Exponential and Logarithmic Models

Copulas and credit models R¨udiger Frey Swiss Banking Institute University of Zurich freyr@isb.unizh.ch Alexander J. McNeil Department of Mathematics ETH Zurich mcneil@math.ethz.ch Mark A. Nyfeler Investment Office RTC UBS Zurich mark.nyfeler@ubs.com October 2001 1 Introduction In this article we focus on the latent variable approach to modelling credit portfolio losses. This methodology underlies all models that descend from Merton’s firm-value model (Merton 1974). In particular, it underlies the most important industry models, such as the model proposed by the KMV corporation and CreditMetrics. In these models default of an obligor occurs if a latent variable, often interpreted as the value of the obligor’s assets, falls below some threshold, often interpreted as the value of the obligor’s liabilities. Dependence between default events is caused by dependence between the latent variables. The correlation matrix of the latent variables is often calibrated by developing factor models that relate changes in asset value to changes in a small number of economic factors. For further reading see papers by Koyluoglu and Hickman (1998), Gordy (2000) and Crouhy, Galai, and Mark (2000). A core assumption of the KMV and CreditMetrics models is the multivariate normality of the latent variables. However there is no compelling reason for choosing a multivariate normal (Gaussian) distribution for asset values. The aim of this article is to show that the aggregate portfolio loss distribution is often very sensitive to the exact nature of the multivariate distribution of the latent variables. This is not simply a question of asset correlation. Even when individual default prob- abilities of obligors and the matrix of latent variable correlations are held fixed, it is still possible to develop alternative models which lead to much heavier-tailed loss distributions. A useful source of alternative models is the family of multivariate normal mixture distribu- tions, which includes Student’s t distribution and the generalized hyperbolic distribution. In most cases it is as easy to base latent variable models on these mixture distributions as it is to base them on the multivariate normal distribution. An elegant way of understanding how a multivariate latent variable distribution de- termines the distribution of the number of defaults in a portfolio is to use the concept of copulas. In this article we show that it is the copula (or dependence structure) of the latent variables that determines the higher order joint default probabilities for groups of obligors, and thus determines the extreme risk that there are many defaults in the portfolio. If we choose alternative latent variable distributions in the normal mixture family then we implicitly work with alternative copulas which often differ markedly from the copula of a Gaussian distribution. Some of these copulas, such as the t copula, possess tail dependence and, in contrast to the multivariate normal, they have a much greater tendency to generate 1 simultaneous extreme values (Embrechts, McNeil, and Straumann 1999). This effect is highly important in latent variable models, since simultaneous low asset values will lead to many joint defaults and past experience shows that realistic credit risk models need to be able to give sufficient weight to scenarios where many joint defaults occur. This article may be understood as a model risk study in the context of latent variable models. Individual default probabilities and asset correlations are insufficient to determine the portfolio loss distribution, since they do not fix the copula of the latent variables. For large portfolios of tens of thousands of counterparties there remains considerable model risk. Risk managers who employ the latent variable methodology should be aware of this. 2 Latent Exponential and Logarithmic Models Exponential and Logarithmic Models By: OpenStaxCollege A nuclear research reactor inside the Neely Nuclear Research Center on the Georgia Institute of Technology campu (credit: Georgia Tech Research Institute) We have already explored some basic applications of exponential and logarithmic functions In this section, we explore some important applications in more depth, including radioactive isotopes and Newton’s Law of Cooling Modeling Exponential Growth and Decay In real-world applications, we need to model the behavior of a function In mathematical modeling, we choose a familiar general function with properties that suggest that it will 1/32 Exponential and Logarithmic Models model the real-world phenomenon we wish to analyze In the case of rapid growth, we may choose the exponential growth function: y = A0ekt where A0 is equal to the value at time zero, e is Euler’s constant, and k is a positive constant that determines the rate (percentage) of growth We may use the exponential growth function in applications