Financial Risk: Credit Risk, Lecture 1


 Leonard Horton
 2 years ago
 Views:
Transcription
1 Financial Risk: Credit Risk, Lecture 1 Alexander Herbertsson Centre For Finance/Departent of Econoics School of Econoics, Business and Law, University of Gothenburg Eail: Financial Risk, Chalers University of Technology, Göteborg Sweden Noveber 13, 2012 Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
2 Content of lecture Short discussion of the iportant coponents of credit risk Study different static portfolio credit risk odels. Discussion of the binoial loss odel Discussion of the ixed binoial loss odel Study of a ixed binoial loss odel with a beta distribution Study of a ixed binoial loss odel with a logitnoral distribution A short discussion of ValueatRisk and Expected shortfall Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
3 Definition of Credit Risk Credit risk the risk that an obligor does not honor his payents Exaple of an obligor: A copany that have borrowed oney fro a bank A copany that has issued bonds. A household that have borrowed oney fro a bank, to buy a house A bank that has entered into a bilateral financial contract (e.g an interest rate swap) with another bank. Exaple of defaults are A copany goes bankrupt. As copany fails to pay a coupon on tie, for soe of its issued bonds. A household fails to pay aortization or interest rate on their loan. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
4 Credit Risk Credit risk can be decoposed into: arrival risk, the risk connected to whether or not a default will happen in a given tieperiod, for a obligor tiing risk, the risk connected to the uncertainness of the exact tiepoint of the arrival risk (will not be studied in this course) recovery risk. This is the risk connected to the size of the actual loss if default occurs (will not be studied in this course, we let the recovery be fixed) default dependency risk, the risk that several obligors jointly defaults during soe specific tie period. This is one of the ost crucial risk factors that has to be considered in a credit portfolio fraework. The coing two lectures focuses only on default dependency risk. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
5 Portfolio Credit Risk is iportant Modelling dependence between default events and between credit quality changes is, in practice, one of the biggest challenges of credit risk odels., David Lando, Credit Risk Modeling, p Default correlation and default dependency odelling is probably the ost interesting and also the ost deanding open proble in the pricing of credit derivatives. While any singlenae credit derivatives are very siilar to other noncredit related derivatives in the defaultfree world (e.g. interestrate swaps, options), basket and portfolio credit derivative have entirely new risks and features., Philipp Schönbucher, Credit derivatives pricing odels, p Epirically reasonable odels for correlated defaults are central to the credit riskanageent and pricing systes of ajor financial institutions., Darrell Duffie and Kenneth Singleton, Credit Risk: Pricing, Measureent and Manageent, p Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
6 Portfolio Credit Risk is iportant Portfolio credit risk odels differ greatly depending on what types of portfolios, and what type of questions that should be considered. For exaple, odels with respect to risk anageent, such as credit ValueatRisk (VaR) and expected shortfall (ES) odels with respect to valuation of portfolio credit derivatives, such as CDO s and basket default swaps In both cases we need to consider default dependency risk, but......in risk anageent odelling (e.g. VaR, ES), the tiing risk is ignored, and one often talk about static credit portfolio odels,...while, when pricing credit derivatives, tiing risk ust be carefully odeled (not treated here) The coing two lectures focuses only on static credit portfolio odels, Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
7 Literature The slides for the coing two lectures are rather selfcontained, except for soe results taken fro Hult & Lindskog. The content of the lecture today and the next lecture is partly based on aterials presented in Lecture notes by Henrik Hult and Filip Lindskog (Hult & Lindskog) Matheatical Modeling and Statistical Methods for Risk Manageent, however, these notes are no longer public available, instead see e.g the book Hult, Lindskog, Haerlid and Rehn: Risk and portfolio analysis  principles and ethods. Quantitative Risk Manageent by McNeil A., Frey, R. and Ebrechts, P. (Princeton University Press) Credit Risk Modeling: Theory and Applications by Lando, D. (Princeton University Press) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
8 Static Models for hoogeneous credit portfolios Today we will consider the following static odes for a hoogeneous credit portfolio: The binoial odel The ixed binoial odel To understand ixed binoial odels, we give a short introduction of conditional expectations After this we look at two different ixed binoial odels. We also shortly discuss ValueatRisk and Expected shortfall Next lecture we consider a ixed binoial odel inspired by the Merton fraework. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
9 The binoial odel for independent defaults Consider a hoogeneous credit portfolio odel with obligors, and where we each obligor can default up to fixed tie point, say T. Each obligor have identical credit loss at a default, say l. Here l is a constant. Let X i be a rando variable such that { 1 if obligor i defaults before tie T X i = 0 otherwise, i.e. if obligor i survives up to tie T We assue that the rando variables X 1, X 2,... X are i.i.d, that is they are all independent with identical distribution. Furtherore P [X i = 1] = p so that P [X i = 0] = 1 p. The total credit loss in the portfolio at tie T, called L, is then given by L = lx i = l X i = ln where N = i=1 i=1 thus, N is the nuber of defaults in the portfolio up to tie T. Since l is a constant, we have P [L = kl] = P [N = k], so it is enough to study the distribution of N. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36 i=1 X i (1)
10 The binoial odel for independent defaults, cont. Since X 1, X 2,...X are i.i.d with with P [X i = 1] = p we conclude that N = i=1 X i is binoially distributed with paraeters and p, that is N Bin(, p). This eans that P [N = k] = ( ) p k (1 p) k k Recalling the binoial theore (a + b) = ( k=0 k) a k b k we see that ( ) P [N = k] = p k (1 p) k = (p + (1 p)) = 1 k k=0 k=0 proving that Bin(, p) is a distribution. Furtherore, E[N ] = p since [ ] E[N ] = E X i = i=1 E[X i ] = p. i=1 Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
