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Lecture 2

Portfolio Theory1I. Very Simple WorldTwo assets:One risky asset (e.g., S&P 500 Index)One risk-free (T-Bill) asset

How much of your wealth will you allocate to each asset?

2Step 3

Identifying Investment Opportunity Set3Portfolio Return and RiskProportion of wealth:in risky asset = y; in risk-free asset = (1-y)

Portfolio (C) return: rC = W1 r1 + W2 r2 = y rp + (1 y) rf = rf + y (rp rf)

Portfolio (C) Variance: C2 = W1212 + W22 22 + 2W1W2Cov(r1,r2)= y2 p2Standard Deviation: C = y p y = (C / p)

Substituting: rC = rf + (C / p) (rp rf)

Investment opportunity set is a straight line, called the capital allocation line (CAL)Infinite combinations are possible. Which one is optimal?4Numerical Problem (old exam)Risk-free asset is expected to yield 7%. Expected return & risk of the risky portfolio is 15% & 22%. Describe and plot the investment options available to the investor.Go to Return&Risk.xls; Tab to Corr and Portfolio Risk.Plug in: E(rp) = r1= 15%; E(rf) = r2= 7%; 1 = 22%; 2 = 0%; Corr = 0.(a) w1=0, w2=1 (b) w1=0.1, w2=0.9 (c) w1=0.2, w2=0.8 and so on.Note portfolio return and standard deviation in each case. Plot it CAL_OneRisky.xls

Portfolio C: Let us invest y% of the wealth in the risky portfolio and (1-y%) in the riskfree asset.rC = 7% + y (15% - 7%) ; C = y (22%)5Graphically:E(r)E(rp) = 15%rf = 7% p = 22%0PF ) S = 8/22E(rp) - rf = 8%CAL6Dominant PortfolioE(r)0PFQWould you rather hold portfolios of F and P, or F and Q?7Step 4

Optimal Asset Allocation(Involves some calculus!)8Optimal Asset AllocationCombining investor risk tolerance with investment options in the mean-variance framework

(In English) Objective: Choose y that will maximize Investor UtilitySubject to: Available investment opportunity (CAL)

(Not English)Objective: Choose y that will max. U = E(rC) - .005 A sC2Subject to: E (rC) = rf + y (E(rp) rf) and sC2 = y2 p2

Solving: y* = (E(rp) rf) / (0.01 A p2)

9Example:Numerical Problem (old exam) Risk-free asset: rf = 7% , Risky asset: rp = 15% , p = 22% , Weight = yLet us assume the standard utility function

If A = 4: y* = (15 7) / (0.01*4*222) = 0.41Invest $0.41 in Risky Asset and lend $0.59 at Risk-free rate E (rC*) = 10.28% sC* = 9.02% U*=8.65

If A = 2 (Aggressive): y* = (157) / (0.01*2*222) = 0.83 E (rC*) = 13.61% sC* = 18.18% U*=10.30

If A = 8 (Conservative): y* = (157) / (0.01*8*222) = 0.21 E (rC*) = 8.65% sC* = 4.55% U*=7.8210Example: SummaryInvestor TypeAllocation in risky asset (y*)Allocation in risk-free assetOptimal portfolio return (E(rC))Optimal portfolio risk (C)Optimal investor utility (U*)Conservative (A=8)21%79%8.65%4.55%7.82Moderate (A=4)41%59%10.28%9.02%8.65Aggressive (A=2)83%17%13.61%18.18%10.3011When A = 1: y* = (15 7) / (0.01*1*222) = 1.65 E (rC) = 20.22% sC = 36.3%

What is the investment strategy when y* > 1 ?Borrow at the risk-free rate and invest in stock.For each $1 of your wealth: Borrow $0.65 at the risk-free rateInvest $1.65 in the risky assetSee CAL_OneRisky.xls: Optimal-OneRisky A=1

Note that leverage increases investment risk!Example: Using Leverage12CAL with Higher Borrowing Rate

E(r)9%7%) S = .36) S = .27Pp = 22%Lending rate = 7%, Borrowing rate = 9%13CAL with Risk PreferencesE(r)7%PLenderBorrowerp = 22% The lender has a larger A when compared to the borrower14An investor who can tolerate more risk (low value of A) will optimally invest a larger proportion of her portfolio in risky assets (higher y*).How to measure risk tolerance? What about life-cycle effect?

