Trading the randomness  Designing an optimal trading strategy under a drifted random walk price model


 Melissa Goodman
 3 years ago
 Views:
Transcription
1 Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore the applicatio of probability mathematics i tradig fiacial markets. To achieve that, the author will build a drifted radom walk model to simulate the price movemets of stocks, ad the based o a set of assumptios about trasactio cost ad tradig advatage, establish a tradig system. The author the will examie the expected payoff from differet tradig strategies uder this system ad coclude a optimal tradig strategy i this model. The appedix will the provide a Matlab program to simulate this model ad test all coclusios i this paper. Cotet Table Part I. Backgroud ad itroductio 2 Part II. Itroducig the variables ad the assumptios 3 Part III. A simple case, o drift (µ 0 = 0) 5 Part IV. Full model with the drift µ 0 ad the edge α 6 Part V. A review o assumptios ad limitatios of the model 8 Appedix. tradigradom.m 9
2 2 Part I. Backgroud ad itroductio Modelig stock price with some kid of radom walk process is ot a ew idea. It was most famously oted by Priceto Ecoomist Burto Malkiel s book A Radom Walk Dow Wall Street, i which Malkiel argued that movemets of stock price behave radomly ad therefore the game of tradig stocks is essetially a martigale 2. However, some other ecoomists ad may market participats believe otherwise. They argue that stock price does ot behave completely like a radom walk, ad have suggested other price models. Oe price model developed by Adrew W. Lo ad Archie Craig MacKilay is the basis of my price model i this paper. It ca be expressed as: X t = µ + X t where X t is the price of the stock at time t µ is the tred force drift ε t is the radom disturbace term 3. To uderstad the above formula, oe ca thik of stock price movemet i a treded market. Each discrete movemet is radom (represeted by ε t ), but there is also a tred force (represeted by µ) that drives the price towards certai directio. Θ (Chart.) For example, as the chart (Chart.) above shows, each price movemet is radom (as there are ups ad dows i a radom sequece); however, as the blue arrow shows, overall there is a directio (or tred force) that price oscillates aroud. This tred directio could be drive by fudametal factors (e.g. i the example of a stock, the performace ad quality of the compay) or mometum factors (e.g. if may people are buyig the same stock, the stock price is drive up, which i tur attracts Burto G. Malkiel, A radom walk dow Wall Street : icludig a lifecycle guide to persoal ivestig. New York, Norto: C Martigale: a model of a fair game where o kowledge of past evets ca help to predict future wiigs. Martigale (probability theory), Wikipedia. 3 Radom Walk Hypothesis, Wikipedia.
3 3 more buyers ad drives price further up a classic explaatio of fiacial bubbles). Therefore, a skilled trader who uderstads those factors well could profit by guessig correctly this tred directio, i other words, predictig the correct µ. Part II. Itroducig the variables ad the assumptios To start with, I will defie the variables i this model mathematically. ) Stock price X t The variable X t represets the stock price at time t. It is the exact price at which the trader ca buy or sell at. Agai, i this model: X t = µ + X t. As i this model we care oly about the relative price movemets, we could set X 0 = 0. Hece, we ca t t t t derive that X = X 0 + µ t = µ t. We will further adjust the formula for this variable below i this model to icorporate some other elemets. 2) The time t I this model time t is a discrete iteger from 0 owards, ad therefore, the price movemets is also a discrete process. 3) The drift µ I this model the drift µ will be of a costat size µ 0 represetig the part of price movemets due to the tred force i a uit time. As show i Chart., oe could easily derive the relatioship betwee µ 0 ad the tred lie agle Θ: µ 0 = taθ. However, the directio of the tred could go either way with equal probability. More details o how to model the sig of µ will be discussed below. 4) The radom disturbace term ε t As said, i this paper, the stock price will be modeled with a simple radom walk with a drift, so the radom disturbace term ε t will behave as a simple radom walk variable with a costat step size ε 0, so its probability distributio fuctio is: ε t = ε 0 P(ε t ) = 2 2 ε t = ε 0 Besides parameters i the origial price model (X t, t, µ, ε), I will also itroduce several ew variables ito this price model below. 5) The tred legth A tred i fiacial markets does ot last forever. It dies out after certai time as the fudametal factors behid it are fully reflected i the price or as the mometum eds. Hece, is used to represet how log (i the same uit of t) the tred lasts.
