Financial Analysis, Planning and
Forecasting
Theory and Application
Chapter 7
Risk Estimation and Diversification
By
Alice C. Lee
San Francisco State University
John C. Lee
J.P. Morgan Chase
Cheng F. Lee
Rutgers University
1
Outline











7.1 Introduction
7.2 Risk classification
7.3 Portfolio analysis and applications
7.4 The market rate of return and market risk premium
7.5 Determination of commercial lending rates
7.6 The dominance principle and performance evaluation
7.7 Summary
Appendix 7A. Estimation of market risk premium
Appendix 7B. Normal distribution and Value at Risk
Appendix 7C. Derivation of Minimum-Variance Portfolio
Appendix 7D. Sharpe Performance Approach to Derive
Optimal Weight
2
7.1 Risk classification
 Total
risk=Business risk + Financial risk
3
7.1 Risk classification
7.1A : Method
n
Expected ROI  X   X i Pi
(7.1)
i 1
ROI: return on investment
n
Variance of ROI     ( Xi  X ) Pi
2
2
i 1
(7.2)
where Xi is ROI and Piis the probability in the ith state of economy
Coefficient of variation (CV) 

X
(7.3)
4
7.1 Risk classification
7.1B : Example
Table 7-1
State of
Economy
State Occurring
(pi)
ROI (A1) of
GM(XAI)
ROI (Bi) of
Ford (XBi)
Boom
.20
15%
14%
Normal
.60
12%
11%
Recession
.20
4%
3%
GM
Ford
Interval
Chance of Occurrence
.11 – .037 < X < .11 + .037
68.27%
.11 – .074 < X < .11 + .074
95.45%
.11 – .111 < X < .11 + .111
99.73%
.10 – .037 < X < .10 + .037
68.27 %
.11 – .074 < X < .11 + .074
95.45%
.11 – .111 < X < .11 + .111
99.73%
5
7.2 Portfolio analysis and applications
 Expected
rate of return on a portfolio
 Variance and standard deviations of a
portfolio
 The efficient portfolios
 Corporate application of diversification
6
7.3 Portfolio Analysis and Application
R P  Wa R a  Wb Rb  Wc Rc
 (.4) (.1)  (.3) (.05)  (.3) (.12)
 .091
R P : expected return rate for portfolio p
Wa ,b,c : the propositions invested in securities a,b, and c.
n


R p  Wi Ri  where Wi  1
i 1
i 1


1 N
COV (W1 R1 ,W2 R2 )   (W1 R1t  W1 R1 )(W2 R2t  W2 R 2 )
N t 1
W1W2 N

 ( R1t  R 1 )( R2t  R 2 )
N t 1
 W1W2Cov( R1 , R2 )
n
(7.4)
(7.5)
7
7.3 Portfolio Analysis and Applications


2
1 N
  VAR (W1 R1t  W2 R2t )   W1R1t  W2 R2t   W1 R1  W2 R 2 

N t 1 
 W12 12  W22 22  2W1W2COV ( R1 , R2 )
2
P
(7.6)
P 
1 N
2
(
R

R
)
P

pt
N t 1
(7.7)
R P : expected return rate for portfolio p
Rpt : the portfolio's rate of return in period t
Wi : the propositions invested in security i
8
7.3 Portfolio analysis and applications
Figure 7-2 Two Portfolios with same mean
and different variance
9
7.3 Portfolio analysis and applications
 P  W12VAR( R1 )  W22VAR( R2 )  2W1W2 COV ( R1 , R2 )
1
2
COV( R1 , R2 )  121 2
(7.8)
(7.9)
Find W1to minimize  p by Appendix 7.C
 2 ( 2 12 1 )
W1  2
2
 1   2  212 1 2
p 
n
2 2
W
 i i 
i 1
n
(7.10)
 WW   
j 1,i  j
(7.11)
n
i 1
i
j
ij
i
j
10
7.3 Portfolio analysis and applications
Figure 7-3 The Correlation Coefficients
11
7.3 Portfolio analysis and applications
Efficient Portfolios
 W12 12

