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42-447: The Z-score is a linear combination of four or five common business ratios, weighted by coefficients. The coefficients were estimated by identifying a set of firms which had declared bankruptcy and then collecting a matched sample of firms which had survived, with matching by industry and approximate size (assets). Altman applied the statistical method of discriminant analysis to a dataset of publicly held manufacturers. The estimation

84-469: A 1 ,..., a n are scalars, then the linear combination of those vectors with those scalars as coefficients is There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, as in the assertion "the set of all linear combinations of v 1 ,..., v n always forms a subspace". However, one could also say "two different linear combinations can have

126-470: A and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field , with some generalizations given at the end of the article. Let V be a vector space over the field K . As usual, we call elements of V vectors and call elements of K scalars . If v 1 ,..., v n are vectors and

168-758: A 1990 study he also defined the term "distressed debt securities," pertaining to firms whose bonds yield more than 10% above the "risk-free" government bond rate. Altman holds a B.A. in Economics, ( CCNY , 1963); an MBA ( UCLA , 1965); and a Ph.D. in Finance (UCLA, 1967). Altman was inducted into the Fixed Income Society's Hall of Fame in 2001 and was an inaugural inductee into the Turnaround Management's Hall of Fame in 2008. Altman has been awarded Honorary Doctorates from Lund University (Sweden) in 2011 and

210-548: A linear combination of p 1 , p 2 , and p 3 ? To find out, consider an arbitrary linear combination of these vectors and try to see when it equals the desired vector x  − 1. Picking arbitrary coefficients a 1 , a 2 , and a 3 , we want Multiplying the polynomials out, this means and collecting like powers of x , we get Two polynomials are equal if and only if their corresponding coefficients are equal, so we can conclude This system of linear equations can easily be solved. First,

252-461: A time. Altman's primary improvement was to apply a statistical method, discriminant analysis, which could take into account multiple variables simultaneously. In its initial test, the Altman Z-score was found to be 72% accurate in predicting bankruptcy two years before the event, with a Type II error (false negatives) of 6% (Altman, 1968). In a series of subsequent tests covering three periods over

294-534: A vector subspace is also an affine subspace, a convex cone, and a convex set, but a convex set need not be a vector subspace, affine, or a convex cone. These concepts often arise when one can take certain linear combinations of objects, but not any: for example, probability distributions are closed under convex combination (they form a convex set), but not conical or affine combinations (or linear), and positive measures are closed under conical combination but not affine or linear – hence one defines signed measures as

336-470: A way to make sense of certain infinite linear combinations, using the topology of V . For example, we might be able to speak of a 1 v 1  + a 2 v 2  + a 3 v 3  + ⋯, going on forever. Such infinite linear combinations do not always make sense; we call them convergent when they do. Allowing more linear combinations in this case can also lead to a different concept of span, linear independence, and basis. The articles on

378-418: Is a subset of V , we may speak of a linear combination of vectors in S , where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set S (and the coefficients must belong to K ). Finally, we may speak simply of a linear combination , where nothing is specified (except that the vectors must belong to V and the coefficients must belong to K ); in this case one

420-590: Is a Professor of Finance, Emeritus, at New York University 's Stern School of Business . He is best known for the development of the Altman Z-score for predicting bankruptcy which he published in 1968. Professor Altman is a leading academic on the High-Yield and Distressed Debt markets and is the pioneer in the building of models for credit risk management and bankruptcy prediction . Altman used to teach "Bankruptcy and Reorganization" and "Credit Risk Management" in

462-412: Is a customized version of the discriminant analysis technique of R. A. Fisher (1936). William Beaver's work, published in 1966 and 1968, was the first to apply a statistical method, t -tests to predict bankruptcy for a pair-matched sample of firms. Beaver applied this method to evaluate the importance of each of several accounting ratios based on univariate analysis, using each accounting ratio one at

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504-481: Is appropriate for the given module. This is simply a matter of doing scalar multiplication on the correct side. A more complicated twist comes when V is a bimodule over two rings, K L and K R . In that case, the most general linear combination looks like where a 1 ,..., a n belong to K L , b 1 ,..., b n belong to K R , and v 1 ,…, v n belong to V . Edward I. Altman Edward I. Altman (born June 5, 1941)

546-452: Is equivalent, by subtracting these ( c i := a i − b i {\displaystyle c_{i}:=a_{i}-b_{i}} ), to saying a non-trivial combination is zero: If that is possible, then v 1 ,..., v n are called linearly dependent ; otherwise, they are linearly independent . Similarly, we can speak of linear dependence or independence of an arbitrary set S of vectors. If S

588-403: Is interesting to consider the set of all linear combinations of these vectors. This set is called the linear span (or just span ) of the vectors, say S = { v 1 , ..., v n }. We write the span of S as span( S ) or sp( S ): Suppose that, for some sets of vectors v 1 ,..., v n , a single vector can be written in two different ways as a linear combination of them: This

630-507: Is linearly independent and the span of S equals V , then S is a basis for V . By restricting the coefficients used in linear combinations, one can define the related concepts of affine combination , conical combination , and convex combination , and the associated notions of sets closed under these operations. Because these are more restricted operations, more subsets will be closed under them, so affine subsets, convex cones, and convex sets are generalizations of vector subspaces:

672-517: Is no reason that n cannot be zero ; in that case, we declare by convention that the result of the linear combination is the zero vector in V . Let the field K be the set R of real numbers , and let the vector space V be the Euclidean space R . Consider the vectors e 1 = (1,0,0) , e 2 = (0,1,0) and e 3 = (0,0,1) . Then any vector in R is a linear combination of e 1 , e 2 , and  e 3 . To see that this

714-453: Is probably referring to the expression, since every vector in V is certainly the value of some linear combination. Note that by definition, a linear combination involves only finitely many vectors (except as described in the § Generalizations section. However, the set S that the vectors are taken from (if one is mentioned) can still be infinite ; each individual linear combination will only involve finitely many vectors. Also, there

756-432: Is so, take an arbitrary vector ( a 1 , a 2 , a 3 ) in R , and write: Let K be the set C of all complex numbers , and let V be the set C C ( R ) of all continuous functions from the real line R to the complex plane C . Consider the vectors (functions) f and g defined by f ( t ) := e and g ( t ) := e . (Here, e is the base of the natural logarithm , about 2.71828..., and i

798-426: Is the imaginary unit , a square root of −1.) Some linear combinations of f and g  are: On the other hand, the constant function 3 is not a linear combination of f and g . To see this, suppose that 3 could be written as a linear combination of e and e . This means that there would exist complex scalars a and b such that ae + be = 3 for all real numbers t . Setting t = 0 and t = π gives

840-503: Is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories. From this point of view, we can think of linear combinations as the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that all possible algebraic operations in a vector space are linear combinations. The basic operations of addition and scalar multiplication, together with

882-839: The Altman models nor other balance sheet-based models are recommended for use with financial companies. This is because of the opacity of financial companies' balance sheets and their frequent use of off-balance sheet items. Modern academic default and bankruptcy prediction models rely heavily on market-based data rather than the accounting ratios predominant in the Altman Z-score. Z-score bankruptcy model: Zones of discrimination: Z-score bankruptcy model (non-manufacturers): Z-score bankruptcy model (emerging markets): Zones of discrimination: Altman, Edward I. (July 2000). "Predicting Financial Distress of Companies" (PDF) . Stern.nyu.edu : 15–22. Altman, Edward I. (September 1968). "Financial Ratios, Discriminant Analysis and

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924-464: The Importance of a Credit Culture by Dr. Edward I Altman, Stern School of Business , New York University. Linear combination In mathematics , a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by , where

966-912: The Prediction of Corporate Bankruptcy". Journal of Finance . 23 (4): 189–209. doi : 10.1111/j.1540-6261.1968.tb00843.x . S2CID   154437292 . Altman, Edward I. (May 2002). "Revisiting Credit Scoring Models in a Basel II Environment" (PDF) . Prepared for "Credit Rating: Methodologies, Rationale, and Default Risk", London Risk Books 2002 . Archived from the original (PDF) on 2006-09-18 . Retrieved 2007-08-08 . Eidleman, Gregory J. (1995-02-01). "Z-Scores – A Guide to Failure Prediction" . The CPA Journal Online . Fisher, Ronald Aylmer (1936). "The Use of Multiple Measurements in Taxonomic Problems" . Annals of Eugenics . 7 (2): 179. doi : 10.1111/j.1469-1809.1936.tb02137.x . hdl : 2440/15227 . The Use of Credit Scoring Modules and

1008-488: The Risk Management Open Enrollment program for Stern Executive Education. He also teaches in the school's MBA programs and has been a Stern faculty member since 1967. The Altman Z-score is a multivariate formula for a measurement of the financial health of a company and a powerful diagnostic tool that forecasts the probability of a company entering bankruptcy . Studies measuring the effectiveness of

1050-706: The Warsaw School of Economics in 2014. He also was named Honorary Professor by the University of Buenos Aires (Argentina) in 1996 and Vigo University (Spain) in 2017. He was named one of the "100 Most Influential People in Finance" by the Treasury & Risk Management magazine in 2005. He is also a co-founder of the International Risk Management Conference, which celebrated its 10th anniversary in 2017. He

1092-524: The Z-Score have shown that the model has an 80%–90% reliability. Altman's equation did an excellent job at distinguishing bankrupt and non-bankrupt firms and is used by a large number of investment managers and hedge funds in their investment strategies and management. In addition, his Z-score model has been used by management of distressed companies to avoid having to file for bankruptcy reorganization. Altman's models have been used by banking institutions throughout

1134-448: The coefficient of each v i ; trivial modifications such as permuting the terms or adding terms with zero coefficient do not produce distinct linear combinations. In a given situation, K and V may be specified explicitly, or they may be obvious from context. In that case, we often speak of a linear combination of the vectors v 1 ,..., v n , with the coefficients unspecified (except that they must belong to K ). Or, if S

1176-682: The crucial problem is to make an inference in the reverse direction, i.e., from ratios to failures.” From about 1985 onwards, the Z-scores gained wide acceptance by auditors, management accountants, courts, and database systems used for loan evaluation (Eidleman). The formula's approach has been used in a variety of contexts and countries, although it was designed originally for publicly held manufacturing companies with assets of more than $ 1 million. Later variations by Altman were designed to be applicable to privately held companies (the Altman Z'-score) and non-manufacturing companies (the Altman Z"-score). Neither

1218-403: The equations a + b = 3 and a + b = −3 , and clearly this cannot happen. See Euler's identity . Let K be R , C , or any field, and let V be the set P of all polynomials with coefficients taken from the field K . Consider the vectors (polynomials) p 1  := 1, p 2  := x + 1 , and p 3  := x + x + 1 . Is the polynomial x  − 1

1260-405: The existence of an additive identity and additive inverses, cannot be combined in any more complicated way than the generic linear combination: the basic operations are a generating set for the operad of all linear combinations. Ultimately, this fact lies at the heart of the usefulness of linear combinations in the study of vector spaces. If V is a topological vector space , then there may be

1302-417: The first equation simply says that a 3 is 1. Knowing that, we can solve the second equation for a 2 , which comes out to −1. Finally, the last equation tells us that a 1 is also −1. Therefore, the only possible way to get a linear combination is with these coefficients. Indeed, so x  − 1 is a linear combination of p 1 , p 2 , and  p 3 . On the other hand, what about

