Implied (risk neutral) probabilities, betting odds and prediction markets

Save this PDF as:

Size: px
Start display at page:

Download "Implied (risk neutral) probabilities, betting odds and prediction markets"

Transcription

1 Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT - We show that the well known euvalence between the "fundamental theorem of asset prcng" and the "dutch book argument" conssts, n realty, of two parts: the frst concernng the exstence of a rsk free nvestment, and the second the possblty of arbtrages. We argument that the "mpled probabltes" of an odds system may be used to rate the bookmaker's farness, and brefly dscuss the sgnfcance of the same probabltes as defned by the prces n the predcton markets. KEYWORDS - Dutch books; arbtrage; rsk neutral probabltes; mpled probabltes; bettng odds; predcton markets. Frst draft: May 20, 2006 Ths draft: September, Reference s to the well known analogy between the "dutch book" argument, accordng to whch a system of bettng odds allows no sure wns f and only f the "mpled probabltes" are formally coherent, and the fundamental theorem of asset prcng (FTAP). Ths last states that n a market where one euro may be alternatvely nvested to gve the random amount S or the certan amount r, no arbtrage s possble f and only f a (rsk neutral: "r.n.") probablty dstrbuton exsts for whch () E[S] = r and for whch the non zero probablty events are the ones consdered as really possble. Moreover, there s only one such dstrbuton f and only f the market s "complete". 2. We consder the typcal (although not the most general) case, n whch an odds system = ( ) correspondng to some "partton of realty" {E } s gven: a famly of n mutually exclusve events s defned, one of whch must necessarly occur; and someone offers to pay (>) euro f event E occurs to anyone that has bet one euro on ths fact. Our man concern wll be the (eventual) relaton between and Prob(E ), but let us start wth askng ourselves whch logc (f any does) underles the choce of the odds. Now, suppose that (for =,..., n) the bettors have altogether placed c euro on the event E. If the bookmaker wants to keep for hmself the fracton k of the total ncome, hs eulbrum wll be safe f the odds satsfy the eualtes: (2) k c = c =... = n c n. So, f the bookmaker has an opnon about what the ratos c /c wll be, he can use (2) to determne the odds he can offer. He wll namely put:

2 (3) h = k ch Note, that the odds may be progressvely adusted, accordng the way the bettors actually share ther bets among the events. From (3) (or (2)), t obvously follows that, for the eulbrum to hold, the rato between the odds must be eual to the nverse of that between the bets: c. (4) c = c (for every, ). 3. It s worthwhle notng that the parameter k has a value every one can calculate (as the nverse of the sum of the nverse of the odds), and may be used to rate the bookmaker's farness. Here are ust two examples. A. The odds the offcal talan bettng agency ("Sna") offered (on may 25, 2006) for the wnners of the three of the prelmnary round robns of the football world champonshp were: r.r. A r.r. B r.r. E Germany,25 England,50 Italy,80 Costarca 25 Paraguay 8,50 Ghana Poland 4,50 Trn. & Tobago 50 Usa 7,50 Ecuador 2 Sweden 3 Chech rep. 2,80 0,8729 0,8790 0,8796 In the last row, the correspondng value of k appears. B. For the sake of a comparson, here are the odds some australan bookmakers offered (on the same may 25) about the wnner of next year's poltcal electons: Coalton Labor k Centrebet,60 2,20 0,9263 IASbet,75 2,05 0,944 Sportngbet,70 2,05 0,9248 Sportsbet,67 2,0 0, We leave now the ueston of the bookmakerss' farness asde, and note that the bets portfolo made up by bettng, for every, (5) a = euro on the event E, has cost, and the wnnng that t yelds s n any case h h 2

