Investments

Investment Analysis and Portfolio ManagementRisk Aversion and Capital Allocation to Risky Assets (Ref. Ch 5-6, BKM)

Main Points:Utility conceptOne risk-free and one risky asset caseInvestment opportunity setSolutions12Risk and Risk AversionMost investors are risk averse. That is bearing risk typically must be companied by a reward in the form of a risk premium.

SpeculationTaking considerable risk for a commensurate gain (positive risk premium)Gamble Bet on an uncertain outcome for enjoymentRisk averse investors are willing to consider:risk-free assetsspeculative positions with positive risk premiumsAllocating Capital Between Risky & Risk-Free AssetsPossible to split investment funds between safe and risky assetsRisk free asset, rf, proxy: ________________________Risky asset or portfolio, rp: _______________________

Example: Your total wealth is $10,000. Arbitrarily you put $2,500 in risk free T-Bills and $7,500 in a stock portfolio . The stock portfolio composition is as follows: Weight of Stock A= 35%Weight of Stock B= 40% Weight of Stock C= 25% T-bills or money market fundrisky portfolio3What percentage of your holdings are in the risk-free asset? 25%. What percent are in the risky asset 75%. Your entire portfolio includes both risky and risk-free assetsThe composition of your complete (ultimate) portfolio is as follows:

Note: The composition of the risky portfolio stays constant.

Allocating Capital Between Risky & Risk-Free AssetsAssetsWeightsWrf2,500/10,000=0.2500WA(7,500/10,000)x0.35=0.2625WB0.75x0.40=0.3000WC0.75x0.25=0.1875Total1.00004How much should be invested in the risky asset? 10%, 50%, 100%, 110%? 110 is taking a loan and investing (on margin)Assuming the risky portfolio has been optimized, Depending on your level of risk you must merely choose between your weights of the risk free and the risky portfolio.Issue:Assuming the risky portfolio has been optimized, depending on your level of risk you must merely choose between your weights of the risk-free and risky portfolio.

In setting weights we will:Examine risk and return tradeoff.Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assetsAllocating Capital Between Risky & Risk-Free Assets5How much should be invested in the risky asset? 10%, 50%, 100%, 110%? 110 is taking a loan and investing (on margin)Assuming the risky portfolio has been optimized, Depending on your level of risk you must merely choose between your weights of the risk free and the risky portfolio.Expected Return and Risk CombinationsE(rc) = yE(rp) + (1 - y)rfc = yrp + (1-y)rfWhere:E(rc) = Return for complete or combined portfolio, a linear combination of component returnsc = Standard deviation of the complete portfolio, a linear combination of component stds. However, this is NOT generally the case for the of combinations of two or more risky assets6ExampleGiven E(rp), rf, rp, and rf let us define a (combined) portfolio with various ys and ask:What are the return for the complete portfolio, E(rC), and the standard deviation, C, for each y?

7AssetsE(r)Std% investedRisk-free5%0%(1-y)Risky14%22%(y)Combined??Example8E(rC) = yE(rp) + (1 - y)rf c = yrp + (1-y)rfy(1-y)E(rc)Std(rc)010.0500.10.90.0590.0220.20.80.0680.0440.30.70.0770.0660.50.50.0950.110.70.30.1130.1540.750.250.11750.1650.90.10.1310.198100.140.221.2-0.20.1580.264

E(r)E(rp) = 14%rf = 5%22%0PFPossible CombinationssE(rp) = 11.75%16.5%y =.75y = 1y = 0Capital Allocation Line (CAL)y=1.29Expected return on the vertical axis, and standard deviation of total portfolio on the horizontal axis. With all of your money in the risk free you will have 7% return and no standard deviation. With all in the risky asset will have 15 expected return and 22% standard deviation. Can move along this line.E(r)E(rp) = 14%rf = 5% = 22%0PFrp ) Slope = 9/22E(rp) - rf = 9%CAL(CapitalAllocationLine)sP or combinations of P & Rf offer a return per unit of risk of 9/22.10This is the complete Capital Allocation Line.The slope says that for an increase in return of 8% we will have an increase in the SD of our portfolio by 22%. The risk premium is also shown.Risk Aversion and AllocationGreater levels of risk aversion lead investors to choose larger proportions of the risk free rate

Lower levels of risk aversion lead investors to choose larger proportions of the portfolio of risky assets

Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

Q: How to choose the optimum complete portfolio?

1112Table: Available Risky Portfolios (Risk-free Rate = 5%)How to choose among the above portfolios?

