Credit Portfolio Management phần 2 pdf

NOTE 1. If the mark-to-market (MTM) model is used, then b(PD) is given by: If a default mode (DM) model is used, then it is given by: b(PD) = 7.6752 × PD 2 – 1.9211 × PD + 0.0774, for PD < 0.05 b(PD) = 0, for PD > 0.05 bPD PD PD PD () .() .() . = ×− +×− 0235 1 047 1 044 The Revolution in Credit—Capital Is the Key 23 PART One The Credit Portfolio Management Process CHAPTER 2 Modern Portfolio Theory and Elements of the Portfolio Modeling Process T he argument we made in Chapter 1 is that the credit function must trans- form into a loan portfolio management function. Behaving like an asset manager, the bank must maximize the risk-adjusted return to the loan port- folio by actively buying and selling credit exposures where possible, and otherwise managing new business and renewals of existing facilities. This leads immediately to the realization that the principles of modern portfolio theory (MPT)—which have proved so successful in the management of eq- uity portfolios—must be applied to credit portfolios. What is modern portfolio theory and what makes it so desirable? And how can we apply modern portfolio theory to portfolios of credit assets? MODERN PORTFOLIO THEORY What we call modern portfolio theory arises from the work of Harry Markowitz in the early 1950s. (With that date, I’m not sure how modern it is, but we are stuck with the name.) As we will see, the payoff from applying modern portfolio theory is that, by combining assets in a portfolio, you can have a higher expected re- turn for a given level of risk; or, alternatively, you can have less risk for a given level of expected return. Modern portfolio theory was designed to deal with equities; so throughout all of this first part, we are thinking about equities. We switch to loans and other credit assets in the next part. The Efficient Set Theorem and the Efficient Frontier Modern portfolio theory is based on a deceptively simple theorem, called the Efficient Set Theorem: 27 An investor will choose her/his optimal portfolio from the set of port- folios that: 1. Offer maximum expected return for varying levels of risk. 2. Offer minimum risk for varying levels of expected return. Exhibit 2.1 illustrates how this efficient set theorem leads to the effi- cient frontier. The dots in Exhibit 2.1 are the feasible portfolios. Note that the different portfolios have different combinations of return and risk. The efficient frontier is the collection of portfolios that simultaneously maxi- mize expected return for a given level of risk and minimize risk for a given level of expected return. The job of a portfolio manager is to move toward the efficient frontier. Expected Return and Risk In Exhibit 2.1 the axes are simply “expected return” and “risk.” We need to provide some specificity about those terms. 28 THE CREDIT PORTFOLIO MANAGEMENT PROCESS EXHIBIT 2.1 The Efficient Set Theorem Leads to the Efficient Frontier In modern portfolio theory, when we talk about return, we are talk- ing about expected returns. The expected return for equity i would be written as E[R i ] = µ i where µ i is the mean of the return distribution for equity i. In modern portfolio theory, risk is expressed as the standard deviation of the returns for the security. Remember that the standard deviation for equity i is the square root of its variance, which measures the dispersion of the return distribution as the expected value of squared deviations about the mean. The variance for equity i would be written as 1 The Effect of Combining Assets in a Portfolio—Diversification Suppose that we form a portfolio of two equities—equity 1 and equity 2. Suppose further that the percentage of the portfolio invested in equity 1 is w 1 and the percentage invested in equity 2 is