Markowitz Mean-variance

Sequence Of MPT Material Basic Return vs RiskOptimizing the Risky Portfolio

Risk Aversion / Utility Function Allocate Between Risk-free asset and Risky Asset (CAL)Stocks vs BondsMarkowitz Mean-Variance Asset Allocation Going In Reverse: the CAPMCalculates theoretical expected return for individual assets basedupon covariance with the market and the risk-free return.

Evolution of Variance Measures in Studying MPTIndividual Assets/Asset Class Variance(Return for Period Mean Return) ^2

Paired Covariances(Asset 1 Return Mean) x (Asset 2 Return Mean)

Portfolio VarianceFor 2 assets: Asset 1 Variance + Asset 2 Variance + 2 x Weighted Covariance

Background AssumptionsAs an investor you want to maximize the returns for a given level of risk.Your portfolio includes all of your assets, not just financial assetsThe relationship between the returns for assets in the portfolio is important.A good portfolio is not simply a collection of individually good investments.

Markowitz Portfolio TheoryDerives the expected rate of return for a portfolio of assets and an expected risk measureMarkowitz demonstrated that the variance of the rate of return is a meaningful measure of portfolio risk under reasonable assumptionsThe portfolio variance formula shows how to effectively diversify a portfolio

Markowitz Portfolio TheoryAssumptionsInvestors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period.Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.Investors estimate the risk of the portfolio on the basis of the variability of expected returns.

Markowitz Portfolio TheoryAssumptionsInvestors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only.For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected returns, investors prefer less risk to more risk.

Markowitz Portfolio TheoryUnder these five assumptions, a single asset or portfolio of assets is efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.

Alternative Measures of RiskVariance or standard deviation of expected return (main focus)Based on deviations from the mean returnLarger values indicate greater riskOther measuresRange of returnsReturns below expectationsSemivariance measures deviations only below the mean

Characteristics of Probability Distributions1) Mean: most likely value2) Variance or standard deviation3) Skewness

* If a distribution is approximately normal, the distribution is described by characteristics 1 and 2.

Single Factor Mean-Variance Model Expected Return

Expected Variance

Portfolio Standard Deviation Formula

Portfolio Standard Deviation Formula

Portfolio Standard Deviation CalculationThe portfolio standard deviation is a function of:The variances of the individual assets that make up the portfolio The covariances between all of the assets in the portfolioThe larger the portfolio, the more the impact of covariance and the lower the impact of the individual security variance

Constructing Risky Portfolios E(rp) still as easy as ever simple weighted average of asset E(r)s Variance of the risky portfolios for two assets = p2 = wA12 x A12 + wA22 x A22 + 2 x wA1 x wA2 x COVAR(A1,A2) Another case of the general formula presented as a special case (Its a global conspiracy) The general rule is just to take the weighted average of all of the covariances for all possible paired sets in the portfolio

Generalize The Portfolio Variance CalculationNow with the general rule:p2 = wA12 x A12 + wA22 x A22 + 2 x wA1 x wA2 x COVAR(A1,A2) Covar(A1,A1) = A12 Note that there are now 4 terms (2 x 2) for 2 assetsp2 = wA1 x wA1 x Covar (A1,A1) + wA2 x wA2 x Covar (A2,A2) +wA1 x wA2 x Covar (A1,A2) + wA1 x wA2 x Covar (A1,A2)

The General Formula Is Easier To Use In A Worksheet Environment We use a covariance matrix Then we need to weight each covariance term for the asset weighting in the portfolioNote that this looks just like the Mean-Variance Worksheet

Sheet1

Asset 1Asset 2

Asset 1

Asset 2

Sheet2

Sheet3

Adding Asset Weights Here is a bordered covariance matrix for a portfolio with 60% asset 1 and 40% asset 2 the asset weights are the bordersNote that this matrix is also on the Mean-Variance Worksheet

Sheet1

Asset 1Asset 2

Asset 1

Asset 2

Asset 1Asset 2

.60.40

Asset 1.60

Asset 2.40

Sheet2

Sheet3

One Final HintRevisiting Covariance vs Correlation Coefficient The formula to calculate 1,2 = Covar (A1,A2) / A1 x A2 If we know 1,2 and need to calculate Covar(A1,A2) :Covar (A1,A2) = A1,A2 x A1 x A2

This formula is used in B6 to C12 and B16 to H22 to calculate the covariance matrix. B16 to H22 should look familiar.