We can assign a welfare or utility score to competing portfolios on the basis of expected return and risk of those portfolios.Many particular scoring systems are legitimate.A particular one used by CFA is as follows:

Figure 6.1 The Trade-off Between Risk and Returns of a Potential Investment Portfolio, P

14Utility FunctionU = utilityE (r) = expected return on the asset or portfolioA = coefficient of risk aversions2 = variance of returns = a scaling factor

15 Evaluating investments by using utility scores, Investors with Varying Degree of Risk Aversion

Risk aversion level is: A=2, choice is Portfolio H, Risk aversion level is: A= 3.5, choice is Portfolio M, Risk aversion level is: A= 5, choice is Portfolio M,

The Indifference Curve AE(r)stdIndifference CurveQEqually preferred portfolios will lie on a curve called the indifference curve, which connects all portfolios with the same utility value.16Indifference Curves

Investors want the most return for the least risk.Hence indifference curves higher and to the left are preferred.I2I1I3U = E[r] - 1/2Ap2

17A and Indifference Curves

Indifference curves describe different combinations of return and risk that provide equal utility (U) or satisfaction. U = E[r] - 1/2Ap2 A and indifference curves: The greater the A the steeper the indifference curve and all else equal, such investors will invest less in risky assets.

The smaller the A the flatter the indifference curve and all else equal, such investors will invest more in risky assets.18This is not in the chapter but it should be. Note this is just one version of U, many forms are possible.E(r)rf = 5%0PFCAL(CapitalAllocationLine)sA=3A=3QS19This is the complete Capital Allocation Line.The slope says that for an increase in return of 8% we will have an increase in the SD of our portfolio by 22%. The risk premium is also shown.E(r)rf = 5%0PFCAL(CapitalAllocationLine)sA=2A=3ST20This is the complete Capital Allocation Line.The slope says that for an increase in return of 8% we will have an increase in the SD of our portfolio by 22%. The risk premium is also shown.Setting up the ProblemThe investment opportunity:Risk-free rate = 5%Risky rp: E(rp) = 14%, std(rp) =22%A mean-variance investor:U = E[r] - 1/2Ap2The optimal portfolio selection:Invest a portion y of the total wealth in the risky asset, leaving the rest in the risk-free asset.Optimal portfolio? Max U(rc)21Portfolio Return and Risk FormulasPortfolio return:ry = wfrf + wprpry = (1-y)rf + yrpry = rf + y(rp rf)

Portfolio excess return:ry - rf = y(rp rf)

Portfolio variance:

22

23Portfolio ConstructionExpected return of the complete portfolio: ry = rf + y(rp rf)E[ry] = rf + y(E[rp] rf)E[ry] = 0.05 + y(0.14 0.05)E[ry] = 0.05 + y(0.09)

The risk of the complete portfolio:sy = yspsy = y(0.22)

24Portfolio Construction Example (Contd.)Rearrange sy=ysP and substitute in E(r).

25The Capital Allocation Line

Forming the Optimization Problem Which combination is optimal if our objective is to maximize our utility function? max U(ry) = E(ry) 0,5 A y2 WhereE(Ry) = rf + y [E(rp) rf ]y2 = y 2p2 Substitute these two equations in U(ry) and get:

max U(ry) = rf + y [E(rp) rf ] 0,5 A y 2p226Ex:Forming the Optimization Problemmax U(ry) = rf + y [E(rp) rf ] 0,5 A y 2p2

For the previous example the function to maximize is: Max U(ry) = 0.05 + y [0.14 0.05] 0,5 A y 2(0.22)2

Max U(y) = 0.05 + 0.09y 0,5A(0.222)(y2)

Choice variable: y

ways to solve an optimization problem:Numerical / GraphicalAnalytical.

27Numerical Solution: Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4y(1-y)E(rc)Std(rc)U010.0500.05000.10.90.0590.0220.05800.20.80.0680.0440.06410.30.70.0770.0660.06830.40.60.0860.0880.07050.4650.5350.091850.10230.07090.50.50.0950.110.07080.60.40.1040.1320.06920.70.30.1130.1540.06560.80.20.1220.1760.06000.90.10.1310.1980.0526100.140.220.04321.1-0.10.1490.2420.03191.2-0.20.1580.2640.018628Analytical SolutionSet f(y) = 0 and calculate the value of choice variable. Verify if f(y*) < 0, then y* is truly the optimal solution.Taking derivatives:

1. Look for y* that satisfies f (y*) = 0:

2. Check for optimality of y*: f (y*) < 0 ?29

Second phase:Optimal Complete PortfolioWe know that: y* = 0.465The funds would be allocated among the assets as follows:53.5% in risk-free asset, 46.5% in risky portfolio in fixed proportions. Assume that the risky portfolio composition is known as 54% in E and 46% in B. Then overall allocation:

Risk-free asset: 53.5%E= 0.465 * 0.54 = 0.251 or 25.1%B= 0.465 * 0.46 = 0.214 or 21.4%TOTAL 100.0%

4.3.201530Determinants of Portfolio WeightsMore generally, the optimal solution can be expressed as

4.3.201531

A more risk-averse investor (with a larger risk-aversion coefficient A) will invest less in the risky asset.

If the risk premium, [E(rp)-rf], of the original risky asset decreases, a risk-averse investor will reduce his holdings in the risky asset accordingly assume A constant.

If the original asset is risky (with Var(Rp)>0), but pays zero risk premium, then no risk-averse investor will hold the risky asset.