2 2
W
2 2



W20W1 20,1
W1W20 1,20 




2 2
W20 20 
Under the mean-variance framework, a security or
portfolio is efficient if
E(A) > E(B) and var(A) = var(B)
Or
E(A) = E(B) and var(A) < var (B).
12
7.3 Portfolio analysis and applications
Figure 7-4 Efficient Frontier in Portfolio Analysis
13
7.3 Portfolio analysis and applications
WMRK 
.001645  .001069
 .1007 by Equation (7.10)
.006206  .001645  2(.001069)
 2 ( 2 12 1 )
W1  2
 1   22  212 1 2
WJNJ  1  WMRK  .8993
Table 7-2 Variance-Covariance Matrix
MRK
JNJ
2
 MRK
 MRK , JNJ
.006206
.001069
MRK
 JNJ ,MRK
2
 JNJ
JNJ
.001069
.001645
14
7.3 Portfolio analysis and applications
R MRK  0.002101, R JNJ  0.004233
E ( RP )  (.1007)(.002101)  (.8993)(.004233)
 .004018
 p2  (.1007) 2 (.006206)  (.8993) 2 (.001645)  2(.1007)(.8993)(.001069)
 .001587
 MRK , JNJ
12 

 JNJ  MRK
.001069
1
2
(.006206) (.001645)
.001069
.001069


(.078781)(.040554) .003195
 .334695
1
2
15
7.4 The market rate of return and market risk premium
Table 7-3 Market Returns and T-bill Rates by Quarters
I t  I t 1
I t 1
(A)
Market Return
Annualized 3month T-bill
Rates
(B)
Quarterly 3month T-bill
Rates
(A-B)
Quarterly Rates
Premium
Year
Quarter
S & P 500
Index
03
4
164.92
04
1
159.17
-3.49
9.52
2.38
-5.87
2
153.17
-3.77
9.87
2.47
-6.24
3
166.09
8.44
10.37
2.59
5.84
4
167.23
.69
8.06
2.02
-1.33
1
180.65
8.02
8.52
2.13
5.89
2
191.84
6.19
6.95
1.74
4.46
05
16
7.4 The market rate of return and market risk premium
Table 7-3 Market Returns and T-bill Rates by Quarters (Cont’d)
Year
06
Mean
Annualized 3month T-bill Rates
(B)
Quarterly 3month T-bill
Rates
(A-B)
Quarterly
Rates
Premium
Quarter
S & P 500
Index
(A)
Market
Return
3
182.07
-5.09
7.10
1.78
-6.87
4
211.27
16.04
7.10
1.78
14.26
1
238.90
13.08
6.56
1.64
11.44
2
250.84
5.00
6.21
1.55
3.45
3
231.32
-7.78
5.21
1.30
-9.08
4
242.17
4.69
5.53
1.38
3.31
195.36
3.50
7.58
1.90
1.61
17
7.5
Determination of commercial lending rates
Table 7.4
Economic condition
Rf
(A)
Probability
(B)
EBIT
(C)
Probability
(D)
Rp
Boom
10%
.25
$2.5m
1.5
.5
.40
.30
.30
2%
3
5
Normal
9
.50
$2.5m
1.5
.5
.40
.30
.30
2
3
5
Poor
8
.25
$2.5m
1.5
.5
.40
.30
.30
2
3
5
18
7.5
Determination of commercial lending rates
Table 7-5
Economic
condition
(A)
Rf
(B)
Probability
(C)
Rp
(D)
Probability
(B X D)
Probability of
Occurrence
(A + C)
Lending
Rate
Boom
10%
.25
2%
3
5
.40
.30
.30
.100
.075
.075
12%
13
15
Normal
9
.50
2
3
5
.40
.30
.30
.200
.150
.150
11
12
14
Poor
8
.25
2
3
5
.40
.30
.30
.100
.075
.075
1.000
10
11
13
19
7.5
Determination of commercial lending rates
R  (.100)(12%)  (.75)(13)  (.075)15  (.200)11  (.150)(12) 
(.150)(14)  (.100)10  (.074)(13%)
 12.2%
  (.100)(12  12.2) 2  (.075)(13  12.2) 2  (.075)(15  12) 2 
(.200)(11  12.2) 2  (.150)(12  12.2) 2  (.150)(14  12.2) 2 
(.100)(10  12.2)  (.075)(11  12.2)  (.075)(13  12.2)
 .004  .048  .588  .288  .006  .436  .484  .108  .048
 1.43%
2
2

