Risk and Return
Professor Thomas Chemmanur
1. Risk Aversion

ASSET – A:
PROB = 0.5
$100
EXPECTED PAYOFF
= 0.5(100) + 0.5(1)
= $50.50
PROB = 0.5
$1



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
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 - Pb ) + D
r=
Pb



Pe = END OF PERIOD PRICE
Pb = BEGINNING OF PERIOD PRICE
D = CASH DISTRIBUTIONS DURING THE PERIOD
3
EXPECTED UTILITY MAXIMIZATION


IF RETURNS ARE NORMALLY DISTRIBUTED, RISKAVERSE 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
Solution to Problem

EXPECTED RETURN
n
r =
pr
s s
s=1
r1 = 0.3(100) + 0.4(15) + 0.3(-70) = 15%
r2 = 0.3(10) + 0.4(15) + 0.3(20) = 15%

VARIANCE,
n
 =  ps (rs  r )
2
s=1
2
5
Solution to Problem
12 = 0.3(100 - 15)2 + 0.4(0)2 + 0.3(-70 - 15)2
= 4335(%)2

STANDARD DEVIATION,
1 = 4335 = 65.84%

SIMILARLY,
 22 = 0.3(10 - 15)2 + 0.4(0) 2 + 0.3(20 - 15) 2
= 15(%)
2
 2 = 15 = 3.872%
6
Solution to Problem

COVARIANCE BETWEEN ASSETS 1 & 2
n
 1,2 =  ps (r1S  r1 )(r2 S  r2 )
s=1

= 0.3(100-15)(10-15) + 0.4(15-15)*
(15-15) +0.3(-70-15)(20-15)
= -255(%)2

CORRELATION CO-EFFICIENT
1,2
1,2
-255
=
=
= -1.00
1 2
(65.84)(3.872)
7
Solution to Problem

PORTFOLIO MEAN AND VARIANCE
PORTFOLIO WEIGHTS Xi , i = 1,…, N.
X1 = 0.5 OR 50% X2 = 0.5 OR 50%
rP = x1r1 + x 2 r2
= 0.5(15) + 0.5(15)
= 15%
 P 2 = x12 12 + x 2 2 2 2 + 2x1x 2 12
= 0.52 (4335) + 0.52(15) + 2(0.5)(0.5)(-255)
= 960(%)2
 P = 960 = 30.98
8
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
Optimal Two-Asset Portfolios

CASE (1)
rP
1,2  -1
r2
rP *
r1
1
2
P
10
Optimal Two-Asset Portfolios

CASE (2)
rP
r2
rP
1,2  +1
1,2' < 1,2
*
r1
 P*  1
2
11
Optimal Two-Asset Portfolios

CASE (3)
rP
12  +1
r2
r1
1

2
P
DIVERSIFICATION IS POSSIBLE ONLY IF THE TWO
ASSET RETURNS ARE LESS THAN PERFECTLY
POSITIVELY CORRELATED.
12
MEAN AND VARIANCE OF AN N-ASSET PORTFOLIO
rP = x1r1 + x 2 r2 + ... + x N rN
 P 2 = [x12 12 + x 2 2 2 2 + ... + x N 2 N 2 ]
+ 2[x1x 2 12 + x1x 3 13 + x 2 x 3 23 + similar
terms for all possible pairs of the N assets]

IF N = 3
 P 2 = [x1212 + x 22 22 + ... + x 32 32 ]
+ 2[x1x 212 + x1x 313 + x 2 x 3 23 ]

NOTE THAT
 ij
 ij =  i j ij , SINCE ij 
 i j
13
PROBLEM – 1
 1 = 6% x1 = 1/3
r2  8%  2 = 3% x 2 = 1/3
r3  12%  3 = 2%
12 = 0.5 13 = 0.6  23 = 1
r1  14%
rP = x1r1 + x 2 r2 + x 3 r3
= 13 (14) +
1
3
(8) +
1
3
(12)
= 11.33%
14
PROBLEM – 1
 P 2 = [( 13 ) 2 62 + ( 13 ) 2 32 + ( 13 ) 2 22 ]
+ 2[( 13 )( 13 )(6)(3)(0.5) + ( 13 )( 13 )(6)(2)(0.6)
+ ( 13 )( 13 )(3)(2)(1) ]
= 10.36(%)2
 P = 10.36 = 3.22%
15
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)
rM  15%,
rF  5%,
 M = 16%, x M = 0.5
 F = 0,  MF = 0, x F = 0.5
rP  0.5(15) + 0.5(5) = 10%
 P 2 = (0.5) 2162 + 0 + 0 = 64(%) 2
 P = 64 = 8%
16
PROBLEM – 2 (B)