3 (6) h h = ( ) = k. So, the same uantty k we saw before may also be seen as a rsk free accumulaton factor (albet, usually smaller than ). 5. We fnally approach to our man goal: nvestgatng the relaton (f any exsts) between odds and probabltes. The dea of nterpretng bet's odds n terms of subectve probabltes goes back to Ramsey and de Fnett: one should be wllng to sell or buy at the prce p the rght to receve one euro f an event occurs f p (or: because p) s, n hs opnon, that event's probablty. The dffcultes arsng from the fact that ths obvously mples the operators to be rsk neutral are usually gnored, or overcome by assumng the stakes small enough for ths to be approxmately true. In our framework, t s therefore natural to call the uanttes (7) p = / (=, 2,, n) mpled probabltes (we mean, of course: mpled by the odds system ). As >, every p s n (0, ), but nothng nsures that P = p s 2. Now, t s straghtforward to see that P = p = k; so we can use our k to normalze the mpled probabltes. The products (8) p * = kp sum up to, and are therefore, formally, probabltes. Puttng (7) and (8) together one gets: (9) p * = k ( =, 2,, n) whch expresses the result that k s the *-expected value of the wnnngs for a bettor that has placed euro on E (for every ). The classcal dutch book argument may now be stated by wrtng: (0) k = p * = p P =. Some comments are necessary. If /P = k > (the odds are "hgh") we are n presence of a "real" dutch book: whoever bets accordng the proportons (5) wll receve more than the amount pad. If, conversely, /P = k < (the odds are "low") thngs are much more uestonable, and one can thnk about a dutch book "su geners". By sellng the portfolo (a ), an operator receves more than what he wll have to pay. The problem s, that to adopt ths strategy, he must be allowed to accept bets n the exact uanttes he wshes: and ths s not possble for a Statng, as we have done, that the probabltes mpled n an odds system are ust the "rsk neutral" ones the FTAP deals about, s maybe an up-to-date way to express the same concept. 2 As we are consderng bets on mutually exclusve events, ths s the only condton of coherence we must care about. 3

4 professonal bookmaker (note that, when P s dfferent from, acceptng a bet on some event cannot be replcated by bettng on all the others). Ths possblty must therefore be specfcally postulated, ust as the one of sellng short the rsk free asset n a fnancal market Let us go back to the (9) and note that, as k may be seen as an accumulaton factor (the one assocated wth the odds system ) each of those n eualtes s perfectly analogous to the eualty (), whch expresses the FTAP. It looks therefore possble to generalze the "dutch book argument" n the followng terms. Just as n the framework of the FTAP, suppose there are at hand two possbltes to nvest money: bettng accordng the odds, or nvestng at the rskless rate ρ (ths could be, eventually assocated wth a dfferent system of bet odds; f ρ =, we get agan the classcal dutch book argument). Fndng the "rsk neutral" probabltes, that s some probabltes accordng to whch the two alternatves are - n expected value - ndfferent, means lookng for a soluton of the system of n+ euatons: () n p = p * = * = ρ ( =, 2,..., n) where ( ) and ρ are gven and (p *) are the unknown. A necessary and suffcent condton for t to admt a sngle soluton s gven by: (2) ρ = h h. (2) s also a condton of "no arbtrage": ths s obvous, snce t expresses the eualty between two rskless factors (n the rght, the one assocated wth appears). The analoges wth the FTAP may be completed as follows. If nstead of () we have to deal wth a system as: (3) n p = p * * = = ρ ( =, 2,..., m < n) then there s - generally - more than one soluton (t s the case of a non complete fnancal market): n m among the events E are consdered as actually possble, but no bet s accepted on them. No odd s gven for them, and the correspondng mpled probabltes cannot be defned. If, on the other hand, the system looks lke: 3 Note also, that n order to be able to speak of proper "arbtrages", one must (n both stuatons) add the hypothess that the tme doesn't flow: or, euvalently, that there s a rsk free rate of 0. Then, f k > one borrows the money to bet; f k <, one sells the bets, and ust keeps the money. 4

