Fi8000 Valuation of Financial Assets Spring Semester 2010 Dr. Isabel Tkatch Assistant Professor of Finance

Investment Strategies Lending vs. Borrowing (risk-free asset) Lending: a positive proportion is invested in the risk-free asset (cash outflow in the present: CF0 < 0, and cash inflow in the future: CF1 > 0) Borrowing: a negative proportion is invested in the risk-free asset (cash inflow in the present: CF0 > 0, and cash outflow in the future: CF1 < 0)

Lending vs. Borrowing A A Lend B Borrow C rf rf

Investment Strategies A Long vs. Short position in the risky asset Long: A positive proportion is invested in the risky asset (cash outflow in the present: CF0 < 0, and cash inflow in the future: CF1 > 0) Short: A negative proportion is invested in the risky asset (cash inflow in the present: CF0 > 0, and cash outflow in the future: CF1 < 0)

Long vs. Short E(R) A B STD(R) Long A and Short B Long A and Long B Short A and Long B B STD(R)

Investment Strategies Passive risk reduction: The risk of the portfolio is reduced if we invest a larger proportion in the risk-free asset relative to the risky one The perfect hedge: The risk of asset A is offset (can be reduced to zero) by forming a portfolio with a risky asset B, such that ρAB=(-1) Diversification: The risk is reduced if we form a portfolio of at least two risky assets A and B, such that ρAB<(+1) The risk is reduced if we add more risky assets to our portfolio, such that ρij<(+1)

One Risky Fund and one Risk-free Asset: Passive Risk Reduction Reduction in portfolio risk B Increase of portfolio Risk C rf rf

Two Risky Assets with ρAB=(-1): The Perfect Hedge Minimum Variance is zero Pmin B STD(R)

The Perfect Hedge – an Example What is the minimum variance portfolio if we assume that μA=10%; μB=5%; σA=12%; σB=6% and ρAB=(-1)?

The Perfect Hedge – Continued What is the expected return μmin and the standard deviation of the return σmin of that portfolio?

Diversification: the Correlation Coefficient and the Frontier ρAB=(-1) -1<ρAB<1 ρAB=+1 B STD(R)

Diversification: the Number of Risky assets and the Frontier STD(R)

Diversification: the Number of Risky assets and the Frontier STD(R)

Diversification: the Number of Risky assets and the Frontier STD(R)

Diversification: the Number of Risky assets and the Frontier STD(R)

Capital Allocation: n Risky Assets State all the possible investments – how many possible investments are there? Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V efficient? Present your results in the μ-σ (mean – standard-deviation) plane.

The Expected Return and the Variance of the Return of the Portfolio wi = the proportion invested in the risky asset i (i=1,…n) p = the portfolio of n risky assets (wi invested in asset i) Rp = the return of portfolio p μp = the expected return of portfolio p σ2p = the variance of the return of portfolio p

The Set of Possible Portfolios in the μ-σ Plane E(R) The Frontier i STD(R)

The Set of Efficient Portfolios in the μ-σ Plane E(R) The Efficient Frontier i STD(R)

Capital Allocation: n Risky Assets The investment opportunity set: {all the portfolios {w1, … wn} where Σwi=1} The Mean-Variance (M-V or μ-σ ) efficient investment set: {only portfolios on the efficient frontier}

The case of n Risky Assets: Finding a Portfolio on the Frontier Optimization: Find the minimum variance portfolio for a given expected return Constraints: A given expected return; The budget constraint.

The case of n Risky Assets: Finding a Portfolio on the Frontier

Capital Allocation: n Risky Assets and a Risk-free Asset State all the possible investments – how many possible investments are there? Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V efficient? Present your results in the μ-σ (mean – standard-deviation) plane.

The Expected Return and the Variance of the Return of the Possible Portfolios wi = the proportion invested in the risky asset i (i=1,…n) p = the portfolio of n risky assets (wi invested in asset i) Rp = the return of portfolio p μp = the expected return of portfolio p σ2p = the variance of the return of portfolio p

The Set of Possible Portfolios in the μ-σ Plane (only n risky assets) E(R) The Frontier i STD(R)

The Set of Possible Portfolios in the μ-σ Plane (risk free asset included) rf STD(R)

The Set of Efficient Portfolios in the μ-σ Plane The Capital Market Line: μp= rf + [(μm-rf) / σm]·σp μ m i rf σ

The Separation Theorem The asset allocation process of the risk-averse investors can be separated into two stages: 1.Choose the optimal portfolio of risky assets m (The allocation between risky securities is identical for all the investors) 2.Choose the optimal allocation of funds between the risky portfolio m and the risk-free asset rf – choose a portfolio on the CML (The allocation between the risky portfolio and the risk free asset is personal and depends on the risk preferences of each investor)

Capital Allocation: n Risky Assets and a Risk-free Asset The investment opportunity set: {all the portfolios {w0, w1, … wn} where Σwi=1} The Mean-Variance (M-V or μ-σ ) efficient investment set: {all the portfolios on the Capital Market Line - CML}

n Risky Assets and One Risk-free Asset: Finding the Market Portfolio Optimization: Find the minimum variance portfolio for a given expected return Constraints: A given expected return; The budget constraint.

n Risky Assets and One Risk-free Asset: Finding the Market Portfolio

n Risky Assets and One Risk-free Asset: Finding the Market Portfolio

Example Find the market portfolio if there are only two risky assets, A and B, and a risk-free asset rf. μA=10%; μB=5%; σA=12%; σB=6%; ρAB=(-0.5) and rf=4%

Example Continued

Example Continued

Practice Problems BKM 7th Ed. Ch. 7: 1-13, 17-22, 25-26 BKM 8th Ed. Ch. 7: 4-19 CFA: 4-6, 10-11 Mathematics of Portfolio Theory: Read and practice parts 11-13