Capm

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Capm

  1. 1. Portfolio Theory and Capital Asset Pricing Model Prof. Ashok Thampy IIMB
  2. 2. Markowitz Portfolio Theory• Combining stocks into portfolios can reduce standard deviation, below the level obtained from a simple weighted average calculation.• Correlation coefficients make this possible.• The various weighted combinations of stocks that create this standard deviations constitute the set of efficient portfolios. portfolios
  3. 3. Markowitz Portfolio Theory  Expected Returns and Standard Deviations vary given different weighted combinations of the stocksExpected Return (%) Reebok 35% in Reebok Coca Cola Standard Deviation
  4. 4. Efficient Frontier•Each half egg shell represents the possible weighted combinations for twostocks.•The composite of all stock sets constitutes the efficient frontierExpected Return (%) Standard Deviation
  5. 5. Efficient Frontier•Lending or Borrowing at the risk free rate ( rf) allows us to exist outside theefficient frontier. Expected Return (%) T r ow i ng B or g din L en rf S Standard Deviation
  6. 6. Efficient FrontierReturn B A Risk (measured as σ )
  7. 7. Efficient FrontierReturn B AB A Risk
  8. 8. Efficient FrontierReturn B N AB A Risk
  9. 9. Efficient Frontier Goal is to moveReturn up and left. WHY? B ABN AB N A Risk
  10. 10. Efficient FrontierReturn Low Risk High Risk High Return High Return Low Risk High Risk Low Return Low Return Risk
  11. 11. Efficient FrontierReturn Low Risk High Risk High Return High Return Low Risk High Risk Low Return Low Return Risk
  12. 12. Portfolio Risk Expected Portfolio Return = (x 1 r1 ) + ( x 2 r2 )Portfolio Variance = x 1σ 1 + x 2σ 2 + 2( x 1x 2ρ 12σ 1σ 2 ) 2 2 2 2
  13. 13. Portfolio Risk nExpected Portfolio Return = ∑ (x iri ) i=1 n Portfolio Variance = ∑ (x x σ i , j =1 i j ij )
  14. 14. Portfolio RiskThe shaded boxes contain variance terms; the remaindercontain covariance terms. 1 2 3 To calculateSTOCK 4 portfolio 5 variance add 6 up the boxes N 1 2 3 4 5 6 N STOCK
  15. 15. Limits of Diversification 2 2 1  1 Portfolio Variance = N   x average variance +(N2 − N ).  .(average covariance ) N  N Portfolio variance =(1/N) x average variance +(1 - 1/N) x average covariance As the number of stocks in the portfolio becomes very large, the portfolio variance tends towards the average covariance.
  16. 16. Portfolio DiversificationSuppose you make a portfolio constructed by taking equalProportions of n assets; that is xi = 1/n for each i. thenThe corresponding portfolio return and variance is : 1 n Expected Portfolio Return = ∑(ri ) n i=1 n 2 1 σ Portfolio Variance = 2 n ∑(σ ij ) = n i, j=1
  17. 17. Question : Find the minimum variance portfolio σ 2 = xaσ a + xb σ b2 + 2 xa xbσ ab p 2 2 2 σ 2 = xaσ a + (1 − xa ) 2 σ b2 + 2 xa (1 − xa )σ ab p 2 2 ∂σ 2 p = 2 xaσ a − 2(1 − xa )σ b2 + 2(1 − 2 xa )σ ab = 0 2 ∂xa Solving this we get : σ b2 − σ ab σ a − σ ab 2 xa = 2 and xb = 2 σ a + σ b − 2σ ab 2 σ a + σ b2 − 2σ ab
  18. 18. The one-fund theorem: There is a single fund F of risky assetssuch that any efficient portfolio can be constructed as acombination of the fund F and the risk free asset. Return FMarket Return = rm Efficient Portfolio Risk Free Return, = rrf Risk
  19. 19. Capital Market Line ReturnMarket Return = rp . Efficient Portfolio Risk Free Return = rf σp Risk Slope = (rp-rf)/ σ p The portfolio that maximizes the Slope gives the efficient portfolio.
  20. 20. The capital market line is mathematically expressed as Follows: rM − rfr = rf + σ σMwhere rM and σ M are the expected valuesand standard deviation of the market rateof return, and r and σ are the expected valueand the standard deviation of the rate of returnof an arbitrary efficient asset.
  21. 21. Capital Asset Pricing ModelThe CAPM : If the market portfolio M is efficient, the expectedrate of return, ri of any asset i satisfies : σ iMri − rf = β i (rM − rf ) where β i = 2 σM
  22. 22. Portfolio BetaBeta of a portfolio is the weighted average beta of individualAssets in the portfolio.
  23. 23. Security Market Line ReturnMarket Return = rm . Efficient Portfolio Risk Free Return = rf 1.0 BETA
  24. 24. Security Market LineReturn SMLrf BETA 1.0 SML Equation = rf + B ( rm - rf )
  25. 25. Systematic and Unsystematic Risk ri = rf + β i (rM − rf ) + ε i E (ε i ) = 0 Cov (ε i , σ M ) = 0 Var (ri ) = σ = β i Var (rM ) + Var (ε i ) 2 2 i Variance = systematic risk + unsystematic risk
  26. 26. Testing the CAPM Beta vs. Average Risk PremiumAvg Risk Premium1931-65 SML 30 20 Investors 10 Market Portfolio 0 Portfolio Beta 1.0
  27. 27. Testing the CAPM Beta vs. Average Risk PremiumAvg Risk Premium In the period 1966-91, return1966-91 has not been proportionate to beta 30 as predicted by the CAPM-SML. 20 SML Investors 10 Market 0 Portfolio Portfolio Beta 1.0

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