11 The binoial odel for independent defaults, cont The portfolio credit loss distribution in the binoial odel bin(50,0.1) probability nuber of defaults lexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
12 The binoial odel for independent defaults, cont. The binoial distribution have very thin tails, that is, it is extreely unlikely to have any losses (see figure). For exaple, if p = 5% and = 50 we have that P [N 8] = 1.2% and for p = 10% and = 50 we get P [N 10] = 5.5% The ain reason for these sall nubers (even for large individual default probabiltes) is due to the independence assuption. To see this, recall that the variance of a rando variable Var(X) easures [ the degree of the deviation of X around its ean, i.e. Var(X) = E (X E[X]) 2]. Since X 1, X 2,...X are independent we have that ( ) Var(N ) = Var X i = i=1 Var(X i ) = p(1 p) (2) i=1 where the second equality is due the independence assuption. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
13 The binoial odel for independent defaults, cont. Furtherore, by Chebyshev s inequality we have that for any rando varialbe X, and any c > 0 it holds P [ X E[X] c] Var(X) c 2 So if p = 5% and = 50 we have that Var(N ) = 50p(1 p) = and and E[N ] = 50p = 2.5 iplying that having say, 6 ore, or less losses than expected, is saller or equal than 6.6%, since by Chebyshev s inequality P [ N 2.5 6] = 6.6% 36 Hence, the probability of having a total nuber of losses outside the interval 2.5 ± 6, i.e. outside the interval [0, 8.5], is saller than 6.6%. In fact, one can show that the deviation of the average nuber of defaults in the portfolio, N, fro the constant p (where p = E[ ] N ) goes to zero as. Thus, N converges towards a constant as (the law of large nubers). Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
14 Independent defaults and the law of large nubers By applying Chebyshev s inequality to the rando variable N together with Equation (2) we get [ ] N P p ε Var( N ) 1 Var(N ε 2 = 2 ) p(1 p) p(1 p) ε 2 = 2 ε 2 = ε 2 and we conclude that P [ N p ε] 0 as. Note that this holds for any ε > 0. This result is called the weak law of large nubers, and says that the N average nuber of defaults in the portfolio, i.e., converges (in probability) to the constant p which is the individual default probability. One can also show the so called strong law of large nubers, that is [ ] N P p when = 1 and we say that N converges alost surely to the constant p. In these lectures we write N p to indicate alost surely convergence. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
15 Independent defaults lead to unrealistic loss scenarios We conclude that the independence assuption, or ore generally, the i.i.d assuption for the individual default indicators X 1, X 2,... X iplies that the average nuber of defaults in the portfolio N converges to the constant p alost surely. Given the recent credit crisis, the assuption of independent defaults is ridiculous. It is an epirical fact, observed any ties in the history, that defaults tend to cluster. Hence, the fraction of defaults in the portfolio N will often have values uch bigger than the constant p. Consequently, the epirical (i.e. observed) density for N will have uch ore fatter tails copared with the binoial distribution. We will therefore next look at portfolio credit odels that can produce ore realistic loss scenarios, with densities for N that have fat tails, and which not iplies that the average nuber of defaults in the portfolio N converges to a constant with probability 1, when. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
16 Conditional expectations Before we continue this lecture, we need to introduce the concept of conditional expectations Let L 2 denote the space of all rando variables X such that E [ X 2] < Let Z be a rando variable and let L 2 (Z) L 2 denote the space of all rando variables Y such that Y = g(z) for soe function g and Y L 2 Note that E[X] is the value µ that iniizes the quantity E [ (X µ) 2]. Inspired by this, we define the conditional expectation E[X Z] as follows: Definition of conditional expectations For a rando variable Z, and for X L 2, the conditional expectation E[X Z] is the rando variable Y L 2 (Z) that iniizes E [ (X Y ) 2]. Intuitively, we can think of E[X Z] as the orthogonal projection of X onto the space L 2 (Z), where the scalar product X, Y is defined as X, Y = E[XY ]. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
17 Properties of conditional expectations For a rando variable Z it is possible to show the following properties 1. If X L 2, then E[E[X Z]] = E[X] 2. If Y L 2 (Z), then E[YX Z] = Y E[X Z] 3. If X L 2, we define Var(X Z) as Var(X Z) = E [ X 2 Z ] E[X Z] 2 and it holds that Var(X) = E[Var(X Z)] + Var(E[X Z]). Furtherore, for an event A, we can define the conditional probability P [A Z] as P [A Z] = E[1 A Z] where 1 A is the indicator function for the event A (note that 1 A is a rando variable). An exaple: if X {a, b}, let A = {X = a}, and we get that P [X = a Z] = E [ 1 {X=a} Z ]. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
18 The ixed binoial odel The binoial odel is also the starting point for ore sophisticated odels. For exaple, the ixed binoial odel which randoizes the default probability in the standard binoial odel, allowing for stronger dependence. The econoic intuition behind this randoizing of the default probability p(z) is that Z should represent soe coon background variable affecting all obligors in the portfolio. The ixed binoial distribution works as follows: Let Z be a rando variable on R with density f Z (z) and let p(z) [0, 1] be a rando variable with distribution F(x) and ean p, that is F(x) = P [p(z) x] and E[p(Z)] = p(z)f Z (z)dz = p. (3) Let X 1, X 2,...X be identically distributed rando variables such that X i = 1 if obligor i defaults before tie T and X i = 0 otherwise. Furtherore, conditional on Z, the rando variables X 1, X 2,...X are independent and each X i have default probability p(z), that is P [X i = 1 Z] = p(z) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