4 4 6) Trader s edge α As the paper discussed above, if the price treds towards a certai directio (drift µ 0), the a skilled trader might be able to gai a edge i tradig by havig a good guess about the directio of the drift. I other words, if the trader ca guess correctly whether µ is positive or egative for more tha half of the time, the she might be able to develop a system that provides a positive expected retur. I this model, we will assig a value α to represet this trader s edge. The trader will guess correctly the value of µ for a probability of α; i the other sceario with a probability of α, I assume that the trader still guess the correct absolute value of µ but with the opposite sig (hece µ). It is easy to see that, by symmetry, if µ is positive ad price is tredig up, but the trader guesses wrog ad decides to shortsell (essetially bettig the price goes dow), it is equivalet to the trader buys whe the price is tredig dow (with a egative µ). Therefore, we could coveietly covert all scearios ito oe that the trader always buys whe eterig a positio, but with µ beig positive whe the trader guesses correctly ad µ beig egative whe the trader guesses wrog. Therefore, give a fixed size of drift µ 0, the drift µ i this case could be modeled with a probability distributio fuctio: P(µ) = 2 + α µ = µ 0 2 α µ = µ 0 7) Profit Limit L ad Stop Loss S To protect their capital, traders ofte set a Stop Loss (S) that whe price goes agaist their pla ad the loss exceeds this poit, they will exit their positio with a loss. As stated above, i this model we coverted all sceario ito oe that trader buys whe eterig a positio. Hece whe X t <= X eter  S, the trader takes i a loss of S ad exits the positio. While a Profit Limit (T) is exactly the opposite, it defies a gai that the trader is satisfied with ad beyod which is uwillig to take further risk. Hece, i this model it represets that whe X t >= X eter + T, the trader takes i a gai of T ad exits the positio. Now it is importat to ote that, if the trader eters the positio at X eter ad exits at X exit it is essetially the same as eterig at 0 ad exits at X exit  X eter. exit X t = µ + X t, so X exit X eter = µ(exit eter ). As ε t are all idepedet ad have the same distributio fuctio, exit eter E( ε t ), so E(X exit X eter ) = µ(exit eter ) + E( t= 0 t= eter+ exit E( ε t )is o differet from t= eter+ exit eter ε t ). I other words, we could reset X eter =X 0 =0 ad X exit = X exiteter (of course with ew radom values for X t ) ad maitai E(X exit X eter ). Hece, i this model, we would simply reset X t to 0 after eterig/reeterig a trade, ad exit the trade whe X t <= S (for a loss) or X t >= L (for a gai). Hece, we ca get a ew formula for X t : t= 0
5 5 X t = t t µ t t eter t eter where teter is the eter time of the curret trade. 8) Trasactio cost C As tradig stocks ivolve a trasactio cost (or ofte called a commissio fee ), this variable C deotes a fixed trasactio cost for each trade. 9) Outcome of the trade O i The Outcome of the ith trade is deoted as O i. This could be a gai or loss. As itroduced above, if the price moves over the Profit Limit L or the Stop Loss S, the trader will take a gai of L or a loss of S respectively. Aother case that the trader will exit the trader ad take a gai/loss i this model is whe the tred arrives to a ed (t=), the trader will have to exit her positio at X i order to prevet the risk of a sharp reversal (a big price chage i the uwated directio). These described rules alog with the trasactio cost C gives us: L C X t >= L O i = S C X t <= S X C t = 0) Total profit of the strategy P This variable P accouts for the sum of gais ad losses of all trades durig the legth of the trade. Its value is the best measuremet of how successful a tradig strategy is. It is give by: P = O + O 2 + O O is the total umber of trades doe withi time. Part III. A simple case, o drift (µ 0 = 0) We start with a special case of the model: µ 0 = 0, the µ = 0 ad X t = X t. I this case, stock price is ideed a radom walk. This situatio happes whe there is a flat market ad price oly oscillates horizotally ad does ot seem to move cosistetly towards either directio (as i Chart 3.). (Chart 3.)