5 (4) m p = p * * = = ρ ( =, 2,..., m n) then the resultng mpled probabltes are not euvalent to the real ones. Even f (2) hold, arbtrages are possble, as the followng examples show. - Suppose m = 2, n = 3, = 2 = 2, ρ =, so that p = p 2 = /2. The bets portfolo ( on E, on E 2 ) costs 2 and yelds 2 f E or E 2 occurs, but nothng f E 3 does. By sellng t and holdng the money one realzes an arbtrage. - Suppose m = 3, n = 2, = 2 = 3 = 3, ρ =, so that p = p 2 = p 3 = /3. The bets portfolo ( on E, on E 2 ) costs 2 and yelds 3 f E or E 2 occurs: that s, n any really possble case. but nothng f E 3 does. By borrowng 2 and buyng that portfolo, one realzes an arbtrage. A last observaton concernng system (): f we don't consder the possblty of the alternatve nvestment, and wrte t ust as n p * = = p * = k ( =, 2,..., n) wth k gven by (6), then - trvally - the (unue) soluton always exsts, and s gven by the normalzed mpled probabltes. Our concluson s that the classcal formulaton of the "dutch book argument" may be slghtly mproved, by statng that the sad normalzed mpled probabltes always exst, and have exactly the meanng of the "rsk neutral" probabltes the FTAP deals wth: they are the probabltes, wth respect to whch bettng on any of the E 's gves the same mean result as nvestng at the rate k. 7. Intepretng, as we have done, the mpled probabltes as "rsk neutral" ones s probably the most modern way to express the hypothess of rsk ndfference we are forced to formulate, f we want to read odds as probabltes. The ueston remans open, of whch the nformatve, "real" content of such probabltes s; and t s an up-to-date ueston, as the recent brth of the "predcton markets" shows. We refer to the most common type of predcton markets: that n whch a contract payng f a gven event E occurs may be bought or sold at prce p. The stuaton s the same of that of a bets market, n whch a bookmaker operates free of charge (k = ) and sells bets on E at the odd /p. p s therefore the "mpled probablty" of E, and s - as one sees - gven by the market (predcton markets have been realzed to make the best of the dea that "odds are probabltes": after all, observng the real behavour of a bettor s much more drect and less based than ust askng hm at what prce he would be ndfferent between bettng on an event or on ts complementary). The consensus looks to be that the prce on such a market "represents the market's expectaton of the probablty that an event wll occur" (Wolfers-Ztzewtz, 2004). These authors are aware of the fact that rsk neutralty must be assumed, but argue that "the sums wagered n predcton markets are typcally small enough that assumng that nvestors are not averse to the dosyncratc rsk nvolved seems reasonable". Two obectons present themselves. 5

6 The frst. To make well ther duty, the predcton markets should attract people endowed wth better than average knowledge, and "force" them to reveal what they know. But ths only works f bettng s consdered as an nterestng nvestment opportunty. It s ndeed dffcult to accept the dea of an expert usng hs sklls for ust a symbolc remuneraton. The second. Ths s n our opnon more mportant, and concerns the very logc of the bettors' behavor. We llustrate t wth a smple example. Let 0,667 be the prce of a contract that pays f E occurs: to buy t means bettng 2/3 on E; sellng t means bettng /3 on none at the odd. Thngs go exactly as n a bets market where the bookes act free of charge, and apply the odds E =,5, none = 3. Can we really thnk that ths means that, n the market's opnon, that Prob(E) = 0,667 and Prob(nonE) = 0,333? Suppose that all the operators share the opnon that Prob(E) = 2Prob(nonE); on the other hand, none = 2 E. In such a stuaton, every operator who s wllng to bet on E s rsk neutral, and s therefore ndfferent between rskng hs money on ether of the events. It look necessary to deduce that as many euro wll be betted on E as on none. Ths contradcts the fact that, for the market's eulbrum, for every euro betted on none one must be on E. We are left wth the uncomfortable concluson that the (so called) market opnon cannot be the opnon of all the partcpants n the market. Bblography N.H. BINGHAM - R. KIESEL, Rsk Neutral Valuaton, Sprnger (998) M.L. EATON - D.A. FREEDMAN, Dutch Book aganst some 'Obectve' Pror, Bernoull (0) 2004 D.P. ELLERMAN, Arbtrage Theory: a Mathematcal Introducton; SIAM Revew (26) J. WOLFERS - E. ZITZEWITZ, Predcton Markets, J. of Economc Perspectves (8)

