Review Portfolio Theory

1 ⎡ E (rm ) − rf ⎤ xm = ⎢ ⎥ 2 A ⎣ σm ⎦ x f = 1 − xm
λ2 = rf
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Opportunities: Many Risky Assets
• • • The opportunity set is defined by the means, variances and covariances, and the budget constraint. We find portfolios on the minimum-variance frontier p f by minimizing the variance
The return on a portfolio is a simple weighted average of the returns on the individual assets • The expected return on a portfolio
E(rp ) = ∑ x j E (rj ) ),
We find the optimal portfolio by maximizing the investor's MV utility function, defined in terms of risk tolerance, subject to the budget constraint. That i Th is, we choose xf, xm, and λ to maximize h d i i
Opportunities: Two Assets One Riskfree
The opportunity set is a straight line when one of the assets is riskless
E ( rp ) = x f rf + xm E (rm ) x f = 1 − xm E ( rp ) = rf + xm ⎡ E (rm ) − rf ⎤ ⎣ ⎦ σ p = xm σ m xm = σ p / σ m ⎡ E ( rm ) − rf E ( rp ) = rf + ⎢ σm ⎣ ⎤ ⎥ σp ⎦
i j
σ 2 = ∑ xi cov (ri , rp ) p
i
• • •
The variance of the return on the portfolio is a weighted average of the p g g covariance of an asset's return with the return on the optimal portfolio An asset contributes the covariance of its return with the return on the optimal portfolio to variance of the return on the p p p portfolio ( (risk) ) This is the insight that means covariance with the optimal portfolio or beta (and not variance) is the proper measure of risk for an asset
If If
ρij = 1 ρij = −1
σ p = xi σ i + x j σ j σ p = xi σ i − x j σ j ρij = 0, σ i = 0, σ p = x j σ j
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If asset i is riskless
In each of the three cases, the opportunity set becomes a straight line
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However, some (fairly) simple manipulation shows oweve , so e ( a y) s p e a pu a o s ows
σ 2 = ∑ xi ∑ x j σ ij p
i j
σ 2 = ∑ xi cov (ri , ∑ x j rj ) p
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Two Asset Cases
Correlation coefficient
ρij ≡
σ ij σi σ j
−1 ≤ ρij ≤ 1
E(rp ) = xi E(ri )+x j E(rj ), where xi +x j = 1
σ 2 = xi2 σ i2 + 2 xi x j σ ij + x 2 σ 2 p j j = xi2 σ i2 + 2 xi x j ρijj σ i σ j + x 2 σ 2 j j
j
where here
∑x
j
j
=1
The expected return on a portfolio is a simple weighted average of the expected returns on the individual assets Or an asset contributes its expected return to the portfolio's expected return (reward) ( d)
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Separation Property: One Riskfree and Many Risky Assets
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Portfolio Theory
• The return on a portfolio
rps = ∑ x j rjs or rp = ∑ x j rj ,
j j
where
∑x
j
j
=1
U = tE ( rp ) − 1 σ 2 t 2 p
An Indifference curve holds utility constant, say at U = U*
U* 1 2 E ( rp ) = + σp 2t t
Mean Variance Utility Function in terms of Risk Aversion
Review of Portfolio Theory
• Almost all economic problems involve making optimal choices given • a set of preferences and • some opportunities subject to • a budget constraint • Preferences are defined in terms of the investor's subjective tradeoff between the expected return and variance of a portfolio • Preferences can be captured in terms of a MV utility function and MV indifference curves
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Preferences: An Investor’s Indifference Curves
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Preferences: Indifference Curves for Two Different Investors
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Preferences: An Investor’s Indifference Curves Cannot Cross
⎡ E (rm ) − rf ⎤ xm = t ⎢ ⎥ 2 σm ⎣ ⎦ x f = 1 − xm
λ1 = t f tr
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Alternatively, we find the optimal portfolio by maximizing the investor's MV utility function, defined in terms of risk aversion, subject to the budget constraint. constraint
2 2 max L = t ( x f rf + xm E (rm )) − 1 xm σ m + λ1 (1 − x f − xm ) 2
We find the derivatives with respect to xf, xm, and λ1, and set them equal to zero to find
2 2 max L = ( x f rf + xm E (rm )) − 1 Axmσ m + λ2 (1 − x f − xm ) 2
We find the derivatives with respect to xf, xm, and λ1, and set them equal to zero to find
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Opportunities: Two Assets One Riskfree
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Opportunities: Two Risky Assets
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Preferences: Mean Variance Utility Functions
Mean Variance Utility Function in terms of Risk Tolerance
U = E ( rp ) − 1 Aσ 2 p 2
An Indifference curve holds utility constant, say at U = U*
1 E ( rp ) = U * + Aσ 2 p 2
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Optimal Portfolio: One Risky One Riskfree Asset
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The Variance of the Return on a Portfolio