Diversification And HedgingDiversificationHedging1.0 0.0 -1.0Correlation Coefficient

Implications for Portfolio FormationAssets differ in terms of expected rates of return, standard deviations, and correlations with one anotherWhile portfolios give average returns, they give lower riskDiversification works!Even for assets that are positively correlated, the portfolio standard deviation tends to fall as assets are added to the portfolio

Implications for Portfolio FormationCombining assets together with low correlations reduces portfolio risk moreThe lower the correlation, the lower the portfolio standard deviationNegative correlation reduces portfolio risk greatlyCombining two assets with perfect negative correlation reduces the portfolio standard deviation to nearly zero

The Efficient FrontierThe efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of returnFrontier will be portfolios of investments rather than individual securitiesExceptions being the asset with the highest return and the asset with the lowest risk (This is true for Minimum Variance Frontier, not the Efficient Frontier)

Efficient Frontier and Alternative PortfoliosC

The Efficient Frontier and Portfolio SelectionAny portfolio that plots inside the efficient frontier (such as point C) is dominated by other portfoliosFor example, Portfolio A gives the same expected return with lower risk, and Portfolio B gives greater expected return with the same riskWould we expect all investors to choose the same efficient portfolio?No, individual choices would depend on relative appetites return as opposed to risk

The Portfolio Standard Deviation

Optimization What is it? What do we optimize? The efficient frontier is a series of portfolios optimized (lowest variance) for a series of possible E(r)s We graph some sub-optimal portfolios (below the kink) The endpoints are the lowest and highest E(r) assets includable in the portfolio

The Two-Asset Example Easiest to graph and comprehend Easy to optimize (formula-based)The graph line is set up by taking evenly-spaced mixes of two assets Low end of the graph is 100% the smaller E(r) asset, top end is 100% the larger E(r) Graph is same style as our Utility graph

The Efficient Frontier and Investor UtilityAn individual investors utility curve specifies the trade-offs she is willing to make between expected return and riskEach utility curve represent equal utility; curves higher and to the left represent greater utility (more return with lower risk)The interaction of the individuals utility and the efficient frontier should jointly determine portfolio selection

The Efficient Frontier and Investor UtilityThe optimal portfolio has the highest utility for a given investorIt lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility

Selecting an Optimal Risky Portfolio

Utility / Risk AversionUtility = E(r new asset) - .005 x Aversion x 2(new asset)U = E(r new asset) - .005 x A x 2(new asset)

If Utility > Risk Free Return, Asset Fits Within Your Risk Profile

Utility CalculationsFor a Risk Aversion Level of 4:Risk-Free Rate = 5.00

Vanguard S&P 500 (VFINX):

U = 11.24 - .005 x 4 x 430.13 = 2.64

Fidelity Japan (FJPNX):

U = 8.21 - .005 x 4 x 2660.23 = -45.00

Utility CurveGraph line represents the portfolios with utility equal to sample PortRisk aversion level (A) is constant

Chart1

3.74

5.24

6.74

8.24

9.74

11.24

12.74

14.24

15.74

17.24

18.74

Indifference Asset/Port

E(r )