6 6 I this case, it is clear that the trader s edge α becomes irrelevat. Uder this assumptio, ituitively it is very questioable that the trader could develop a tradig system with a positive expected value without a edge ad with a trasactio cost. I fact, we ca proof that without a trasactio cost, o matter how the trader sets her Profit Limit L ad Stop Loss S, all her trades will have a expected value of 0. Coclusio 3.: If µ 0 =0 ad C=0, the for ay i, E(O i ) = 0 regardless of S, L, ad. E(O i ) = E(X exit ) = t= 0 E(X t ) P(t = exit) E(X t ) = E(X t ) + E(ε t ) = E(X t ) =...= E(X 0 ) = X 0 = 0 For t (0,) Hece, E(O i ) = 0. It is oteworthy to poit out that S ad L oly affects P(t = exit) but ot E(X t ), which ituitively explais this coclusio. With a positive C, the it is easy to see E(O i ) = 0 C = C. From here, it is also easy to see that E(P) = E( O i ) = E(O i ) = C = C (Coclusio 3.2). This coclusio ca also be verified by ruig the Matlab program tradigradom.m with the parameter drift set to 0 (e.g. try ruig tradigradom(00, 0, 0.4, 0, 0, 00, 3, 00000) ). As this coclusio suggests, i a flat market, the trader is always expected to lose o matter how she trades; ad the more she trades, the more she is expected to lose. Thus, i such markets, traders are better off ot tradig at all. Part IV. Full model with the drift µ 0 ad the edge α Now to complete the model, let us itroduce the drift variable µ 0 ad the trader s edge variable α, ad the model become much more complicated to solve. Agai, we try to start with a simpler exceptio. If the trader forgets to set her Profit Limit ad Stop Loss, or sets them to be too far from the origial price X 0 so that either of them could be effectively reached withi the tred legth, the what would happe? Mathematically, this problem could be expressed mathematically as: if Max(X 0, X, X ) < L ad Mi(X 0, X, X ) > S, what is E(P)? I such a case, a trade will ot be exited o a Profit Limit or o a Stop Loss. The oly way left for a trade to be exited i this case is whe t arrives at the tred legth ad the stock is sold at X. Therefore, the whole strategy will cosist of a sigle trade with a outcome O = X C. Thus, E(P) = E(O ) = E(X C) = E(X ) C, sice E(X ) = E( µ t ) = E( µ t ) + E( ε t ) = E(µ t ) + E(ε t ),
7 7 ad E(µ t ) = ( 2 + α)µ 0 + ( 2 α)( µ 0) = 2αµ 0, E(ε t ) = 2 ε ( ε 0) = 0, we get E(X ) = E(µ t ) + E(ε t ) = 2αµ = 2αµ 0. Hece, we arrive that E(P) = E(X ) C = 2αµ 0 C. (Coclusio 4.) This formula illustrates the tradeoff of the strategy: how much a trader expected to gai from the size of the tred (µ 0 ) ad her probability advatage to beefit from that tred (2α), versus her trasactio cost (C). We ca verify this coclusio by ruig the Matlab program tradigradom.m with the appropriate parameters (e.g. try ruig tradigradom(00, 0.6, 0.4, 0., 0000, 0000, 3, 00000) ). Now fially, we could examie the effect of Profit Limit (L) ad Stop Loss (S). Let us first assume that whe a trader is stopped out, she immediately eters the positio agai. What would the trader s expected profit be? l s E(P) = E( O i ) = E(O i ) = (L C) + ( S C) + (X C) P r = l(l C) + s( S C) + (X C) P r = (ll ss + P r X ) (l + s + P r )C (Coclusio 4.2) where l deotes the umber of trades that reaches L, s deotes the oes that eds at S, ad P r deotes the probability that there will be a residual trade (the last trade exited i betwee L ad S whe the tred eds, it could be avoided whe the payoff X C is clearly egative. To calculate l, s, ad P r is beyod the scope of this paper (iterested readers are welcome to explore these values with tradigradom.m). However, with oe more reasoable assumptio, we could approximate E(P) above. That assumptio is, µ 0 ad ε 0 are both small. This is a reasoable assumptio i modelig stock market as stock prices chage i very small time itervals (ofte a small fractio of secod), ad therefore each icremetal chage (the combied result of µ t ad ε t ) is also very small. I accout of that, let us examie what happes whe the trader exits her positio at time t due to either a Profit Limit or Stop Loss: If trader exits her positio at t due to a Profit Limit L, the we kow X t < L ad X t >= L, so X t L = X t + µ t L = µ t + (X t L) <= µ t <= µ 0 + ε 0. Similarly, we kow whe a trader exits at t due to a Stop Loss S, the X t > S ad X t <= S ( S) X t = S (X t + µ t ) = (µ t ) + ( S X t ) <= (µ t ) <= µ 0 + ε 0. Therefore, we see that if we replace S or L with X t (the price at exit time t), the error is bouded by ±(µ 0 +ε 0 ), i other words, very small. (Coclusio 4.3) Also, we will use P r = i this approximatio, as we will have a very high probability to ed up with a residual (small or big). Hece, we ca approximate the expected profit E(P) by: E(P) = E( O i ) = E(O i ) E(X ith exit C) = E(X ith exit ) C ith.exit ith.exit = E( µ t ) C = E( µ t ) C = E(X ) C ith.eter ith.eter
