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

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

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

 

e b

b

(P - P ) + D =

Pr

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

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%

55

Solution to Problem

EXPECTED RETURN

VARIANCE,

n

s ss=1

= r p r1 = 0.3(100) + 0.4(15) + 0.3(-70) = 15%r

2 = 0.3(10) + 0.4(15) + 0.3(20) = 15%r

n2 2

s ss=1

= ( )p r r

66

Solution to Problem

STANDARD DEVIATION,

SIMILARLY,

2 2 2 21

2

= 0.3(100 - 15) + 0.4(0) + 0.3(-70 - 15)

= 4335(%)

1 = 4335 = 65.84%

2 2 2 22

2

= 0.3(10 - 15) + 0.4(0) + 0.3(20 - 15)

= 15(%)

2 = 15 = 3.872%

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

n

1,2 s 1S 1 2 2s=1

= ( )( )Sp r r r r

1,21,2

1 2

-255 = = = -1.00

(65.84)(3.872)

88

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

1 1 2 2 = x + xPr r r

2 2 2 2 21 1 2 2 1 2 12 = x + x + 2x xP

= 960 = 30.98P

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 = +1 PERFECTLY POSITIVELY CORRELATED

RETURNS

-1 1,2 +1

MOST STOCKS HAVE POSITIVELY CORRELATED (IMPERFECTLY) RETURNS.

1010

Optimal Two-Asset Portfolios

CASE (1)

Pr

2r

1r

*Pr

1 2

1,2 -1

P

1111

Optimal Two-Asset Portfolios

CASE (2)

Pr

2r

1r

*Pr

1 2

1,2 +1

*P

1,2 1,2' <

1212

Optimal Two-Asset Portfolios

CASE (3)

DIVERSIFICATION IS POSSIBLE ONLY IF THE TWO ASSET RETURNS ARE LESS THAN PERFECTLY POSITIVELY CORRELATED.

12 +1 Pr

2r

1r

1 2 P

1313

MEAN AND VARIANCE OF AN N-ASSET PORTFOLIO

IF N = 3

NOTE THAT

1 1 2 2 N N = x + x + ... + xPr r r r

2 2 2 2 2 2 21 1 2 2 N N

1 2 12 1 3 13 2 3 23

= [x + x + ... + x ]

+ 2[x x + x x + x x + similar

terms for all possible pairs of the N assets]

P

2 2 2 2 2 2 21 1 2 2 3 3

1 2 12 1 3 13 2 3 23

= [x + x + ... + x ]

+ 2[x x + x x + x x ]P

= , SINCE ijij i j ij ij

i j

1414

PROBLEM – 1

1 1 1

2 2 2

3 3

12 13 23

1 1 2 2 3 3

1 1 13 3 3

14% = 6% x = 1/3

8% = 3% x = 1/3

12% = 2%

= 0.5 = 0.6 = 1

= x + x + x

= (14) + (8) + (12)

= 11.33%

P

r

r

r

r r r r

1515

PROBLEM – 1

2 2 2 2 2 2 21 1 13 3 3

1 1 1 13 3 3 3

1 13 3

2

= [( ) 6 + ( ) 3 + ( ) 2 ]

+ 2[( )( )(6)(3)(0.5) + ( )( )(6)(2)(0.6)

+ ( )( )(3)(2)(1) ]

= 10.36(%)

P

= 10.36 = 3.22%P

1616

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)

M M

F MF F

2 2 2 2

15%, = 16%, x = 0.5

5%, = 0, = 0, x = 0.5

0.5(15) + 0.5(5) = 10%

= (0.5) 16 + 0 + 0 = 64(%)

= 64 = 8%

M

F

P

P

P

r

r

r

1717

PROBLEM – 2 (B)

SINCE YOU ARE BORROWING AN AMOUNT EQUAL TO YOUR WEALTH W AT THE RISK-FREE RATE,

NOTICE THAT

-WF Wx = = -1 W + W

Wx = = +2M

Fx + x = -1 + 2 = 1M

Pr = (-1)(5) + 2(15) = 25%2 2 2 2 2

F F M F M MF

2 2 2

= [x + x + 2 x x ]

= 2 (16) = 1024(%)

= 1024 = 32%

P M

P

1818

OPTIMAL PORTFOLIO CHOICE BY MV INVESTORS

PICK x1, x2, ….., xN TO

SUBJECT TO THE RESTRICTIONS:

(CANNOT INVEST MORE THAN AVAILABLE WEALTH, INCLUDING BORROWING, ETC.)

PMAXIMIZE r

P Pˆ(1) (RISK-TOLERANCE)

1 2 N(2) x + x + .... x = 1

1919

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.

P

Pr

MINVARIANCE

r

MIN-VARIANCE

**

*

*

**

*

*

*

EFFICIENT FRONTIER

2020

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.

P

Pr

Fr

M *

TEFFICIENT SET IS THE STRAIGHT LINE:rFMT

2121

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.

2222

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

ij

2323

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

2424

SYSTEMATIC AND UNSYSTEMATIC RISK

TOTAL RISK ( 2 OR ) = SYSTEMATIC (ß OR im/ m2 )

+ UNSYSTEMATIC RISK

SINCE UNSYSTEMATIC RISK IS DIVERSIFIABLE, ONLY SYSTEMATIC OR MARKET RISK IS “PRICED”

i IS THE APPROPRIATE MEASURE OF SYSTEMATIC RISK

2

( , )

( )i m im

im m

Cov R R

Var R

2525

THE CAPITAL ASSET PRICING MODEL

[ ]i F i m FR R R R

F

M

EXPECTED RETURN ON ANY ASSET i

= RISK FREE RATE [T-BILLS OR OTHER TREASURY BONDS]

= EXPECTED RETURN ON THE MARKET PORTFOLIO

iR

R

R

i : BETA OF ith STOCK

SECURITY MARKET LINE

m = 1

RF

M R

iR

M FSLOPE: ( - )R R

2626

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:

0.04 0.08i ir

1 0.04 (0.5)(0.08) 0.08 8%r

2 0.04 2(0.08) 0.2 20%r

2727

Problem 3

PROBLEM 4

0

0

$2 g = 0.04 r = 0.08

2(1+g) 2(1.04)P = $52 / SHARE

r-g 0.08 0.04

D

1 1

2 2

6%; 0.5

0.5( - ) 6% (1)

12%; 1.5

1.5( - ) 12% (2)

F m F

F m F

r

r r r

r

r r r

2828

Problem 4

SUBTRACTING (1) FROM (2),

FROM (1),

- 6%m Fr r

0.5(6) 6%

6 - 0.5(6) 3%

9%

F

F

m

r

r

r

2929

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):

itR

i

ERROR TERM: uit “BEST” STRAIGHT

LINE THAT EXPLAINS THE DATA

mtR

SLOPE = i

it i i mt itR R u