If the risk premium is negative, a risk-averse investor will start shorting the asset.32The Capital Allocation Line where lending and borrowing rates are the same

33Previously CAL assumed that lending and borrowing rates are the same for the investor. This may not hold, then we have to use two different risk-fee rates.Lending and Borrowing portfolios?Lend at rf=5% and borrow at rf=7%Lending range slope = 9/22 = 0.41Borrowing range slope = 7/22 = 0.318CAL kinks at PThe Capital Allocation Line where lending and borrowing rates are NOT the same34The Capital Allocation Line where lending and borrowing rates are NOT the same

Asset_AllocErf5%A=4Erp14%Std_rf0%Std_rp22%y(1-y)E(rc)Std(rc)U010.0500.05000.10.90.0590.0220.05800.20.80.0680.0440.06410.30.70.0770.0660.06830.40.60.0860.0880.07050.4650.5350.091850.10230.07090.50.50.0950.110.07080.60.40.1040.1320.06920.70.30.1130.1540.06560.80.20.1220.1760.06000.90.10.1310.1980.0526100.140.220.04321.1-0.10.1490.2420.03191.2-0.20.1580.2640.0186

Arbitrary_PortfoliosExample: Diversification effectLong Only PortfolioTwo-risky asset portfolioAssetsStdE(r)wA 0.120.080.5B0.20.130.5P0.13114877050.105Cov(A,B)0.0072Corr0.3

This portfolio has expected return half-way between the expected returns on Assets A and B,but the portfolio standard deviation is less than half-way between the asset standard deviationsExample: Diversification effectLong-Short PortfolioTwo-risky asset portfolioAssetsStdE(r)wA 0.120.08-0.5B0.20.131.5P0.28774989140.105Cov(A,B)0.0072Corr0.3Long-Short Portfolio has both a higher expected return and standard deviation than Asset B

N2_DiversificationExample: Diversification effectTwo-risky asset portfolioPortfolio Frontier:Assets120.05E(R) 0.080.13Std0.120.2Cov(1,2)0.0072 Corr0.3 w1w2E(R)Std -0.51.50.1550.2880.155-0.21.20.1400.2340.14 -0.11.10.1350.2170.135 010.1300.2000.13all in #20.10.90.1250.1840.125 0.30.70.1150.1550.115 0.40.60.1100.1420.11 0.50.50.1050.1310.1050.60.40.1000.1230.1 0.80.20.0900.1150.09MVP0.90.10.0850.1160.085 MVPCall Solver100.0800.1200.08all in #1wAwBE(R)StdConstraintValue1.1-0.10.0750.1270.075 0.820.180.0890.114111.2-0.20.0700.1370.07

Note: Individual variances and correlation coefficients are held constant in this graph.

Corr_effExample: Diversification effect - Risky Portfolio - Case of 2 Assetsrf=0.05Asset12 E(R)0.080.13SD0.120.2Cov0.0072 Corr Coeff0.3

w1w2E(R)SDE(R)Sratio-0.61.60.1600.3060.1600.359-0.51.50.1550.2880.1550.365-0.41.40.1500.2700.1500.371-0.31.30.1450.2520.1450.378-0.21.20.1400.2340.1400.385-0.11.10.1350.2170.1350.392010.1300.2000.1300.4000.10.90.1250.1840.1250.4080.30.70.1150.1550.1150.4200.40.60.1100.1420.1100.423
Odabasi: Odabasi:Highest Sharpe Ratio for Corr=0,3 case0.50.50.1050.1310.1050.4190.60.40.1000.1230.1000.4080.80.20.0900.1150.0900.3490.90.10.0850.1160.0850.303100.0800.1200.0800.2501.1-0.10.0750.1270.0750.1961.2-0.20.0700.1370.0700.1461.3-0.30.0650.14939879520.0650.1001.4-0.40.0600.16297239030.0600.0611.5-0.50.0550.17776388830.0550.0281.6-0.60.0500.1934941860.050-0.000

Corr_EffectsExample: Diversification effect - Risky Portfolio - Case of 2 Assetsrf=0.05Asset12 E(R)0.080.13SD0.120.2Cov0.0072Correlation CoefficientsCorr-1.00.00.31.0Sharpe Ratio

