CHAPTER TWELVE ARBITRAGE PRICING THEORY --Arti Pradhan
Background <ul><li>Estimating expected return with the Asset Pricing Models of Modern Finance </li></ul><ul><ul><li>CAPM <...
Market Index on Efficient Set Corresponding Security Market Line Expected Return Risk (Return Variability) Market Index A ...
Market Index Market Index Inside Efficient Set Corresponding Security Market Cloud Expected Return Risk (Return Variabilit...
FACTOR MODELS <ul><li>ARBITRAGE PRICING THEORY (APT) </li></ul><ul><ul><li>is an equilibrium factor model of security retu...
Curved Relationship Between Expected Return and Interest Rate Beta -15% -5% 5% 15% 25% 35% Expected Return -3 -1 1 3 Inter...
<ul><li>Two stocks </li></ul><ul><ul><li>A: E(r) = 4%;  Interest-rate beta = -2.20 </li></ul></ul><ul><ul><li>B: E(r) = 26...
The Arbitrage Pricing Theory <ul><li>Two different stocks </li></ul><ul><ul><li>C: E(r) = 15%;  Interest-rate beta = -1.00...
<ul><li>No-arbitrage condition for asset pricing </li></ul><ul><ul><li>If risk-return relationship is non-linear, you can ...
APT Relationship Between Expected Return and Interest Rate Beta  -15% -5% 5% 15% 25% 35% Expected Return -3 -1 1 3 Interes...
FACTOR MODELS <ul><li>ARBITRAGE PRICING THEORY (APT) </li></ul><ul><ul><li>Three Major Assumptions: </li></ul></ul><ul><ul...
FACTOR   MODELS <ul><li>MULTIPLE-FACTOR MODELS </li></ul><ul><ul><li>FORMULA </li></ul></ul><ul><li>r i  = a i  + b i1  F ...
FACTOR MODELS <ul><li>SECURITY PRICING </li></ul><ul><ul><li>FORMULA: </li></ul></ul><ul><li>r i  =   0  +   1  b 1  +  ...
FACTOR MODELS <ul><li>  where  r  is the return on security i </li></ul><ul><li>  is the risk free rate </li></ul><ul>...
FACTOR   MODELS <ul><li>hence </li></ul><ul><ul><li>a stock’s expected return is equal to the risk free rate plus k risk p...
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12 apt

  1. 1. CHAPTER TWELVE ARBITRAGE PRICING THEORY --Arti Pradhan
  2. 2. Background <ul><li>Estimating expected return with the Asset Pricing Models of Modern Finance </li></ul><ul><ul><li>CAPM </li></ul></ul><ul><ul><ul><li>Strong assumption - strong prediction </li></ul></ul></ul>
  3. 3. Market Index on Efficient Set Corresponding Security Market Line Expected Return Risk (Return Variability) Market Index A B C Market Beta Expected Return x x x x x x x x x x x x x x x x x x x x x x x x
  4. 4. Market Index Market Index Inside Efficient Set Corresponding Security Market Cloud Expected Return Risk (Return Variability) Expected Return Market Beta
  5. 5. FACTOR MODELS <ul><li>ARBITRAGE PRICING THEORY (APT) </li></ul><ul><ul><li>is an equilibrium factor model of security returns </li></ul></ul><ul><ul><li>Principle of Arbitrage </li></ul></ul><ul><ul><ul><li>the earning of riskless profit by taking advantage of differentiated pricing for the same physical asset or security </li></ul></ul></ul><ul><ul><li>Arbitrage Portfolio </li></ul></ul><ul><ul><ul><li>requires no additional investor funds </li></ul></ul></ul><ul><ul><ul><li>no factor sensitivity </li></ul></ul></ul><ul><ul><ul><li>has positive expected returns </li></ul></ul></ul><ul><ul><li>Example … </li></ul></ul>