SINCE YOU ARE BORROWING AN AMOUNT EQUAL TO
YOUR WEALTH W AT THE RISK-FREE RATE,
xF =

-W
W
= -1
xM =
W+W
W
= +2
NOTICE THAT
x F + x M = -1 + 2 = 1
rP = (-1)(5) + 2(15) = 25%
 P 2 = [x F2 F2 + x M 2 M 2 + 2 x F x M MF ]
= 22 (16) 2 = 1024(%) 2
 P = 1024 = 32%
17
OPTIMAL PORTFOLIO CHOICE BY MV INVESTORS


PICK x1, x2, ….., xN TO MAXIMIZE rP
SUBJECT TO THE RESTRICTIONS:
(1)  P  ˆ P (RISK-TOLERANCE)
(2) x1 + x 2 + .... x N = 1

(CANNOT INVEST MORE THAN AVAILABLE WEALTH,
INCLUDING BORROWING, ETC.)
18
OPTIMAL PORTFOLIO CHOICE BY MV INVESTORS

SOLUTION WITH NO RISK-FREE ASSET
rP
EFFICIENT
FRONTIER
*
*
rMIN
VARIANCE
*
*
*
*
*
*
*
 MIN-VARIANCE

P
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.
19
OPTIMAL PORTFOLIO CHOICE BY MV INVESTORS

SOLUTION WITH RISK FREE LENDING / BORROWING
T
rP
EFFICIENT SET IS
THE STRAIGHT LINE:
M*
rFMT
rF
P

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.
20
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
RISK OF A WELL-DIVERSIFIED PORTFOLIO

WHAT HAPPENS WHEN YOU INCREASE THE NUMBER
OF STOCKS IN A PORTFOLIO?
P
 ij
No. of Assets in a Portfolio

IT CAN BE SHOWN THAT THE TOTAL PORTFOLIO
VARIANCE GOES TOWARD THE AVERAGE
COVARIANCE BETWEEN TWO STOCKS AS N  
22
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
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”
Cov( Ri , Rm )  im
i 
 2
Var ( Rm )
m

i IS THE APPROPRIATE MEASURE OF SYSTEMATIC
RISK
24
THE CAPITAL ASSET PRICING MODEL
Ri  RF  i [ Rm  RF ]
Ri  EXPECTED RETURN ON ANY ASSET i
RF = RISK FREE RATE [T-BILLS OR OTHER TREASURY BONDS]
RM = EXPECTED RETURN ON THE MARKET PORTFOLIO
Ri
SECURITY MARKET LINE
RM
SLOPE: ( RM - RF )
RF
m = 1
i : BETA OF ith STOCK
25
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:
ri  0.04  0.08i
STOCK 1:
STOCK 2:
r1  0.04  (0.5)(0.08)  0.08  8%
r2  0.04  2(0.08)  0.2  20%
26
Problem 3
D0  $2
g = 0.04
r = 0.08
2(1+g)
2(1.04)
P0 =

 $52 / SHARE
r-g
0.08  0.04
PROBLEM 4
r1  6%; 1  0.5 
rF  0.5(rm - rF )  6%
(1)
r2  12%;  2  1.5 
rF  1.5(rm - rF )  12%
(2)
27
Problem 4

SUBTRACTING (1) FROM (2),
rm - rF  6%

FROM (1),
rF  0.5(6)  6%
rF  6 - 0.5(6)  3%
rm  9%
28
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):
Rit
ERROR
TERM: uit





i






SLOPE = i


“BEST” STRAIGHT
LINE THAT EXPLAINS
THE DATA


Rit   i  i Rmt  uit
Rmt
29