Credit Portfolio Management phần 10 potx

Risk Contribution Another important concept in modern portfolio theory is the idea of the risk contribution of an asset to a portfolio. It is important to remember that it can be defined in more than one way, but the most common definition is what we call marginal standard deviation—the amount of variation that a particular asset (call it “A”) adds to the portfolio. If the asset has a weight or number of shares of w A , then its risk contribution is defined by: (A.30) The fraction on the right-hand side after “w A ” contains the “partial derivative” symbol (“ ∂ ”). All this means is that (in the fraction) we are calculating the change in the portfolio standard deviation with respect to a very small change in the weight (i.e., dollar amount or number of shares) of the particular asset. Another way of saying this is that this derivative tells us the sensitivity of the portfolio standard deviation with respect to asset A. Calculating RC A in this way is equivalent to adding up all the elements of the row (or column, since it is symmetric) corresponding to asset “A” of the covariance matrix we introduced earlier and dividing this sum by the portfolio standard deviation. This confirms that the sum of all the risk contributions (RC i ) is equal to the portfolio standard deviation—that is, (A.31) To derive equation A.31, we need to apply equation A.30 to equation A.29. First, we note that so that ∂σ ∂σ ∂σ ∂ p Ap p A ww = 1 2 2 ∂σ ∂ σ ∂σ ∂ p A p p A ww 2 2= σ pi i N RC= = ∑ 1 RC w w AA p A = ∂σ ∂ Appendix 311 Now we can use the expression for the portfolio variance equation A.29 to calculate and so we have that (A.32) To show that equation A.32 obeys equation A.31, simply sum over all the assets in the portfolio: Derivation of the Default Event Correlation Formula We now turn to the derivation of the equation for default event correlation as a function of a joint probability equation A.19. Assume two binomial random variables X and Y (that can have values of 0 or 1) have a joint probability distribution p(x,y). Assume a joint probability p(1,1) = J. Start with definition of correlation (see equation A.15): (A.33) We now try to calculate cov[X,Y] using its definition calculated using the joint probability density for discrete random variables: ρ σσ xy xy XY = cov[ , ] RC w wVV wVV ww V V ww VV A A N A p j j N Aj i i N iA j N p Aj j N A N Aj Ai i N A N iA p ==== ==== ∑∑∑∑ ∑∑∑∑ =+         =+         = 1111 1111 2 1 2 1 2 σ σ σ cov[ , ] cov[ , ] cov[ , ] cov[ , ] σσσ σ pp p 22 + () = RC w wVV wVV A A p j j N Aj i i N iA =+         == ∑∑ 2 11 σ cov[ , ] cov[ , ] ∂σ ∂ p A j j N Aj i i N iA w wVV wVV 2 11 =+ == ∑∑ cov[ , ] cov[ , ] 312 STATISTICS FOR CREDIT PORTFOLIO MANAGEMENT where p(x i, y j ) = P(X = x i and Y = y j ). Let the marginal probabilities be µ x and µ y : µ x = p(1,0) + p(1,1) µ y = p(0,1) + p(1,1) Using the above definition for covariance, cov[X,Y] = p(0,0)(0– µ x )(0– µ y ) + p(0,1)(0 – µ x )(1 – µ y ) + p(1,0)(1 – µ x )(0 – µ y )+ p(1,1)(1 – µ x )(1 – µ y ) Now recall that: p(1,1) = J p(1,0) = µ x – J p(0,1) = µ y – J so p(0,0) = 1 – µ x – µ y + J Inserting these into the preceding yields: cov[X,Y] = (1– µ x – µ y +J)(– µ x )(– µ y ) + ( µ y –J)(– µ x )(1 – µ y ) + ( µ x –J)(1– µ x )(– µ y ) + J(1– µ x )(1 – µ y ) and just doing the algebra, and after canceling terms, we get cov[X,Y] = J – µ x µ y cov[ , ] ( )( ) ( )( ) ()( )( ) cov[ , ] XY J J JJ XY J J x x xy x y y x xy x x xy x y xy xy yx xy xy xy xy =− + + − − + − − + +−−+ +−−+ =−−+ −++ µ µ µµ µ µ µ µ µµ µ µ µµ µ µ µµ µµ µµ µµ µµ µµ µµ 2 22 2 1 µµµµ µµ µµ µ µµ µ µ µµ xxy xy xy y xy x y xy J JJ JJJJ − −+ +− +−−+ 2 cov[ , ] [( )( )] ( )( ) ( , ) , XY E X Y x y px y x y ijixjy ij ≡− −= − − µµ µ µΣ Appendix 313 Inserting this into equation A.33 yields equation A.19 for the default event correlation ( ρ xy ). IMPORTANT PROBABILITY DISTRIBUTIONS We now turn our attention to the specifics of the probability distributions that we have encountered in this book and that are useful to understand in credit portfolio management. Normal By far the most common of any distribution, the normal distribution is also called the bell curve or the Gauss distribution, after the prominent mathematician and physicist, Carl Friedrich Gauss, of the early eighteenth century. Many distributions converge to the normal one when certain limits are taken. For example, the normal distribution is the limiting distribution of the binomial one when the number of trials tends to infinity (more on that later). The normal distribution is given by the density function (A.34) A plot of this function is shown in Exhibit A.16. fx e x () () = −− 1 2 2 2 2 πσ µ σ 314 STATISTICS FOR CREDIT PORTFOLIO MANAGEMENT EXHIBIT A.16 Comparison of the Normal Distribution to the Binomial Distribution for the Same Parameters as in Exhibit A.4 8% default rate, 100 loans 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Number of Defaults Probability Binomial Normal 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Note that the normal distribution extends out to negative values and is perfectly symmetric about the mean (=8%). The binomial and normal are close approximations to each other, but there are times when one is not ap- propriate and the other will be, for example, in the modeling of probabilities. The fact that the normal permits negative values means that it will not be used (typically) to describe the probability of default. Upper/Lower Bounds The normal distribution has no lower or upper bounds. It has a non-zero value all the way up (down) to infinity (minus infinity). Parameters The bell curve has two parameters that describe it com- pletely—the mean ( µ ) and the standard deviation ( σ ) or variance (= σ 2 ). Principal Applications The normal distribution has applications every- where—in every science, social science (e.g., economics and finance), engi- neering, and so on. It is the most basic (nontrivial) distribution, and many other distributions are compared to it. Lognormal The lognormal distribution is a variation on the normal distribution, in which the natural logarithm of a random variable is normally distributed. When plotted against the random variable itself, the distribution is asym- metric and allows only positive (or zero) values. The lognormal distribution is given by the following density function: (A.35) Note the similarity to the normal distribution. Here the parameters µ and σ do not refer to the mean and standard deviation of the distribution (see Parameters, below), but are the parameters of the corresponding normal distribution. Exhibit A.17 shows a plot of the lognormal distribution compared to the normal distribution. Upper/Lower Bounds The lower bound is zero, and the upper is infinity. Parameters The lognormal distribution has two parameters directly related to the corresponding normal distribution. Suppose y is a normally distributed random variable with mean µ and variance σ 2 (that is, y ~ fx x e x () (ln( ) ) = −− 1 2 2 2 2 πσ µ σ Appendix 315 N( µ , σ 2 )). Then the variable x = e y is log-normally distributed with a mean equal to exp( µ + σ 2 /2) and a variance equal to exp(2 µ + σ 2 )[exp( σ 2 ) – 1]. 8 Principal Applications In econometrics, the lognormal distribution has been particularly useful in modeling size distributions, such as the distribution of firm sizes in an industry or the distribution of income in a country. In financial applications it is widely used in modeling the behavior of stock prices, which are always positive (greater than zero). This implies that the logarithm of the stock price is normally distributed. This assumption is used in the Black–Scholes option pricing formula and theory. Binomial The binomial distribution comes out of the answer to the question: “What is the probability that I will get v number of aces after tossing a die n times? The binomial distribution gives the probability that v events will occur in n trials, given an event has a probability p of occurring, and is explicitly written as: (A.36) fv n vn v pp np vnv , () ! !