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1344-620: The linear closure. Linear and affine combinations can be defined over any field (or ring), but conical and convex combination require a notion of "positive", and hence can only be defined over an ordered field (or ordered ring ), generally the real numbers. If one allows only scalar multiplication, not addition, one obtains a (not necessarily convex) cone ; one often restricts the definition to only allowing multiplication by positive scalars. All of these concepts are usually defined as subsets of an ambient vector space (except for affine spaces, which are also considered as "vector spaces forgetting

1386-584: The next 31 years (up until 1999), the model was found to be approximately 80–90% accurate in predicting bankruptcy one year before the event, with a Type II error (classifying the firm as bankrupt when it does not go bankrupt) of approximately 15–20% (Altman, 2000). This overstates the predictive ability of the Altman Z-score, however. Scholars have long criticized the Altman Z-score for being “largely descriptive statements devoid of predictive content ... Altman demonstrates that failed and non-failed firms have dissimilar ratios, not that ratios have predictive power. But

1428-425: The origin"), rather than being axiomatized independently. More abstractly, in the language of operad theory , one can consider vector spaces to be algebras over the operad R ∞ {\displaystyle \mathbf {R} ^{\infty }} (the infinite direct sum , so only finitely many terms are non-zero; this corresponds to only taking finite sums), which parametrizes linear combinations:

1470-568: The polynomial x  − 1? If we try to make this vector a linear combination of p 1 , p 2 , and p 3 , then following the same process as before, we get the equation However, when we set corresponding coefficients equal in this case, the equation for x  is which is always false. Therefore, there is no way for this to work, and x  − 1 is not a linear combination of p 1 , p 2 , and  p 3 . Take an arbitrary field K , an arbitrary vector space V , and let v 1 ,..., v n be vectors (in V ). It

1512-401: The same value" in which case the reference is to the expression. The subtle difference between these uses is the essence of the notion of linear dependence : a family F of vectors is linearly independent precisely if any linear combination of the vectors in F (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is

1554-426: The sub-operads where the terms sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by R n {\displaystyle \mathbf {R} ^{n}} being or the standard simplex being model spaces, and such observations as that every bounded convex polytope

1596-570: The various flavors of topological vector spaces go into more detail about these. If K is a commutative ring instead of a field, then everything that has been said above about linear combinations generalizes to this case without change. The only difference is that we call spaces like this V modules instead of vector spaces. If K is a noncommutative ring, then the concept still generalizes, with one caveat: since modules over noncommutative rings come in left and right versions, our linear combinations may also come in either of these versions, whatever

1638-575: The vector ( 2 , 3 , − 5 , 0 , … ) {\displaystyle (2,3,-5,0,\dots )} for instance corresponds to the linear combination 2 v 1 + 3 v 2 − 5 v 3 + 0 v 4 + ⋯ {\displaystyle 2\mathbf {v} _{1}+3\mathbf {v} _{2}-5\mathbf {v} _{3}+0\mathbf {v} _{4}+\cdots } . Similarly, one can consider affine combinations, conical combinations, and convex combinations to correspond to

1680-459: The world in order to quantify the probability of default of their lending portfolios. He compared owning the debt vs. equity for major companies that went bankrupt. The debt did better. One columnist wrote that a particular case was a goldmine for the debt and a landmine for the stock. In the 1990s Altman and his PhD student Professor Edith Hotchkiss coined the term Chapter 22 , which refers to companies which file for bankruptcy more than once. In

1722-408: Was as follows: Altman found that the ratio profile for the bankrupt group fell at −0.25 avg, and for the non-bankrupt group at +4.48 avg. Altman's work built upon research by accounting researcher William Beaver and others. In the 1930s and on, Mervyn and others had collected matched samples and assessed that various accounting ratios appeared to be valuable in predicting bankruptcy. Altman Z-score

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1764-445: Was originally based on data from publicly held manufacturers, but has since been re-estimated based on other datasets for private manufacturing, non-manufacturing and service companies. The original data sample consisted of 66 firms, half of which had filed for bankruptcy under Chapter 7 . All businesses in the database were manufacturers, and small firms with assets of < $ 1 million were eliminated. The original Z-score formula

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