  6. 6. Curved Relationship Between Expected Return and Interest Rate Beta -15% -5% 5% 15% 25% 35% Expected Return -3 -1 1 3 Interest Rate Beta A B C D E F
  7. 7. <ul><li>Two stocks </li></ul><ul><ul><li>A: E(r) = 4%; Interest-rate beta = -2.20 </li></ul></ul><ul><ul><li>B: E(r) = 26%; Interest-rate beta = 1.83 </li></ul></ul><ul><ul><li>Invest 54.54% in E and 45.46% in A </li></ul></ul><ul><ul><li>Portfolio E(r) = .5454 * 26% + .4546 * 4% = 16% </li></ul></ul><ul><ul><li>Portfolio beta = .5454 * 1.83 + .4546 * -2.20 = 0 </li></ul></ul><ul><ul><li>With many combinations like this, you can create a risk-free portfolio with a 16% expected return. </li></ul></ul>The Arbitrage Pricing Theory
  8. 8. The Arbitrage Pricing Theory <ul><li>Two different stocks </li></ul><ul><ul><li>C: E(r) = 15%; Interest-rate beta = -1.00 </li></ul></ul><ul><ul><li>D: E(r) = 25%; Interest-rate beta = 1.00 </li></ul></ul><ul><ul><li>Invest 50.00% in E and 50.00% in A </li></ul></ul><ul><ul><li>Portfolio E(r) = .5000 * 25% + .4546 * 15% = 20% </li></ul></ul><ul><ul><li>Portfolio beta = .5000 * 1.00 + .5000 * -1.00 = 0 </li></ul></ul><ul><ul><li>With many combinations like this, you can create a risk-free portfolio with a 20% expected return. Then sell-short the 16% and invest the proceeds in the 20% to arbitrage. </li></ul></ul>
  9. 9. <ul><li>No-arbitrage condition for asset pricing </li></ul><ul><ul><li>If risk-return relationship is non-linear, you can arbitrage. </li></ul></ul><ul><ul><li>Attempts to arbitrage will force linearity in relationship between risk and return. </li></ul></ul>The Arbitrage Pricing Theory
  10. 10. APT Relationship Between Expected Return and Interest Rate Beta -15% -5% 5% 15% 25% 35% Expected Return -3 -1 1 3 Interest Rate Beta A B C D E F
  11. 11. FACTOR MODELS <ul><li>ARBITRAGE PRICING THEORY (APT) </li></ul><ul><ul><li>Three Major Assumptions: </li></ul></ul><ul><ul><ul><li>capital markets are perfectly competitive </li></ul></ul></ul><ul><ul><ul><li>investors always prefer more to less wealth </li></ul></ul></ul><ul><ul><ul><li>price-generating process is a K factor model </li></ul></ul></ul>
  12. 12. FACTOR MODELS <ul><li>MULTIPLE-FACTOR MODELS </li></ul><ul><ul><li>FORMULA </li></ul></ul><ul><li>r i = a i + b i1 F 1 + b i2 F 2 +. . . </li></ul><ul><li> + b iK F K + e i </li></ul><ul><li>where r is the return on security i </li></ul><ul><li>b is the coefficient of the factor </li></ul><ul><li>F is the factor </li></ul><ul><li>e is the error term </li></ul>
  13. 13. FACTOR MODELS <ul><li>SECURITY PRICING </li></ul><ul><ul><li>FORMULA: </li></ul></ul><ul><li>r i =  0 +  1 b 1 +  2 b 2 +. . .+  K b K </li></ul><ul><li>where </li></ul><ul><li>r i = r RF +(  1  r RF  b i1  2  r RF )b i2 +  </li></ul><ul><li> r RF  b iK </li></ul>
  14. 14. FACTOR MODELS <ul><li> where r is the return on security i </li></ul><ul><li>  is the risk free rate </li></ul><ul><li>b is the factor </li></ul><ul><li>e is the error term </li></ul>
  15. 15. FACTOR MODELS <ul><li>hence </li></ul><ul><ul><li>a stock’s expected return is equal to the risk free rate plus k risk premiums based on the stock’s sensitivities to the k factors </li></ul></ul>
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