( )! ()= − − − 1 316 STATISTICS FOR CREDIT PORTFOLIO MANAGEMENT EXHIBIT A.17 Comparison of the Lognormal Distribution with the Normal Distribution with the Same Mean (= 8) and Standard Deviation (= 2.713) Number of Defaults Probability Normal Lognormal 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0246810 12 14 16 18 20 22 24 26 where the exclamation mark “!” denotes the factorial symbol (e.g., 5! = 5⋅4⋅3⋅2⋅1 = 120, and by definition, 0! = 1). This distribution converges to the normal distribution when n tends to infinity. It converges more rapidly if p is close to 1 / 2 . Note that default probabilities are much smaller than 1 / 2 , reflecting the fact that default probability distributions (and loss distributions, for that matter) are far from normally distributed. Upper/Lower Bounds The lower bound is zero and the upper is n, the total number of “trials” (e.g., rolls of the die, spins of the wheel, names in the loan portfolio, and so on). See Exhibit A.18. Note the different vertical scales. Parameters The binomial distribution has two parameters: the total number of “trials,” n (e.g., the number of loans in a portfolio), and the probability of an event to occur, p (e.g., the probability of default). The expected value of the distribution is given by np, and the variance is given by np(1 – p). Principal Applications In credit risk modeling, the binomial distribution is used sometimes as a starting point for more complex models (which re- quire the incorporation of correlation effects between obligors). For example, the normal (Gauss) distribution can be derived from the binomial when taking the limit that n goes to infinity. Appendix 317 EXHIBIT A.18 Binomial Distribution with Three Different Probabilities of Default. (Panel A: 1%, Panel B: 3%, Panel C: 8%.) 100 loans, 1% default rate 0 4 6 8 10 12 14 16 18 Number of Defaults Probability 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Panel A Poisson The Poisson distribution is the mathematical distribution governing a random variable in which one counts “rare” events, but at a definite average rate. This is called a “Poisson process”—a process in which discrete events are observable in an area of opportunity—a continuous interval (of time, length, surface area, etc.)—in such a manner that if we shorten the area of 318 STATISTICS FOR CREDIT PORTFOLIO MANAGEMENT Panel B 100 loans, 3% default rate 0 4 6 8 10 12 14 16 18 Number of Defaults Probability 0 0.05 0.1 0.15 0.2 0.25 2 Panel C EXHIBIT A.18 (Continued) 100 loans, 8% default rate 0 2 4 6 8 10 12 14 16 18 Number of Defaults Probability 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 opportunity enough, we obtain three conditions: (1) the probability of observing that exactly one success in the interval is stable; (2) the probability of observing that more than one success in the interval is 0; (3) the occur- rence that a success in any one interval is statistically independent of that in any other interval. Examples include finding the probability of the number of: Radioactive decays per second Stoppages on a production line Deaths per month due to a per week disease Flaws in a bolt of fabric Imperfections per square meter Customers arriving at a service in rolls of metals station per minute Telephone calls per hour Requests arriving at a server received by an office computer per second Cashews per can of mixed nuts Accidents at a particular Bacteria in a given culture intersection per month per