1
2 2
20
7.6 The dominance principle and performance evaluation
Figure 7-5 Distribution of Leading Rate(R)
21
7.6 The dominance principle and performance evaluation
Figure 7-6 The Dominance Principle in
Portfolio Analysis
22
7.6 The dominance principle and performance evaluation
Table 7-6
The example of Sharpe Performance Measure
Smith Fund
Jones Fund
Average return ( R )
18%
16%
Standard deviation (σ)
20%
15%
Risk-free rate = Rf = 9.5%
RS  Rf
.18  .095
SPS 

 .425
S
.2
R J  R f .16  .095
SPJ 

 .433
J
.15
23
7.7 Summary
In Chapter 7, we defined the basic concepts of risk and
risk measurement. Based on the relationship of risk
and return, we demonstrated the efficient portfolio
concept and its implementation, as well as the
dominance principle and performance measures.
Interest rates and market rates of return were used as
measurements to show how the commercial lending
rate and the market risk premium are calculated.
24
Appendix 7A. Estimation of market risk premium
Table 7A-1 Summary Statistics of Annual Returns (1926-2006)
Geometric
Mean
Arithmetic
Mean
Standard
Deviation
S&P 500 Index
6.02%
7.90%
19.39%
U.S. Treasury Bills (3 Month)
3.79%
3.83%
3.45%
Long-Term Corporate Bonds (20 Year)
6.44%
7.04%
3.05%
Long-Term Government Bonds (20 Year)
5.24%
5.60%
9.04%
Series
Sources:
(1) The Center for Research in Security Prices, Wharton School of Business, The
University of Pennsylvania.
(2) Federal Reserve Economic Data, The Federal Reserve Bank of St. Louis.
25
Appendix 7A. Estimation of market risk premium
Exhibit 7A-1: Derived Series: Summary Statistics of Annual
Component Returns (1926-2006)
Series
Geometric
Mean
Arithmetic
Mean
Standard
Deviation
Equity risk premia (stockbills)
2.13%
4.06%
19.48%
Default premia
(LT corps-LT govts.)
1.1
1.44
8.13
Horizon premia
(LT govts. - bills)
1.34
1.76
9.4
Real interest rates (bills –
inflation)
.63
.71
4.03
Distribution
26
Appendix 7A. Estimation of market risk premium
Exhibit 7A-2: Simulated Total Return Distributions of Common Stock
(1977-2000) by Geometric Average Annual Rates
27
Appendix 7B. Normal distribution and Value at Risk
f x ( x) 
1
 2
( x )2
e
2 2
Figure 7B-1
Probability Density Function for a Normal Distribution, Showing the
Probability That a Normal Random Variable Lies between a and b
(Shaded Area)
28
Appendix 7B. Normal distribution and Value at Risk
Figure 7B-2
Probability Density
Function of Normal
Random Variables
with Equal
Variances: Mean 2 is
Greater Than 1.
Figure 7B-3
Probability Density
Functions of Normal
Distributions with Equal
Means and Different
Variances
29
Appendix 7B. Normal distribution and Value at Risk
Table 7B-1
Probability, P, That a Normal Random Variable with Mean
Standard Deviation σ lies between K – σ and K – σ.
P
K/ σ
.50
.674
.60
.842
.70
1.036
.80
1.281
.90
1.645
.95
1.960
 and
Mean  =12. If the investor may believe there is a 50% chance
that the actual return will be between 10.5% and 13.5%.
K=(13.5-10.5)/2=1.5 and K/ σ=0.674
2
Then σ=1.5/0.674=2.2255,  =4.95
30
Appendix 7B. Normal distribution and Value at Risk
Figure 7B-5
For a Normal Random Variable with Mean 12, Standard Deviation
4.95, the Probability is .5 of a Value between 10.5 and 13.5
31
Appendix 7C. Derivation of Minimum-Variance Portfolio
Min . 
w
2 1
p
2
P
  w12 12  w22 22  2w1w2Cov( R1 , R2 )
(7.C.2)
 w12 12  (1  w1 )2  22  2w1 (1  w1 )Cov( R1 , R2 )
 w12 12   22  2w1 22  w12 22  2w1Cov( R1 , R2 )  2w12Cov( R1 , R2 )
By taking partial derivative of  p2 with respect to w1, we obtain
 p2
w1
0
2w1  12  2 22  2w1  22  2Cov( R1 , R2 )  4 w1 Cov( R1 , R2 )  0
[ 12   22  2Cov( R1 , R2 )]w1   22  Cov( R1 , R2 )
 22  Cov( R1 , R2 )
w1  2
 1   22  2Cov( R1 , R2 )
32
Appendix 7D. Sharpe Performance Approach to Derive
Optimal Weight
Max S p 
w1
E ( Rp )  R f
p
where
E(R )
= expected rates of return for portfolio P.
R
= risk free rates of return
S p = Sharpe performance measure
 p as defined in equation (7.C.1) of
Appendix C
p
f
33
Appendix 7D. Sharpe Performance Approach to Derive
Optimal Weight
S p
0
(7.D.1)
E ( R p )  w1 E ( R1 )  w2 E ( R2 )
(7.D.2)
w1
 p  [w   w   2w1w2Cov( R1 , R2 )]
2
1
2
1
2
2
2
2
w1  w2  1
1/ 2
(7.D.3)
(7.D.4)
34
Appendix 7D. Sharpe Performance Approach to Derive
Optimal Weight
E ( R p )  R f  w1 E ( R1 )  w2 E ( R2 )  R f  w1E ( R1 )  (1  w1 ) E ( R2 )  R f  f ( w1 )
 p  [ w12 12  w22 22  2w1w2Cov( R1 , R2 )]1/ 2
 [ w12 12  (1  w1 ) 2  22  2w1 (1  w1 )Cov( R1 , R2 )]1/ 2  g ( w2 )
f ( w1 )