19 The ixed binoial odel Since P [X i = 1 Z] = p(z) we get that E[X i Z] = p(z), because E[X i Z] = 1 P [X i = 1 Z] + 0 (1 P [X i = 1 Z]) = p(z). Furtherore, note that E[X i ] = p and thus p = E[p(Z)] = P [X i = 1] since P [X i = 1] = E[X i ] = E[E[X i Z]] = E[p(Z)] = where the last equality is due to (3). One can show that 1 0 p(z)f Z (z)dz = p. Var(X i ) = p(1 p) and Cov(X i, X j ) = E [ p(z) 2] p 2 = Var(p(Z)) (4) Next, letting all losses be the sae and constant given by, say l, then the total credit loss in the portfolio at tie T, called L, is L = lx i = l X i = ln where N = i=1 i=1 thus, N is the nuber of defaults in the portfolio up to tie T ] Again, since P [ L = kl = P [N = k], it is enough to study N. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36 i=1 X i
20 The ixed binoial odel, cont. However, since the rando variables X 1, X 2,...X now only are conditionally independent, given the outcoe Z, we have ( ) P [N = k Z] = p(z) k (1 p(z)) k k so since P [N = k] = E[P [N = k Z]] = E [( k) p(z) k (1 p(z)) k] it holds that ( ) P [N = k] = p(z) k (1 p(z)) k f Z (z)dz. (5) k Furtherore, since X 1, X 2,... X no longer are independent we have that ( ) Var(N ) = Var X i = Var(X i ) + Cov(X i, X j ) (6) i=1 i=1 and by hoogeneity in the odel we thus get i=1 j=1,j i Var(N ) = Var(X i ) + ( 1)Cov(X i, X j ). (7) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
21 The ixed binoial odel, cont. So inserting (4) in (7) we get that Var(N ) = p(1 p) + ( 1) ( E [ p(z) 2] p 2). (8) Next, it is of interest to study how our portfolio will behave when, that is when the nuber of obligors in the portfolio goes to infinity. Recall that Var(aX) = a 2 Var(X) so this and (8) iply that ( ) N Var = Var(N ) p(1 p) 2 = + ( 1)( E [ p(z) 2] p 2). We therefore conclude that ( ) N Var E [ p(z) 2] p 2 as (9) Note especially the case when p(z) is a constant, say p, so that p = p. Then we are back in the standard binoial loss odel and E [ p(z) 2] p 2 = p 2 p 2 = 0 so Var ( N ) 0, i.e. the average nuber of defaults in the portfolio converge to a constant (which is p) as the portfolio size tend to infinity (this is the law of large nubers.) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
22 The ixed binoial odel, cont. So in the ixed binoial odel, we see fro (9) that the law of large nubers do not hold, i.e. Var ( N ) does not converge to 0. Consequently, the average nuber of defaults in the portfolio, i.e. not converge to a constant as. N, does This is due to the fact that the rando variables X 1, X 2,... X, are not independent. The dependence aong the X 1, X 2,... X, is created by Z. However, conditionally on Z, we have that the law of large nubers hold (because if we condition on Z, then X 1, X 2,... X are i.i.d with default probability p(z)), that is given a fixed outcoe of Z then N p(z) as (10) and since a.s convergence iplies convergence in distribution (10) iplies that for any x [0, 1] we have [ ] N P x P [p(z) x] when. (11) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
23 The ixed binoial odel, cont. Note that (11) can also be verified intuitive fro (10) by aking the following observation. Fro (10) we have that [ ] { N P θ 0 if p(z) > θ Z as 1 if p(z) θ that is, Next, recall that [ ] N P θ Z 1 {p(z) θ} as. (12) [ ] [ [ ]] N P θ N = E P θ Z so (12) in (13) renders [ ] N P θ E [ ] 1 {p(z) θ} = P [p(z) θ] = F(θ) as where F(x) = P [p(z) x], i.e. F(x) is the distribution function of the rando variable p(z). (13) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
24 Large Portfolio Approxiation (LPA) Hence, fro the above rearks we conclude the following iportant result: Large Portfolio Approxiation (LPA) for ixed binoial odels For large portfolios in a ixed binoial odel, the distribution of the average nuber of defaults in the portfolio converges to the distribution of the rando variable p(z) as, that is for any x [0, 1] we have [ ] N P x P [p(z) x] when. (14) The distribution P [p(z) x] is called the Large Portfolio Approxiation (LPA) to the distribution of N. The above result iplies that if p(z) has heavy tails, then the rando variable N will also have heavy tails, as, which then iplies a strong default dependence in the credit portfolio. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
25 The ixed binoial odel: the beta function One exaple of a ixing binoial odel is to let p(z) = Z where Z is a beta distribution, Z Beta(a, b), which can generate heavy tails. We say that a rando variable Z has beta distribution, Z Beta(a, b), with paraeters a and b, if it s density f Z (z) is given by f Z (z) = 1 β(a, b) za 1 (1 z) b 1 a, b > 0, 0 < z < 1 (15) where β(a, b) denotes the beta function which satisfies the recursive relation β(a + 1, b) = a β(a, b). a + b Also note that (15) iplies that P [0 Z 1] = 1, that is Z [0, 1] with probability one. Furtherore, since p(z) = Z, the distribution of N converges to the distribution of the beta distribution, i.e [ ] N P x 1 x z a 1 (1 z) b 1 dz as β(a, b) 0 Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
26 The ixed binoial odel: the beta function, cont. 14 Two different beta densities 12 a=1,b=9 a=10,b=90 10 density x lexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
27 The ixed binoial odel: the beta function, cont. If Z has beta distribution with paraeters a and b, one can show that E[Z] = a a + b and Var(Z) = ab (a + b) 2 (a + b + 1). Consider a ixed binoial odel where p(z) = Z has beta distribution with paraeters a and b. Then, by using (5) one can show that ( ) β(a + k, b + k) P [N = k] =. (16) k β(a, b) It is possible to create heavy tails in the distribution P [N = k] by choosing the paraeters a and b properly in (16). This will then iply ore realistic probabilities for extree loss scenarios, copared with the standard binoial loss distribution (see figure on next page). Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
28 The ixed binoial odel: the beta function, cont The portfolio credit loss distribution in the standar and ixed binoial odel ixed binoial 50, beta(1,9) binoial(50,0.1) probability nuber of defaults lexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
29 Mixed binoial odels: logitnoral distribution Another possibility for ixing distribution p(z) is to let p(z) be a logitnoral distribution. This eans that p(z) = exp ( (µ + σz)) where σ > 0 and Z N(0, 1), that is Z is a standard noral rando variable. Note that p(z) [0, 1]. Furtherore, if 0 < x < 1 then p 1 (x) is well defined and given by p 1 (x) = 1 ( ( ) ) x ln µ. (17) σ 1 x The ixing distribution F(x) = P [p(z) x] = P [ Z p 1 (x) ] for a logitnoral distribution is then given by F(x) = P [ Z p 1 (x) ] = N(p 1 (x)) for 0 < x < 1 where p 1 (x) is given as in Equation (17) and N(x) is the distribution function of a standard noral distribution. Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
