Risk and Return

1. Risk and Return Professor Thomas Chemmanur

2. $100 PROB = 0.5 PROB = 0.5 $1 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? 2

3. 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 Pe = END OF PERIOD PRICE Pb = BEGINNING OF PERIOD PRICE D = CASH DISTRIBUTIONS DURING THE PERIOD 3

4. 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 STATE PROB KELLY Vs. WATER (S) (ps) PROD (r1S) (r2S) BOOM 0.3 100% 10% NORMAL 0.4 15% 15% RECESSION 0.3 -70% 20% 4

5. Solution to Problem EXPECTED RETURN VARIANCE, 5

6. Solution to Problem STANDARD DEVIATION, SIMILARLY, 6

7. 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 7

8. Solution to Problem PORTFOLIO MEAN AND VARIANCE PORTFOLIO WEIGHTS Xi , i = 1,…, N. X1 = 0.5 OR 50% X2 = 0.5 OR 50% = 0.5(15) + 0.5(15) = 15% = 0.52 (4335) + 0.52(15) + 2(0.5)(0.5)(-255) = 960(%)2 8

9. 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 = +1 PERFECTLY POSITIVELY CORRELATED RETURNS -1 1,2 +1 MOST STOCKS HAVE POSITIVELY CORRELATED (IMPERFECTLY) RETURNS. 9

10. Optimal Two-Asset Portfolios CASE (1) 10

11. Optimal Two-Asset Portfolios CASE (2) 11

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

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

14. PROBLEM – 1 14

15. PROBLEM – 1 15

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

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

18. OPTIMAL PORTFOLIO CHOICE BY MV INVESTORS PICK x1, x2, ….., xN TO SUBJECT TO THE RESTRICTIONS: (CANNOT INVEST MORE THAN AVAILABLE WEALTH, INCLUDING BORROWING, ETC.) 18

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 σP HE IS WILLING TO TAKE. EFFICIENT FRONTIER * * * * * * * * * 19

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 rF. T M * EFFICIENT SET IS THE STRAIGHT LINE: rFMT 20

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. 21

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   P No. of Assets in a Portfolio 22

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 23

24. SYSTEMATIC AND UNSYSTEMATIC RISK TOTAL RISK ( 2OR  ) = SYSTEMATIC (ß OR im/ m2 ) + UNSYSTEMATIC RISK SINCE UNSYSTEMATIC RISK IS DIVERSIFIABLE, ONLY SYSTEMATIC OR MARKET RISK IS “PRICED” iIS THE APPROPRIATE MEASURE OF SYSTEMATIC RISK 24

25. THE CAPITAL ASSET PRICING MODEL SECURITY MARKET LINE RF i : BETA OF ith STOCK m = 1 25

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: 26

27. Problem 3 PROBLEM 4 27

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

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: uit  “BEST” STRAIGHT LINE THAT EXPLAINS THE DATA     SLOPE = i           29