S p
g ( w2 ) f ( w1 ) g ( w2 )  f ( w1 ) g ( w2 )


0
2
w1
w2
[ g ( w2 )]
(7.D.5)
(7.D.6)
(7.D.7)
35
Appendix 7D. Sharpe Performance Approach to Derive
Optimal Weight
f '( w1 ) 
g '( w1 ) 
f ( w1 )
 E ( R1 )  E ( R2 )
w1
(7.D.8)
g ( w1 )
w1
1
1
1
2 2
2 2
2
  [ w1  1  (1  w1 )  2  2 w1 (1  w1 )Cov( R1 , R2 )]
2
 [2 w1 12  2 w1 22  2 22  2Cov( R1 , R2 )  4 w1Cov( R1, R2 )]
 [ w1 12  w1 22   22  Cov( R1 , R2 )  2 w1Cov( R1 , R2 )]
 [w12 12  (1  w1 ) 2  22  2 w1 (1  w1 )Cov( R1 , R2 )]1/ 2
(7.D.9)
36
Appendix 7D. Sharpe Performance Approach to Derive
Optimal Weight
f '( w1 ) g ( w1 )  f ( w1 ) g '( w1 )  0
(7.D.10)
f '( w1 ) g ( w1 )  f ( w1 ) g '( w1 )
[ E ( R1 )  E ( R2 )]  [ w   (1  w1 )   2 w1 (1  w1 )Cov( R1 , R2 )]
2
1
2
1
2
2
2
1
2
 [ w1 E ( R1 )  (1  w1 ) E ( R2 )  R f ]
 [ w1 12  w1 22   22  Cov( R1 , R2 )  2w1Cov( R1 , R2 )]
1
 [ w   (1  w1 )   2w1 (1  w1 )Cov( R1 , R2 )]
2
1
2
1
2
2
2
2
(7.D.11)37
Appendix 7D. Sharpe Performance Approach to Derive
Optimal Weight
[ E ( R1 )  E ( R2 )]
 [ w12 12  (1  w1 ) 2  22  2w1 (1  w1 )Cov ( R1 , R2 )]
 [ w1 E ( R1 )  (1  w1 ) E ( R2 )  R f ]
 [ w1 12  w1 22   22  Cov ( R1 , R2 )  2w1Cov ( R1 , R2 )]
(7.D.12)
38
Appendix 7D. Sharpe Performance Approach to Derive
Optimal Weight
Left hand side of equation (D12):
[ E ( R1 )  E ( R2 )]  [ w12 12  (1  w1 ) 2  22  2 w1 (1  w1 )Cov( R1 , R2 )]
 [ E ( R1 )  E ( R2 )]
 [ w12 12   22  2w1 22  w12 22  2w1Cov( R1 , R2 )  2w12Cov( R1 , R2 )]
 [ E ( R1 )  E ( R2 )]  w12 [ 12   22  2Cov( R1 , R2 )]  2w1[Cov( R1 , R2 )   22 ]   22 
 [ E ( R1 )  E ( R2 )]  w12 [ 12   22  2Cov( R1 , R2 )]
 [ E ( R1 )  E ( R2 )]  2w1[Cov( R1 , R2 )   22 ]
 [ E ( R1 )  E ( R2 )]   22
 [ E ( R1 )  E ( R2 )]  [ 12   22  2Cov( R1 , R2 )]w12
 2[ E ( R1 )  E ( R2 )]  [Cov( R1 , R2 )   22 ]w1  [ E ( R1 )  E ( R2 )]   22