w1w2E(R)SDSDSDSDw1for Corr=0,3-0.61.60.1600.3920.3280.3060.248-0.60.359-0.51.50.1550.3600.3060.2880.240-0.50.365-0.41.40.1500.3280.2840.2700.232-0.40.371-0.31.30.1450.2960.2620.2520.224-0.30.378-0.21.20.1400.2640.2410.2340.216-0.20.385-0.11.10.1350.2320.2200.2170.208-0.10.392010.1300.2000.2000.2000.200
Odabasi: Odabasi:In the Box: The Case of No Short Sales.Notice that for the same return the SD of the portfolio decreases with the correlation coefficient. Look at the the portfolio with E(R)=0,100.4000.10.90.1250.1680.1800.1840.1920.10.4080.30.70.1150.1040.1450.1550.1760.30.4200.40.60.1100.0720.1290.1420.1680.40.423
Odabasi: Odabasi:Highest Sharpe Ratio for Corr=0,3 case0.50.50.1050.0400.1170.1310.1600.50.4190.60.40.1000.0080.1080.1230.1520.60.4080.80.20.0900.0560.1040.115
Odabasi: Odabasi:Minimum Variance Portfolio for the case of Corr= 0,30.1360.80.3490.90.10.0850.0880.1100.1160.1280.90.303100.0800.1200.1200.1200.12010.2501.1-0.10.0750.1520.1340.1270.1121.10.1961.2-0.20.0700.1840.1490.1370.1041.20.1461.3-0.30.0650.2160.1670.14939879520.0961.30.1001.4-0.40.0600.2480.1860.16297239030.0881.40.0611.5-0.50.0550.280.2060.1780.081.50.0281.6-0.60.0500.3120.2260.1930.0721.6-0.000

Optimal_risky_portfolioExample: Diversification effectTwo-risky asset portfolioAssets12rf=0.05E(R) 0.080.13Std0.120.2Cov(AB)72Correlation Coefficients -110.3Sharpew1w2E(R)StdStdStd Ratio for Corr=0,3-0.21.20.1400.2640.2160.2340.140.385-0.11.10.1350.2320.2080.2170.1350.392010.1300.2000.2000.2000.130.4000.10.90.1250.1680.1920.1840.1250.4080.30.70.1150.1040.1760.1550.1150.4200.40.60.1100.0720.1680.1420.110.4230.60.40.1000.0080.1520.1230.10.4080.80.20.0900.0560.1360.1150.090.3490.90.10.0850.0880.1280.1160.0850.303100.0800.1200.1200.1200.080.2501.1-0.10.0750.1520.1120.1270.0750.1961.2-0.20.0700.1840.1040.1370.070.146

Power_of_diversificationPOWER OF DIVERSIFICATION N=2, EQUAL WEIGHTSrf=0.05Asset12 E(R)0.080.13SD0.20.2STD of equally weighted portfolio:0.1414213562Cov0.0072 Corr Coeff0.0

w1w2E(R)SDE(R)Sratio-0.61.60.1600.3420.1600.322-0.51.50.1550.3160.1550.332-0.41.40.1500.2910.1500.343-0.31.30.1450.2670.1450.356-0.21.20.1400.2430.1400.370-0.11.10.1350.2210.1350.385010.1300.2000.1300.4000.10.90.1250.1810.1250.4140.30.70.1150.1520.1150.4270.40.60.1100.1440.1100.416
Odabasi: Odabasi:Highest Sharpe Ratio for Corr=0,3 case0.50.50.1050.1410.1050.3890.60.40.1000.1440.1000.3470.80.20.0900.1650.0900.2430.90.10.0850.1810.0850.193100.0800.2000.0800.1501.1-0.10.0750.2210.0750.1131.2-0.20.0700.2430.0700.0821.3-0.30.0650.26683328130.0650.0561.4-0.40.0600.29120439560.0600.0341.5-0.50.0550.3162277660.0550.0161.6-0.60.0500.34176014980.050-0.000

Power_of_diversification_n3PORTFOLIO SELECTION WITH 3 RISKY SECURITIES AND A RISK FREE SECURITY

ParametersE(R_1)0.136annual returnvolatility_10.2SDcorrelation_1,20E(R_2)0.15annual returncorrelation_1,30volatility_20.2SDcorrelation_2,30

