Risk and Return Professor Thomas Chemmanur

22 1. Risk Aversion ASSET – A: EXPECTED PAYOFF = 0.5(100) + 0.5(1) = $50.50 ASSET – B:PAYS $50.50 FOR SURE WHICH ASSET WILL A RISK AVERSE INVESTOR CHOOSE? RISK NEUTRAL INVESTORS  INDIFFERENT BETWEEN A AND B. RISK LOVING? PROB = 0.5 $100 $1

33 Certainty Equivalent THE CERTAINTY EQUIVALENT OF A RISK AVERSE INVESTOR  THE AMOUNT HE OR SHE WILL ACCEPT FOR SURE INSTEAD OF A RISKY ASSET. THE MORE RISK-AVERSE THE INVESTOR, THE LOWER HIS CERTAINTY EQUIVALENT. RETURN FROM ANY ASSET P e = END OF PERIOD PRICE P b = BEGINNING OF PERIOD PRICE D = CASH DISTRIBUTIONS DURING THE PERIOD

44 EXPECTED UTILITY MAXIMIZATION IF RETURNS ARE NORMALLY DISTRIBUTED, RISK- AVERSE INDIVIDUALS CAN MAXIMIZE EXPECTED UTILITY BASED ONLY ON THE MEAN, VARIANCE, AND COVARIANCE BETWEEN ASSET RETURNS. PROBLEM STATEPROB KELLY Vs. WATER (S) (p s ) PROD (r 1S ) (r 2S ) BOOM %10% NORMAL %15% RECESSION %20%

55 Solution to Problem EXPECTED RETURN VARIANCE,

66 Solution to Problem STANDARD DEVIATION, SIMILARLY,

77 Solution to Problem COVARIANCE BETWEEN ASSETS 1 & 2 = 0.3(100-15)(10-15) + 0.4(15-15)* (15-15) +0.3(-70-15)(20-15) = -255(%) 2 CORRELATION CO-EFFICIENT

88 Solution to Problem PORTFOLIO MEAN AND VARIANCE PORTFOLIO WEIGHTS X i, i = 1,…, N. X 1 = 0.5 OR 50% X 2 = 0.5 OR 50% = 0.5(15) + 0.5(15) = 15% = (4335) (15) + 2(0.5)(0.5)(-255) = 960(%) 2

99 Choosing Optimal Portfolios IN A MEAN-VARIANCE FRAMEWORK, THE OBJECTIVE OF INDIVIDUALS WILL BE MAXIMIZE THEIR EXPECTED RETURN, WHILE MAKING SURE THAT THE VARIANCE OF THEIR PORTFOLIO RETURN (RISK) DOES NOT EXCEED A CERTAIN LEVEL.  1,2 = -1 PERFECTLY NEGATIVELY CORRELATED RETURNS  1,2 = +1PERFECTLY POSITIVELY CORRELATED RETURNS -1   1,2  +1 MOST STOCKS HAVE POSITIVELY CORRELATED (IMPERFECTLY) RETURNS.

10 Optimal Two-Asset Portfolios CASE (1)

11 Optimal Two-Asset Portfolios CASE (2)

12 Optimal Two-Asset Portfolios CASE (3) DIVERSIFICATION IS POSSIBLE ONLY IF THE TWO ASSET RETURNS ARE LESS THAN PERFECTLY POSITIVELY CORRELATED.

13 MEAN AND VARIANCE OF AN N-ASSET PORTFOLIO IF N = 3 NOTE THAT

14 PROBLEM – 1

15 PROBLEM – 1

16 RISKY ASSETS WITH LENDING AND BORROWING NOTE THAT, FOR THE RISK-FREE ASSET,  F = 0. FURTHER, WHILE “LENDING” IMPLIES THAT X F > 0, “BORROWING” IMPLIES THAT X F < 0. PROBLEM – 2 (A)

17 PROBLEM – 2 (B) SINCE YOU ARE BORROWING AN AMOUNT EQUAL TO YOUR WEALTH W AT THE RISK-FREE RATE, NOTICE THAT

18 OPTIMAL PORTFOLIO CHOICE BY MV INVESTORS PICK x 1, x 2, ….., x N TO SUBJECT TO THE RESTRICTIONS: (CANNOT INVEST MORE THAN AVAILABLE WEALTH, INCLUDING BORROWING, ETC.)