30 Correlations in ixed binoial odels Recall the definition of the correlation Corr(X, Y ) between two rando variables X and Y, given by Corr(X, Y ) = Cov(X, Y ) Var(X) Var(Y) where Cov(X, Y ) = E[XY ] E[X] E[Y] and Var(X)) = E [ X 2] E[X] 2. Furtherore, also recall that Corr(X, Y ) ay soeties be seen as a easure of the dependence between the two rando variables X and Y. Now, let us consider a ixed binoial odel as presented previously. We are interested in finding Corr(X i, X j ) for two pairs i, j in the portfolio (by the hoogeneousportfolio assuption this quantity is the sae for any pair i, j in the portfolio where i j). Below, we will therefore for notational convenience siply write ρ X for the correlation Corr(X i, X j ). Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
31 Correlations in ixed binoial odels, cont. Recall fro previous slides that P [X i = 1 Z] = p(z) where p(z) is the ixing variable. Furtherore, we also now that Cov(X i, X j ) = E [ p(z) 2] p 2 and Var(X i ) = p(1 p) (18) where p = E[p(Z)]. Thus, the correlation ρ X in a ixed binoial odels is then given by ρ X = E[ p(z) 2] p 2 p(1 p) where p = E[p(Z)] = P [X i = 1] is the default probability for each obligor. Hence, the correlation ρ X in a ixed binoial is copletely deterined by the fist two oents of the ixing variable p(z), that is E[p(Z)] and E [ p(z) 2]. Exercise 1: Show that P [X i = 1, X j = 1] = E [ p(z) 2] where i j. Exercise 2: Show that Var(X i ) = E[p(Z)] (1 E[p(Z)]). (19) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
32 ValueatRisk Recall the definition of ValueatRisk Definition of ValueatRisk Given a loss L and a confidence level α (0, 1), then VaR α (L) is given by the sallest nuber y such that the probability that the loss L exceeds y is no larger than 1 α, that is VaR α (L) = inf {y R : P [L > y] 1 α} where F L (x) is the distribution of L. = inf {y R : 1 P [L y] 1 α} = inf {y R : F L (y) α} Note that ValueatRisk is defined for a fixed tie horizon, so the above definition should also coe with a tie period, e.g, if the loss L is over one day, then we talk about a oneday VaR α (L). In credit risk, one typically consider VaR α (L) for the loss over one year. Note that if F L (x) is continuous, then VaR α (L) = F 1 L (α) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
33 ValueatRisk for static credit portfolios Consider the sae type of hoogeneous static credit portfolio odels as studied previously today, with obligors and where each obligor can default up to tie T. Each obligor have identical credit loss l at a default, where l is a constant. The total credit loss in the portfolio at tie T is then given by L = ln where N is the nuber of defaults in the portfolio up to tie T. Note that the individual loss l is given by ln where N is the notional of the individual loan and l is the loss as a fraction of N (i.e l [0, 1]) By linearity of VaR (see in lecture notes by H&L) we can without loss of generality assue that N = 1, so that l = l, since VaR α (cl) = cvar α (L) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
34 ValueatRisk for static credit portfolios, cont. If p(z) is a ixing variable with distribution F(x) we know that [ ] N P x F(x) as which iplies that P [L x] = P [ N x ] ( x ) F l l as Hence, if F(x) is continuous, and if is large, we have the following approxiation forula for VaR α (L) where L denotes the loss L. VaR α (L) l F 1 (α) (20) Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
35 Expected shortfall The expected shortfall ES α (L) is defined as and one can show that ES α (L) = E[L L VaR α (L)] ES α (L) = 1 1 α 1 α VaR u (L)du. Hence, for the sae static credit portfolio as on the two previous slides, we have the following approxiation forula for ES α (L) (when is large) ES α (L) l 1 α 1 α F 1 (u)du where L denotes the loss L and where we used (20). Here, F(x) is the continuous distribution of the ixing variable p(z). Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
36 Thank you for your attention! Alexander Herbertsson (Univ. of Gothenburg) Financial Risk: Credit Risk, Lecture 1 Noveber 13, / 36
IV Approximation of Rational Functions 1. IV.C Bounding (Rational) Functions on Intervals... 4
Contents IV Approxiation of Rational Functions 1 IV.A Constant Approxiation..................................... 1 IV.B Linear Approxiation....................................... 3 IV.C Bounding (Rational)
More informationFixedIncome Securities and Interest Rates
Chapter 2 FixedIncoe Securities and Interest Rates We now begin a systeatic study of fixedincoe securities and interest rates. The literal definition of a fixedincoe security is a financial instruent
More informationExample: Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?
Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent,
More informationFactor Model. Arbitrage Pricing Theory. Systematic Versus NonSystematic Risk. Intuitive Argument
Ross [1],[]) presents the aritrage pricing theory. The idea is that the structure of asset returns leads naturally to a odel of risk preia, for otherwise there would exist an opportunity for aritrage profit.
More informationAnalysis of the purchase option of computers
Analysis of the of coputers N. Ahituv and I. Borovits Faculty of Manageent, The Leon Recanati Graduate School of Business Adinistration, TelAviv University, University Capus, RaatAviv, TelAviv, Israel
More informationSAMPLING METHODS LEARNING OBJECTIVES
6 SAMPLING METHODS 6 Using Statistics 66 2 Nonprobability Sapling and Bias 66 Stratified Rando Sapling 62 6 4 Cluster Sapling 64 6 5 Systeatic Sapling 69 6 6 Nonresponse 62 6 7 Suary and Review of
More informationGuidelines for calculating sample size in 2x2 crossover trials : a simulation study
J.Natn.Sci.Foundation Sri Lanka 011 39 (1): 7789 RESEARCH ARTICLE Guidelines for calculating saple size in x crossover trials : a siulation study N.M. Siyasinghe and M.R. Sooriyarachchi Departent of Statistics,
More informationQuality evaluation of the modelbased forecasts of implied volatility index
Quality evaluation of the odelbased forecasts of iplied volatility index Katarzyna Łęczycka 1 Abstract Influence of volatility on financial arket forecasts is very high. It appears as a specific factor