Recurrence. 1 Definitions and main statements

Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

Section 5.4 Annuities, Present Value, and Amortization

Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

Using Series to Analyze Financial Situations: Present Value

2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

1. Math 210 Finite Mathematics

1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

An Alternative Way to Measure Private Equity Performance

An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143

1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

Simple Interest Loans (Section 5.1) :

Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

The Cox-Ross-Rubinstein Option Pricing Model

Fnance 400 A. Penat - G. Pennacc Te Cox-Ross-Rubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te no-arbtrage

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

An Overview of Financial Mathematics

An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money

Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important

How Much to Bet on Video Poker

How Much to Bet on Vdeo Poker Trstan Barnett A queston that arses whenever a gae s favorable to the player s how uch to wager on each event? Whle conservatve play (or nu bet nzes large fluctuatons, t lacks

Finite Math Chapter 10: Study Guide and Solution to Problems

Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

+ + + - - This circuit than can be reduced to a planar circuit

MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia

To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate

Formula of Total Probability, Bayes Rule, and Applications

1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.

FINANCIAL MATHEMATICS

3 LESSON FINANCIAL MATHEMATICS Annutes What s an annuty? The term annuty s used n fnancal mathematcs to refer to any termnatng sequence of regular fxed payments over a specfed perod of tme. Loans are usually

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

Beating the Odds: Arbitrage and Wining Strategies in the Football Betting Market

Beatng the Odds: Arbtrage and Wnng Strateges n the Football Bettng Market NIKOLAOS VLASTAKIS, GEORGE DOTSIS and RAPHAEL N. MARKELLOS* ABSTRACT We examne the potental for generatng postve returns from wagerng

Estimating the Effect of the Red Card in Soccer

Estmatng the Effect of the Red Card n Soccer When to Commt an Offense n Exchange for Preventng a Goal Opportunty Jan Vecer, Frantsek Koprva, Tomoyuk Ichba, Columba Unversty, Department of Statstcs, New

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio

Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of

OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004

OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

7.5. Present Value of an Annuity. Investigate

7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

PERRON FROBENIUS THEOREM

PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()

A Probabilistic Theory of Coherence

A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want

The Short-term and Long-term Market

A Presentaton on Market Effcences to Northfeld Informaton Servces Annual Conference he Short-term and Long-term Market Effcences en Post Offce Square Boston, MA 0209 www.acadan-asset.com Charles H. Wang,

Interest Rate Fundamentals

Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts

Financial Mathemetics

Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

Cautiousness and Measuring An Investor s Tendency to Buy Options

Cautousness and Measurng An Investor s Tendency to Buy Optons James Huang October 18, 2005 Abstract As s well known, Arrow-Pratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets

Communication Networks II Contents

8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

BERNSTEIN POLYNOMIALS

On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.

Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When

Chapter 11 Practice Problems Answers

Chapter 11 Practce Problems Answers 1. Would you be more wllng to lend to a frend f she put all of her lfe savngs nto her busness than you would f she had not done so? Why? Ths problem s ntended to make

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

The Stock Market Game and the Kelly-Nash Equilibrium

The Stock Market Game and the Kelly-Nash Equlbrum Carlos Alós-Ferrer, Ana B. Ana Department of Economcs, Unversty of Venna. Hohenstaufengasse 9, A-1010 Venna, Austra. July 2003 Abstract We formulate the