39
Appendix 7D. Sharpe Performance Approach to Derive
Optimal Weight
Right hand side of equation (D12)
[ w1 E ( R1 )  (1  w1 ) E ( R2 )  rf ]  [ w1 12  w1 22   22  Cov( R1 , R2 )  2w1Cov( R1 , R2 )]
 {w1[ E ( R1 )  E ( R2 )  w1 E ( R2 )  R f ]}  [ w1 12  w1 22  2 w1Cov( R1 , R2 )   22  Cov( R1 , R2 )]
 {w1[ E ( R1 )  E ( R2 )]  [ E ( R2 )  R f ]}  {w1[ 12   22  2Cov( R1 , R2 )]  Cov ( R1 , R2 )   22 }
 w1[ E ( R1 )  E ( R2 )]  w1[ 12   22  2Cov( R1 , R2 )]  w1[ E ( R1 )  E ( R2 )]  [Cov( R1 , R2 )   22 ]
 [ E ( R2 )  R f ]  w1[ 12   22  2Cov( R1 , R2 )]  [ E ( R2 )  R f ]  [Cov( R1 , R2 )   22 ]
 [ E ( R1 )  E ( R2 )]  [ 12   22  2Cov( R1 , R2 )]w12
 [ E ( R1 )  E ( R2 )]  [Cov( R1 , R2 )   22 ]w1
 [ E ( R2 )  R f ]  [ 12   22  2Cov( R1 , R2 )]w1
 [ E ( R2 )  R f ]  [Cov( R1 , R2 )   22 ]
40
Appendix 7D. Sharpe Performance Approach to Derive
Optimal Weight
[ E ( R1 )  E ( R2 )]  [Cov( R1 , R2 )   22 ]w1  [ E ( R1 )  E ( R2 )]   22
 [ E( R2 )  R f ] [12   22  2Cov( R1 , R2 )]w1  [ E( R2 )  R f ] [Cov( R1, R2 )   22 ]
(7.D.13)
[ E( R1 )  E( R2 )]2 22 2 [ E( R2 )  R f ] [Cov( R1, R2 )   22 ]
 [ E( R2 )  R f ] [1   2  2Cov( R1, R2 )]w1  [ E( R1 )  E( R2 )] [Cov( R1, R2 )   22 ]w1
(7.D.14)
41
Appendix 7D. Sharpe Performance Approach to Derive
Optimal Weight
[ E ( R1 )  E ( R2 )  E ( R2 )  R f ]   22  [ E ( R2 )  R f ]Cov( R1 , R2 )
 {[ E ( R2 )  R f ] 12  [ E ( R2 )  R f ] 22  [ E ( R2 )  R f ][2Cov( R1 , R2 )]
 [ E ( R1 )  E ( R2 )]Cov( R1 , R2 )  [ E ( R1 )  E ( R2 )] 22 }w1
 [ E ( R1 )  R f ] 22  [ E ( R2 )  R f ] 12  [ E ( R1 )  R f  E ( R2 )  R f ]Cov( R1 , R2 )] w1
w1 
[ E ( R1 )  R f ] 22  [ E ( R2 )  R f ]Cov( R1 , R2 )
[ E ( R1 )  R f ] 22  [ E ( R2 )  R f ]12  [ E ( R1 )  R f  E ( R2 )  R f ]Cov( R1 , R2 )
42