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Empirical Validation of APT

Lecture 12 27 November 2010

APT: a solution to CAPM shortcomings z

Actual SML is flatted than predicted by CAPM

z

Factors other than β influence stock returns: –

–

–

Size of the firm (can be

measured by market capitalization) Firm’s perspectives (can be measured by book-to-market ratio) Momentum (average stock’s return over the past six months)

APT: a reminder z z z

Factor model + absence of arbitrage = APT Returns are determined by the sensitivity to a common set of factors The expected return of a security is determined by the following equation:

E (Ri ) = R f + β 1i λ1 + β 2i λ 2 + ... + β ki λ k –

Where λk = E(RPk) – Rf (risk premium of kth factor)

Estimating the factors z

Factor analysis –

z

Using macroeconomic variables to generate factors –

z

A statistical procedure aimed at finding factorreplicating portfolios

Macro variables are proxies for the factors

Using characteristic-sorted portfolios to estimate factors –

Portfolios of securities with some common characteristic are proxies for the factors

Factor analysis: intuition z

Consider 2 stocks whose returns follow a onefactor model: – –

z

From historical data, we can obtain their variances, and covariance between returns –

z

R1 = a1 + b1F + ε1 R2 = a2 + b2F + ε2

(σ1)2, (σ2)2, σ12

Therefore, we can find the factor properties, and the sensitivities of the two stock’s returns –

b1, b2, (σF)2

Factor analysis: intuition z

Once we’ve found the sensitivities of individual securities to the risk factor, we can construct a factor-replicating portfolio RP1 – –

z

RP1 = A + F Factor risk-premium = λ1 = E(RP1) – Rf = A – Rf

The same logic can be applies to a sample of N securities whose returns are described by a (N-1) factor model

Factor analysis: pros and cons z

Advantages: –

z

RPs found in the process of factor analysis provide the best possible explanation of the covariance between the stocks’ returns estimated from historical data

Disadvantages: – –

Assumes covariances do not change over time Doesn’t name the economic variables to which factors are linked

Factor analysis: empirical findings z z

Roll & Ross (1980) tried to establish the # of factors Were forced to estimate factors on small number of stocks – –

z

Results: – – –

z

42 groups 30securities each In 88.1% of the groups there was at least 1 factor with non-zero risk premium In 57.1% - at least 2 factors In 33% - at least 3

Conclusions: –

At least 3 factors are important for APT, but probably no more than 4

Macroeconomic variables approach z

Identify the complete set of possible factor affecting the returns: changes (unanticipated) in – – – – – –

z

Limit the # of significant factors –

z

Unemployment Inflation Interest rates spreads Oil prices Credit spreads Etc. Usually no more than 5

Find out which factors are significant –

By regressing the stocks returns on different sets of factors

Macroeconomic variables approach: pros and cons z

Advantages: – –

z

Very intuitive Names the exact factors that determine the stocks returns

Disadvantages: – – –

Potentially important factors may be difficult to quantify (e.g., political changes) ‘Factors’ are constructed as ‘unanticipated’ changes, which might be difficult to measure Multicollinearity issues

An example z z z z

Let the only factor affecting the returns be Fint, the news on interest rates Before the FOMC meeting, the market participants expect the FED not to change the base rate: E(Fint) = 0 After the meeting, B. Bernanke announces that the interest rate is raised by 25 b.p. => Fint = 0.25 > 0 What should the sensitivities to Fint be for: – – –

Fixed income securities? Stocks? Commodities?