More informationDependent Events and Operational Risk
Dependent Events and Operational Risk Miro R. Powojowski, Diane Reynolds and Hans J.H. Tuenter Most people acknowledge that soe correlation between operational losses exists. Modelling it, however, is
More informationUse of extrapolation to forecast the working capital in the mechanical engineering companies
ECONTECHMOD. AN INTERNATIONAL QUARTERLY JOURNAL 2014. Vol. 1. No. 1. 23 28 Use of extrapolation to forecast the working capital in the echanical engineering copanies A. Cherep, Y. Shvets Departent of finance
More informationReliability Constrained Packetsizing for Linear Multihop Wireless Networks
Reliability Constrained acketsizing for inear Multihop Wireless Networks Ning Wen, and Randall A. Berry Departent of Electrical Engineering and Coputer Science Northwestern University, Evanston, Illinois
More informationOnline Bagging and Boosting
Abstract Bagging and boosting are two of the ost wellknown enseble learning ethods due to their theoretical perforance guarantees and strong experiental results. However, these algoriths have been used
More informationPricing Asian Options using Monte Carlo Methods
U.U.D.M. Project Report 9:7 Pricing Asian Options using Monte Carlo Methods Hongbin Zhang Exaensarbete i ateatik, 3 hp Handledare och exainator: Johan Tysk Juni 9 Departent of Matheatics Uppsala University
More informationThis paper studies a rental firm that offers reusable products to price and qualityofservice sensitive
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol., No. 3, Suer 28, pp. 429 447 issn 523464 eissn 5265498 8 3 429 infors doi.287/so.7.8 28 INFORMS INFORMS holds copyright to this article and distributed
More informationPlane Trusses. Section 7: Flexibility Method  Trusses. A plane truss is defined as a twodimensional
lane Trusses A plane truss is defined as a twodiensional fraework of straight prisatic ebers connected at their ends by frictionless hinged joints, and subjected to loads and reactions that act only at
More informationMINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3UNIFORM HYPERGRAPHS
MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3UNIFORM HYPERGRAPHS JIE HAN AND YI ZHAO Abstract. We show that for sufficiently large n, every 3unifor hypergraph on n vertices with iniu
More informationExtendedHorizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona Network
2013 European Control Conference (ECC) July 1719, 2013, Zürich, Switzerland. ExtendedHorizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona
More informationF3 Financial Strategy. Examiner s Answers
Strategic Level Paper F3 Financial Strategy March 2014 exaination Exainer s Answers Question One Rationale Question One focusses on an external party a hotel chain, which is hoping to set up a hotel in
More informationEvaluating Inventory Management Performance: a Preliminary DeskSimulation Study Based on IOC Model
Evaluating Inventory Manageent Perforance: a Preliinary DeskSiulation Study Based on IOC Model Flora Bernardel, Roberto Panizzolo, and Davide Martinazzo Abstract The focus of this study is on preliinary
More informationarxiv:0805.1434v1 [math.pr] 9 May 2008
Degreedistribution stability of scalefree networs Zhenting Hou, Xiangxing Kong, Dinghua Shi,2, and Guanrong Chen 3 School of Matheatics, Central South University, Changsha 40083, China 2 Departent of
More informationEndogenous CreditCard Acceptance in a Model of Precautionary Demand for Money
Endogenous CreditCard Acceptance in a Model of Precautionary Deand for Money Adrian Masters University of Essex and SUNY Albany Luis Raúl RodríguezReyes University of Essex March 24 Abstract A creditcard
More information1 Interest rates and bonds
1 Interest rates and bonds 1.1 Copounding There are different ways of easuring interest rates Exaple 1 The interest rate on a oneyear deposit is 10% per annu. This stateent eans different things depending
More informationPERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO
Bulletin of the Transilvania University of Braşov Series I: Engineering Sciences Vol. 4 (53) No.  0 PERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO V. CAZACU I. SZÉKELY F. SANDU 3 T. BĂLAN Abstract:
More informationIrrigation Economics
Irrigation Econoics The Hydrologic Cycle Rain Evaporation Irrigation Runon Agricultural Production Runoff Deep Percolation to Groundwater The use of irrigation water depends on: Econoics (prices and
More informationModeling operational risk data reported above a timevarying threshold
Modeling operational risk data reported above a tievarying threshold Pavel V. Shevchenko CSIRO Matheatical and Inforation Sciences, Sydney, Locked bag 7, North Ryde, NSW, 670, Australia. eail: Pavel.Shevchenko@csiro.au
More informationLEASING, LEMONS, AND MORAL HAZARD
LEASING, LEMONS, AND MORAL HAZARD by Justin P. Johnson Johnson Graduate School of Manageent Cornell University Sage Hall Ithaca, NY 14853 jpj25@cornell.edu and Michael Waldan Johnson Graduate School of
More informationConstruction Economics & Finance. Module 3 Lecture1
Depreciation: Construction Econoics & Finance Module 3 Lecture It represents the reduction in arket value of an asset due to age, wear and tear and obsolescence. The physical deterioration of the asset
More informationThe public private partnership paradox
The public private partnership paradox Stephen Gray * UQ Business School, University of Queensland Jason Hall UQ Business School, University of Queensland Grant Pollard Value Decisions ABSTRACT A public
More informationFisher Information and CramérRao Bound. 1 Fisher Information. Math 541: Statistical Theory II. Instructor: Songfeng Zheng
Math 54: Statistical Theory II Fisher Information CramérRao Bound Instructor: Songfeng Zheng In the parameter estimation problems, we obtain information about the parameter from a sample of data coming
More informationThe Circle Method on the Binary Goldbach Conjecture
The Circle Method on the Binary Goldbach Conjecture Jeff Law April 3, 2005 Matheatics Departent Princeton University Princeton, J 08544 The Goldbach Conjecture The Goldbach Conjecture, appears to be very
More informationIntroduction to Unit Conversion: the SI
The Matheatics 11 Copetency Test Introduction to Unit Conversion: the SI In this the next docuent in this series is presented illustrated an effective reliable approach to carryin out unit conversions
More informationFuzzy Sets in HR Management