involving doubling time, the time it takes for a quantity to double Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time In some applications, however, as we will see when we discuss the logistic equation, the logistic model sometimes fits the data better than the exponential model On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably choose the exponential decay model Again, we have the form y = A0ekt where A0 is the starting value, and e is Euler’s constant Now k is a negative constant that determines the rate of decay We may use the exponential decay model when we are calculating half-life, or the time it takes for a substance to exponentially decay to half of its original quantity We use half-life in applications involving radioactive isotopes In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph Exponential growth and decay graphs have a distinctive shape, as we can see in [link] and [link] It is important to remember that, although parts of each of the two graphs seem to lie on the x-axis, they are really a tiny distance above the x-axis 2/32 Exponential and Logarithmic Models A graph showing exponential growth The equation is y = 2e3x A graph showing exponential decay The equation is y = 3e − 2x Exponential growth and decay often involve very large or very small numbers To describe these numbers, we often use orders of magnitude The order of magnitude is the power of ten, when the number is expressed in scientific notation, with one digit to 3/32 Exponential and Logarithmic Models the left of the decimal For example, the distance to the nearest star, Proxima Centauri, measured in kilometers, is 40,113,497,200,000 kilometers Expressed in scientific notation, this is 4.01134972 × 1013 So, we could describe this number as having order of magnitude 1013 A General Note Characteristics of the Exponential Function, y = A0ekt An exponential function with the form y = A0ekt has the following characteristics: • • • • • • • • one-to-one function horizontal asymptote: y = domain: ( – ∞, ∞) range: (0, ∞) x intercept: none y-intercept: (0, A0) increasing if k > (see [link]) decreasing if k < (see [link]) An exponential function models exponential growth when k > and exponential decay when k < Graphing Exponential Growth A population of bacteria doubles every hour If the culture started with 10 bacteria, graph the population as a function of time 4/32 Exponential and Logarithmic Models When an amount grows at a fixed percent per unit time, the growth is exponential To find A0 we use the fact that A0 is the amount at time zero, so A0 = 10 To find k, use the fact that after one hour (t = 1) the population doubles from 10 to 20 The formula is derived as follows 20 = 10ek ⋅ = ek ln2 = k Divide by 10 Take the natural logarithm t so k = ln(2) Thus the equation we want to graph is y = 10e(ln2)t = 10(eln2) = 10 · 2t The graph is shown in [link] The graph of y = 10e(ln2)t Analysis The population of bacteria after ten hours is 10,240 We could describe this amount is being of the order of magnitude 104 The population of bacteria after twenty hours is 10,485,760 which is of the order of magnitude 107, so we could say that the population has increased by three orders of magnitude in ten hours 5/32 Exponential and Logarithmic Models Half-Life We now turn to exponential decay One of the common terms associated with exponential decay, as stated above, is half-life, the length of time it takes an exponentially decaying quantity to decrease to half its original amount Every radioactive isotope has a half-life, and the process describing the exponential decay of an isotope is ... MỤC LỤC Mục lục Chuyên đề Công thức mũ, lũy thừa logarit 1.1 Công thức mũ lũy thừa 1.2 Công thức logarit Chuyên đề Hàm số lũy thừa, hàm số mũ hàm số logarit 11 2.1 Tập xác định hàm số 11 2.2 Đạo hàm giá trị nhỏ nhất, giá trị lớn hàm số - Tiếp tuyến đồ thị hàm số 13 2.3 Tính đơn điệu, cực trị đồ thị hàm số 15 Chuyên đề Phương trình mũ phương trình logarit 23 3.1 Phương trình mũ logarit 23 3.2 Phương pháp đưa số 24 3.3 Phương pháp logarit hóa mũ hóa 25 3.4 Phương pháp đặt ẩn phụ 26 CHUYÊN ĐỀ CÁC CÔNG THỨC MŨ - LŨY THỪA VÀ LOGARIT 1.1 Công thức mũ lũy thừa 1.1.1 Bài tập tự luận 1.1.2 Câu hỏi trắc nghiệm khách quan Cấp độ nhận biết thông hiểu Câu Tính giá trị biểu thức A = 625 A 14 B 12 Câu Kết phép tính A = 16 A 40 B 32 −1 + 16 − 2−2 64 C 11 −0,75 D 10 + 0, 25− C −24 −0,25 Câu Kết phép tính B = 27 + − 250,5 16 A B C 16 D 257 D 54 Câu Cho x, y hai số thực dương m, n hai số thực tùy ý Đẳng thức sau sai? n m A (x ) = x n.m m n B x x = x m+n xm C n = y Câu Cho a, b > 0; m, n ∈ N∗ Hãy tìm khẳng định đúng? x y m−n D (xy)n = xn y n Th.S Trần Quang Thạnh √ n Sđt: 0935-29-55-30 m B an : bm = (a : b)m−n am = a n √ √ n k a = n+k a C A D an bn = (a.b)n √ √ Câu Rút gọn biểu thức P = a a √ 3+1 A P = a B P = a 3−1 3+2 với a > √ C P = a2 3+1 D P = a 2√ Câu Cho a số thực dương, biểu thức a a viết dạng lũy thừa với số mũ hữu tỉ A a B a Câu Cho f (x) = D a √ √ x x Khi f (0, 09) A 0,1 B 0,2 Câu Biểu thức 11 C a C 0,3 D 0,4 √ √ √ x x x5 , x > viết dạng lũy thừa với số mũ hữu tỉ A x B x C x D x √ 4 a3 b2 Câu 10 Rút gọn √ , với a,b số thực dương ta a12 b6 A a2 b B ab2 C a2 b2 Câu 11 Cho biểu thức A = (a + 1)−1 + (b + 1)−1 Nếu a = + D a.b √ −1 b = − √ −1 giá trị A A B C D √ Câu 12 Cho biểu thức P = x x x x, x > Mệnh đề đúng? A P = x Câu 13 Biểu thức C = 15 B P = x 10 15 B x Câu 14 Cho biểu thức D = A D = x 13 D x 16 √ x x2 x3 , với x > Mệnh đề đúng? B D = x 24 Câu 16 Cho biểu thức P = C x 16 √ a Câu 15 Rút gọn biểu thức E = (1 + a2 )−1 √ √ A B 2a A P = x D P = x √ x x x x (x > 0) viết dạng lũy thừa số mũ hữu tỉ A x 18 13 C P = x 10 C D = x D D = x √ 2 − a−2 − −1 : (với a = 0, a = ±1) a a−3 C a D a √ x x2 x3 , với x > Mệnh đề đúng? 13 B P = x 24 C P = x D P = x Câu 17 Cho số thực a, b, α (a > b > 0, α = 1) Mệnh đề sau đúng? a α aα A (a + b)α = aα + bα B = −α C (a − b)α = aα − bα D (ab)α = aα bα b b AMS-LATEX Trang Th.S Trần Quang Thạnh Sđt: 0935-29-55-30 √ 4 a3 b2 Câu 18 Cho a, b số dương Rút gọn biểu thức P = √ kết a12 b6 A ab2 B a2 b C ab D a2 b2 1√ 1√ a3 b + b3 a √ √ Câu 19 Cho số thực dương a b Rút gọn biểu thức P = √ − ab 6 a+ b A B −1 C D −2 Câu 20 Cho số thực dương a Biểu thức thu gọn biểu thức P = A B a + C 2a −n −n a a− + a 3 là: a a + a− D a −n a +b a − b−n − (với ab = 0, a = ±b) −n − b−n −n + b−n a a a n bn 2an bn 3an bn 4an bn A 2n B C D b − a2n b2n − a2n b2n − a2n b2n − a2n √ Câu 22 Cho a > Viết biểu thức P = a a6 dạng lũy thừa với số mũ hữu tỷ Câu 21 Rút gọn biểu thức F = A P = C P = a7 B P = a D P = a6 Câu 23 Trong khẳng định sau, khẳng định sai? A Nếu a > ax > ay x > y B Nếu a > ax ≤ ay x ≤ y C Nếu < a < ax > ay x > y D Nếu < a = ax = ay x = y 7 x y + x.y Câu 24 Cho x, y > 0, rút gọn P = √ √ x+ 6y √ √ C P = x.y A P = x + y B P = x + y √ Câu 25 Cho a > 0, rút gọn P = a √ 5−2 √ a1− a B P = a √ D P = √ xy 5+2 3−2 C P = D P = a2 a √ Câu 26 Giả sử a số thực dương, khác Biểu thức a a viết dạng aα Khi 11 A α = B α = C α = D α = 3 √ Câu 27 Đưa biểu thức A = a a a lũy thừa số < a = 1ta biểu thức A P = Lathi, B.P. “Ordinary Linear Differential and Difference Equations” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press LLC, 1999 c  1999byCRCPressLLC 2 Ordinary Linear Differential and Difference Equations B.P. Lathi California State University, Sacramento 2.1 Differential Equations Classical Solution • MethodofConvolution 2.2 Difference Equations Initial Conditions andIterativeSolution • Classical Solution • MethodofConvolution References 2.1 Differential Equations Afunctioncontainingvariablesandtheirderivativesiscalledadifferentialex pression,andanequation involvingdifferentialexpressionsiscalledadifferentialequation. Adifferentialequationisanordinary differential equation if it contains only one independent variable; it is a partial differential equation if it contains more than one independentvariable. Weshall deal here only withordinary differential equations. In the mathematical texts, the independent variable is generally x, which can be anything such as time, distance, velocity, pressure, and so on. In most of the applications in control systems, the independent variable is time. For this reason we shall use here independent variable t for time, although it canstand for any other variable as well. The following equation  d 2 y dt 2  4 + 3 dy dt + 5y 2 (t) = sint is an ordinary differential equation of second order because the highest derivative is of the second order. An nth-order differential equation is linear ifit is of the form a n (t) d n y dt n + a n−1 (t) d n−1 y dt n−1 +···+a 1 (t) dy dt + a 0 (t)y(t) = r(t) (2.1) where the coefficients a i (t) are not functions of y(t). If these coefficients (a i ) are constants, the equation is linear with constant coefficients. Many engineering (as well as nonengineering) systems can be modeled by these equations. Systems modeled by these equations are known as linear time- invariant (LTI) systems. In this chapter we shall deal exclusively with linear differential equations with constant coefficients. Certain other forms of differential equations are dealt with elsewhere in this volume. c  1999 by CRC Press LLC Role of Auxiliary Conditions in Solution of Differential Equations We now show that a differential equation does not, in general, have a unique solution unless some additional constraints (or conditions) on the solution are known. This fact should not come as a surprise. A function y(t) has a unique derivative dy/dt, but for a given derivative dy/dt there are infinite possible functions y(t).Ifwearegivendy/dt , it is impossible to determine y(t) uniquely unless an additional piece of information about y(t) is given. For example, the solution of a differential equation dy dt = 2 (2.2) obtained by integrating both sides of the equation is y(t) = 2t + c (2.3) for any value of c. Equation 2.2 specifies a