w 2 . The expected return for the portfolio is E[R p ] = w 1 E[R 1 ] + w 2 E[R 2 ] That is, the expected return for the portfolio is simply the weighted sum of the expected returns for the two equities. The variance for our two-equity portfolio is where things begin to get interesting. The variance of the portfolio depends not only on the variances of the individual equities but also on the covariance between the returns for the two equities ( σ 1,2 ): Since covariance is a term about which most of us do not have a mental picture, we can alternatively write the variance for our two-equity portfolio in terms of the correlation between the returns for equities 1 and 2 ( ρ 1,2 ): σσσ ρσσ p ww ww 2 1 2 1 2 2 2 2 2 1 21212 2=++ , σσσ σ p ww ww 2 1 2 1 2 2 2 2 2 1212 2=++ , σ iii EER R 22 =−[( [ ] ) ] Modern Portfolio Theory and Elements of the Portfolio Modeling Process 29 This boring-looking equation turns out to be very powerful and has changed the way that investors hold equities. It says: Unless the equities are perfectly positively correlated (i.e., unless ρ 1,2 = 1) the riskiness of the portfolio will be smaller than the weighted sum of the riskiness of the two equities that were used to create the portfolio. That is, in every case except the extreme case where the equities are perfectly positively correlated, combining the equities into a portfolio will result in a “diversification effect.” This is probably easiest to see via an example. Example: The Impact of Correlation Consider two equities—Bristol-Meyers Squibb and Ford Motor Company. Using historical data on the share prices, we found that the mean return for Bristol-Meyers Squibb was 15% yearly and the mean return for Ford was 21% yearly. Using the same data set, we calculated the standard deviation in Bristol-Myers Squibb’s return as 18.6% yearly and that for Ford as 28.0% yearly. E(R BMS ) = µ BMS = 15% E(R F ) = µ F = 21% σ BMS = 18.6% σ F = 28.0% The numbers make sense: Ford has a higher return, but it is also more risky. Now let’s use these equities to create a portfolio with 60% of the portfolio invested in Bristol-Myers Squibb and the remaining 40% in Ford Motor Company. The expected return for this portfolio is easy to calculate: Expected Portfolio Return = (0.6)15 + (0.4)21 = 17.4% The variance of the portfolio depends on the correlation of the returns on Bristol-Meyers Squibb’s equity with that of Ford ( ρ BMS, F ): The riskiness of the portfolio is measured by the standard deviation of the portfolio re- turn—the square root of the variance. The question we want to answer is whether the riskiness of the portfolio (the portfolio standard deviation) is larger, equal to, or smaller than the weighted sum of the risks (the standard deviations) of the two equities: Weighted Sum of Risks = (0.6)18.6 + (0.4)28.0 = 22.4% To answer this question, let’s look at three cases. Variance of Portfolio Return=+ + (.)( .) (.)( ) ( )( . )( . )( )( . )( . ) , 0 6 18 6 0 4 28 20604 186280 22 22 ρ BMS F 30 THE CREDIT PORTFOLIO MANAGEMENT PROCESS CASE 1: THE RETURNS ARE UNCORRELATED ( ρ BMS,F = 0): Variance of Portfolio Returns = (0.6) 2 (18.6) 2 + (0.4) 2 (28) 2 + 0 = 250.0 In this case, the riskiness of portfolio is less than the weighted sum of the risks of the two equities: Standard Deviation of Portfolio = 15.8% yearly < 22.4% If the returns are uncorrelated, combining the assets into a portfolio will generate a large diversification effect. C ASE 2: THE RETURNS ARE PERFECTLY POSITIVELY CORRELATED ( ρ BMS,F = 1): In this extreme case, the riskiness of portfolio is equal to the weighted sum of the risks of the two equities: Standard Deviation of Portfolio = 22.4% yearly The only case in which there will be no diversification effect is when the returns are per- fectly positively correlated. C ASE 3: THE RETURNS ARE PERFECTLY NEGATIVELY CORRELATED ( ρ BMS,F = –1): In this extreme case, not only is the riskiness of portfolio less than the weighted sum of the risks of the two equities, the portfolio is riskless: Standard Deviation of Portfolio = 0% yearly If the returns are perfectly negatively correlated, there will be a combination of the two as- sets that will result in a zero risk portfolio. From Two Assets to N Assets Previously we noted that, for a two-asset portfolio, the variance of the portfolio is Variance of Portfolio Returns=+ +− =+−= (.)( .) (.)( ) ( )( . )( . )( )( . )( . ) 0 6 18 6 0 4 28 20604 1186280 124 6 125 4 250 0 0 22 22 Variance of Portfolio Returns=+ + = (.)( .) (.)( ) ( )( . )( . )( )( . )( . ) . 0 6 18 6 0 4 28 206041186280 500 0 22 22 Modern Portfolio Theory and Elements of the Portfolio Modeling Process 31 This two-asset portfolio variance is portrayed graphically in Exhibit 2.2. The term in the upper-left cell shows the degree to which equity 1 varies with itself (the variance of the returns for equity 1); and the term in the lower-right cell shows the degree to which equity 2 varies with it- self (the variance of the returns for equity 2). The term in the upper-right shows the degree to which the returns for equity 1 covary with those for equity 2, where the term ρ 1,2 σ 1 σ 2 is the covariance of the returns for eq- uities 1 and 2. Likewise, the term in the upper-right shows the degree to which the returns for equity 2 covary with those for equity 1. (Note that ρ 1,2 = ρ 2,1 .) Exhibit 2.3 portrays the portfolio variance for a portfolio of N equi- ties. With our two-equity portfolio, the variance–covariance matrix con- tained 2 × 2 = 4 cells. An N-equity portfolio will have N × N = N 2 cells in its variance–covariance matrix. In Exhibit 2.3, the shaded boxes on the diagonal are the variance terms. The other boxes are the covariance terms. There are N variance terms and N 2 – N covariance terms. If we sum up all the cells (i.e., we sum the i rows and the j columns) we get the variance of the portfolio returns: The Limit of Diversification—Covariance We have seen that, if we combine equities in a portfolio, the riskiness of the portfolio is less than the weighted sum of the riskiness of the individual eq- uities (unless the equities are perfectly positively correlated). How far can we take this? What is the limit of diversification? σσ pijij j N i N ww 2 11 = == ∑∑ , σσσ ρσσ p ww ww 2 1 2 1 2 2 2 2 2 1 21212 2=++ , 32 THE CREDIT PORTFOLIO MANAGEMENT PROCESS EXHIBIT 2.2 Graphical Representation of Variance for Two-Equity Portfolio Equity 1 Equity 2 Equity 1 w 1 2 σ 1 2 w 1 w 2 ρ 1, 2 σ 1 σ 2 Equity 2 w 2 w 1 ρ 2, 1 σ 2 σ 1 w 2 2 σ 2 2 [...]... (1991 20 00), N.R Adjusted 0 0.11 0. 12 0.76 4.99 17.79 40.96 0.3 12. 32 Y3 0.06 0.08 0.11 0. 62 4. 32 16.74 40.36 0 .24 11. 72 Y3 0 0.17 0.18 1. 32 7.35 21 .76 46.68 0.49 15.49 Y4 0.13 0.11 0.16 1.03 5.9 19.78 44.55 0.38 14.08 Y4 0 0 .27 0 .29 2. 08 9. 72 25.1 48.81 0.76 18.3 Y5 0.13 0.15 0 .25 1.49 7.44 22 .15 47.73 0.53 16.08 Y5 0 0 .27 0.38 2. 39 12. 06 27 .57 52. 26 0.88 20 .73 Y6 0.13 0.19 0.33 1.87 9. 12 24 .21 49. 52. .. 86.96% 84.83% 83.89% 90.61% 88.10% 92. 38% 83. 62% 95 .21 % 93 .24 % 87.85% 90.39% 97.78% 93.43% 96.39% 96.35% 96. 82% 96.75% 96 .20 % 92. 45% 92. 62% 92. 47% 97. 02% 96.78% 91 .29 % 98.40% 97.35% 97.08% 97 .29 % 0.9760 0.9788 0.9649 0.9795 0.9634 0.9763 0.9744 0.9308 0.9595 0.9 523 0.969 0.9 726 0.93 62 0.9778 0.9703 0.9705 0.9667 27 0 137 166 137 22 0 123 184 29 4 149 93 168 24 9 28 7 188 340 131 406 North America only North... 49. 52 0.66 17.94 Y6 0.13 0 .24 0.44 2. 12 10. 52 26.69 50.94 0.78 19.83 Y7 EXHIBIT 3.3 Historical Default Probabilities from Standard & Poor’s Risk Solutions’ CreditPro 0.13 0. 