Macroeconomic variables approach: empirical findings z

Chen, Roll & Ross (1986) –

Important factors: z z z

–

Less significant factors: z z

–

Changes in GDP growth rate Changes in default risk premium (spread b/w YTM on AAA and BBB rated bonds) Changes in the slope of the YC (spread b/w LT and ST bonds rates) Unexpected changes in the price level (difference b/w actual and expected inflation rates) Changes in expected inflation (T-bill yield)

Unimportant: z

The return on market index, when added to the regression, could not explain the expected returns

Macroeconomic variables approach: empirical findings z

Chan, Chen & Hsieh (1985); Jagannathan & Wang (1996) –

–

Identified default spread and labor income as systematic factors associated with positive riskpremiums Research on explaining the small firm effect z z z

Small cap stock returns appeared to be highly correlated with changes in the spread b/w BBB and default-free bonds The spread seems to be a fairly good predictor of future market returns Betas are higher when the spread is high

Characteristic-sorted portfolios approach z

Identify the possible characteristics: – – – –

z z

Size Forward-looking indicators (B/M) Backward-looking indicators (stock’s performance during the past 6-12 months) Etc.

Group portfolios of stocks that satisfy the above characteristics Calculate the risk premiums associated with the respective characteristic (risk factor)

Characteristic-sorted portfolios approach: pros and cons z

Advantages: –

–

z

The approach is based NOT on cross-correlations between individual securities, but on the correlation with some common characteristic => does not require the constant correlation assumption Uses portfolios returns, which are highly unpredictable, but observable, to calculate factor risk premiums

Disadvantages: –

Relationships found in historical data are not guaranteed to have explanatory power in the future

Characteristic-sorted portfolios approach: empirical findings z

Fama & French (1993) –

Suggested a 3-factor model composed of the following three zero-cost (self-financing) portfolios: z z z

–

Value-weighted index portfolio (long) + T-bills (short) High book-to-market stocks (long) + low book-to-market stocks (short) Small-firm stocks (long) + large-firm stocks (long)

Estimated risk-premiums for each factor (b/w 1963 and 1994) z

5.2%; 3.2%; 5.4%

Characteristic-sorted portfolios approach: empirical findings –

Estimated the factor sensitivities for stocks in different industry groups:

Characteristic-sorted portfolios approach: empirical findings 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% 0,7

0,8

0,9

1

Risk-premium (multi-factor)

1,1

1,2

Risk-premium (CAPM)

1,3

APT vs. CAPM – 1 z

Desired portfolio: –

–

z

CAPM: two-fund separation theorem states that all investors will form their desired portfolios by combining only 2 assets – the risk-free asset and the market portfolio APT: in a complete market the desired portfolio can be formed by combining the factor-replicating portfolios and the risk-free asset

Securities pricing under both theories: –

Assets with identical risk exposures must bring identical returns => otherwise arbitrage is possible

APT vs. CAPM – 2 z

Sources of risk: – –

z

CAPM: fluctuations of the market portfolio returns APT: fluctuations in various risk factors (macroeconomic, financial, political, etc.)

Assumptions: APT is less restrictive – – –

It doesn’t require investors to have homogenous expectations CAPM – a pure theoretical framework APT – a practical framework

Assumptions (APT) z z z

In equilibrium there are no arbitrage opportunities Returns of risky assets can be described by a factor model Financial markets are frictionless – –

z

No transaction costs Perfect information, etc.

There is a large number of securities => investors hold well-diversified portfolios => specific risks are diversified away; the only source of risk arises from fluctuations in factors determining the return

APT vs. CAPM: Summary z

APT solves the major problem of CAPM –

z

APT doesn’t name the particular factors that should be used to determine the expected returns –

z

It does not require to determine the market portfolio

As opposed to CAPM, which specifies the particular source of risk

Both models state that only systematic (nondiversifiable) risk should be rewarded

A formal link b/w CAPM and APT z

Consider a one-factor model, where F if other than the return on the market portfolio –

z

Ri = ai + biF + εi

Assuming CAPM holds, and there is some correlation (ρ) b/w F and Market portfolio, the stock’s β can be calculated as following:

