The investor has two alternatives investment theory

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The investor has two alternatives:
Investment theory, class #2 Page 15 of 22 ® All rights reserved to Tal Mofkadi Invest δ in the market portfolio Invest δ in stock A Additional return: ] ) ( [ ) ( f m r r E r E = Δ δ New portfolio risk: { ( ) ( 2 2 2 2 2 2 0 0 2 2 2 2 2 ) 2 ( ) 1 ( , cov 1 2 ) 1 ( m m m f rf m r M σ δ δ σ σ δ σ δ δ σ δ σ δ σ + + = + = + + + = 43 42 1 and since δ is very small, we can omit 2 δ and get that the additional portfolio risk is: 2 2 2 m σ δ σ = Δ The compensation on risk for the individual is therefore: 2 2 2 2 ) ( 2 ] ) ( [ ) ( m f m m f m r r E r r E r E σ σ δ δ σ = = Δ Δ Additional return: ] ) ( [ ) ( f A r r E r E = Δ δ New portfolio risk: ( ) { ( ) ( ) ) , ( 1 2 1 , 1 2 , 2 ) , ( 1 2 1 2 2 2 2 0 0 0 2 2 2 2 2 2 M A Cov r M Cov r A Cov M A Cov A m f f rf A m + + = + + + = δ σ δ σ σ δ δ δ δ σ δ σ δ σ σ 43 42 1 43 42 1 and since δ is very small, we can omit 2 δ and get that the additional portfolio risk is: ) , ( 2 2 M A COV = Δ δ σ The compensation on risk for the individual is therefore: ) , ( 2 ) ( ) , ( 2 ] ) ( [ ) ( 2 M A COV r r E M A COV r r E r E f A f A = = Δ Δ δ δ σ Key: the two “compensation on risk” should equal, and we get: ) , ( 2 ) ( 2 ) ( ) ( 2 2 M A Cov r r E r r E r E f A M f M = = Δ Δ σ σ If, for example, the RHS is larger than the LHS, the investor should sell short M and buy A, and increase his return and not the risk. From this we get:
Investment theory, class #2 Page 16 of 22 ® All rights reserved to Tal Mofkadi [ ] f M M f A r r E M A Cov r r E + = ) ( ) , ( ) ( 2 σ and if we define 2 ) , ( M A M A Cov σ β = , we get: [ ] f M A f A r r E r r E + = ) ( ) ( β 3.4 Calculating Beta If we use a linear regression (OLS) of the form: [ ] f M A f A r r E r r E = ) ( ) ( β or alternatively: ( ) { { ) ( ) ( 1 M r A r E r E A f A β β β α + = . This is also known as the “market model” Statistics tell us that in this model: ( ) ( ) ( ) ( ) x Var Y X Cov X n X Y X n Y X i i i , 2 2 = = β . Technically, this is done using statistical program or by using Excel’s regression capabilities or “Slope” worksheet function. Will be seen in the tutorial! 3.4.1 The meaning of Beta Beta is a measure of the additional risk of an asset to the overall market portfolio. In other words, Beta is a proxy of an asset’s risk due to market “shocks”. For example, if the beta of asset i is 0.5 this means that when the market portfolio absorbs a shock of 1% (i.e. r m -r f is grater in 1%) asset i expect to absorb a shock of 0.5% (i.e. r i -r f is expected to be grater in 0.5%). This makes asset i less risky than the market portfolio (why?) and therefore we should expect it to yield lower rate of return. Note that the beta of a portfolio is a weighted sum of its asset’s betas, that is: = = n i i i p x 1 β β
Investment theory, class #2 Page 17 of 22 ® All rights reserved to Tal Mofkadi 3.5 Variance decomposition For any asset, the total variance can be expressed as follows: () ( ) ( ) { risk Specific 2 risk Systematic 2 2 risk Specific risk Systematic 2 risk Total ε σ σ β ε β + = + = 3 2 1 3 2 1 43 42 1 3 2 1 m i i Var m Var i Var 3.6 The SML (security market line) [ ] f M A f A r r E r r E + = ) ( ) ( β Note that: 1. 1 = m β since 1 1 2 = = m m m m σ σ σ β 2. 0 = rf β since 0 0 0 2 = = m m rf σ σ β Thus we can graphically present the SML formula: CML Stock A Standard deviation- σ Expected return - E(r) R f Specific risk Market risk Stupidity Dreams
Investment theory, class #2
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