Acta Polytechnica Hungarica Vol. 8, No. 3, 2011 Fuzzy Sets in HR Manageent Blanka Zeková AXIOM SW, s.r.o., 760 01 Zlín, Czech Republic blanka.zekova@sezna.cz Jana Talašová Faculty of Science, Palacký Univerzity,
More informationMachine Learning Applications in Grid Computing
Machine Learning Applications in Grid Coputing George Cybenko, Guofei Jiang and Daniel Bilar Thayer School of Engineering Dartouth College Hanover, NH 03755, USA gvc@dartouth.edu, guofei.jiang@dartouth.edu
More informationSemiinvariants IMOTC 2013 Simple semiinvariants
Seiinvariants IMOTC 2013 Siple seiinvariants These are soe notes (written by Tejaswi Navilarekallu) used at the International Matheatical Olypiad Training Cap (IMOTC) 2013 held in Mubai during AprilMay,
More informationCalculating the Return on Investment (ROI) for DMSMS Management. The Problem with Cost Avoidance
Calculating the Return on nvestent () for DMSMS Manageent Peter Sandborn CALCE, Departent of Mechanical Engineering (31) 453167 sandborn@calce.ud.edu www.ene.ud.edu/escml/obsolescence.ht October 28, 21
More informationMulticriteria analysis for behavioral segmentation
Paper Multicriteria analysis for behavioral segentation Janusz Granat Abstract Behavioral segentation is a process of finding the groups of clients with siilar behavioral patterns The basic tool for segentation
More informationMobile Software Reliability Measurement Using Growth Model in Testing *
International Journal of u and e Service, Science and Technology 37 Mobile Software Reliability Measureent Using Growth Model in Testing * HaengKon Ki 1, EunJu Park 1 1 Dept. of Coputer inforation
More informationOpenGamma Documentation Bond Pricing
OpenGaa Docuentation Bond Pricing Marc Henrard arc@opengaa.co OpenGaa Docuentation n. 5 Version 2.0  May 2013 Abstract The details of the ipleentation of pricing for fixed coupon bonds and floating rate
More informationPerformance Analysis of WorkConserving Schedulers for Minimizing Total FlowTime with Phase Precedence
Perforance Analysis of WorkConserving Schedulers for Miniizing Total lowtie with Phase Precedence Yousi Zheng, Prasun Sinha, Ness B Shroff, Departent of Electrical and Coputer Engineering, Departent
More informationInternational Journal of Management & Information Systems First Quarter 2012 Volume 16, Number 1
International Journal of Manageent & Inforation Systes First Quarter 2012 Volue 16, Nuber 1 Proposal And Effectiveness Of A Highly Copelling Direct Mail Method  Establishent And Deployent Of PMOSDM Hisatoshi
More informationExpected Value. Let X be a discrete random variable which takes values in S X = {x 1, x 2,..., x n }
Expected Value Let X be a discrete random variable which takes values in S X = {x 1, x 2,..., x n } Expected Value or Mean of X: E(X) = n x i p(x i ) i=1 Example: Roll one die Let X be outcome of rolling
More informationADJUSTING FOR QUALITY CHANGE
ADJUSTING FOR QUALITY CHANGE 7 Introduction 7.1 The easureent of changes in the level of consuer prices is coplicated by the appearance and disappearance of new and old goods and services, as well as changes
More informationLecture L9  Linear Impulse and Momentum. Collisions
J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9  Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,
More information5.7 Chebyshev Multisection Matching Transformer
/9/ 5_7 Chebyshev Multisection Matching Transforers / 5.7 Chebyshev Multisection Matching Transforer Reading Assignent: pp. 555 We can also build a ultisection atching network such that Γ f is a Chebyshev
More informationHomework Zero. Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis. Complete before the first class, do not turn in
Homework Zero Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis Complete before the first class, do not turn in This homework is intended as a self test of your knowledge of the basic statistical
More informationReading about this material is essential. The book will add a lot
Advanced Volatility Add a little wind and we get a little increase in volatility. Add a hurricane and we get a huge increase in volatility. (c) 0060, Gary R. Evans. For educational uses only. May not
More informationHW 2. Q v. kt Step 1: Calculate N using one of two equivalent methods. Problem 4.2. a. To Find:
HW 2 Proble 4.2 a. To Find: Nuber of vacancies per cubic eter at a given teperature. b. Given: T 850 degrees C 1123 K Q v 1.08 ev/ato Density of Fe ( ρ ) 7.65 g/cc Fe toic weight of iron ( c. ssuptions:
More informationPure Bending Determination of StressStrain Curves for an Aluminum Alloy
Proceedings of the World Congress on Engineering 0 Vol III WCE 0, July 68, 0, London, U.K. Pure Bending Deterination of StressStrain Curves for an Aluinu Alloy D. TorresFranco, G. UrriolagoitiaSosa,
More informationResidual Cutting Method for Elliptic Boundary Value Problems: Application to Poisson's Equation
JOURNAL OF COMPUTATIONAL PHYSICS 137, 247264 (1997) ARTICLE NO. CP975807 Residual Cutting Method for Elliptic Boundary Value Probles: Application to Poisson's Equation Atsuhiro Taura, Kazuo Kikuchi, and
More informationCOMBINING CRASH RECORDER AND PAIRED COMPARISON TECHNIQUE: INJURY RISK FUNCTIONS IN FRONTAL AND REAR IMPACTS WITH SPECIAL REFERENCE TO NECK INJURIES
COMBINING CRASH RECORDER AND AIRED COMARISON TECHNIQUE: INJURY RISK FUNCTIONS IN FRONTAL AND REAR IMACTS WITH SECIAL REFERENCE TO NECK INJURIES Anders Kullgren, Maria Krafft Folksa Research, 66 Stockhol,
More information2.8 Expected values and variance
y 1 b a 0 y 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 a b (a) Probability density function x 0 a b (b) Cumulative distribution function x Figure 2.3: The probability density function and cumulative distribution function
More informationMetric of Carbon Equity: Carbon Gini Index Based on Historical Cumulative Emission per Capita
ADVANCES IN CLIMATE CHANGE RESEARCH 2(3): 134 140, 2011 www.cliatechange.cn DOI: 10.3724/SP.J.1248.2011.00134 POLICY FORUM Metric of Carbon Equity: Carbon Gini Index Based on Historical Cuulative Eission
More informationDeveloping Countries in Competition for Foreign Direct Investment
Seinar in International Econoics 5 Marc 2015 Developing Countries in Copetition for Foreign Direct Investent oran Vukšić Institute of Public Finance, Zagreb Tis seinar series is an activity in te fraework
More informationPhysics 211: Lab Oscillations. Simple Harmonic Motion.
Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.