liter Firms defaulting in a portfolio Typing errors per page of loans per year Cases of a rare disease per year The distribution was invented by the French mathematician Simeon- Denise Poisson (1781–1840) and was first applied to describe the probability of a particular number of Prussian soldiers being killed by being kicked by horses. Actuaries use a Poisson distribution to model events like a hurricane’s striking a specific location on the eastern seaboard of the United States. The Poisson distribution is related to a “rare” event (though rare is a relative term) in which the time of arrival is exponentially distributed— that is, the probability of arrival time decreases exponentially (i.e., as e –rt , where r is some average arrival rate and t is time) with increasing time. The Poisson distribution describes the probability of having v “arrivals” (e.g., defaults) in a fixed time interval, as already discussed. Later on, the gamma distribution is discussed, and it is also part of the same family. The gamma distribution describes the probability distribution of the time of the k th “arrival” (if k = 1 then the gamma distribution becomes the expo- nential distribution). The Poisson distribution is a limiting form of the binomial distribution, that being when the probability of an event is very small (e.g., default events), and the number of “trials” n (e.g., the number of names in a portfolio) is large. It turns out that this distribution, p µ (n), that is, the probability of n events occurring in some time interval, is equal to: Appendix 319 (A.37) where µ is the expected mean number of events per unit of time and the exclamation mark (“!”) is the factorial symbol or operation (e.g., 5! = 5⋅4⋅3⋅2⋅1 = 120, and by definition, 0! = 1). Upper/Lower Bounds The lower bound is zero and the upper is infinity. Exhibit A.19 shows the Poisson distribution for the same parameters as in Exhibit A.16. Notice how close the Poisson distribution is to the binomial. It is a little flatter near the peak and a little more fat-tailed than the binomial distribution. Parameters The Poisson distribution takes only one parameter: the average of the distribution, µ . It turns out that the variance is equal to the mean, so the standard deviation is . Principal Applications There is an incredibly large array of applications for the Poisson distribution. In physics it is well-known for being the distribution governing radioactive decay. In credit portfolio management, it is used as the starting point to model default event probability distributions in the Credit Risk+ model, described in Chapter 3. µ pn e n n µ µ µ () ! = − 320 STATISTICS FOR CREDIT PORTFOLIO MANAGEMENT EXHIBIT A.19 Comparison of the Poisson Distribution to the Binomial 100 loans, 8% default rate Number of Defaults Probability Poisson Binomial 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0246810121416 18 20 22 24 26 [...]... characteristics of, generally, 102 103 coefficient, generally, 297, 301 in Credit Risk+, 107 108 , 146 impact of, 30–31 implications of, 297–298 as implicit factor, 105 106 macro factor model, 106 model, 152–153 Moody’s–KMV model, 103 105 , 111 RiskMetrics Group’s model, 105 Covariance, 29, 32–34, 38, 254, 296–297, 309 Credit assets, 34–38 Credit cards, 225 Credit conversion factor (CCF), 19–20 Credit default swap... comparison, 154 structural, 110 133 types of, overview, 109 – 110 used by financial institutions, 161 CreditPortfolioView (McKinsey and Company), 40, 109 , 149, 134–136, 151–153, 161 CreditPro (S&P Risk Solutions), 44, 46–47, 97, 138 Credit quality, in test portfolio, 154–155 Credit rating, 223 Credit ratings agencies, 39 Credit Rating System (CRS) (Fitch Risk Management) , 62–63, 72–74 Credit Research Database... 