8 8 = 2αµ 0 C. (Coclusio 4.4) Note that this is clearly worse off tha the result we get from the o Profit Limit or Stop Loss strategy (Coclusio 4.) as it oly icrease tradig cost but does ot improve tradig gais. Hece, we arrive our fial coclusio: the theoretical optimal tradig strategy i this model is to: predict the directio of the tred, buy (or shortsell) the stock i the begiig accordig to the predicted tred directio, ad exit the positio whe the tred eds. This simple strategy gives the best expected profit of: E(P) = 2αµ 0 C. (Coclusio 4.5) Part V. A review o assumptios ad limitatios of the model The fial coclusio about optimal strategy (Coclusio 4.5) seems a little surprisig. Those who are more familiar with stock tradig will ofte hear the importace of Stop Losses ad Profit Limits. To uderstad this discrepacy, it is importat to realize that all coclusios are arrived uder all the assumptios of this model discussed above. Some key assumptios I would like to reemphasize are: ) The magitude of the drift µ 0, the disturbace term ε 0, ad the trader s edge α are all costats. Hece, the trader caot lear from her gais or losses i this model, which elimiates a importat fuctio of Stop Losses ad Profit Limits i real tradig. 2) The trader trades cotiuously i this model. Immediately after the trader exits a positio, she eters agai. This assimilates more with Program/Algorithmic tradig, while real huma traders ofte take reflective breaks or looks for optimal eter poits durig their tradig. 3) The drifts µ t ad the disturbace ε t are small. This is probably the most importat assumptio made (also probably the most doubtful oe) to achieve Coclusio 4.5. Because of the small icremet price chage, Stop Loss s fuctio of protectig traders from sharp price movemets agaist them is miimized. While i real world tradig, a trader who sets a stop loss could be protected from a large differece betwee X exit ad S. These assumptios reveal some limitatios of this model. However, this paper ad the coclusios still serve reasoably well as a exploratory effort towards fidig a optimal tradig strategy.
9 9 Appedix: tradigradom.m fuctio avg_profit = tradigradom(tred_legth, drift, disturbace, edge, stop, limit, cost, ru_time) profit = 0; for r = :ru_time x = 0; radd = rad(); if (radd < (.5 + edge)) directio = ; else directio = ; ed; t = 0; tradig = ; while (t < tred_legth) tradig = ; rade = radi(2)*23; x = x + directio * drift + rade * disturbace; if (x <= stop) profit = profit stop cost; tradig = 0; ed; if (x >= limit) profit = profit + limit cost; tradig = 0; ed; t = t + ; ed; if (tradig == ) profit = profit + x  cost; ed; ed; avg_profit = profit / ru_time; ed
Chapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationDivide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016
CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationKey Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationDescriptive statistics deals with the description or simple analysis of population or sample data.
Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small
More informationLecture Notes CMSC 251
We have this messy summatio to solve though First observe that the value remais costat throughout the sum, ad so we ca pull it out frot Also ote that we ca write 3 i / i ad (3/) i T () = log 3 (log ) 1
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationTime Value of Money. First some technical stuff. HP10B II users
Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationA Resource for Freestanding Mathematics Qualifications Working with %
Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs 16 000 ad its value falls by 1%
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationDomain 1: Designing a SQL Server Instance and a Database Solution
Maual SQL Server 2008 Desig, Optimize ad Maitai (70450) 18004186789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a
More informationDivide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015
CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision  A measure of agreement between repeated measurements (repeatability).
INTRODUCTION This laboratory ivestigatio ivolves makig both legth ad mass measuremets of a populatio, ad the assessig statistical parameters to describe that populatio. For example, oe may wat to determie
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More information3. Continuous Random Variables
Statistics ad probability: 31 3. Cotiuous Radom Variables A cotiuous radom variable is a radom variable which ca take values measured o a cotiuous scale e.g. weights, stregths, times or legths. For ay
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationSimple Annuities Present Value.
Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX9850GB PLUS to efficietly compute values associated with preset value auities.
More informationPreSuit Collection Strategies
PreSuit Collectio Strategies Writte by Charles PT Phoeix How to Decide Whether to Pursue Collectio Calculatig the Value of Collectio As with ay busiess litigatio, all factors associated with the process
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More informationEstimating the Mean and Variance of a Normal Distribution
Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers
More informationStandard Errors and Confidence Intervals
Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5yearold boys. If we assume
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea  add up all
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More informationMocks.ie Maths LC HL Further Calculus mocks.ie Page 1
Maths Leavig Cert Higher Level Further Calculus Questio Paper By Cillia Fahy ad Darro Higgis Mocks.ie Maths LC HL Further Calculus mocks.ie Page Further Calculus ad Series, Paper II Q8 Table of Cotets:.
