Efficient Diversification Chapter 6

Portfolio Risk So far we’ve been focusing on individual assets, but what happens when we combine individual assets into a portfolio?

Diversification A B A&B Time Time Time

Portfolio v Single Asset Investing everything in a single asset exposes you to that investment’s total risk A portfolio allows us to spread out our investment reducing the impact of swings in a single asset →Assets are unlikely to all move in the same direction Intuition: “Don’t put all your eggs in one basket”

Historical Non-Diversifiers People on record for saying: “Put all your eggs in one basket and watch it closely” Mark Twain: Died penniless Andrew Carnegie: Died known as “The Richest Man in the World” →Definition of Risk

Diversification and Portfolio Risk Market/Systematic/Non-diversifiable Risk Risk factors common to whole economy Affect all firms, economy wide risk Unique/Firm-Specific/Nonsystematic/ Diversifiable Risk Risk that can be eliminated by diversification Affect individual or small groups of firms (industries) What does σ measure?

The Wonders of Diversifying

Risk versus Diversification

How Diversification Works Reduces/eliminates unsystematic risk. Why can’t we diversify away systematic risk?

Aside: Why Diversified Investors Set Prices Investor A is undiversified, while B has a diversified portfolio. Both investors want to buy a share of Facebook. Who gets the share? Which investor uses a 10% discount rate? 20%? Facebook will pay a constant $20 dividend Which investor offers a higher price for the share? (Remember: Price = Div / discount rate)

Measuring Co-Movement Covariance: measures how two variables move in relation to one another Positive: The two variables move up together or down together Ex: Height and Weight Negative: When one moves up the other moves down Ex: Sleep and Coffee consumption

Calculation Cov (rS, rB) = σSB = Σ p(i)*(rS(i)– μS)*(rB(i) – μb) σXY = p1*(X1 – μS)*(Y1 – μB)+p2*(X2 – μS)*(Y2 – μB)+….+pN*(XN – μS)*(YN – μB) Note that σss = Σ p(i)*(Xs(i) – μs)*(Xs(i) – μs) Which implies? How does a risky asset’s return move with the risk free?

Covariance Strength If the covariance of asset 1 and 2 is -1,000, and between asset 1 and 3 is 500. Which asset is more closely related to 1?

Correlation Coefficient Measures the strength of the covariance relation It is a standardization of covariance Bounded by -1 & 1 1 means the two are perfectly positively correlated -1 means the two are perfectly negatively correlated

Correlation Coefficient Formula ρSB = σSB / (σS * σB) ρSB – Correlation Coefficient σSB – Covariance between S and B σS – Std Dev of S σB – Std Dev of B

Comparing Strength When determining the strength of a correlation all we care about is the absolute value of the correlation coefficient If ρ13 is -0.8, and ρ23 is 0.5, which asset is more correlated with 3?

Correlation Coefficient Example σ13 is -1,000; σ23 is 500 σ1 is 10; σ2 is 1,000; σ3 is 250 Is 1 or 2 more strongly correlated with 3?

Covariance & Correlation Aside The covariance between a risky asset and a risk free is 0. Why?

Portfolio Return & Risk Portfolio Return: weighted average of component returns A stock’s weight is the percent of the portfolio it represents Expected Return: weighted average of component expected returns Variance: depends on covariance between the assets

Portfolio Return Example If a portfolio comprised of Google {E(r) = 25%} and Acme {E(r) = 10%} has an E(r) = 20%, how much is invested in Google and Acme?

Portfolio Variance Formulas Depends primarily on covariances between the assets Two Stock Portfolio σP2 = (wBσB)2+(wSσS)2+2wBwSσSB σP2 = (wBσB)2+(wSσS)2+2wBwSσSσBρSB Three Stock Portfolio σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+ 2wBwSσSB + 2wBwCσBC + 2wSwCσSC σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+ 2wBwSσSσBρSB + 2wBwCσBσCρBC + 2wSwCσSσCρSC

Portfolio Variance Formulas Depends primarily on covariances between the assets Two Stock Portfolio σP2 = (wBσB)2+(wSσS)2+2wBwSσSB σP2 = (wBσB)2+(wSσS)2+2wBwSσSσBρSB Three Stock Portfolio σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+ 2wBwSσSB + 2wBwCσBC + 2wSwCσSC σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+ 2wBwSσSσBρSB + 2wBwCσBσCρBC + 2wSwCσSσCρSC

Portfolio Variance Example Two stocks A and B have expected returns of 10% and 20%. In the past, A and B have had std dev of 15% and 25%, respectively, with a correlation coefficient of 0.2. You decide to invest 30% in A and the rest in B. Calculate the portfolio return and portfolio risk. Has diversification been of any use? Explain.