More informationData Set Generation for Rectangular Placement Problems
Data Set Generation for Rectangular Placeent Probles Christine L. Valenzuela (Muford) Pearl Y. Wang School of Coputer Science & Inforatics Departent of Coputer Science MS 4A5 Cardiff University George
More informationarxiv: v1 [physics.edph] 11 Aug 2009
Geoetrical siplification of the dipoledipole interaction forula arxiv:0908.1548v1 [physics.edph] 11 Aug 2009 Ladislav Kocbach and Suhail Lubbad Dept. of Physics and Technology, University of Bergen,
More informationJoint Distribution and Correlation
Joint Distribution and Correlation Michael Ash Lecture 3 Reminder: Start working on the Problem Set Mean and Variance of Linear Functions of an R.V. Linear Function of an R.V. Y = a + bx What are the properties
More informationInvention of NFV Technique and Its Relationship with NPV
International Journal of Innovation and Applied Studies ISSN 20289324 Vol. 9 No. 3 Nov. 2014, pp. 11881195 2014 Innovative Space of Scientific Research Journals http://www.ijias.issrjournals.org/ Invention
More informationNote on a generalized wage rigidity result. Abstract
Note on a generalized age rigidity result Ariit Mukheree University of Nottingha Abstract Considering Cournot copetition, this note shos that, if the firs differ in labor productivities, the equilibriu
More informationAlgorithmica 2001 SpringerVerlag New York Inc.
Algorithica 2001) 30: 101 139 DOI: 101007/s0045300100030 Algorithica 2001 SpringerVerlag New York Inc Optial Search and OneWay Trading Online Algoriths R ElYaniv, 1 A Fiat, 2 R M Karp, 3 and G Turpin
More informationApplying Multiple Neural Networks on Large Scale Data
0 International Conference on Inforation and Electronics Engineering IPCSIT vol6 (0) (0) IACSIT Press, Singapore Applying Multiple Neural Networks on Large Scale Data Kritsanatt Boonkiatpong and Sukree
More informationChapter 8: Newton s Third law
Warning: My approach is a soewhat abbreviated and siplified version of what is in the text, yet just as coplete. Both y treatent and the text s will prepare you to solve the sae probles. Restating Newton
More informationA quantum secret ballot. Abstract
A quantu secret ballot Shahar Dolev and Itaar Pitowsky The Edelstein Center, Levi Building, The Hebrerw University, Givat Ra, Jerusale, Israel Boaz Tair arxiv:quantph/060087v 8 Mar 006 Departent of Philosophy
More informationSoftware Quality Characteristics Tested For Mobile Application Development
Thesis no: MGSE201502 Software Quality Characteristics Tested For Mobile Application Developent Literature Review and Epirical Survey WALEED ANWAR Faculty of Coputing Blekinge Institute of Technology
More information6. Time (or Space) Series Analysis
ATM 55 otes: Tie Series Analysis  Section 6a Page 8 6. Tie (or Space) Series Analysis In this chapter we will consider soe coon aspects of tie series analysis including autocorrelation, statistical prediction,
More informationResearch Article Performance Evaluation of Human Resource Outsourcing in Food Processing Enterprises
Advance Journal of Food Science and Technology 9(2): 964969, 205 ISSN: 20424868; eissn: 20424876 205 Maxwell Scientific Publication Corp. Subitted: August 0, 205 Accepted: Septeber 3, 205 Published:
More informationMarkov Models and Their Use for Calculations of Important Traffic Parameters of Contact Center
Markov Models and Their Use for Calculations of Iportant Traffic Paraeters of Contact Center ERIK CHROMY, JAN DIEZKA, MATEJ KAVACKY Institute of Telecounications Slovak University of Technology Bratislava
More informationAn Integrated Approach for Monitoring Service Level Parameters of SoftwareDefined Networking
International Journal of Future Generation Counication and Networking Vol. 8, No. 6 (15), pp. 1974 http://d.doi.org/1.1457/ijfgcn.15.8.6.19 An Integrated Approach for Monitoring Service Level Paraeters
More informationHOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES?
HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Abstract. We copare the option pricing forulas of Louis Bachelier and BlackMertonScholes
More informationESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET
ESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET Francisco Alonso, Roberto Blanco, Ana del Río and Alicia Sanchis Banco de España Banco de España Servicio de Estudios Docuento de
More informationChaotic Structures in Brent & WTI Crude Oil Markets: Empirical Evidence
www.ccsenet.org/ijef International Journal of Econoics and Finance Vol. 3, No. 5; October 2011 Chaotic Structures in Brent & WTI Crude Oil Markets: Epirical Evidence LingYun HE Center for Futures and
More informationExperiment 2 Index of refraction of an unknown liquid  Abbe Refractometer
Experient Index of refraction of an unknown liquid  Abbe Refractoeter Principle: The value n ay be written in the for sin ( δ +θ ) n =. θ sin This relation provides us with one or the standard ethods
More informationPosition Auctions and Nonuniform Conversion Rates
Position Auctions and Nonunifor Conversion Rates Liad Blurosen Microsoft Research Mountain View, CA 944 liadbl@icrosoft.co Jason D. Hartline Shuzhen Nong Electrical Engineering and Microsoft AdCenter
More informationPREDICTION OF POSSIBLE CONGESTIONS IN SLA CREATION PROCESS
PREDICTIO OF POSSIBLE COGESTIOS I SLA CREATIO PROCESS Srećko Krile University of Dubrovnik Departent of Electrical Engineering and Coputing Cira Carica 4, 20000 Dubrovnik, Croatia Tel +385 20 445739,
More informationInformation Processing Letters
Inforation Processing Letters 111 2011) 178 183 Contents lists available at ScienceDirect Inforation Processing Letters www.elsevier.co/locate/ipl Offline file assignents for online load balancing Paul
More informationSOME APPLICATIONS OF FORECASTING Prof. Thomas B. Fomby Department of Economics Southern Methodist University May 2008
SOME APPLCATONS OF FORECASTNG Prof. Thoas B. Foby Departent of Econoics Southern Methodist University May 8 To deonstrate the usefulness of forecasting ethods this note discusses four applications of forecasting
More informationAn Improved Decisionmaking Model of Human Resource Outsourcing Based on Internet Collaboration
International Journal of Hybrid Inforation Technology, pp. 339350 http://dx.doi.org/10.14257/hit.2016.9.4.28 An Iproved Decisionaking Model of Huan Resource Outsourcing Based on Internet Collaboration
More informationThe United States was in the midst of a
A Prier on the Mortgage Market and Mortgage Finance Daniel J. McDonald and Daniel L. Thornton This article is a prier on ortgage finance. It discusses the basics of the ortgage arket and ortgage finance.
More informationFactored Models for Probabilistic Modal Logic
Proceedings of the TwentyThird AAAI Conference on Artificial Intelligence (2008 Factored Models for Probabilistic Modal Logic Afsaneh Shirazi and Eyal Air Coputer Science Departent, University of Illinois
More informationThe Research of Measuring Approach and Energy Efficiency for Hadoop Periodic Jobs
Send Orders for Reprints to reprints@benthascience.ae 206 The Open Fuels & Energy Science Journal, 2015, 8, 206210 Open Access The Research of Measuring Approach and Energy Efficiency for Hadoop Periodic
More informationPage 1. Algebra 1 Unit 5 Working with Exponents Per
Algera 1 Unit 5 Working with Exponents Nae Per Ojective 1: Siplifying Expressions with Exponents (Powers INSIDE parentheses) Exponents are a shortcut. They are a quicker way of writing repeated ultiplication.
More informationSalty Waters. Instructions for the activity 3. Results Worksheet 5. Class Results Sheet 7. Teacher Notes 8. Sample results. 12
1 Salty Waters Alost all of the water on Earth is in the for of a solution containing dissolved salts. In this activity students are invited to easure the salinity of a saple of salt water. While carrying
More informationABSTRACT KEYWORDS. Comonotonicity, dependence, correlation, concordance, copula, multivariate. 1. INTRODUCTION
MEASURING COMONOTONICITY IN MDIMENSIONAL VECTORS BY INGE KOCH AND ANN DE SCHEPPER ABSTRACT In this contribution, a new easure of coonotonicity for diensional vectors is introduced, with values between
More informationPneumatic Nozzle Flapper Control of an. Array of Cantilevers Using a. Single Thermal MicroActuator
Pneuatic Nozzle Flapper Control of an Array of Cantilevers Using a Single Theral MicroActuator Murad AbuKhalaf Instructors: Professor Edward S. Kolesar, Texas Christian University Professor Frank L. Lewis,
More informationConstruction of Graeco Sudoku Square Designs of Odd Orders
onfring International Journal of Data Mining, Vol, No, June 0 37 Construction of Graeco Sudoku Square Designs of Odd Orders J Subraani Abstract The Sudoku puzzle typically consists of a ninebynine
More informationElectric Forces between Charged Plates
CP.1 Goals of this lab Electric Forces between Charged Plates Overview deterine the force between charged parallel plates easure the perittivity of the vacuu (ε 0 ) In this experient you will easure the
More informationPOL 571: Expectation and Functions of Random Variables
POL 571: Expectation and Functions of Random Variables Kosuke Imai Department of Politics, Princeton University March 10, 2006 1 Expectation and Independence To gain further insights about the behavior
More informationProgram Interference in MLC NAND Flash Memory: Characterization, Modeling, and Mitigation
Progra Interference in MLC NAND Flash Meory: Characterization, Modeling, and Mitigation Yu Cai, Onur Mutlu, Erich F. Haratsch 2 and Ken Mai. Data Storage Systes Center, Departent of Electrical and Coputer
More informationAnalysis of symbolic sequences using the JensenShannon divergence
PHYSICAL REVIEW E, VOLUME 65, 041905 Analysis of sybolic sequences using the JensenShannon divergence Ivo Grosse, 1,2 Pedro BernaolaGalván, 2,3 Pedro Carpena, 2,3 Raón RoánRoldán, 4 Jose Oliver, 5 and
More information( C) CLASS 10. TEMPERATURE AND ATOMS
CLASS 10. EMPERAURE AND AOMS 10.1. INRODUCION Boyle s understanding of the pressurevolue relationship for gases occurred in the late 1600 s. he relationships between volue and teperature, and between
More informationModeling Parallel Applications Performance on Heterogeneous Systems
Modeling Parallel Applications Perforance on Heterogeneous Systes Jaeela AlJaroodi, Nader Mohaed, Hong Jiang and David Swanson Departent of Coputer Science and Engineering University of Nebraska Lincoln
More informationThe AGA Evaluating Model of Customer Loyalty Based on Ecommerce Environment
6 JOURNAL OF SOFTWARE, VOL. 4, NO. 3, MAY 009 The AGA Evaluating Model of Custoer Loyalty Based on Ecoerce Environent Shaoei Yang Econoics and Manageent Departent, North China Electric Power University,
More informationEfficient Key Management for Secure Group Communications with Bursty Behavior
Efficient Key Manageent for Secure Group Counications with Bursty Behavior Xukai Zou, Byrav Raaurthy Departent of Coputer Science and Engineering University of NebraskaLincoln Lincoln, NE68588, USA Eail:
More informationDefinition The covariance of X and Y, denoted by cov(x, Y ) is defined by. cov(x, Y ) = E(X µ 1 )(Y µ 2 ).
Correlation Regression Bivariate Normal Suppose that X and Y are r.v. s with joint density f(x y) and suppose that the means of X and Y are respectively µ 1 µ 2 and the variances are 1 2. Definition The
More informationEndogenous Market Structure and the Cooperative Firm
Endogenous Market Structure and the Cooperative Fir Brent Hueth and GianCarlo Moschini Working Paper 14WP 547 May 2014 Center for Agricultural and Rural Developent Iowa State University Aes, Iowa 500111070
More informationLecture L263D Rigid Body Dynamics: The Inertia Tensor
J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L63D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall
More informationASIC Design Project Management Supported by Multi Agent Simulation
ASIC Design Project Manageent Supported by Multi Agent Siulation Jana Blaschke, Christian Sebeke, Wolfgang Rosenstiel Abstract The coplexity of Application Specific Integrated Circuits (ASICs) is continuously
More information 265  Part C. Property and Casualty Insurance Companies
Part C. Property and Casualty Insurance Copanies This Part discusses proposals to curtail favorable tax rules for property and casualty ("P&C") insurance copanies. The syste of reserves for unpaid losses
More informationCalculation Method for evaluating Solar Assisted Heat Pump Systems in SAP 2009. 15 July 2013
Calculation Method for evaluating Solar Assisted Heat Pup Systes in SAP 2009 15 July 2013 Page 1 of 17 1 Introduction This docuent describes how Solar Assisted Heat Pup Systes are recognised in the National
More informationInsurance Spirals and the Lloyd s Market
Insurance Spirals and the Lloyd s Market Andrew Bain University of Glasgow Abstract This paper presents a odel of reinsurance arket spirals, and applies it to the situation that existed in the Lloyd s
More information