ρ β i = bi 2 σM

z

The case can be extrapolated to the case of a multi-factor model

Essential reading z

Brealey, Myers: Principles of Corporate Finance, Seventh Edition –

z

Chapter 8.4

Grinblatt, Titman: Financial Markets and Corporate Strategy, Second Edition –

Chapter 6

View more...
Lecture 12 27 November 2010

APT: a solution to CAPM shortcomings z

Actual SML is flatted than predicted by CAPM

z

Factors other than β influence stock returns: –

–

–

Size of the firm (can be

measured by market capitalization) Firm’s perspectives (can be measured by book-to-market ratio) Momentum (average stock’s return over the past six months)

APT: a reminder z z z

Factor model + absence of arbitrage = APT Returns are determined by the sensitivity to a common set of factors The expected return of a security is determined by the following equation:

E (Ri ) = R f + β 1i λ1 + β 2i λ 2 + ... + β ki λ k –

Where λk = E(RPk) – Rf (risk premium of kth factor)

Estimating the factors z

Factor analysis –

z

Using macroeconomic variables to generate factors –

z

A statistical procedure aimed at finding factorreplicating portfolios

Macro variables are proxies for the factors

Using characteristic-sorted portfolios to estimate factors –

Portfolios of securities with some common characteristic are proxies for the factors

Factor analysis: intuition z

Consider 2 stocks whose returns follow a onefactor model: – –

z

From historical data, we can obtain their variances, and covariance between returns –

z

R1 = a1 + b1F + ε1 R2 = a2 + b2F + ε2

(σ1)2, (σ2)2, σ12

Therefore, we can find the factor properties, and the sensitivities of the two stock’s returns –

b1, b2, (σF)2

Factor analysis: intuition z

Once we’ve found the sensitivities of individual securities to the risk factor, we can construct a factor-replicating portfolio RP1 – –

z

RP1 = A + F Factor risk-premium = λ1 = E(RP1) – Rf = A – Rf

The same logic can be applies to a sample of N securities whose returns are described by a (N-1) factor model

Factor analysis: pros and cons z

Advantages: –

z

RPs found in the process of factor analysis provide the best possible explanation of the covariance between the stocks’ returns estimated from historical data

Disadvantages: – –

Assumes covariances do not change over time Doesn’t name the economic variables to which factors are linked

Factor analysis: empirical findings z z

Roll & Ross (1980) tried to establish the # of factors Were forced to estimate factors on small number of stocks – –

z

Results: – – –

z

42 groups 30securities each In 88.1% of the groups there was at least 1 factor with non-zero risk premium In 57.1% - at least 2 factors In 33% - at least 3

Conclusions: –

At least 3 factors are important for APT, but probably no more than 4

Macroeconomic variables approach z

Identify the complete set of possible factor affecting the returns: changes (unanticipated) in – – – – – –

z

Limit the # of significant factors –

z

Unemployment Inflation Interest rates spreads Oil prices Credit spreads Etc. Usually no more than 5

Find out which factors are significant –

By regressing the stocks returns on different sets of factors

Macroeconomic variables approach: pros and cons z

Advantages: – –

z

Very intuitive Names the exact factors that determine the stocks returns

Disadvantages: – – –

Potentially important factors may be difficult to quantify (e.g., political changes) ‘Factors’ are constructed as ‘unanticipated’ changes, which might be difficult to measure Multicollinearity issues

An example z z z z

Let the only factor affecting the returns be Fint, the news on interest rates Before the FOMC meeting, the market participants expect the FED not to change the base rate: E(Fint) = 0 After the meeting, B. Bernanke announces that the interest rate is raised by 25 b.p. => Fint = 0.25 > 0 What should the sensitivities to Fint be for: – – –

Fixed income securities? Stocks? Commodities?

