1. Financial Analysis, Planning and Forecasting Theory and Application
Chapter 8
Risk Estimation and Diversification By
Cheng F. Lee
Rutgers University, USA
John Lee
Center for PBBEF Research, USA

2. Outline 8.1 Introduction 8.2 Risk classification
8.3 Portfolio analysis and applications
8.4 The market rate of return and market risk premium
8.5 Determination of commercial lending rates
8.6 The dominance principle and performance evaluation
8.7 Summary
Appendix 8A. Estimation of market risk premium
Appendix 8B. The normal distribution
Appendix 8C. Derivation of Minimum-Variance Portfolio
Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

3. Total risk=Business risk + Financial risk
8.1 Risk classification
Total risk=Business risk + Financial risk

4. (8.1) (8.2) (8.3) 8.1 Risk classification 8.1A : Method
ROI: return on investment

5. 8.1 Risk classification Table 8-1 8.1B : Example 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 4% 3%
Interval
Chance of Occurrence GM
.11 – .037 < X <
68.27%
.11 – .074 < X <
95.45%
.11 – .111 < X <
99.73%
Ford
.10 – .037 < X <
68.27 %

6. 8.2 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

7. 8.3 Portfolio Analysis and Application
(8.4)
(8.5)

8. 8.3 Portfolio Analysis and Applications
(8.6)
(8.7)

9. 8.3 Portfolio analysis and applications
Figure 8-2 Two Portfolios with same mean and different variance

10. 8.3 Portfolio analysis and applications
(8.8)
(8.9)
(8.10)
(8.11)

11. 8.3 Portfolio analysis and applications
Figure 8-3 The Correlation Coefficients

12. 8.3 Portfolio analysis and applications
Efficient Portfolios
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).

13. 8.3 Portfolio analysis and applications
Figure 8-4 Efficient Frontier in Portfolio Analysis

14. 8.3 Portfolio analysis and applications
Table 8-2 Variance-Covariance Matrix
MRK
JNJ

15. 8.3 Portfolio analysis and applications

16. 8.4 The market rate of return and market risk premium
Table 8-3 Market Returns and T-bill Rates by Quarters
Year
Quarter
S & P 500
Index
(A)
Market Return
Annualized 3-
month T-bill
Rates
(B)
Quarterly 3-
(A-B)
Quarterly Rates
Premium 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
167.23
.69
8.06
2.02
-1.33 05
180.65
8.02
8.52
2.13
5.89
191.84
6.19
6.95
1.74
4.46

17. 8.4 The market rate of return and market risk premium
Table 8-3 Market Returns and T-bill Rates by Quarters (Cont’d)
Year
Quarter
S & P 500
Index
(A)
Market
Return
Annualized 3-
month T-bill Rates
(B)
Quarterly 3-
month T-bill
Rates
(A-B)
Quarterly
Premium 3
182.07
-5.09
7.10
1.78
-6.87 4
211.27
16.04
14.26 06 1
238.90
13.08
6.56
1.64
11.44 2
250.84
5.00
6.21
1.55
3.45
231.32
-7.78
5.21
1.30
-9.08
242.17
4.69
5.53
1.38
3.31
Mean
195.36
3.50
7.58
1.90
1.61

18. 8.5 Determination of commercial lending rates
Table 8.4
Economic condition Rf
(A)
Probability
(B)
EBIT
(C)
(D) Rp
Boom
10%
.25
$2.5m
1.5 .5
.40
.30 2% 3 5
Normal 9
.50 2
Poor 8

19. 8.5 Determination of commercial lending rates
Table 8-5
Economic
condition
(A) Rf
(B)
Probability
(C) Rp
(D)
(B X D)
Probability of
Occurrence
(A + C)
Lending
Rate
Boom
10%
.25 2% 3 5
.40
.30
.100
.075
12% 13 15
Normal 9
.50 2
.200
.150 11 12 14
Poor 8
1.000 10

20. 8.5 Determination of commercial lending rates

21. 8.6 The dominance principle and performance evaluation
Figure 8-5 Distribution of Leading Rate(R)

22. 8.6 The dominance principle and performance evaluation
Figure 8-6 The Dominance Principle in Portfolio Analysis

23. 8.6 The dominance principle and performance evaluation
Table 8-6
The example of Sharpe Performance Measure
Smith Fund
Jones Fund
Average return ( )
18%
16%
Standard deviation (σ)
20%
15%
Risk-free rate = Rf = 9.5%

24. 8.7 Summary
In Chapter 8, 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.

25. Appendix 8A. Estimation of market risk premium
Table 8A-1 Summary Statistics of Annual Returns ( )
Series
Geometric
Mean
Arithmetic
Standard
Deviation
S&P 500 Index
9.78
11.64
19.30
U.S. Treasury Bills (3 Month)
3.71%
3.75
3.13%
Long-Term Corporate Bonds (20 Year)
7.04%
7.08
3.07%
Long-Term Government Bonds (20 Year)
5.57%
6.00
9.97%
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.

26. Appendix 8A. Estimation of market risk premium
Exhibit 8A-1: Derived Series: Summary Statistics of Annual Component Returns ( )
Series
Geometric Mean
Arithmetic Mean
Standard Deviation
Distribution
Equity risk premia (stock-bills)
5.95%
7.88%
19.34%
Default premia
(LT corps-LT govts.)
0.9
1.39
9.67
Horizon premia
(LT govts. - bills)
1.18
1.63
9.76
Real interest rates (bills – inflation)
0.55
0.64
4.12

27. Appendix 8B. The Normal Distribution
Figure 8B-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 8B. The Normal Distribution
Figure 8B-2
Probability Density
Function of Normal
Random Variables
with Equal
Variances: Mean 2 is Greater Than 1.
Figure 8B-3
Probability Density
Functions of Normal
Distributions with Equal
Means and Different
Variances

29. Appendix 8B. The Normal Distribution
Table 8B-1
Probability, P, That a Normal Random Variable with Mean and
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
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=( )/2=1.5 and K/ σ=0.674
Then σ=1.5/0.674=2.2255, =4.95

30. Appendix 8B. The Normal Distribution
Figure 8B-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 8C. Derivation of Minimum-Variance Portfolio
By taking partial derivative of with respect to w1, we obtain

32. Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight
where
= expected rates of return for portfolio P.
= risk free rates of return
= Sharpe performance measure
as defined in equation (8.C.1) of Appendix C

33. Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

34. Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

35. Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

36. Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

37. Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

38. Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight
Left hand side of equation (8.D12):

39. Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight
Right hand side of equation (8.D.12)

40. Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

41. Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight