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Arbitrage Pricing Models

Arbitrage Pricing Models. Arbitrage Pricing Theory. Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit Since no investment is required, an investor can create large positions to secure large levels of profit

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Arbitrage Pricing Models

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  1. Arbitrage Pricing Models

  2. Arbitrage Pricing Theory Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit • Since no investment is required, an investor can create large positions to secure large levels of profit • In efficient markets, profitable arbitrage opportunities will quickly disappear

  3. Example: Returns with Equal Probability

  4. Arbitrage Example

  5. E(r) P 25.83 D 22.25 s 6.40 8.58 Arbitrage Action and Returns Action: Short 3 shares of D and buy 1 of A, B & C to form portfolio P Returns: You earn a higher rate on the investment than you pay on the short sale

  6. Payoffs (short 300 shares of D. buy 100 shares of A. B & C each)

  7. Example - continued • What will happen in the market?

  8. APT & Well-Diversified Portfolios • F is some macroeconomic factor • For a well-diversified portfolio eP approaches zero • The result is similar to CAPM

  9. E(ri) = Expected Return U Underpriced asset Slope =  = risk premium RFR O Overpriced asset 0 Risk class of assets O and U Factor beta Arbitrage Pricing Theory Line

  10. E(r)(%) E(r)(%) F F Individual Security Portfolio Portfolio & Individual Security Comparison

  11. E(r)% 10 A D 7 6 C Risk Free = 4 Beta for F .5 1.0 Disequilibrium Example

  12. Disequilibrium Example • Short Portfolio C • Use funds to construct an equivalent risk higher return Portfolio D • D is comprised of A & Risk-Free Asset • Arbitrage profit of 1%

  13. E(r) M [E(rM) - rf] Market Risk Premium Risk Free Beta (Market Index) 1.0 APT with Market Index Portfolio

  14. APT and CAPM Compared • APT applies to well diversified portfolios and not necessarily to individual stocks • With APT it is possible for some individual stocks to be mispriced - not lie on the SML • APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio • APT can be extended to multifactor models

  15. A Two-Factor APT Model • The single factor APT can be extended to include more independent risk factors that work together to determine market prices • APT is more flexible than SML • However, APT offers no clues as to what factors are relevant • Research must be done to determine best explanatory factor

  16. Three Highly Diversified Portfolios • Three risk-averse investors form portfolios B, C and D (each contains N assets) with two risk factors • These are arbitrage portfolios, requiring no cash investment • When N is large, unsystematic residual risk is diversified away

  17. Three Highly Diversified Portfolios • The general form of the APT model with two factors is: • The specific APT model for the three portfolios on the previous slide is:

  18. The Arbitrage Portfolio • Consider a mispriced asset • Portfolios S and U have the same risk but different expected returns • Portfolio U is underpriced • Smart investors would buy portfolio U

  19. The Arbitrage Portfolio • It is possible to set up a perfect hedge with portfolios S and U to create a riskless profit opportunity

  20. The k-Dimensional APT Hyperplane • A more elaborate model with k risk factors is: • Salomon Smith Barney uses a multi-factor arbitrage pricing model including factors such as: • The market’s trend or drift • Economic growth • Credit quality • Interest rates • Inflation shock • Small-cap premiums

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