32 0.55 2. 4 11.5 28 .57 51.73 0.91 21 .19 Y8 0.13 0.44 0.73 2. 71 12. 95 29 .4 54.09 1.08 22 .46 Y9 0.13 0.67 0.73 3 .25 13.78 30.45 55.73 1 .26 23 .47 Y10 47 Data Requirements and Sources for Credit Portfolio Management EXHIBIT 3.4 Transition... Products2 Print Media2 Services for Business & Industry2 1 Score Score within within Correlation Sample 1 Notch 2 Notches Coefficient Size 67.08% 82. 81% 88 .20 % 95.48% 96.89% 97 .29 % 0.9709 0.9876 161 22 1 67.78% 80. 32% 80. 12% 78.10% 55.00% 64 .23 % 78.80% 53. 72% 63.76% 69.54% 55.95% 76.47% 57.49% 72. 34% 66.18% 67.08% 73.15% 93.70% 88.15% 93.98% 92. 70% 88.64% 88. 62% 86.96% 84.83% 83.89% 90.61% 88.10% 92. 38%... Solutions’ CreditPro One-Year Transition Matrix All Industries and Countries Pool: ALL (1981 20 00), N R Adjusted AAA AAA AA A BBB BB B CCC AA A BBB BB B CCC D 93.65 0.66 0.07 0.03 0.03 0 0.15 5.83 91. 72 2 .25 0 .26 0.06 0.1 0 0.4 6.95 91.76 4.83 0.44 0. 32 0 .29 0.09 0.49 5.18 89 .25 6.66 0.46 0.88 0.03 0.06 0.49 4.44 83 .23 5. 72 1.91 0 0.09 0 .2 0.8 7.46 83. 62 10 .28 0 0. 02 0.01 0.15 1.04 3.84 61 .23 0 0.01 0.04 0 .24 ... 43 Data Requirements and Sources for Credit Portfolio Management 20 02 SURVEY OF CREDIT PORTFOLIO MANAGEMENT PRACTICES How many rating grades does your system contain? Large Corporates Non-defaulted entities Defaulted entities Average Range Average Range MiddleMarket Corporates Banks Other Financial 13 5 22 3 1–7 12 5 22 3 1–7 13 5 22 3 1–13 13 5 22 3 1–7 Do you employ facility ratings that are separate... Range Yes 10 2 25 62% Indicate the functional responsibility for assigning and reviewing the ratings Assigns Reviews Ratings Rating “Line” (unit with marketing/customer relationship responsibilities) Dedicated credit group other than Credit Portfolio Management Credit Portfolio Management Institution’s Risk Management Group Internal Audit Other Both 40% 13% 8% 25 % 10% 10% 0% 17% 30% 25 % 28 % 40% 83%... encompass a “through the cycle” assessment of 46 0 0. 02 0.03 0 .2 0.83 6.06 27 .07 0.08 4.45 0 0.06 0.07 0.38 2. 39 12. 37 35. 12 0.15 8.53 Y2 0 0.03 0.03 0 .23 0.85 5.78 27 .78 0.09 4.16 0 0.06 0.07 0.46 2. 46 12. 64 35.96 0.18 8.5 Y2 Source: Standard & Poor’s Risk Solutions AAA AA A BBB BB B CCC Inv grade Spec grade Y1 All Industries and Countries Pool: ALL (1995 20 00), N R Adjusted AAA AA A BBB BB B CCC Inv grade... Moody’s–KMV Portfolio Manager™, the RiskMetrics Group’s CreditManager™, CSFB’s Credit Risk+, and McKinsey’s CreditPortfolioView™ In Chapter 4, we describe the various credit portfolio models NOTE 1 The Statistics Appendix contains more detailed explanations of these expressions CHAPTER 3 Data Requirements and Sources for Credit Portfolio Management xhibit 3.1 is repeated from Chapter 2, because it... variables Altman (20 00) claims the following accuracy results from back testing of the model Years Prior Bankruptcies to Event Correctly Predicted 1 2 3 4 5 96 .2% 84.9 74.5 68.1 69.8 Non-Bankruptcies Correctly Predicted 89.7% 93.1 91.4 89.5 82. 1 Note that column 2 is related to Type 1 errors, while column three indicates Type 2 errors  52 THE CREDIT PORTFOLIO MANAGEMENT PROCESS TYPE 1 AND TYPE 2 ERRORS In . 1 and 2 ( ρ 1 ,2 ): σσσ ρσσ p ww ww 2 1 2 1 2 2 2 2 2 1 21 2 12 2=++ , σσσ σ p ww ww 2 1 2 1 2 2 2 2 2 121 2 2= ++ , σ iii EER R 22 =−[( [ ] ) ] Modern Portfolio Theory and Elements of the Portfolio. 6 18 6 0 4 28 20 604 118 628 0 124 6 125 4 25 0 0 0 22 22 Variance of Portfolio Returns=+ + = (.)( .) (.)( ) ( )( . )( . )( )( . )( . ) . 0 6 18 6 0 4 28 20 604118 628 0 500 0 22 22 Modern Portfolio. diversification? σσ pijij j N i N ww 2 11 = == ∑∑ , σσσ ρσσ p ww ww 2 1 2 1 2 2 2 2 2 1 21 2 12 2=++ , 32 THE CREDIT PORTFOLIO MANAGEMENT PROCESS EXHIBIT 2. 2 Graphical Representation of Variance for Two-Equity Portfolio Equity