19 OPTIMAL PORTFOLIO CHOICE BY MV INVESTORS SOLUTION WITH NO RISK-FREE ASSET NOT ALL INVESTORS WILL CHOOSE TO HOLD THE MINIMUM VARIANCE PORTFOLIO. THE PRECISE LOCATION OF AN INVESTOR ON THE EFFICIENT FRONTIER DEPENDS ON THE RISK σ P HE IS WILLING TO TAKE. * * * * * * * * * EFFICIENT FRONTIER

20 OPTIMAL PORTFOLIO CHOICE BY MV INVESTORS SOLUTION WITH RISK FREE LENDING / BORROWING THE SET OF RETURNS YOU CAN GENERATE BY COMBINING A RISK-FREE AND RISKY ASSET LIES ON THE STRAIGHT LINE JOINING THE TWO TO GO ON THE LINE SEGMENT MT, AN INVESTOR WILL BORROW AT THE RISK-FREE RATE r F. M * T EFFICIENT SET IS THE STRAIGHT LINE: r F MT

21 OPTIMAL PORTFOLIO CHOICE BY MV INVESTORS WHEN INVESTORS AGREE ON THE PROBABILITY DISTRIBUTION OF THE RETURNS OF ALL ASSETS: MARKET EQUILIBRIBUM IF INVESTORS AGREE ON THE DISTRIBUTIONS OF ALL ASSETS RETURNS, THEY WILL AGREE ON THE COMPOSITION OF THE PORTFOLIO M: THE “MARKET PORTFOLIO”. IN SUCH A WORLD, INVESTORS WILL ALL INVEST THEIR WEALTH BETWEEN TWO PORTFOLIOS  THE RISK-FREE ASSET AND THE MARKET PORTFOLIO. THE MARKET PORTFOLIO IS THE PORTFOLIO OF ALL RISKY ASSETS IN THE ECONOMY, WEIGHTED IN PROPORTION TO THEIR MARKET VALUE.

22 RISK OF A WELL-DIVERSIFIED PORTFOLIO WHAT HAPPENS WHEN YOU INCREASE THE NUMBER OF STOCKS IN A PORTFOLIO? IT CAN BE SHOWN THAT THE TOTAL PORTFOLIO VARIANCE GOES TOWARD THE AVERAGE COVARIANCE BETWEEN TWO STOCKS AS N   No. of Assets in a Portfolio PP

23 SYSTEMATIC AND UNSYSTEMATIC RISK SYSTEMATIC RISK: THIS IS RISK WHICH AFFECTS A LARGE NUMBER OF ASSETS TO A GREATER OR LESSER DEGREE  THEREFORE, IT IS RISK THAT CANNOT BE DIVERSIFIED AWAY  E.G. RISK OF ECONOMIC DOWNTURN WITH OIL PRICE INCREASE UNSYSTEMATIC RISK: RISK THAT SPECIFICALLY AFFECTS A SINGLE ASSET OR SMALL GROUP OF ASSETS  CAN BE DIVERSIFIED AWAY  E.G. STRIKE IN A FIRM, DEATH OF A CEO, INCREASE IN RAW MATERIALS PRICE

24 SYSTEMATIC AND UNSYSTEMATIC RISK TOTAL RISK (  2 OR  ) = SYSTEMATIC (ß OR  im /  m 2 ) + UNSYSTEMATIC RISK SINCE UNSYSTEMATIC RISK IS DIVERSIFIABLE, ONLY SYSTEMATIC OR MARKET RISK IS “PRICED”  i IS THE APPROPRIATE MEASURE OF SYSTEMATIC RISK

25 THE CAPITAL ASSET PRICING MODEL  i : BETA OF i th STOCK SECURITY MARKET LINE  m = 1 RFRF

26 APPLICATION OF THE CAPM 1. IN ESTIMATING THE COST OF CAPITAL FOR A FIRM 2. AS A BENCHMARK IN PORTFOLIO PERFORMANCE MEASUREMENT PROBLEM – 3 SECURITY MARKET LINE: STOCK 1: STOCK 2:

27 Problem 3 PROBLEM 4

28 Problem 4 SUBTRACTING (1) FROM (2), FROM (1),

29 ESTIMATING BETA WE CAN ESTIMATE BETA FOR EACH STOCK BY FITTING ITS RETURN OVER TIME AGAINST THE RETURN OF THE MARKET PORTFOLIO (S&P 500 INDEX), USING LINEAR REGRESSION (USE EXCEL TO DO THIS):               ERROR TERM: u it “BEST” STRAIGHT LINE THAT EXPLAINS THE DATA SLOPE =  i