More informationSum of Exterior Angles of Polygons TEACHER NOTES
Sum of Exterior Agles of Polygos TEACHER NOTES Math Objectives Studets will determie that the iterior agle of a polygo ad a exterior agle of a polygo form a liear pair (i.e., the two agles are supplemetary).
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationLearning objectives. Duc K. Nguyen  Corporate Finance 21/10/2014
1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the timevalue
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationSection 73 Estimating a Population. Requirements
Sectio 73 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectio
More informationCDs Bought at a Bank verses CD s Bought from a Brokerage. Floyd Vest
CDs Bought at a Bak verses CD s Bought from a Brokerage Floyd Vest CDs bought at a bak. CD stads for Certificate of Deposit with the CD origiatig i a FDIC isured bak so that the CD is isured by the Uited
More informationBasic Measurement Issues. Sampling Theory and AnalogtoDigital Conversion
Theory ad AalogtoDigital Coversio Itroductio/Defiitios Aalogtodigital coversio Rate Frequecy Aalysis Basic Measuremet Issues Reliability the extet to which a measuremet procedure yields the same results
More informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lieup for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More informationAmendments to employer debt Regulations
March 2008 Pesios Legal Alert Amedmets to employer debt Regulatios The Govermet has at last issued Regulatios which will amed the law as to employer debts uder s75 Pesios Act 1995. The amedig Regulatios
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationLecture 7: Borel Sets and Lebesgue Measure
EE50: Probability Foudatios for Electrical Egieers JulyNovember 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,
More informationHypothesis Tests Applied to Means
The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationThe Forgotten Middle. research readiness results. Executive Summary
The Forgotte Middle Esurig that All Studets Are o Target for College ad Career Readiess before High School Executive Summary Today, college readiess also meas career readiess. While ot every high school
More informationSection 7: Free electron model
Physics 97 Sectio 7: ree electro model A free electro model is the simplest way to represet the electroic structure of metals. Although the free electro model is a great oversimplificatio of the reality,
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More information0,1 is an accumulation
Sectio 5.4 1 Accumulatio Poits Sectio 5.4 BolzaoWeierstrass ad HeieBorel Theorems Purpose of Sectio: To itroduce the cocept of a accumulatio poit of a set, ad state ad prove two major theorems of real
More informationA Guide to the Pricing Conventions of SFE Interest Rate Products
A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios
More informationI. Why is there a time value to money (TVM)?
Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios
More informationPlugin martingales for testing exchangeability online
Plugi martigales for testig exchageability olie Valetia Fedorova, Alex Gammerma, Ilia Nouretdiov, ad Vladimir Vovk Computer Learig Research Cetre Royal Holloway, Uiversity of Lodo, UK {valetia,ilia,alex,vovk}@cs.rhul.ac.uk
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More informationCovariance and correlation
Covariace ad correlatio The mea ad sd help us summarize a buch of umbers which are measuremets of just oe thig. A fudametal ad totally differet questio is how oe thig relates to aother. Stat 0: Quatitative
More informationFourier Series and the Wave Equation Part 2
Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries
More informationB1. Fourier Analysis of Discrete Time Signals
B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationInstallment Joint Life Insurance Actuarial Models with the Stochastic Interest Rate
Iteratioal Coferece o Maagemet Sciece ad Maagemet Iovatio (MSMI 4) Istallmet Joit Life Isurace ctuarial Models with the Stochastic Iterest Rate NiaNia JI a,*, Yue LI, DogHui WNG College of Sciece, Harbi
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributiofree statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More information3.2 Introduction to Infinite Series
3.2 Itroductio to Ifiite Series May of our ifiite sequeces, for the remaider of the course, will be defied by sums. For example, the sequece S m := 2. () is defied by a sum. Its terms (partial sums) are
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationRecursion and Recurrences
Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationSolving Inequalities
Solvig Iequalities Say Thaks to the Authors Click http://www.ck12.org/saythaks (No sig i required) To access a customizable versio of this book, as well as other iteractive cotet, visit www.ck12.org CK12
More informationSwaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps
Swaps: Costat maturity swaps (CMS) ad costat maturity reasury (CM) swaps A Costat Maturity Swap (CMS) swap is a swap where oe of the legs pays (respectively receives) a swap rate of a fixed maturity, while
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More information