An increase in risky assets volatility (e.g., higher next month) will lower allocation in the risky assetEstimate of future volatility?

An increase in risk premium (E(rp)-rf) will increase allocation in the risky assetForecast future risk premium?

This simple derivation motivates the importance of the risk premium in the asset allocation decision.Key Economic Interpretations (old exam)15II. More Complex World Two risky assets:Stocks (e.g., S&P500 Index)Bonds (e.g., Barclays Aggregate Bond Index)

No risk-free asset

How much of your wealth will you allocate to each asset?

16

Bonds Produced Greater IncomePercentage of total return 19702014Past performance is no guarantee of future results. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. 2015 Morningstar. All Rights Reserved. Bonds Produced Greater Income 19702014Bonds have supplied a higher, more stable income stream relative to stocks. Bond investors generally receive income at fixed intervals, which can be used to offset cash obligations or increase portfolio liquidity. This image illustrates the relative cash income produced by bonds and stocks since 1970. Over this period, bonds provided significantly higher cash income compared with stocks. If you are in a position where you require a reliable stream of income, bonds can be an excellent investment vehicle to fulfill that need.Government bonds are guaranteed by the full faith and credit of the United States government as to the timely payment of principal and interest, while stocks are not guaranteed and have been more volatile than bonds. About the dataStocks are represented by the Ibbotson Large Company Stock Index. Bonds are represented by the 20-year U.S. government bond. An investment cannot be made directly in an index. The values shown in the image are calculated as a percentage of each assets average (arithmetic) annual total return over the period 19702014. Total return is comprised of income return, reinvestment of income return, and capital appreciation.

Adding a Bond Allocation to Diversify19702014Past performance is no guarantee of future results. Risk is measured by standard deviation. Risk and return are based on annual data over the period 19702014. Portfolios presented are based on modern portfolio theory. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. 2015 Morningstar. All Rights Reserved.Adding a Bond Allocation to Diversify Historically, adding bonds to a portfolio comprised of stocks and cash equivalents has reduced portfolio volatility without sacrificing the level of return. This image illustrates the risk and return profiles of two hypothetical investment portfolios since 1970. Although both portfolios had the same level of return, the portfolio containing bonds assumed less riskmeaning it experienced less volatility.Notice that by diversifying the original portfolio to include bonds, the overall risk of the portfolio was reduced without having to sacrifice the 9.9% return. Because stocks, bonds, and cash equivalents generally do not react identically to the same economic or market stimuli, combining these assets can often produce a more appealing risk and return tradeoff. Diversification does not eliminate the risk of experiencing investment losses. Government bonds and Treasury bills are guaranteed by the full faith and credit of the United States government as to the timely payment of principal and interest, while stocks are not guaranteed and have been more volatile than the other asset classes. About the data Stocks are represented by the Ibbotson Large Company Stock Index. Bonds are represented by the five-year U.S. government bond and cash by the 30-day U.S. Treasury bill. An investment cannot be made directly in an index. Risk is measured by standard deviation. Standard deviation measures the fluctuation of returns around the arithmetic average return of the investment. The higher the standard deviation, the greater the variability (and thus risk) of the investment returns. Return is measured by the arithmetic average annual return. The data assumes reinvestment of all income and does not account for taxes or transaction costs.

Step 3

Identifying Investment Opportunity Set(More Calculus!)19Complex World:TWO RISKY ASSETS

RISKY ASSETS (1 and 2):Expected Return = E(r1) and E(r2)Standard Deviation = 1 and 2Correlation between risky assets = 12Risky asset proportion in portfolio = w1 and w2

Portfolio Return: rp = W1 r1 + W2 r2 Variance: p2 = W1212 + W22 22 + 2W1W2Cov(r1,r2) p2 = W1212 + W22 22 + 2W1W21212

Objective: Identify the set of investment optionsSee CAL_TwoRisky.xls : IOS-TwoRisky20Asset Correlations and Portfolio Parameters Correlation coefficientPortfolio return (doesnt depend on correlation)

Portfolio variance & standard deviation (depend on correlation)

=1(straight line)W1 r1 + W2 r2

Var = W1212 + W22 22 + 2W1W212Std. Dev. = (W11 + W2 2 )-1