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

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

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

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

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 < .11 + .037 68.27% .11 – .074 < X < .11 + .074 95.45% .11 – .111 < X < .11 + .111 99.73% Ford .10 – .037 < X < .10 + .037 68.27 %

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

8.3 Portfolio Analysis and Application (8.4) (8.5)

8.3 Portfolio Analysis and Applications (8.6) (8.7)

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

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

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

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).

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

8.3 Portfolio analysis and applications Table 8-2 Variance-Covariance Matrix MRK JNJ .006206 .001069 .001645

8.3 Portfolio analysis and applications

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

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

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

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

8.5 Determination of commercial lending rates

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

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

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%

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.

Appendix 8A. Estimation of market risk premium Table 8A-1 Summary Statistics of Annual Returns (1926-2006) 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.

Appendix 8A. Estimation of market risk premium Exhibit 8A-1: Derived Series: Summary Statistics of Annual Component Returns (1926-2006) 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

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)

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

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=(13.5-10.5)/2=1.5 and K/ σ=0.674 Then σ=1.5/0.674=2.2255, =4.95

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

Appendix 8C. Derivation of Minimum-Variance Portfolio By taking partial derivative of with respect to w1, we obtain

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

Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

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

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

Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight

Appendix 8D. Sharpe Performance Approach to Derive Optimal Weight