Calculations wA = 30% wB = A = 10% B = 20% A = 15% B = 25% AB = 0.2 Portfolio Return = Portfolio Variance = Portfolio Standard Deviation = Weighted Average Standard Deviation

Remarks on Diversification Diversification reduces the p from 22% to 18.9% What happens if AB = 1? What happens as AB approaches -1?

Portfolio Example An investor wants to maximize his return, but doesn’t want the risk of his portfolio to exceed 15%. He has two options, a stock fund with a standard deviation of 35% and T-Bills, how much does he invest in the stock fund?

Individual Stock Allocation Your offer a portfolio comprised of 70% stock A and 30% stock B. If an investor has half their wealth invested in your portfolio, how much of her wealth is in stock A?

Which Stock do you Prefer? Stock A :  = 10%;  = 2% Stock B :  = 10%;  = 3% Stock C :  = 12%;  = 2%

Fundamental Premise of Portfolio Theory Rational investors prefer the highest expected return at the lowest possible risk Investors want Lower Risk & Higher Returns → Mean-Variance Criterion If E(rA) ≥ E(rB) and σA ≤ σB Portfolio A dominates portfolio B How can investors lower risk without sacrificing return?

Investment Opportunity Set The set of possible risk-return pairings offered by the portfolios that can be formed from the available securities

Investment Opportunity Set Example

Investment Opportunity Set Continued return Stock Min Var. Bonds P

Changing the Correlation Coefficient Artificial Risk Free Asset (-1) How does the correlation coefficient affect the diversification benefit?

What portfolios do we prefer? return Efficient Portfolios: -Maximize risk premium for any level of standard deviation -Minimize standard deviation for any level of return -Maximize Sharpe ratio for any standard deviation or risk premium P

Including More Assets Possible Portfolios Efficient Portfolios return

Including a risk free asset? Capital Allocation Line, Capital Market Line A risk free asset changes our investment opportunity set and our efficient portfolio Optimal Risky Portfolio (O) return rf P

Which Efficient Portfolio? O, the Optimal Efficient Portfolio It offers the best risk return trade off Highest SHARPE RATIO (ri - rf) / i All portfolios on the CAL will have the same Sharpe Ratio as O

Optimal Risky Portfolio Given two risky asset

Capital Allocation Line The set of efficient portfolios available once we include a risk free asset Why only two assets? O dominates all other risky assets All other are either too risky or offer to low a return Rf is used to adjust for risk tolerance

Tobin’s Separation Property Implies investing can be separated into two tasks Find O (Optimal risky portfolio) Determining where we want to be on the CAL? Known as Asset Allocation

Asset Allocation How do we allocate our portfolio across asset classes Bogle: This the most fundamental investment question Basically: How much risk do we want? We can adjust our risk level through our risk free investments

Finding Your Spot Notionally there is a formula for finding an investors preferred location on the CAL, which is based on an investors level of risk aversion Risk Aversion measures how investors feel about risk An investor that is more risk averse will hold more of the risk free asset As risk aversion decreases (investor becomes more risk tolerant, risk loving) the investor will hold more O

Measuring Risk Aversion We measure risk aversion with A The risk premium an investor demands for investing everything in a portfolio given its risk The Sharpe Ratio an investor requires to put everything into a single diversified portfolio

How Much O Should an Investor Hold Y is the amount of an investors portfolio invested in the Optimal Risky Portfolio To find Y divide the Available Sharpe Ratio (which is offered by O) by that demanded by the investors (A)

What Does Our Portfolio Look Like? Where are we on the CAL? What risk premium do you demand to invest all your money in a diversified portfolio with a σ of 20%? If you have $250,000 to invest, how much of your total investment portfolio is invested in O? O’s risk premium is 11% and its σ is 18% Where is the rest of your portfolio invested?

Moving on the CAL As risk averse increases Y falls → Hold more T-Bills Purchasing T-Bills is lending the government money As investors become more risk tolerant (loving) Y increase → Hold more of O We can move past O by shorting T-Bills Borrow T-Bills → Sell them →Use proceeds to buy O Latter return the T-Bill plus interest

Composition of our portfolio (C)

Our Portfolio (C)

Alternative: Passive Investing Based on the idea that securities are fairly priced Steps involved Buy a portfolio of assets that tracks the broad economy Relax Advantages: Simple, Low Cost, Outperform the average active strategy