E(R_3)0.14annual returnvolatility_30.2SD

R_f0.02maximum SR1.0565increment0.1

w_1w_2w_3volatilityE(R_p)Efficient frontierSharpe ratio1.00000.00000.00000.20000.13600.13600.5800security 11.00000.00000.20000.15000.15000.6500security 21.00000.20000.14000.14000.6000security 30.00000.0200risk free1.50001.5000-2.00000.58310.14900.63610.2212various portfolios below1.40001.5000-1.90000.55930.14940.61090.23141.30001.5000-1.80000.53590.14980.58620.24221.20001.5000-1.70000.51300.15020.56200.25381.10001.5000-1.60000.49070.15060.53850.26611.00001.5000-1.50000.46900.15100.51560.27930.90001.5000-1.40000.44810.15140.49340.29320.80001.5000-1.30000.42800.15180.47220.30790.70001.5000-1.20000.40890.15220.45200.32330.60001.5000-1.10000.39090.15260.43300.33920.50001.5000-1.00000.37420.15300.41530.35550.40001.5000-0.90000.35890.15340.39920.37170.30001.5000-0.80000.34530.15380.38480.38750.20001.5000-0.70000.33350.15420.37230.40240.10001.5000-0.60000.32370.15460.36200.4158-0.00001.5000-0.50000.31620.15500.35410.4269-0.10001.5000-0.40000.31110.15540.34870.4352-0.20001.5000-0.30000.30850.15580.34600.4401-0.30001.5000-0.20000.30850.15620.34600.4414-0.40001.5000-0.10000.31110.15660.34870.4390-0.50001.50000.00000.31620.15700.35410.4332-0.60001.50000.10000.32370.15740.36200.4244-0.70001.50000.20000.33350.15780.37230.4132-0.80001.50000.30000.34530.15820.38480.4003-0.90001.50000.40000.35890.15860.39920.3862-1.00001.50000.50000.37420.15900.41530.3715-1.10001.50000.60000.39090.15940.43300.3566-1.20001.50000.70000.40890.15980.45200.3419-1.30001.50000.80000.42800.16020.47220.3276-1.40001.50000.90000.44810.16060.49340.3138-1.50001.50001.00000.46900.16100.51560.30061.50001.4000-1.90000.55930.14800.61090.22891.40001.4000-1.80000.53520.14840.58540.23991.30001.4000-1.70000.51150.14880.56040.25181.20001.4000-1.60000.48830.14920.53590.26461.10001.4000-1.50000.46560.14960.51190.27831.00001.4000-1.40000.44360.15000.48870.29300.90001.4000-1.30000.42240.15040.46630.30870.80001.4000-1.20000.40200.15080.44470.32540.70001.4000-1.10000.38260.15120.42430.34290.60001.4000-1.00000.36440.15160.40500.36110.50001.4000-0.90000.34760.15200.38720.37980.40001.4000-0.80000.33230.15240.37110.39850.30001.4000-0.70000.31870.15280.35680.41660.20001.4000-0.60000.30720.15320.34460.43350.10001.4000-0.50000.29800.15360.33480.4483-0.00001.4000-0.40000.29120.15400.32770.4602-0.10001.4000-0.30000.28710.15440.32330.4682-0.20001.4000-0.20000.28570.15480.32180.4719-0.30001.4000-0.10000.28710.15520.32330.4710-0.40001.40000.00000.29120.15560.32770.4657-0.50001.40000.10000.29800.15600.33480.4564-0.60001.40000.20000.30720.15640.34460.4439-0.70001.40000.30000.31870.15680.35680.4292-0.80001.40000.40000.33230.15720.37110.4129-0.90001.40000.50000.34760.15760.38720.3959-1.00001.40000.60000.36440.15800.40500.3787-1.10001.40000.70000.38260.15840.42430.3617-1.20001.40000.80000.40200.15880.44470.3453-1.30001.40000.90000.42240.15920.46630.3296-1.40001.40001.00000.44360.15960.48870.3147-1.50001.40001.10000.46560.16000.51190.30071.50001.3000-1.80000.53590.14700.58620.23701.40001.3000-1.70000.51150.14740.56040.24911.30001.3000-1.60000.48740.14780.53500.26221.20001.3000-1.50000.46390.14820.51010.27641.10001.3000-1.40000.44090.14860.48580.29171.00001.3000-1.30000.41860.14900.46220.30820.90001.3000-1.20000.39700.14940.43940.32600.80001.3000-1.10000.37630.14980.41760.34490.70001.3000-1.00000.35670.15020.39680.36510.60001.3000-0.90000.33820.15060.37740.38610.50001.3000-0.80000.32120.15100.35940.40780.40001.3000-0.70000.30590.15140.34320.42950.30001.3000-0.60000.29260.15180.32910.45050.20001.3000-0.50000.28140.15220.31730.46980.10001.3000-0.40000.27280.15260.30820.4861-0.00001.3000-0.30000.26680.15300.30190.4984-0.10001.3000-0.20000.26380.15340.29870.5057-0.20001.3000-0.10000.26380.15380.29870.5072-0.30001.30000.00000.26680.15420.30190.5029-0.40001.30000.10000.27280.15460.30820.4935-0.50001.30000.20000.28140.15500.31730.4797-0.60001.30000.30000.29260.15540.32910.4628-0.70001.30000.40000.30590.15580.34320.4439-0.80001.30000.50000.32120.15620.35940.4240-0.90001.30000.60000.33820.15660.37740.4039-1.00001.30000.70000.35670.15700.39680.3841-1.10001.30000.80000.37630.15740.41760.3651-1.20001.30000.90000.39700.15780.43940.3471-1.30001.30001.00000.41860.15820.46220.3302-1.40001.30001.10000.44090.15860.48580.3144-1.50001.30001.20000.46390.15900.51010.29961.50001.2000-1.70000.51300.14600.56200.24561.40001.2000-1.60000.48830.14640.53590.25891.30001.2000-1.50000.46390.14680.51010.27331.20001.2000-1.40000.44000.14720.48490.28911.10001.2000-1.30000.41670.14760.46020.30621.00001.2000-1.20000.39400.14800.43620.32490.90001.2000-1.10000.37200.14840.41310.34510.80001.2000-1.00000.35100.14880.39080.36700.70001.2000-0.90000.33110.14920.36980.39030.60001.2000-0.80000.31240.14960.35010.41480.50001.2000-0.70000.29530.15000.33200.44020.40001.2000-0.60000.28000.15040.31580.46570.30001.2000-0.50000.26680.15080.30190.49020.20001.2000-0.40000.25610.15120.29060.51220.10001.2000-0.30000.24820.15160.28220.5302-0.00001.2000-0.20000.24330.15200.27710.5425-0.10001.2000-0.10000.24170.15240.27530.5479-0.20001.20000.00000.24330.15280.27710.5458-0.30001.20000.10000.24820.15320.28220.5367-0.40001.20000.20000.25610.15360.29060.5216-0.50001.20000.30000.26680.15400.30190.5022-0.60001.20000.40000.28000.15440.31580.4800-0.70001.20000.50000.29530.15480.33200.4565-0.80001.20000.60000.31240.15520.350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weighted 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N3_DiversificationPORTFOLIO SELECTION WITH 3 RISKY SECURITIES AND A RISK FREE SECURITY

ParametersE(R_1)0.136annual returnvolatility_10.154SDcorrelation_1,20.27E(R_2)0.15annual returncorrelation_1,30.54volatility_20.23SDcorrelation_2,30.11

E(R_3)0.14annual returnvolatility_30.18SD

R_f0.02maximum SR0.9004increment0.1

w_1w_2w_3volatilityE(R_p)Efficient frontierSharpe ratio1.00000.00000.00000.15400.13600.13600.7532security 11.00000.00000.23000.15000.15000.5652security 21.00000.18000.14000.14000.6667security 30.00000.0200risk free1.50001.5000-2.00000.47740.14900.44980.2702various portfolios below1.40001.5000-1.90000.46590.14940.43950.27781.30001.5000-1.80000.45470.14980.42940.28551.20001.5000-1.70000.44380.15020.41960.29341.10001.5000-1.60000.43320.15060.41010.30151.00001.5000-1.50000.42300.15100.40090.30970.90001.5000-1.40000.41320.15140.39200.31800.80001.5000-1.30000.40380.15180.38350.32640.70001.5000-1.20000.39480.15220.37540.33490.60001.5000-1.10000.38630.15260.36780.34330.50001.5000-1.00000.37820.15300.36060.35160.40001.5000-0.90000.37080.15340.35380.35980.30001.5000-0.80000.36390.15380.34760.36770.20001.5000-0.70000.35750.15420.34190.37530.10001.5000-0.60000.35190.15460.33680.3825-0.00001.5000-0.50000.34680.15500.33230.3892-0.10001.5000-0.40000.34250.15540.32840.3953-0.20001.5000-0.30000.33890.15580.32510.4007-0.30001.5000-0.20000.33600.15620.32260.4053-0.40001.5000-0.10000.33390.15660.32070.4091-0.50001.50000.00000.33260.15700.31950.4119-0.60001.50000.10000.33200.15740.31900.4138-0.70001.50000.20000.33230.15780.31920.4147-0.80001.50000.30000.33330.15820.32010.4147-0.90001.50000.40000.33510.15860.32170.4136-1.00001.50000.50000.33770.15900.32400.4117-1.10001.50000.60000.34100.15940.32700.4088-1.20001.50000.70000.34500.15980.33070.4052-1.30001.50000.80000.34980.16020.33490.4008-1.40001.50000.90000.35520.16060.33980.3958-1.50001.50001.00000.36130.16100.34530.39031.50001.4000-1.90000.45240.14800.42730.28301.40001.4000-1.80000.44070.14840.41680.29131.30001.4000-1.70000.42940.14880.40660.30001.20001.4000-1.60000.41840.14920.39670.30881.10001.4000-1.50000.40770.14960.38710.31791.00001.4000-1.40000.39740.15000.37780.32710.90001.4000-1.30000.38750.15040.36890.33650.80001.4000-1.20000.37800.15080.36040.34600.70001.4000-1.10000.36900.15120.35230.35550.60001.4000-1.00000.36050.15160.34460.36500.50001.4000-0.90000.35260.15200.33740.37440.40001.4000-0.80000.34520.15240.33080.38360.30001.4000-0.70000.33840.15280.32470.39240.20001.4000-0.60000.33230.15320.31920.40090.10001.4000-0.50000.32680.15360.31430.4088-0.00001.4000-0.40000.32210.15400.31000.4160-0.10001.4000-0.30000.31820.15440.30650.4224-0.20001.4000-0.20000.31500.15480.30360.4279-0.30001.4000-0.10000.31260.15520.30150.4325-0.40001.40000.00000.31110.15560.30010.4359-0.50001.40000.10000.31040.15600.29950.4382-0.60001.40000.20000.31050.15640.29960.4393-0.70001.40000.30000.31150.15680.30040.4392-0.80001.40000.40000.31330.15720.30210.4379-0.90001.40000.50000.31590.15760.30440.4356-1.00001.40000.60000.31930.15800.30750.4321-1.10001.40000.70000.32350.15840.31130.4278-1.20001.40000.80000.32850.15880.31580.4226-1.30001.40000.90000.33410.15920.32090.4166-1.40001.40001.00000.34050.15960.32660.4100-1.50001.40001.10000.34740.16000.33280.40301.50001.3000-1.80000.42770.14700.40510.29701.40001.3000-1.70000.41590.14740.39450.30631.30001.3000-1.60000.40440.14780.38410.31601.20001.3000-1.50000.39320.14820.37410.32601.10001.3000-1.40000.38250.14860.36440.33621.00001.3000-1.30000.37210.14900.35500.34670.90001.3000-1.20000.36210.14940.34600.35740.80001.3000-1.10000.35260.14980.33750.36820.70001.3000-1.00000.34360.15020.32930.37900.60001.