318–320 Portfolio beta, 248 See also Beta Portfolio diversification: credit derivatives, 206–208 effect of, 29–31 limits of, 32–34 two-asset portfolio, 31–32 Portfolio Management Data (PMD) (S&P), 92–94, 95, 100 101 Portfolio Manager (Moody’s–KMV), 102 105 , 107 –119, 148–155, 157–158, 160–161, 163–178, 257 Portfolio value distribution, 116–118, 174–176 PricewaterhouseCoopers, 270, 275 Pricing: credit. .. taxonomy of, 193–201 CreditEdge, 66 Credit enhancements, in securitization, 238–239 Credit event, 197 Credit exposures, 19, 39, 193, 202, 204–206 Credit function, changes in, 1–5 CreditGrades (RiskMetrics Group), 84–85 Credit- linked note, 199–200 335 Index CreditManager (RiskMetrics Group), characteristics of, 40, 103 , 105 , 109 , 116, 119–133, 145, 148–155, 157–160, 254, 257 CreditMetrics, 119 CreditModel (S&P... model, 212 single-period, pricing of, 216 Credit derivative market: composition of, 201–202 evolution of, 201–202 size of, 202–203 Credit derivatives: credit asset management, 203–208 defined, 193 global, 202–203 implications of, 13, 15–16 importance to portfolio management, 208 macrolevel management, 204 market, see Credit derivative market micromanagement, 203 portfolio diversification and, 206–208 pricing,... 62–63, 72–74 Credit Research Database (CRD) (Moody’s), 44–45, 65 Credit risk: capital, 39, 245–246, 249 implications of, 206–208 regulatory capital and, 17 uncertainty and, 209 Credit Risk+ (Credit Suisse First Boston), 37, 107 109 , 141–154, 157, 159, 161 CreditServer, 119 Credit spreads, 85–91, 166, 196 See also specific types of credit spreads Creditworthiness, 184, 205 Crossover loans, 191 CSLT (Chase),... structural models, 210 211 First-to-default basket, 200–201 Fitch Risk Management (FRM): Credit Rating System (CRS), see Credit Rating System default probabilities, 45, 47 Loan Loss Database, 92, 94–95, 100 Floating-rate financing, 185 Ford Motor Company, 30–31, 86 Forward credit spreads, 217–218 Forward spread curve, 126 Freddie Mac, 225 Fully committed syndication, 184 Fully diversified portfolio, 34, 38... Hayt “How the Market Values Credit Derivatives.” RMA Journal, March 2000 Smithson, Charles and Gregory Hayt Credit Derivatives: Implications for Bank Portfolio Management. ” RMA Journal, April 2000 Smithson, Charles and Hal Holappa Credit Derivatives: What Are These Youthful Instruments and Why Are They Used?” Risk, December 1995 Smithson, Charles, Hal Holappa, and Shaun Rai Credit Derivatives: A Look... 110, 119, 167 Merton model, 21, 210 211 Merton–Perold approximation, 249–251 Microsoft, 297–299, 305 Modeled volatility, 71 Modern Portfolio Theory (MPT): basic statistics applications, 306– 310 covariance, 29, 32–34, 38 credit assets and, 34–40 Efficient Frontier, 27 Efficient Set Theorem, 27–28 expected return and risk, 28–29 portfolio diversification, 29–31 two-asset portfolio, 29, 31–32 Modes, 9 Monte... 105 , 109 , 116, 119–133, 145, 148–155, 157–160, 254, 257 CreditMetrics, 119 CreditModel (S&P Risk Solutions), analytics, 54–59, 72–74, 121 Credit Monitor (Moody’s–KMV), 37, 66–67, 71, 75–82, 110, 114, 122, 145, 154–155 Credit portfolio modeling process, 38–40 Credit portfolio models: academic comparisons, 153 actuarial, 141–148 characteristics of, generally, 21–23 calibration of, 156–161 design comparisons, . STATISTICS FOR CREDIT PORTFOLIO MANAGEMENT Panel B 100 loans, 3% default rate 0 4 6 8 10 12 14 16 18 Number of Defaults Probability 0 0.05 0.1 0.15 0.2 0.25 2 Panel C EXHIBIT A.18 (Continued) 100 loans,. FOR CREDIT PORTFOLIO MANAGEMENT EXHIBIT A.16 Comparison of the Normal Distribution to the Binomial Distribution for the Same Parameters as in Exhibit A.4 8% default rate, 100 loans 0 2 4 6 8 10. distribution governing radioactive decay. In credit portfolio management, it is used as the starting point to model default event probability distributions in the Credit Risk+ model, described in Chapter

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