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

Today ☺ Portfolio Theory ☺ The Mean-Variance Criterion ☺ Capital Allocation ☺ The Mathematics of Portfolio Theory

Nation’s Financial Industry Gripped by Fear NY Time, September 15, 2008 By BEN WHITE and JENNY ANDERSON ‘Fear and greed are the stuff that Wall Street is made of.’

The Mean-Variance Criterion (M-V or μ-σ criterion) STD(R) – “fear” E(R) - “g reed ” ☺ ☺

Capital Allocation - Data There are three (risky) assets and one risk-free asset in the market. The risk-free rate is rf = 1%, and the distribution of returns of risky assets is normal with the following parameters AssetABC Expected Return 5.6%4.2%1.7% Standard Deviation of the Return 2.5%5.0%2.1%

Capital Allocation: n mutually exclusive assets State all the possible investments. Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V efficient (i.e. which assets can not be thrown out of the set of desirable investments by a risk-averse investor who uses the M-V rule)? Present your results on the μ-σ (mean – standard-deviation) plane.

The Mean-Variance Criterion (M-V or μ-σ criterion) Let A and B be two (risky) assets. All risk- averse investors prefer asset A to B if { μ A ≥ μ B and σ A < σ B } or if { μ A > μ B and σ A ≤ σ B } Note that these rules apply only when we assume that the distribution of returns is normal.

The Expected Return and the STD of Return ( μ-σ plane) rf C A B

Capital Allocation: n mutually exclusive assets The investment opportunity set: {rf, A, B, C} The Mean-Variance (M-V or μ-σ ) efficient investment set: {rf, A, C} Note that investment B is not in the efficient set since investment A dominates it (one dominant investment is enough).

Capital Allocation: One Risky Asset (A) and One 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 on the μ-σ (mean – standard-deviation) plane.

The Expected Return and STD of Return of the Portfolio α = the proportion invested in the risky asset A p = the portfolio with α invested in the risky asset A and (1- α) invested in the risk-free asset rf and (1- α) invested in the risk-free asset rf R p = the return of portfolio p μ p = the expected return of portfolio p σ p = the standard deviation of return of portfolio p R p = α· R A + (1- α)·rf μ p = E[ α· R A + (1- α)·rf ] = α·μ A + (1- α)·rf σ 2 p = V[ α· R A + (1- α)·rf ] = (α·σ A ) 2 Or σ p = α·σ A

Capital Allocation: One Risky Asset and One Risk-free Asset The investment opportunity set: { all portfolios with proportion α invested in A and (1- α ) invested in the risk-free asset rf } The Mean-Variance (M-V or μ-σ ) efficient investment set: { all the portfolios in the opportunity set }

The Capital Allocation Line

The Expected Return and the STD of Return ( μ-σ plane) rf C A B A

The Capital Allocation Line (CAL): Four Basic Investment Strategies rf C A B A P1P1 P2P2

Portfolios on the CAL Portfolioα E(R p ) = μ p Std(R p ) = σ p rf01.00%0.00% P1P1P1P %0.625% A15.60%2.50% P2P2P2P %3.75%

Capital Allocation: n Mutually Exclusive Risky Asset and One 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 on the μ-σ (mean – standard-deviation) plane.

The Expected Return and the STD of Return ( μ-σ plane) rf C A B

Capital Allocation: One Risky Asset and One Risk-free Asset The investment opportunity set: {all the portfolios with proportion α invested in the risky asset j and (1- α ) invested in the risk-free asset, (j = A or B or C)} The Mean-Variance (M-V or μ-σ ) efficient investment set: {all the portfolios with proportion α invested in the risky asset A and (1- α ) invested in the risk-free asset – (why A?)}

Capital Allocation: Two 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 on the μ-σ (mean – standard-deviation) plane.

The Expected Return and STD of Return of the Portfolio w A = the proportion invested in the risky asset A w B = (1- w A ) = the proportion invested in the risky asset B p = the portfolio with w A invested in the risky asset A and (1- w A ) invested in the risky asset B (1- w A ) invested in the risky asset B R p = the return of portfolio p μ p = the expected return of portfolio p σ p = the standard deviation of the return of portfolio p R p = w A ·R A + (1-w A )·R B μ p = E[ w A ·R A + (1-w A )·R B ] σ 2 p = V[ w A ·R A + (1-w A )·R B ]

Two Risky Assets: The Investment Opportunity Set STD(R p ) E(R p ) B A

Two Risky Assets: The M-V Efficient Set (Frontier) STD(Rp) E(R p ) B A

Two Mutually Exclusive Risky Assets: The M-V Efficient Set STD(R) E(R) B A

Two Risky Assets: The M-V Efficient Set (Frontier) STD(R) E(R) B A

Two Risky Assets: The M-V Efficient Set (Frontier) STD(R) E(R) B A

Two Risky Assets: The M-V Efficient Set (Frontier) STD(R) E(R) B A P

Capital Allocation: Two Risky Assets The investment opportunity set: {all the portfolios on the frontier: with proportion w A invested in the risky asset A and (1- w A ) invested in the risky asset B} The Mean-Variance (M-V or μ-σ ) efficient investment set: {all the portfolios on the efficient frontier}

Two Risky Assets: The M-V Efficient Set (Frontier) STD(R) E(R) B A P1P1 P2P2 P3P3 P min

Portfolios on the Efficient Frontier w A = the proportion invested in the risky asset A w B = (1- w A ) = the proportion invested in the risky asset B What is the value of w A for each one of the portfolios indicated on the graph? - Assume that μ A =10%; μ B =5%; σ A =12%; σ B =6%; ρ AB =(-0.5). What is the investment strategy that each portfolio represents? How can you find the minimum variance portfolio? What is the expected return and the std of return of that portfolio?

Portfolios on the Frontier Portfolio wAwAwAwA E(R p ) = μ p Std(R p ) = σ p P1P1P1P %16.57% A110.00%12.00% P2P2P2P %4.06% P min ??? B05.00%6.00% P3P3P3P %13.08%

The Minimum Variance Portfolio

Practice Problems BKM 7th Ed. Ch. 6: 15-18, 20-21, 25, 32, 34-35; BKM 8th Ed. Ch. 6: 15-18, 26-27, 21, CFA: 6, 8-9; Mathematics of Portfolio Theory: Read and practice parts 6-10.