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- 1. Portfolio Theory and Capital Asset Pricing Model Prof. Ashok Thampy IIMB
- 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. 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. 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. 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. Efficient FrontierReturn B A Risk (measured as σ )
- 7. Efficient FrontierReturn B AB A Risk
- 8. Efficient FrontierReturn B N AB A Risk
- 9. Efficient Frontier Goal is to moveReturn up and left. WHY? B ABN AB N A Risk
- 10. Efficient FrontierReturn Low Risk High Risk High Return High Return Low Risk High Risk Low Return Low Return Risk
- 11. Efficient FrontierReturn Low Risk High Risk High Return High Return Low Risk High Risk Low Return Low Return Risk
- 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. Portfolio Risk nExpected Portfolio Return = ∑ (x iri ) i=1 n Portfolio Variance = ∑ (x x σ i , j =1 i j ij )
- 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. 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. 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. 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. 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. 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. 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. 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. Portfolio BetaBeta of a portfolio is the weighted average beta of individualAssets in the portfolio.
- 23. Security Market Line ReturnMarket Return = rm . Efficient Portfolio Risk Free Return = rf 1.0 BETA
- 24. Security Market LineReturn SMLrf BETA 1.0 SML Equation = rf + B ( rm - rf )
- 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. Testing the CAPM Beta vs. Average Risk PremiumAvg Risk Premium1931-65 SML 30 20 Investors 10 Market Portfolio 0 Portfolio Beta 1.0
- 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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