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And portfolio variance
Covariance And portfolio variance
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Review question Define the internal rate of return.
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Answer: The internal rate of return of a project is r such that, given the cash flows CFt of the project,
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The states of nature model
Time zero is now. Time one is the future. At time one the possible states of the world are s = 1,2,…,S. Mutually exclusive, collectively exhaustive states. This IS the population. No sampling.
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The states of nature model
States s = 1,2,…,S. Probabilities ps Asset j Payoffs Rj,s Expected rate of return
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= rate of return on j in state s
= probability of state s = expectation of rate on j
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Variance and standard deviation
Form deviations Take their expectation.
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Covariance Form the product of the deviations
(positive if they both go in the same direction) and take the expectation of that.
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Covariance It measures the tendency of two assets to move together.
Variance is a special case -- the two assets are the same. Variance = expectation of the square of the deviation of one asset. Covariance = expectation of the product of the deviations of two assets.
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Correlation coefficient
Like covariance, it measures the tendency of two assets to move together. It is scaled between -1 and +1.
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Correlation coefficient
= covariance divided by the product of the standard deviations. Size of deviations is lost.
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Intuition from correlation coefficients
= 1, always move the same way and in proportion. = -1, always move in opposite directions and in proportion. = 0, no tendency either way.
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Portfolio Risk and Return
Portfolio weights x and 1-x on assets A and B.
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An amazing fact Mixing a risky asset with a safe asset
is often safer than the safe asset.
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Variance of portfolio return
Diversification effects
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Portfolio risk and return,
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Portfolio deviation Deviation squared Remember
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Portfolio variance
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Portfolio variance depends on covariance of the assets.
Positive covariance raises the variance of the portfolio.
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Correlation coefficient
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Portfolio variance depends on covariance of the assets.
Positive covariance raises the variance of the portfolio.
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