1 x. explaining relative price – arbitrage pricing theory

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X. Explaining Relative Price – Arbitrage Pricing Theory

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Assume a two index RGP

1 1 2 2 ... cov( ) 0i i i i i i jR a I I e e e

i i2 Portfolio Expected Return

A 15 1.0 .6 B 14 .5 1.0 C 10 .3 .2

Three points describe a plane

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1 1

2 12

1

But

i

iiP

iP i

iP

X

R X R

X

X

Any weighted average of points on a plane where the weights sum to 1 lie on the plane –any portfolio lies on the plane

Lies above a below – riskless arbitrage

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iMiii eRR

iiiii eIIR ...2211 RGP

iFi BRR

...2211 iiFi BBRR

APT

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What are the ’s?

What are the I’s?

What are the ’s?

ij

j

In general, I’s are systematic influences which have an impact on the return of a large percentage of stocks.

’s are characteristics of individual firms or sensitivities of industrial firms to systematic influences.

’s are the market price of the ’s.

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How to Identify; Five Approaches:

1) Statistical methods for identifying the I’s and ’s simultaneously – factor analysis or principle components analysis.

2) Identify the firm characteristics that are judged as most important – estimate the ’s from multiple regression.

3) Identify the I’s; a priori macro variable.

4) Identify the I’s; a set of portfolios sufficient to capture all influences.

5) Mixtures of the above.

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Identify simultaneously the ’s and the I’s.

Factor Analysis

Let the data speak to the return generating process.

ij

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Conceptually most difficult to understand

Takes return data for each member of a setof securities over time (e.g. monthly returns)and the mathematically determines a set ofIndexes (portfolios) which best explains

returns iCov e 0je

Simple Example 4 stocks (countries) Belgium, Canada, France, U.S. Returrn data – monthly 1979-1988

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C

U

B F1t Bt Ct Ft

Ut

f .67(R R ) .76(R R ) .76(R R )

+ .77(R R )

WHERE 1. R INDICATES RETURN 2. f INDICATES FACTOR VALUE 3. B IS BELGIUM

4. C IS CANADA

5. U IS UNITED STATES

6. F IS FRANCE

7. t IS TIME PERIOD

8. BARS INDICATE MEANS

C2,t Bt Ct

Ut

f .40(R R ) .73(R -R ) .37( )

+ .41(R )

B

U

Ft FR R

R

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FOUR FACTOR MODEL OF THE

JAPANESE ECONOMY

VERSUS ONE FACTOR MARKET INDEX

NRI 400; 20 PORTFOLIOS

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BARRA – MODEL

SIZE

LIQUIDITY

GROWTH

VALUE

FINANCIAL LEVERAGE

INDUSTRY MEMBERSHIP

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M F 1 2 3 4

2

5 M

R -R .0022 1.33I 0.56I 2.29I 0.93I

R 24

(-3.94) (4.96) (1.99) (-2.27)

I (R -

F 1 2 3 4R ) (.0022 1.33I 0.56I 2.29I 0.93I )

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Salomon Brothers Risk Attribute Model

1. Economics Growth = Monthly Changes In Total Industrial Production

2. Credit Quality = Return on High Yield Bonds – Return on Governments (10+Year)

3. Long Term Interest Rates = Yield Change in 30 Year Treasuries

4. Short Term Interest Rates = Yield Change in 3 Month Treasuries

5. Inflation Shock = Realization Inflation – Expected Inflation (CPI)

6. US Dollar = Change in Value of US Dollar Trade Weighted

7. Market (After 6 Factors Removed). S&P

8. Small-Cap Premium = Return Russel 2000 – S&P 500 (seven factors removed)

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PORTFOLIO APPROCH

RETURN ON MUTUAL FUNDS RELATED TO:

1. RETURN ON S&P INDEX

2. RETURN ON SMALL STOCK INDEX

3. RETURN ON BOND INDEX

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FAMA – FRENCH

1. TERM = LONG TERM GOVERNMENT BOND RETURN – T-BILL RATE

2. DEFAULT – LONG TERM CORPORATE BOND RETURN – RETURN ON LONG TERM GOVERNMENT BOND

3. SIZE – RETURN ON PORTFOLIO OF SMALL STOCK -PORTFOLIO OF REGULAR STOCKS

4. BOOK TO MARKET = RETURN ON PORTFOLIO OF HIGH BOOK TO MARKET FIRMS – RETURN ON PORTFOLIO OF LOW BOOL TO MARKET FIRMS

5. RETURN ON MARKET – T-BILL RATE

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NORMAL DURATION

RETURN RELATED TO RETURN ON A MARKET PORTFOLIO

RETURN RELATED TO

1. RETURN ON A 4 YEAR PORTFOLIO OF BONDS (LEVEL OF INTEREST RATE)

2. DIFFERENCE BETWEEN 10 YEAR TREASURY AND 2 YEAH TREASURY (TWIST IN YIELD CURVE)

3. DIFFERENCE BETWEEN 4 YEAR AAA CORPORATE AND 4 YEAH TREASURY (PRICE OF RISK)

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1 1 2 2

1 1

Why Bother

APT and CAPM are not mutually exclusive

If CAPM holds

( )

If APT holds

But if in indexes can be replicated with

portfolios of securities and

CAPM holds

i F M F

i F i i

R R R R

R R

2 2( ) ( )M F M FR R R R

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1 1 2 2

1 1 2 2

( ) ( )

-

( - )

Not Mutually Exclusive

M M

M

M

i F i F i F

i F i i F

i F i F

R R b R R b R R

R R b b R R

R R R R

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Passive Management

1. Better Match an index

2. Match an index + or – certain stocks

3. Passive management with changed sensitivity

a. Pension fund liabilities that go up with inflation will pay a price to have assets that go up with inflation. APT tells investors that the cost of zero inflation exposure is (-4.32x.37) = 1.60% b. Take on oil exposure may be free lunch

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Active Management

1. Make bets e.g. on interest rates or inflation

2. Look for stocks out of equilibrium

3. Long short zero risk portfolios

1 x. explaining relative price – arbitrage pricing theory

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