Indifference Curve At One Aversion Level

Utility Matrix

ExampleVFINXDODIXFJPNXRisk Free

Expected Return22.0011.247.788.215.00

Variance1176.00430.1443.192660.230.00

Conversion Factor0.0050.0050.0050.0050.005

Risk Aversion Levels:116.129.097.56-5.095.00

210.246.947.35-18.395.00

34.364.797.13-31.695.00

4-1.522.646.92-44.995.00

5-7.400.496.70-58.305.00

6-13.28-1.666.48-71.605.00

7-19.16-3.816.27-84.905.00

8-25.04-5.976.05-98.205.00

9-30.92-8.125.84-111.505.00

10-36.80-10.275.62-124.805.00

Utility Matrix

16.129.08937.56405-5.091155

10.246.93867.3481-18.39235

4.364.78797.13215-31.693455

-1.522.63726.9162-44.99465

-7.40.48656.70025-58.295755

-13.28-1.66426.4843-71.59695

-19.16-3.81496.26835-84.898055

-25.04-5.96566.0524-98.19925

-30.92-8.11635.83645-111.500355

-36.8-10.2675.6205-124.80155

Example

VFINX

DODIX

FJPNX

RiskfFree

Risk Aversion Factor

Utility

Utility Indifference Curve

16.129.08937.56405-5.091155

10.246.93867.3481-18.39235

4.364.78797.13215-31.693455

-1.522.63726.9162-44.99465

-7.40.48656.70025-58.295755

-13.28-1.66426.4843-71.59695

-19.16-3.81496.26835-84.898055

-25.04-5.96566.0524-98.19925

-30.92-8.11635.83645-111.500355

-36.8-10.2675.6205-124.80155

Example

VFINX

DODIX

FJPNX

RiskfFree

Risk Aversion Factor

Utility

Sheet3

Asset E( r)11.24= Values you input for any sample asset or portfolio

Asset 2430.14= Formulas/Data input for you

Asset 20.74= Formulas you must determine/input

Aversion Level4.00

Utility Level2.64

Return Increment1.50

Point EqvlntE( r) Eqvlnt

17.433.74

211.415.24

314.326.74

416.748.24

518.859.74

620.7411.24

722.4812.74

824.0914.24

925.6015.74

1027.0217.24

1128.3718.74

Test Case:

Asset E( r)11.24

Asset 2430.14

Asset 20.74

Aversion Level4.00

Asset Utility2.64

Return Increment0.75

Point EqvlntE( r) Eqvlnt

115.587.49Use these figures to check your calculations

216.748.24by inputtitng the orange #s above and

317.828.99create formulas in the blank cells

418.859.74

519.8210.49

620.7411.24

721.6211.99

822.4812.74

923.2913.49

1024.0914.24

1124.8514.99

Sheet3

Indifference Asset/Port

E(r )

