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Lecture 12

Lecture 12. Arbitrage Pricing Theory. Pure Arbitrage. A pure (or risk-free) arbitrage opportunity exists when an investor can construct a zero-investment portfolio that yields a sure profit. Zero-investment means that the investor does not have to use any of his or her own money.

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Lecture 12

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  1. Lecture 12 Arbitrage Pricing Theory

  2. Pure Arbitrage • A pure (or risk-free) arbitrage opportunity exists when an investor can construct a zero-investment portfolio that yields a sure profit. • Zero-investment means that the investor does not have to use any of his or her own money.

  3. Pure Arbitrage • One obvious case is when a violation of the law of one price occurs. • Example: The exchange rate is $1.50/£ in New York and $1.48/£ in London.

  4. Arbitrage Pricing Theory • The APT is based on the premise that equilibrium market prices ought to be rational in the sense that they rule out risk-free arbitrage opportunities.

  5. Arbitrage Pricing Theory • The APT assumes that: 1. Security returns are a function of one or more macroeconomic factors. 2. All securities can be sold short and the proceeds can be used to purchase other securities.

  6. Single-Factor APT • The return on security i is ri = E(ri) + biF + ei. • E(ri) is the expected return. • F is the factor. • bi measures the sensitivity of rito F. • ei is the firm specific return. • E(ei) = 0 and E(F) = 0.

  7. Well Diversified Portfolios • rP=E(rP) + bPF + eP. • bP = Swibi • eP = Swiei ’0 • s 2(eP) = Swi2 s 2(ei) ’ 0 • sP2 = bP2sF2 + s 2(eP) ’bP2sF2 • sP’bPsF

  8. Single-Factor APT Diversified Portfolio Security i r r i i P i i i i i i i i i i i i i i i i i i i i i i F F i i i i

  9. Single-Factor APT r • Two well diversified portfolios with the same beta must have the same expected return. p A B Factor Realization

  10. Single-Factor APT • The expected return on a well diversified portfolio is a linear function of the portfolio’s beta. E(rP ) = rf + [RP]bP • RP is the risk premium. • rf is the risk-free rate.

  11. Single-Factor APT Expected Return C B 20% i i 15% A 10% i i D 5% 0.5 1.0 1.5 Beta

  12. Single-Factor APT • Let P be a well diversified portfolio. E(rP ) = rf + [RP]bP • RP is the risk premium = E*- rf • E* is the expected return on any well diversified portfolio with b*= 1.0. • rf is the risk-free rate or return on a zero beta portfolio.

  13. Single-Factor APT E[r ] P * E * RP = E - r * f r f 1.0 b P

  14. Single-Factor APT • Risk-free arbitrage applies only to well diversified portfolios. • However, an investor can increase the expected return on her portfolio without increasing systematic risk if individual securities violate the relationship ri = E(ri) + [RP]bi.

  15. Single-Factor APT • Consider the following portfolio which is part of a well diversified portfolio. Amount SecurityInvestedE(ri) i A $20,000 8% 0.6 B $40,000 10% 1.2 C $40,000 13% 1.6 • E(rP) = .2x8+.4x10+.4x13 = 10.8% • P = .2x0.6+.4x1.2+.4x1.6 = 1.24

  16. Single-Factor APT • Sell B and purchase $16,000 of A and $24,000 of C. Amount SecurityInvestedE(ri) i A $36,000 8% 0.6 C $64,000 13% 1.6 • E(rP) = .36x8 + .64x13 = 11.2% • P = .36x0.6 + .64x1.6 = 1.24

  17. Multi-Factor APT • The return on security i is ri = E(ri) + b1iF1+ ... + bkiFk+ei. • E(ri) is the expected return. • Fj is factor j, (j = 1,...,k). • bji measures the sensitivity of rito factor j, (j = 1,...,k). • ei is the firm specific return.

  18. Multi-Factor APT • The return on a well diversified portfolio is rP = E(rP) + b1PF1+ ... + bkPFk. • E(rP) is the expected return. • Fj is factor j, (j = 1,...,k). • bjP measures the sensitivity of rPto factor j, (j = 1,...,k). • eP = Swieig 0.

  19. Multi-Factor APT Diversified Portfolio The relationship between the return on a well diversified portfolio and factor j, holding other factors equal to zero. r P i i i i i i F j

  20. Multi-Factor APT • Arbitrage causes the expected return on a well diversified portfolio to be E[rP] = rf + [RP1]b1P +...+ [RPk]bkP • bjP is the sensitivity of portfolio P to unexpected changes in factor j. • RPj is the risk premium on factor j.

  21. Multi-Factor APT E[r ] P E j RP = E - r j j f r f 1.0 b j Relationship when all other betas are zero.

  22. Multi-Factor APT • Risk-free arbitrage applies only to well diversified portfolios. • However, an investor can increase the expected return on her portfolio without increasing systematic risk if individual securities violate the relationship E[ri] = rf + [RP1]b1i +...+ [RPk]bki

  23. Portfolio Strategy • Portfolio strategy involves choosing the optimal risk-return tradeoff. • The APT can be used to estimate > security expected returns, > security variances, and > covariances between security returns.

  24. Portfolio Strategy • The APT can also be used to refine the measure of risk. • Factor risks can affect investors differently. • The appropriate pattern of factor sensitivities depends upon a variety of considerations unique to the investor.

  25. Portfolio Sensitivities Productivity Beta Portfolios S - Stocks B – Bonds U – Unit Beta Z – Zero Beta h U 1.0 S h B h Z h 1.0 Inflation Beta

  26. Identifying Factors • The biggest problem is identifying the factors that systematically affect security returns. • Theory is silent regarding the factors. • A variety of macroeconomic factors have been used.

  27. Chen, Roll & Ross • Growth rate in industrial production. • Rate of inflation. • Expected rate of inflation. • Spread between long-term and short-term interest rates. • Spread between low-grade and high-grade bonds.

  28. Berry, Burmeister & McElroy • Growth rate in aggregate sales. • Rate of return on the S&P500. • Rate of inflation. • Spread between long-term and short-term interest rates. • Spread between low-grade and high-grade bonds.

  29. Salomon Brothers • Growth rate in GNP. • Rate of inflation. • Rate of interest. • Rate of change in oil prices. • Rate of growth in defense spending.

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