3000-0.90000.33510.15060.32170.38980.50001.3000-0.80000.32720.15100.31460.40040.40001.3000-0.70000.31990.15140.30810.41070.30001.3000-0.60000.31330.15180.30210.42070.20001.3000-0.50000.30740.15220.29680.43000.10001.3000-0.40000.30230.15260.29220.4387-0.00001.3000-0.30000.29790.15300.28830.4464-0.10001.3000-0.20000.29440.15340.28510.4531-0.20001.3000-0.10000.29170.15380.28270.4586-0.30001.30000.00000.29000.15420.28110.4628-0.40001.30000.10000.28910.15460.28030.4656-0.50001.30000.20000.28910.15500.28030.4670-0.60001.30000.30000.29000.15540.28110.4669-0.70001.30000.40000.29180.15580.28270.4654-0.80001.30000.50000.29450.15620.28520.4625-0.90001.30000.60000.29810.15660.28840.4583-1.00001.30000.70000.30240.15700.29230.4530-1.10001.30000.80000.30760.15740.29690.4467-1.20001.30000.90000.31350.15780.30230.4396-1.30001.30001.00000.32010.15820.30820.4317-1.40001.30001.10000.32740.15860.31480.4233-1.50001.30001.20000.33530.15900.32190.41451.50001.2000-1.70000.40330.14600.38320.31241.40001.2000-1.60000.39140.14640.37240.32291.30001.2000-1.50000.37980.14680.36190.33391.20001.2000-1.40000.36850.14720.35180.34521.10001.2000-1.30000.35760.14760.34200.35691.00001.2000-1.20000.34710.14800.33250.36880.90001.2000-1.10000.33700.14840.32350.38100.80001.2000-1.00000.32750.14880.31490.39330.70001.2000-0.90000.31840.14920.30670.40570.60001.2000-0.80000.31000.14960.29910.41800.50001.2000-0.70000.30220.15000.29210.43020.40001.2000-0.60000.29510.15040.28570.44190.30001.2000-0.50000.28870.15080.27990.45310.20001.2000-0.40000.28310.15120.27490.46350.10001.2000-0.30000.27830.15160.27060.4729-0.00001.2000-0.20000.27440.15200.26710.4811-0.10001.2000-0.10000.27140.15240.26440.4879-0.20001.20000.00000.26930.15280.26250.4931-0.30001.20000.10000.26820.15320.26150.4966-0.40001.20000.20000.26810.15360.26140.4983-0.50001.20000.30000.26890.15400.26220.4982-0.60001.20000.40000.27080.15440.26380.4964-0.70001.20000.50000.27350.15480.26630.4928-0.80001.20000.60000.27720.15520.26960.4877-0.90001.20000.70000.28180.15560.27370.4812-1.00001.20000.80000.28720.15600.27860.4736-1.10001.20000.90000.29340.15640.28420.4649-1.20001.20001.00000.30030.15680.29040.4555-1.30001.20001.10000.30800.15720.29730.4455-1.40001.20001.20000.31620.15760.30470.4351-1.50001.20001.30000.32510.15800.31270.42451.50001.1000-1.60000.37950.14500.36170.32941.40001.1000-1.50000.36740.14540.35080.34141.30001.1000-1.40000.35560.14580.34020.35381.20001.1000-1.30000.34410.14620.32990.36671.10001.1000-1.20000.33310.14660.31990.38011.00001.1000-1.10000.32250.14700.31040.39380.90001.1000-1.00000.31240.14740.30130.40780.80001.1000-0.90000.30280.14780.29260.42210.70001.1000-0.80000.29380.14820.28450.43640.60001.1000-0.70000.28540.14860.27700.45060.50001.1000-0.60000.27770.14900.27010.46450.40001.1000-0.50000.27080.14940.26380.47790.30001.1000-0.40000.26460.14980.25830.49050.20001.1000-0.30000.25940.15020.25350.50200.10001.1000-0.20000.25500.15060.24960.5121-0.00001.1000-0.10000.25170.15100.24660.5206-0.10001.10000.00000.24930.15140.24450.5271-0.20001.10000.10000.24790.15180.24320.5316-0.30001.10000.20000.24770.15220.24300.5338-0.40001.10000.30000.24840.15260.24370.5338-0.50001.10000.40000.25020.15300.24530.5315-0.60001.10000.50000.25310.15340.24790.5271-0.70001.10000.60000.25690.15380.25130.5208-0.80001.10000.70000.26170.15420.25560.5128-0.90001.10000.80000.26740.15460.26070.5034-1.00001.10000.90000.27390.15500.26660.4929-1.10001.10001.00000.28120.15540.27320.4816-1.20001.10001.10000.28920.15580.28040.4696-1.30001.10001.20000.29780.15620.28820.4573-1.40001.10001.30000