function whose slope is 2 for all t. Any straight line with a slope of 2 satisfies this equation. Clearly the solution is not unique, but if we place an additional constraint on the solution y(t), then we specify a unique solution. For example, suppose we require that y(0) = 5; then out of all the possible solutions available, only one function has a slope of 2 and an intercept with the vertical axis at 5. By setting t = 0 in Equation 2.3 and substituting y(0) = 5 in the same equation, we obtain y(0) = 5 = c and y(t) = 2t + 5 which is the unique solution satisfying both Equation 2.2 and the constraint y(0) = 5. Inconclusion, differentiation isanirreversibleoperationduringwhichcertain informationislost. Toreversethisoperation,onepieceofinformationabouty(t)mustbeprovidedtorestoretheoriginal y(t). Usingasimilarargument,wecanshowthat,givend 2 y/dt 2 ,wecandeterminey(t)uniquelyonly if two additional Exponential and Logarithmic Equations Exponential and Logarithmic Equations By: OpenStaxCollege Wild rabbits in Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-WesleyChapter 6I/O Streams as an Introduction to Objects and Classes Slide 6- 3Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-WesleyOverview6.1 Streams and Basic File I/O 6.2 Tools for Stream I/O6.3 Character I/O6.4 Inheritance Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley6.1Streams and Basic File I/O Slide 6- 5Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-WesleyI/O StreamsI/O refers to program input and outputInput is delivered to your program via a stream objectInput can be fromThe keyboardA fileOutput is delivered to the output device via a streamobjectOutput can be to The screenA file Slide 6- 6Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-WesleyObjectsObjects are special variables thatHave their own special-purpose functionsSet C++ apart from earlier programming languages Slide 6- 7Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-WesleyStreams and Basic File I/OFiles for I/O are the same type of files used tostore programsA stream is a flow of data.Input stream: Data flows into the programIf input stream flows from keyboard, the program willaccept data from the keyboardIf input stream flows from a file, the program will acceptdata from the fileOutput stream: Data flows out of the programTo the screenTo a file Slide 6- 8Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesleycin And cout StreamscinInput stream connected to the keyboardcout Output stream connected to the screencin and cout defined in the iostream libraryUse include directive: #include <iostream>You can declare your own streams to use with files. Slide 6- 9Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-WesleyWhy Use Files?Files allow you to store data permanently!Data output to a file lasts after the program endsAn input file can be used over and overNo typing of data again and again for testingCreate a data file or read an output file at yourconvenienceFiles allow you to deal with larger data sets Slide 6- 10Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-WesleyFile I/OReading from a fileTaking input from a fileDone from beginning to the end (for now)No backing up to read something again (OK to start over)Just as done from the keyboardWriting to a fileSending output to a fileDone from beginning to end (for now)No backing up to write something again( OK to start over)Just as done to the screen [...]... only to the stream named in the call Slide 6- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Overview 6.1 Streams and Basic File I/O 6.2 Tools for Stream I/O 6.3 Character I/O 6.4 Inheritance Slide 6- 45 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Manipulators  A manipulator is a function called Introduction to Exponential and Logarithmic Functions Introduction to Exponential and Logarithmic Functions By: OpenStaxCollege 1/3 Introduction to Exponential and Logarithmic Functions Electron micrograph of E.Coli bacteria (credit: “Mattosaurus,” Wikimedia Commons) Focus in on a square centimeter of your skin Look closer Closer still If you could look closely enough, you would see hundreds of thousands of microscopic organisms They are bacteria, and they are not only on your skin, but in your mouth, nose, and even your intestines In fact, the bacterial cells in your body at any given moment ... x-axis 2/32 Exponential and Logarithmic Models A graph showing exponential growth The equation is y = 2e3x A graph showing exponential decay The equation is y = 3e − 2x Exponential growth and decay... 2013 follow a logarithmic curve 16/32 Exponential and Logarithmic Models We could restrict the interval from 2000 to 2010, apply regression analysis using a logarithmic model, and use it to predict... we often consider an exponential or logarithmic model, though other models, such as quadratic models, may also be considered In choosing between an exponential model and a logarithmic model, we