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals

FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant

Efficient Project Portfolio as a tool for Enterprise Risk Management

Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

Section 5.3 Annuities, Future Value, and Sinking Funds

Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

General Auction Mechanism for Search Advertising

General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an

Section 2.3 Present Value of an Annuity; Amortization

Secton 2.3 Present Value of an Annuty; Amortzaton Prncpal Intal Value PV s the present value or present sum of the payments. PMT s the perodc payments. Gven r = 6% semannually, n order to wthdraw \$1,000.00

Stock Profit Patterns

Stock Proft Patterns Suppose a share of Farsta Shppng stock n January 004 s prce n the market to 56. Assume that a September call opton at exercse prce 50 costs 8. A September put opton at exercse prce

What is Candidate Sampling

What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

Multiple discount and forward curves

Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of

Extending Probabilistic Dynamic Epistemic Logic

Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )

February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression

Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,

On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

Return decomposing of absolute-performance multi-asset class portfolios. Working Paper - Nummer: 16

Return decomposng of absolute-performance mult-asset class portfolos Workng Paper - Nummer: 16 2007 by Dr. Stefan J. Illmer und Wolfgang Marty; n: Fnancal Markets and Portfolo Management; March 2007; Volume

5. Simultaneous eigenstates: Consider two operators that commute: Â η = a η (13.29)

5. Smultaneous egenstates: Consder two operators that commute: [ Â, ˆB ] = 0 (13.28) Let Â satsfy the followng egenvalue equaton: Multplyng both sdes by ˆB Â η = a η (13.29) ˆB [ Â η ] = ˆB [a η ] = a

Probability and Optimization Models for Racing

1 Probablty and Optmzaton Models for Racng Vctor S. Y. Lo Unversty of Brtsh Columba Fdelty Investments Dsclamer: Ths presentaton does not reflect the opnons of Fdelty Investments. The work here was completed

Performance attribution for multi-layered investment decisions

Performance attrbuton for mult-layered nvestment decsons 880 Thrd Avenue 7th Floor Ne Yor, NY 10022 212.866.9200 t 212.866.9201 f qsnvestors.com Inna Oounova Head of Strategc Asset Allocaton Portfolo Management

Course outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy

Fnancal Tme Seres Analyss Patrck McSharry patrck@mcsharry.net www.mcsharry.net Trnty Term 2014 Mathematcal Insttute Unversty of Oxford Course outlne 1. Data analyss, probablty, correlatons, vsualsaton

The Application of Fractional Brownian Motion in Option Pricing

Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

Forecasting the Direction and Strength of Stock Market Movement

Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

University of Kent Department of Economics Discussion Papers

Unversty of Kent Department of Economcs Dscusson Papers EXPERT ANALYSIS AND INSIDER INFORMATION IN HORSERACE BETTING: REGULATING INFORMED MARKET BEHAVIOR John Person and Mchael A. Smth November 2008 KDPE

2.4 Bivariate distributions

page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

Optimality in an Adverse Selection Insurance Economy. with Private Trading. April 2015

Optmalty n an Adverse Selecton Insurance Economy wth Prvate Tradng Aprl 2015 Pamela Labade 1 Abstract An externalty s created n an adverse selecton nsurance economy because of the nteracton between prvate

MATHEMATICAL ENGINEERING TECHNICAL REPORTS. Sequential Optimizing Investing Strategy with Neural Networks

MATHEMATICAL ENGINEERING TECHNICAL REPORTS Sequental Optmzng Investng Strategy wth Neural Networks Ryo ADACHI and Akmch TAKEMURA METR 2010 03 February 2010 DEPARTMENT OF MATHEMATICAL INFORMATICS GRADUATE

SP Betting as a Self-Enforcing Implicit Cartel

SP Bettng as a Self-Enforcng Implct Cartel by Ad Schnytzer and Avcha Snr Department of Economcs Bar-Ilan Unversty Ramat Gan Israel 52800 e-mal: schnyta@mal.bu.ac.l snrav@mal.bu.ac.l Abstract A large share