Macroeconomic variables approach: empirical findings z

Chen, Roll & Ross (1986) –

Important factors: z z z

–

Less significant factors: z z

–

Changes in GDP growth rate Changes in default risk premium (spread b/w YTM on AAA and BBB rated bonds) Changes in the slope of the YC (spread b/w LT and ST bonds rates) Unexpected changes in the price level (difference b/w actual and expected inflation rates) Changes in expected inflation (T-bill yield)

Unimportant: z

The return on market index, when added to the regression, could not explain the expected returns

Macroeconomic variables approach: empirical findings z

Chan, Chen & Hsieh (1985); Jagannathan & Wang (1996) –

–

Identified default spread and labor income as systematic factors associated with positive riskpremiums Research on explaining the small firm effect z z z

Small cap stock returns appeared to be highly correlated with changes in the spread b/w BBB and default-free bonds The spread seems to be a fairly good predictor of future market returns Betas are higher when the spread is high

Characteristic-sorted portfolios approach z

Identify the possible characteristics: – – – –

z z

Size Forward-looking indicators (B/M) Backward-looking indicators (stock’s performance during the past 6-12 months) Etc.

Group portfolios of stocks that satisfy the above characteristics Calculate the risk premiums associated with the respective characteristic (risk factor)

Characteristic-sorted portfolios approach: pros and cons z

Advantages: –

–

z

The approach is based NOT on cross-correlations between individual securities, but on the correlation with some common characteristic => does not require the constant correlation assumption Uses portfolios returns, which are highly unpredictable, but observable, to calculate factor risk premiums

Disadvantages: –

Relationships found in historical data are not guaranteed to have explanatory power in the future

Characteristic-sorted portfolios approach: empirical findings z

Fama & French (1993) –

Suggested a 3-factor model composed of the following three zero-cost (self-financing) portfolios: z z z

–

Value-weighted index portfolio (long) + T-bills (short) High book-to-market stocks (long) + low book-to-market stocks (short) Small-firm stocks (long) + large-firm stocks (long)

Estimated risk-premiums for each factor (b/w 1963 and 1994) z

5.2%; 3.2%; 5.4%

Characteristic-sorted portfolios approach: empirical findings –

Estimated the factor sensitivities for stocks in different industry groups:

Characteristic-sorted portfolios approach: empirical findings 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% 0,7

0,8

0,9

1

Risk-premium (multi-factor)

1,1

1,2

Risk-premium (CAPM)

1,3

APT vs. CAPM – 1 z

Desired portfolio: –

–

z

CAPM: two-fund separation theorem states that all investors will form their desired portfolios by combining only 2 assets – the risk-free asset and the market portfolio APT: in a complete market the desired portfolio can be formed by combining the factor-replicating portfolios and the risk-free asset

Securities pricing under both theories: –

Assets with identical risk exposures must bring identical returns => otherwise arbitrage is possible

APT vs. CAPM – 2 z

Sources of risk: – –

z

CAPM: fluctuations of the market portfolio returns APT: fluctuations in various risk factors (macroeconomic, financial, political, etc.)

Assumptions: APT is less restrictive – – –

It doesn’t require investors to have homogenous expectations CAPM – a pure theoretical framework APT – a practical framework

Assumptions (APT) z z z

In equilibrium there are no arbitrage opportunities Returns of risky assets can be described by a factor model Financial markets are frictionless – –

z

No transaction costs Perfect information, etc.

There is a large number of securities => investors hold well-diversified portfolios => specific risks are diversified away; the only source of risk arises from fluctuations in factors determining the return

APT vs. CAPM: Summary z

APT solves the major problem of CAPM –

z

APT doesn’t name the particular factors that should be used to determine the expected returns –

z

It does not require to determine the market portfolio

As opposed to CAPM, which specifies the particular source of risk

Both models state that only systematic (nondiversifiable) risk should be rewarded

A formal link b/w CAPM and APT z

Consider a one-factor model, where F if other than the return on the market portfolio –

z

Ri = ai + biF + εi

Assuming CAPM holds, and there is some correlation (ρ) b/w F and Market portfolio, the stock’s β can be calculated as following:

ρ β i = bi 2 σM

z

The case can be extrapolated to the case of a multi-factor model

Essential reading z

Brealey, Myers: Principles of Corporate Finance, Seventh Edition –

z

Chapter 8.4

Grinblatt, Titman: Financial Markets and Corporate Strategy, Second Edition –

Chapter 6

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