The Optimal Risky Portfolio

Lecture No.3– SAPM

Portfolio construction Process

• Capital allocation between risky portfolio and risk-free assets– Depends upon risk aversion and risk-return trade-off

• Asset allocation among asset classes– Broad outlines of portfolio established

• Security selection– Specific securities selected for the portfolio

• Steps 2 and 3 lead to optimal risky portfolio• Optimal risky portfolio is the combination of risky

assets that provides the best risk-return trade-off

Diversification and portfolio risk

• Two sources of risk – – firm-specific risk or unique risk – Market risk or systematic risk (inflation,

business cycles, exchange rates etc )• Diversification can reduce firm-specific risk to zero

if specific risk is independent (known as the insurance principle)

• Diversification cannot eliminate the systematic risk, i.e. risk attributable to market-wide sources

• Hence investors only care about systematic risks• Return on assets compensates only for systematic

risks

Portfolio expected return and risk

• Consider 2 risky assets, a debt mutual fund D and an equity mutual fund E

• If the weights of the 2 assets are Wd and We then portfolio expected return E(rp) is

and portfolio risk (σp) is given as

Correlation and Portfolio Risk

• Expected return of the portfolio is the weighted average of expected returns of component assets with their proportions as weights

• Portfolio variance is driven by the covariance between component assets

• If correlation between assets is 1 (perfect positive correlation) then portfolio standard deviation = weighted average of component standard deviations

• If correlation less than 1, portfolio standard deviation is less than weighted average of component standard deviations

• If correlation is -1 (perfect negative correlation), portfolio variance is lowest – we can construct a zero-variance portfolio

Case of perfect positive correlation

• When rho=1, equation for portfolio variance becomes

• Or• Hence standard deviation of portfolio =

weighted average of component standard deviations

• Thus no benefit from diversification

Correlation > 0 < 1• When assets are less than perfectly positively

correlated we can construct the minimum variance portfolio

• The Minimum Variance Portfolio has a standard deviation less than that of the component assets

• Equation for obtaining weights for Minimum Variance Portfolio for portfolio consisting of 2 assets D and E

• Example

When correlation is zero

• The equation for portfolio variance becomes

• The weights for the minimum variance portfolio are

• and

Case of perfect negative correlation

• For assets with perfect negative correlation

• The weights for the zero variance portfolio are

• and

Example-Portfolio return and risk

• The expected return and risk of the two assets are E(rd)= 0.08, E(re)=0.13, (σd) =0.12 and (σe)=0.20Wd We E( rp) Portfol io Std Deviation for the given correlation

-1 0 0.3 10.00 1.00 0.1300 0.2000 0.2000 0.2000 0.20000.10 0.90 0.1250 0.1680 0.1804 0.1840 0.19200.20 0.80 0.1200 0.1360 0.1618 0.1688 0.18400.30 0.70 0.1150 0.1040 0.1446 0.1547 0.17600.40 0.60 0.1100 0.0720 0.1292 0.1420 0.16800.50 0.50 0.1050 0.0400 0.1166 0.1311 0.16000.60 0.40 0.1000 0.0080 0.1076 0.1226 0.15200.70 0.30 0.0950 0.0240 0.1032 0.1170 0.14400.80 0.20 0.0900 0.0560 0.1040 0.1145 0.13600.90 0.10 0.0850 0.0880 0.1098 0.1156 0.12801.00 0.00 0.0800 0.1200 0.1200 0.1200 0.1200

Observations• For ρ=1, portfolio standard deviation is simply

weighted average of asset standard deviation (no benefit of diversification)

• For ρ=0.30 and ρ=0, portfolio standard deviation – decreases initially as equity component increases

indicating diversification benefit and– increases as portfolio becomes concentrated in equity– we can find the minimum-variance portfolio which

has standard deviation less than that of individual assets

• For ρ=-1, diversification is most effective due to “perfect hedge”– For ρ=-1, we can construct a zero-variance

portfolio

Portfolio expected return and SD

Exhibit 4.5: Mean Standard Deviation Diagram: Portfolios of Two Risky Securities with Arbitrary Correlation,

The Minimum Variance Frontier

• We plot the set of portfolios with the lowest variance at a given level of expected return ( recall the mean variance criterion)

• Above set known as the Mean Standard Deviation frontier or Minimum Variance frontier

• Portion of mean-SD frontier below the global minimum variance portfolio is inefficient

• For each frontier portfolio on the lower portion there exists another frontier portfolio on the upper portion with same σ but a higher E(r)

• Portion above the global minimum variance portfolio is known as the efficient frontier in the absence of a risk-free asset

• Investors will only choose a portfolio on the efficient frontier

The Efficient Frontier

Risky Portfolio with a risk-free asset• Given a risky portfolio of 2 risky assets, a debt

mutual fund and an equity mutual fund and a risk-free asset with return rf

• How do we find the optimal risky portfolio?• Plot CALs starting from the risk-free rate and

passing through the opportunity set of risky assets – debt and equity funds

• The highest CAL will have highest slope, i.e. Max S = (E(rp)-rf)/σp

• Optimal risky portfolio is the tangency point of the highest CAL to the portfolio opportunity set

• Tangency portfolio consists of risky assets only

Risky portfolio + risk-free asset

We draw CALs from the risk-free rate to various portfolios on the efficient frontier

The Optimal risky portfolioWe find the CAL from the risk-free rate to the point of tangency to the efficient frontierPortfolio at tangency point will have highestReward-to-risk ratio

Optimal Risky Portfolio – 2 assets

• Equation for determining weights of optimal risky portfolio with 2 risky assets

And

where RD and RE are excess returns on debt and equity funds i.e. Expected return less the risk-free rate

Optimal Complete Portfolio

• Calculate the weights of the optimal risky portfolio as above

• Compute the E(r ) and σ of the optimal risky portfolio

• We have the risk-free rate and the investor’s degree of risk aversion

• Proportion to be invested in risky portfolio is

• Balance is the proportion invested in risk-free asset

Optimal Complete Portfolio

• Point P where the CAL is tangent to the efficient frontier depicts the optimal risky portfolio • At Points 1 and 2 the indifference curves of 2 different investors are tangent to the CAL • Points 1 and 2 depict the Optimum complete portfolio for those investors

Summary• Note that optimal risky portfolio is the same for all

investors– Formula for computation of weights of optimal risky

portfolio does not include the investor’s degree of risk aversion

• Hence the fund manager will offer the same optimal risky portfolio to all his investors- his job becomes easier !

• The optimal complete portfolio for each investor (the allocation of funds between the risky portion and risk-free portion) will be different

• It will depend on investor’s preferences, i.e. his degree of risk aversion and indifference curve

• More risk averse investors will have lower proportion of the optimal risky portfolio in their complete portfolio than less risk-averse investors