Price Impact Asymmetry of Block Trades: An Institutional Trading Explanation

Prce Impact Asymmetry of Block Trades: An Insttutonal Tradng Explanaton Gdeon Saar 1 Frst Draft: Aprl 1997 Current verson: October 1999 1 Stern School of Busness, New York Unversty, 44 West Fourth Street,

The impact of hard discount control mechanism on the discount volatility of UK closed-end funds

Investment Management and Fnancal Innovatons, Volume 10, Issue 3, 2013 Ahmed F. Salhn (Egypt) The mpact of hard dscount control mechansm on the dscount volatlty of UK closed-end funds Abstract The mpact

Activity Scheduling for Cost-Time Investment Optimization in Project Management

PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta- San Sebastán, September 8 th -10 th 010 Actvty Schedulng

Kiel Institute for World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1120

Kel Insttute for World Economcs Duesternbrooker Weg 45 Kel (Germany) Kel Workng Paper No. Path Dependences n enture Captal Markets by Andrea Schertler July The responsblty for the contents of the workng

Gender differences in revealed risk taking: evidence from mutual fund investors

Economcs Letters 76 (2002) 151 158 www.elsever.com/ locate/ econbase Gender dfferences n revealed rsk takng: evdence from mutual fund nvestors a b c, * Peggy D. Dwyer, James H. Glkeson, John A. Lst a Unversty

Small pots lump sum payment instruction

For customers Small pots lump sum payment nstructon Please read these notes before completng ths nstructon About ths nstructon Use ths nstructon f you re an ndvdual wth Aegon Retrement Choces Self Invested

Hedging Interest-Rate Risk with Duration

FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error

Intra-year Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor

Chapter 15: Debt and Taxes

Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt

Support Vector Machines

Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

Forecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network

700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School

Learning Performance of Prediction Markets with Kelly Bettors

Learnng Performance of Predcton Markets wth Kelly Bettors Alna Beygelzmer IBM Research Hawthorne, NY beygel @ usbmcom John Langford, Davd M Pennock Yahoo! Research New York, NY {jl,pennockd} @ yahoo-nccom

Analysis of Premium Liabilities for Australian Lines of Business

Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton

10.2 Future Value and Present Value of an Ordinary Simple Annuity

348 Chapter 10 Annutes 10.2 Future Value and Present Value of an Ordnary Smple Annuty In compound nterest, 'n' s the number of compoundng perods durng the term. In an ordnary smple annuty, payments are

Combinatorial Agency of Threshold Functions

Combnatoral Agency of Threshold Functons Shal Jan Computer Scence Department Yale Unversty New Haven, CT 06520 shal.jan@yale.edu Davd C. Parkes School of Engneerng and Appled Scences Harvard Unversty Cambrdge,

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

Fixed income risk attribution

5 Fxed ncome rsk attrbuton Chthra Krshnamurth RskMetrcs Group chthra.krshnamurth@rskmetrcs.com We compare the rsk of the actve portfolo wth that of the benchmark and segment the dfference between the two

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

Interest Rate Futures

Interest Rate Futures Chapter 6 6.1 Day Count Conventons n the U.S. (Page 129) Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (n perod) 30/360 Actual/360 The day count conventon

Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the

Bond futures INRODUCION Bond futures contracts are futures contracts that allow nvestor to buy n the future a theoretcal government notonal bond at a gven prce at a specfc date n a gven quantty. Compared

1 Example 1: Axis-aligned rectangles

COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

1 Approximation Algorithms

CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

Exploring decision makers use of price information in an efficient speculative market

Explorng decson makers use of prce nformaton n an effcent speculatve market J.E.V. Johnson, O.D. Jones, L. Tang Abstract We explore the extent to whch the decsons of partcpants n a speculatve market effectvely

"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *

Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789-794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC

Traffic-light a stress test for life insurance provisions

MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax