affecting the whole economy.

Can ignore negative correlation between securities (i.e., the effect of diversification).

So, the SIM optimal portfolio can be significantly inferior to that of the Markowitz model when stocks are correlated.

Question 5

as n increases what happens to total vairance in the single index model?

Answer 5

As n increases, total variance approaches the systematic variance – that is why we use a broad index and remove non-systematic risk.

Question 6

What are the eight main assumptions of CAPM?

Answer 6

Main Assumptions of CAPM:

1) Many investors
Individual wealth is small – price takers, i.e., they cannot dramatically influence the price of any asset to their advantage.

2) Investors plan for only one holding period. Myopic behaviour – no long term, multi-period investing.

3) Investments are limited to a universe of publicly-traded financial assets.
Stocks, bonds, risk-free assets.
Rules out investment in non-traded assets.

4) No transaction costs and taxes paid by investors.

5) All investors are rational mean-variance optimizers. All use the Markowitz portfolio selection model and want to get the highest return for a level of risk.

6) All investors analyse securities the same way. Derive the same input list into the Markowitz model. Homogeneous expectations – same expectations.

7) All investors choose to hold a portfolio of risky assets in proportions identical to the market portfolio (M).

8) M (the portfolio that has the highest Sharpe ratio) will be the tangency portfolio to the optimal Capital Allocation Line (CAL). Capital Market Line is the best attainable Capital Allocation Line (CAL).

Question 7

what is the CAPM?

Answer 7

CAPM: an asset’s risk premium is due to its contribution to the risk of investors’ overall portfolio.

Question 8

what is the Security Market Line?

Answer 8

Expected return-beta relationship can portrayed graphically by the security market line (SML) which Depicts the relationship between beta and expected returns.

The security market line (SML) graphs the individual asset’s risk premiums as a function of their risk.

The relevant measure of risk for individual assets held as parts of well-diversified portfolios is not their standard deviation or variance.

Rather it is the contribution of the asset to the portfolio’s variance, i.e., beta.

Question 9

What is the capital market line (CML) ?

Answer 9

The capital market line (CML) graphs the risk premiums of efficient portfolios (portfolios composed of the market and the riskless asset) as a function of standard deviation.

Question 10

where are overpriced, underpriced and fairly priced assets in th3 security makret line?

Answer 10

The SML provides the required rate of return necessary to compensate investors for both risk as well as the time value of money.

Fairly priced assets should plot exactly on the SML, that is, their expected returns are commensurate with their risk.

Under-priced assets, which provide an expected return in excess of the fair return stipulated by the SML, should be plotted above the SML (A).
That is, given their betas, their expected returns are greater than dictated by the CAPM.

Overpriced stocks plot below.

The difference between the SML and a stock is alpha.

Question 11

what is the difference between the SML and the stock?

Answer 11

The difference between the SML and a stock is alpha.

Question 12

What are the Key implications of CAPM?

Answer 12

Key implications of CAPM:

1) The market portfolio is efficient (i.e., it is the tangency portfolio, and it lies on the minimum variance frontier).

2) Expected returns on all assets are linearly related to their betas, and no other variable has marginal explanatory power.

3) The beta premium (i.e., excess return of the market portfolio) is positive.

4) Assets uncorrelated with the market have expected returns equal to the risk-free interest rate.

Question 13

What does CAPM suggest about the factors driving excess stock returns?

Answer 13

CAPM suggests that excess stock returns are driven by only one systematic risk fact

Question 14

What is the Single Index Model (SIM)?

Answer 14

The Single Index Model (SIM) is used where the market portfolio
𝑀
M is proxied by a broad, value-weighted stock-index portfolio.

Question 15

What is the form of the Single Index Model (SIM)?

Answer 15

The form of the SIM is:

𝑟𝑖−𝑟𝑓=𝛼𝑖+𝛽𝑖 (r𝑚− r𝑓)+𝑒𝑖

Question 16

What is the relationship between firm-specific residuals in the CAPM model?

Answer 16

Each firm-specific residual is uncorrelated across stocks and uncorrelated with the market factor.

Question 17

How is the total risk of a stock determined in the CAPM model?

Answer 17

The total risk of the stock is the sum of the variance of the systematic components and the variance of the firm-specific residual

𝜎𝑖²=𝛽𝑖² 𝜎𝑚²+𝜎𝑖² (𝑒𝑖)

Question 18

What is the expected return formula for a stock in the CAPM model?

Answer 18

The expected return of the stock is:

𝐸(𝑟𝑖)=𝛼𝑖+𝛽𝑖 𝐸(𝑟𝑚)

Question 19

What are the two considerations investors face when forming portfolios?

Answer 19

the two considerations investors face when forming portfolios:

1) Diversify away non-systematic risk.

2) Choose stocks with positive alphas to increase the portfolio’s risk premium.

Question 20

What happens when investors buy positive-alpha stocks and short negative-alpha stocks?

Answer 20

Investors buy positive-alpha stocks, increasing their price, and short negative-alpha stocks, causing their prices to fall, until all alpha values are zero.

Question 21

What is the optimal risky portfolio when all stocks have zero alphas?

Answer 21

The optimal risky portfolio is the market portfolio, when all stocks have zero alphas.

Question 22

What are the key implications of CAPM for testing its validity?

Answer 22

the key implications of CAPM for testing its validity:

1) Expected returns on assets are linearly related to their betas, with no other variable having marginal explanatory power.

2) The beta premium (excess return of the market portfolio) is positive.

3) Assets uncorrelated with the market have expected returns equal to the risk-free rate.

Question 23

How do you test the CAPM? if CAPM is true what should happen? what are two problems with this analysis?

Answer 23

Regress a cross-section of average asset returns on estimates of market betas:

CAPM: 𝐸(𝑟𝑖) =𝑟𝑓+[𝐸(𝑟𝑚)−𝑟𝑓] 𝛽𝑖

Regression:𝑟𝑖 =𝑎+ 𝛾 𝛽𝑖+ 𝑒𝑖

If the CAPM is true:

1) The regression intercept (a) should equal the average risk-free rate.

2) The coefficient on market beta (𝛾) should equal the average excess return on the market (beta premium).

Two problems for this analysis:

1) Estimated market beta subject to measurement errors.

2) Statistical problems with correlated error term (led to the Fama-MacBeth approach

Question 24

How do researchers overcome the problem of measurement errors in market beta when testing CAPM?

Answer 24

Researchers study portfolios instead of individual stocks to minimize the measurement errors associated with market beta.

Question 25

How do researchers increase the range of beta across portfolios?

Answer 25

Researchers sort stocks on pre-ranking beta (estimated over the past 2-5 years) and then form portfolios for analysis.

Question 26

How do Fama & MacBeth (1973) address the second problem of correlated error terms in CAPM testing?

Answer 26

They use a better approach by estimating month-by-month cross-sectional regressions of monthly returns on betas instead of a single cross-section regression.

Question 27

How do Fama & MacBeth (1973) test CAPM using monthly regressions?

Answer 27

They use the time-series means of the monthly slopes and intercepts to test: 1) Average beta premium > 0 2) Intercept = average risk-free interest rate 3) Whether additional variables (e.g., squared beta, residual variances) explain average returns.

Question 28

What did Fama & MacBeth (1973) find in their analysis?

Answer 28

Their evidence is consistent with earlier results, showing little explanatory power in additional variables. This suggests that CAPM holds, as the proxy of the market portfolio used appears to lie on the minimum variance frontier.

Question 29

How does Jensen (1968) test CAPM using time-series regression?

Answer 29

Jensen tests CAPM with the following time-series regression: 𝑟𝑖𝑡− 𝑟𝑓𝑡=𝛼𝑖+𝛽𝑖𝑚 (𝑟𝑚𝑡−𝑟𝑓𝑡 )+𝜀𝑖𝑡

Question 30

What does the "Jensen alpha" refer to?

Answer 30

"Jensen alpha" refers to the estimated intercept 𝛼𝑖 in the time-series regression. If CAPM is true, the intercept should equal 0.

Question 31

What did the evidence from time-series regressions (e.g., Black, Jensen, and Scholes, 1972) reveal about CAPM?

Answer 31

The evidence confirmed the cross-sectional evidence of a somewhat flat relation, showing that the alpha for high beta stocks is negative, while the alpha for low beta stocks is positive.

Question 32

How can time-series tests be applied to portfolios?

Answer 32

Time-series tests can be applied to portfolios by sorting them according to pre-ranking beta and firm characteristics, and then testing if the portfolios’ intercepts are jointly equal to 0.

Question 33

How did evidence from the 1980s and 90s contradict CAPM?

Answer 33

Studies in the 1980s and 90s show that firm characteristics/ratios, such as size, earnings-to-price ratio, debt/equity ratio, and book-to-market ratio, can explain the cross-section of expected returns, contradicting CAPM.

Question 34

What did Fama and French (1992) find regarding the return-beta relation?

Answer 34

Fama and French (1992) confirmed that the return-beta relation is even flatter using more recent data. Returns of portfolios sorted by book-to-market ratio are unrelated to their beta.

Question 35

How do behavioralists view stocks with high book-to-market ratios (B/M)?

Answer 35

Stocks with high B/M are often firms that have fallen on bad times, while low B/M is associated with growth firms.

Question 36

What does sorting firms on book-to-market ratios reveal in terms of investor behavior?

Answer 36

Sorting firms on book-to-market ratios reveals investor overreaction to good and bad times, where investors may overextrapolate past performance.

Question 37

How do overreactions by investors affect stock prices?

Answer 37

Overreaction causes stock prices to be too high for growth (low B/M) firms and too low for distressed (high B/M, value) firms.

Question 38

at happens when the overreaction in stock prices is corrected?

Answer 38

When the overreaction is corrected, value stocks (high B/M) experience high returns, while growth stocks (low B/M) experience low returns.

Question 39

Why do critics argue that CAPM may be inadequate for explaining asset pricing?

Answer 39

Critics argue that CAPM has unrealistic assumptions, such as investors only caring about the mean and variance of one-period portfolio returns, and that more complex models are needed to account for other factors.

Question 40

What are some factors that investors may care about beyond the mean and variance of portfolio returns?

Answer 40

Investors may care about how their portfolio return covaries with labor income and future investment opportunities, which makes the portfolio’s return variance an incomplete measure of risk.

Question 41

What is the issue with market beta as a description of an asset's risk?

Answer 41

Market beta may not fully capture an asset’s risk, as it ignores other factors that influence risk, such as the correlation with labor income or future investment opportunities

Question 42

What do researchers suggest for improving asset pricing models?

Answer 42

Researchers suggest searching for more comprehensive asset pricing models that can better explain average returns by accounting for additional dimensions of risk.

Question 43

What is the Fama and French (1993) 3-factor model?

Answer 43

The 3-factor model proposed by Fama and French (1993) is: 𝐸(𝑟𝑖𝑡)−𝑟𝑓𝑡= 𝛽𝑖𝑚 [𝐸(𝑟𝑚𝑡 )−𝑟𝑓𝑡]+ 𝛽𝑖𝑠 𝐸(𝑆𝑀𝐵𝑡)+ 𝛽𝑖ℎ 𝐸(𝐻𝑀𝐿𝑡) 𝑆𝑀𝐵_𝑡(small minus big) is the difference in returns on diversified portfolios of small and big stocks – size premium. 𝐻𝑀𝐿𝑡 (high minus low) is the difference in returns on diversified portfolios of high and low B/M stocks – value premium

Question 44

What is the time-series regression equation for the 3-factor model?

Answer 44

𝑟_𝑖𝑡−𝑟_𝑓𝑡= 𝛽𝑖𝑚 (𝑟𝑚𝑡−𝑟𝑓𝑡 )+ 𝛽𝑖𝑠 𝑆𝑀𝐵𝑡+ 𝛽𝑖ℎ 𝐻𝑀𝐿𝑡+𝜀𝑡 𝑆𝑀𝐵𝑡(small minus big) is the difference in returns on diversified portfolios of small and big stocks – size premium. 𝐻𝑀𝐿𝑡 (high minus low) is the difference in returns on diversified portfolios of high and low B/M stocks – value premium

Question 45

What is one shortcoming of the Fama-French 3-factor model?

Answer 45

A shortcoming of the FF3 model is that the size and value factors are not motivated by theory; they are rather brute force constructs meant to capture return patterns.

Question 46

What is the likely explanation for the higher returns of small and high-BM stocks in the FF3 model?

Answer 46

The higher returns of small and high-BM stocks likely reflect unidentified state variables producing systematic risk in returns that are not captured by the market returns.

Question 47

What refinement was made to the Fama-French 3-factor model to address its limitations?

Answer 47

A further refinement is the inclusion of the momentum factor (UMDt), which is the difference in returns on diversified portfolios of past winners and past losers, forming the "Carhart 4-factor model."

Question 48

What is the market proxy problem in CAPM testing?

Answer 48

The market proxy problem arises because we cannot observe all assets that trade, making it impossible to properly account for the market portfolio. As a result, we can never truly test CAPM, but instead, we are testing if the proxy for the market portfolio lies on the minimum variance frontier.

Question 49

What is the focus of tests for CAPM given the market proxy problem?

Answer 49

Tests for CAPM must focus on the mean-beta relationship as applied to all observed assets and a stock index portfolio.

Question 50

What are some issues when estimating beta and residual variances in CAPM testing?

Answer 50

Beta may be correlated with residual variances. Error terms in the regression may be correlated across stocks. These issues can introduce bias in the slope of the Security Market Line (SML).

Question 51

Why is alpha and beta variation over time a challenge for CAPM?

Answer 51

Alpha and beta are likely to vary over time, but CAPM does not account for such variation, nor does the standard regression techniques used to test it.

Question 52

How can the issue of time-varying alpha and beta be addressed?

Answer 52

The issue can be addressed using GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models, developed by Robert Engle (Nobel Prize), which allow for modeling time-varying volatility.

Question 53

How might beta vary in response to economic conditions?

Answer 53

Beta may not vary randomly but could react to economic conditions, leading to the concept of a conditional CAPM, where beta changes based on economic factors.

Question 54

How can beta be decomposed in the context of conditional CAPM?

Answer 54

Beta can be decomposed into two components: one that measures sensitivity to changes in corporate profitability (i.e., cash flows) and another that measures sensitivity to discount rates.

Question 55

What are some additional pricing factors that have been shown to explain stock returns beyond CAPM?

Answer 55

Additional pricing factors include profitability and investment factors (Fama and French, 2015), as well as accruals, ESG factors, and more.

Question 56

How is the single-index CAPM viewed in the academic world today?

Answer 56

in the academic world, single-index CAPM is considered somewhat outdated, as more comprehensive models have emerged to explain stock returns.

Question 57

How have academics and the industry approached asset pricing models differently?

Answer 57

Academics have moved toward using multiple-index models in search of a more accurate CAPM, while the industry has generally stuck with the single-index model.

Question 58

What is the "test of the non-testable" in CAPM?

Answer 58

The market portfolio cannot be tested because the true market portfolio cannot be observed, making the test of CAPM untestable in practice.

Question 59

What did Malkiel (1995) find about the performance of mutual funds from 1971-1991?

Answer 59

Malkiel (1995) showed that the distribution of alphas for mutual funds had a mean slightly negative, but statistically indistinguishable from zero, suggesting that mutual funds did not outperform the market index on a risk-adjusted basis.

Question 60

What conclusion does Malkiel’s (1995) study support regarding CAPM?

Answer 60

Malkiel’s study supports the idea that CAPM seems optimal, as mutual funds on average do not appear to outperform the market index after adjusting for risk

Question 61

What is the Single Index Model (SIM) used for in relation to CAPM?

Answer 61

The SIM is a statistical model used to leap from expected returns (CAPM) to realized returns by replacing theoretical market portfolio with an empirical proxy (a market index) and expected values with realized values.

Question 62

What is the equation for expected returns in CAPM?

Answer 62

The equation for expected returns in CAPM is: E(r i )−rf =βi[E(r m)−r f ]

Question 63

What is the equation for realized returns in the Single Index Model (SIM)?

Answer 63

The equation for realized returns in SIM is: 𝑟 𝑖−𝑟𝑓= 𝛼 𝑖 +𝛽𝑖 (𝑟 𝑚− 𝑟𝑓)+ 𝑒𝑖

Question 64

What is the main difference between CAPM and the Single Index Model (SIM)?

Answer 64

The main difference is that SIM replaces the theoretical market portfolio with an empirical market index and uses realized returns instead of expected returns.