Indifference Curve At One Aversion Level

Utility Level Worksheet

Sheet1

ExampleVFINXDODIXFJPNXRisk Free

Expected Return22.0011.247.788.215.0050.002228.00784.00

Variance1176.00430.1443.192660.230.00-20.0022-42.001764.00

Conversion Factor0.0050.0050.0050.0050.0051176

Risk Aversion Levels:116.129.097.56-5.095.00

210.246.947.35-18.395.00

34.364.797.13-31.695.00

4-1.522.646.92-44.995.00

5-7.400.496.70-58.305.00

6-13.28-1.666.48-71.605.00

7-19.16-3.816.27-84.905.00

8-25.04-5.976.05-98.205.00

9-30.92-8.125.84-111.505.00

10-36.80-10.275.62-124.805.00

Sheet1

16.129.08937.56405-5.091155

10.246.93867.3481-18.39235

4.364.78797.13215-31.693455

-1.522.63726.9162-44.99465

-7.40.48656.70025-58.295755

-13.28-1.66426.4843-71.59695

-19.16-3.81496.26835-84.898055

-25.04-5.96566.0524-98.19925

-30.92-8.11635.83645-111.500355

-36.8-10.2675.6205-124.80155

Example

VFINX

DODIX

FJPNX

RiskfFree

Risk Aversion Factor

Utility

Sheet2

16.129.08937.56405-5.091155

10.246.93867.3481-18.39235

4.364.78797.13215-31.693455

-1.522.63726.9162-44.99465

-7.40.48656.70025-58.295755

-13.28-1.66426.4843-71.59695

-19.16-3.81496.26835-84.898055

-25.04-5.96566.0524-98.19925

-30.92-8.11635.83645-111.500355

-36.8-10.2675.6205-124.80155

Example

VFINX

DODIX

FJPNX

RiskfFree

Risk Aversion Factor

Utility

Sheet3

Utility Levels MatrixFull Value Range

Chart2

16.229.08937.56405-5.091155

10.446.93867.3481-18.39235

4.664.78797.13215-31.693455

-1.122.63726.9162-44.99465

-6.90.48656.70025-58.295755

-12.68-1.66426.4843-71.59695

-18.46-3.81496.26835-84.898055

-24.24-5.96566.0524-98.19925

-30.02-8.11635.83645-111.500355

-35.8-10.2675.6205-124.80155

Example

VFINX

DODIX

FJPNX

RiskfFree

Risk Aversion Factor

Utility

Sheet1

ExampleVFINXDODIXFJPNXRisk Free

Expected Return22.0011.247.788.215.00

Variance1156.00430.1443.192660.230.00

Conversion Factor0.0050.0050.0050.0050.005

116.229.097.56-5.095.00

210.446.947.35-18.395.00

34.664.797.13-31.695.00

4-1.122.646.92-44.995.00

5-6.900.496.70-58.305.00

6-12.68-1.666.48-71.605.00

7-18.46-3.816.27-84.905.00

8-24.24-5.976.05-98.205.00

9-30.02-8.125.84-111.505.00

10-35.80-10.275.62-124.805.00

Sheet1

Example

VFINX

DODIX

FJPNX

RiskfFree

Risk Aversion Factor

Utility

Sheet2

Example

VFINX

DODIX

FJPNX

RiskfFree

Risk Aversion Factor

Utility

Sheet3

Utility Levels MatrixTruncated Value Range

Chart3

16.229.08937.56405-5.091155

10.446.93867.3481-18.39235

4.664.78797.13215-31.693455

-1.122.63726.9162-44.99465

-6.90.48656.70025-58.295755

-12.68-1.66426.4843-71.59695

-18.46-3.81496.26835-84.898055

-24.24-5.96566.0524-98.19925

-30.02-8.11635.83645-111.500355

-35.8-10.2675.6205-124.80155

Example

VFINX

DODIX

FJPNX

RiskfFree

Risk Aversion Factor

Utility

Sheet1

ExampleVFINXDODIXFJPNXRisk Free

Expected Return22.0011.247.788.215.00

Variance1156.00430.1443.192660.230.00

Conversion Factor0.0050.0050.0050.0050.005

116.229.097.56-5.095.00

210.446.947.35-18.395.00

34.664.797.13-31.695.00

4-1.122.646.92-44.995.00

5-6.900.496.70-58.305.00

6-12.68-1.666.48-71.605.00

7-18.46-3.816.27-84.905.00

8-24.24-5.976.05-98.205.00

9-30.02-8.125.84-111.505.00

10-35.80-10.275.62-124.805.00

Sheet1

Example

VFINX

DODIX

FJPNX

RiskfFree

Risk Aversion Factor

Utility

Sheet2

Example

VFINX

DODIX

FJPNX

RiskfFree

Risk Aversion Factor

Utility

Sheet3

Investor Differences and Portfolio SelectionA relatively more conservative investor would perhaps choose Portfolio XOn the efficient frontier and on the highest attainable utility curveA relatively more aggressive investor would perhaps choose Portfolio YOn the efficient frontier and on the highest attainable utility curve

Risky Assets Versus The Risk-free AssetAn Example:Risk-free rate = 4%Risky Assets/Portfolio E (r) = 9%Risky Assets/Portfolio = 8%

What is the risk premium?

The Risk-Free Asset Is Not Included When Calculating Optimal Risky Portfolios Because = 0, there is no diversification benefit Therefore, adding the risk-free asset cannot improve the efficient frontier The risk-free asset plays a critical role AFTER the optimal risky portfolios and efficient frontier have been determined Utility can be improved with the subsequent addition of the risk-free asset

Risky Assets Versus the Risk-free AssetCombinations With No BorrowingE (r)4%0RiskyPortfolio8%9%

Risky Assets Versus the Risk-free AssetCombinations With Borrowing at R(f)E (r)4%0RiskyPortfolio8%9%

Risky Assets Versus the Risk-free AssetCombinations With Borrowing at R(f)E (r)4%0RiskyPortfolio8%9%y < 1y > 1

Risky Assets Versus the Risk-free AssetCombinations With Borrowing at R(f) + 1%E (r)4%0RiskyPortfolio8%9%y < 1y > 1

Risky Assets Versus the Risk-free AssetCombinations With Borrowing at R(f) + 1%E (r)4%0RiskyPortfolio8%9%y < 1y > 1This is the Capital Allocation Line CAL

Risky Assets Versus the Risk-free AssetCombinations With Borrowing at R(f) + 1%E (r)4%0RiskyPortfolio8%9%y < 1y > 1You can solve for utility curves that will intersect the CAL at the optimal point

Annualization of Variance / S.D.Quarterly Annual

q2 = .36 x 4 1.44

q = .60 x SQRT(4) 1.20 Note: 4=periods / year

Monthly Annual

m2 = .12 x 12 1.44

m = .3464 x SQRT(12) 1.20