.30710.15660.29650.4448-1.50001.10001.40000.31700.15700.30540.43221.50001.0000-1.50000.35610.14400.34070.34821.40001.0000-1.40000.34390.14440.32960.36181.30001.0000-1.30000.33190.14480.31890.37601.20001.0000-1.20000.32030.14520.30840.39081.10001.0000-1.10000.30920.14560.29840.40621.00001.0000-1.00000.29850.14600.28880.42210.90001.0000-0.90000.28830.14640.27960.43840.80001.0000-0.80000.27870.14680.27090.45500.70001.0000-0.70000.26970.14720.26280.47160.60001.0000-0.60000.26140.14760.25540.48810.50001.0000-0.50000.25390.14800.24860.50420.40001.0000-0.40000.24720.14840.24260.51950.30001.0000-0.30000.24140.14880.23730.53360.20001.0000-0.20000.23650.14920.23300.54620.10001.0000-0.10000.23270.14960.22950.5569-0.00001.00000.00000.23000.15000.22710.5652-0.10001.00000.10000.22840.15040.22560.5710-0.20001.00000.20000.22790.15080.22520.5739-0.30001.00000.30000.22860.15120.22580.5740-0.40001.00000.40000.23040.15160.22740.5712-0.50001.00000.50000.23330.15200.23010.5658-0.60001.00000.60000.23730.15240.23370.5579-0.70001.00000.70000.24230.15280.23820.5481-0.80001.00000.80000.24830.15320.24350.5365-0.90001.00000.90000.25510.15360.24970.5237-1.00001.00001.00000.26280.15400.25660.5099-1.10001.00001.10000.27120.15440.26420.4956-1.20001.00001.20000.28030.15480.27240.4809-1.30001.00001.30000.29000.15520.28110.4662-1.40001.00001.40000.30030.15560.29040.4515-1.50001.00001.50000.31110.15600.30010.43721.50000.9000-1.40000.33350.14300.32020.36891.40000.9000-1.30000.32100.14340.30900.38441.30000.9000-1.20000.30890.14380.29810.40081.20000.9000-1.10000.29720.14420.28760.41791.10000.9000-1.00000.28590.14460.27740.43581.00000.9000-0.90000.27510.14500.26770.45430.90000.9000-0.80000.26490.14540.25850.47340.80000.9000-0.70000.25530.14580.24990.49280.70000.9000-0.60000.24640.14620.24180.51230.60000.9000-0.50000.23820.14660.23450.53150.50000.9000-0.40000.23090.14700.22790.55010.40000.9000-0.30000.22450.14740.22210.56750.30000.9000-0.20000.21910.14780.21730.58330.20000.9000-0.10000.21480.14820.21340.59680.10000.90000.00000.21170.14860.21060.6075-0.00000.90000.10000.20970.14900.20890.6150-0.10000.90000.20000.20900.14940.20820.6190-0.20000.90000.30000.20960.14980.20870.6193-0.30000.90000.40000.21140.15020.21030.6159-0.40000.90000.50000.21440.15060.21300.6091-0.50000.90000.60000.21860.15100.21680.5994-0.60000.90000.70000.22380.15140.22150.5871-0.70000.90000.80000.23010.15180.22720.5728-0.80000.90000.90000.23730.15220.23370.5570-0.90000.90001.00000.24540.15260.24100.5403-1.00000.90001.10000.25420.15300.24890.5231-1.10000.90001.20000.26380.15340.25750.5057-1.20000.90001.30000.27400.15380.26670.4884-1.30000.90001.40000.28470.15420.27630.4714-1.40000.90001.50000.29590.15460.28640.4549-1.50000.90001.60000.30760.15500.29690.43891.50000.8000-1.30000.31160.14200.30050.39161.40000.8000-1.20000.29900.14240.28920.40941.30000.8000-1.10000.28670.14280.27820.42831.20000.8000-1.00000.27490.14320.26750.44821.10000.8000-0.90000.26350.14360.25730.46911.00000.8000-0.80000.25260.14400.24750.49080.90000.8000-0.70000.24240.14440.23820.51320.80000.8000-0.60000.23280.14480.22960.53610.70000.8000-0.50000.22400.14520.22170.55900.60000.8000-0.40000.21600.14560.21450.58150.50000.8000-0.30000.20900.14600.20820.60290.40000.8000-0.20000.20300.14640.20280.62260.30000.8000-0.10000.19820.14680.19840.63980.20000.80000.00000.19460.14720.19520.65370.10000.80000.10000.19230.14760.19310.6636-0.00000.80000.20000.19130.14800.19230.6690-0.1000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weighted 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