Introduction to Financial Markets

Introduction to Financial Econometrics Hypothesis Testing in the Market Model Eric Zivot Department of Economics University of Washington February 29, 2000 1 Hypothesis Testing in the Market Model In this chapter, we illustrate how to carry out some simple hypothesis tests concerning the parameters of the excess returns market model regression. 1.1 A Review of Hypothesis Testing Concepts To be completed. 1.2 Testing the Restriction α =0. Using the market model regression, R t = α + βR Mt + ε t ,t=1, .,T ε t ∼ iid N(0, σ 2 ε ), ε t is independent of R Mt (1) consider testing the null or maintained hypothesis α = 0 against the alternative that α 6=0 H 0 : α =0vs. H 1 : α 6=0. If H 0 is true then the market model regression becomes R t = βR Mt + ε t and E[R t |R Mt = r Mt ]=βr Mt . We will reject the null hypothesis, H 0 : α =0,if the estimated value of α is either much larger than zero or much smaller than zero. Assuming H 0 : α = 0 is true, ˆα ∼ N(0,SE(ˆα) 2 ) and so is fairly unlikely that ˆα will 1 be more than 2 values of SE(ˆα) from zero. To determine how big the estimated value of α needs to be in order to reject the null hypothesis we use the t-statistic t α=0 = b α − 0 d SE( b α) , where b α is the least squares estimate of α and d SE( b α) is its estimated standard error. The value of the t-statistic, t α=0 , gives the number of estimated standard errors that b α is from zero. If the absolute value of t α=0 is much larger than 2 then the data cast considerable doubt on the null hypothesis α =0whereasifitislessthan2thedata are in support of the null hypothesis 1 . To determine how big | t α=0 | needs to be to reject the null, we use the fact that under the statistical assumptions of the market model and assuming the null hypothesis is true t α=0 ∼ Student− t with T − 2 degrees of freedom If we set the significance level (the probability that we reject the null given that the null is true) of our test at, say, 5% then our decision rule is Reject H 0 : α =0atthe5%levelif|t α=0 | >t T −2 (0.025) where t T −2 is the 2 1 2 % critical value from a Student-t distribution with T − 2degrees of freedom. Example 1 Market Model Regression for IBM Consider the estimated MM regression equation for IBM using monthly data from January 1978 through December 1982: b R IBM,t =−0.0002 (0.0068) +0.3390 (0.0888) ·R Mt ,R 2 =0.20, b σ ε =0.0524 where the estimated standard errors are in parentheses. Here b α = −0.0002, which is very close to zero, and the estimated standard error, d SE(ˆα) = 0.0068, is much larger than b α. The t-statistic for testing H 0 : α =0vs. H 1 : α 6=0is t α=0 = −0.0002− 0 0.0068 = −0.0363 so that b α is only 0.0363 estimated standard errors from zero. Using a 5% significance level, t 58 (0.025) ≈ 2and |t α=0 | =0.0363 < 2 so we do not reject H 0 : α = 0 at the 5% level. 1 This interpretation of the t-statistic relies on the fact that, assuming the null hypothesis is true so that α =0, bα is normally distributed with mean 0 and estimated variance d SE(bα) 2 . 2 1.3 Testing Hypotheses about β In the market model regression β measures the contribution of an asset to the vari- ability of the market index portfolio. One hypothesis of interest is to test if the asset has the same level of risk as the Introduction to Financial Markets Introduction to Financial Markets By: OpenStaxCollege Building Home Equity Many people choose to purchase their home rather than rent This chapter explores how the global financial crisis has influenced home ownership (Credit: modification of work by Diana Parkhouse/Flickr Creative Commones) The Housing Bubble and the Financial Crisis of 2007 In 2006, housing equity in the United States peaked at $13 trillion That means that the market prices of homes, less what was still owed on the loans used to buy these houses, 1/4 Introduction to Financial Markets equaled $13 trillion This was a very good number, since the equity represented the value of the financial asset most U.S citizens owned However, by 2008 this number had gone down to $8.8 trillion, and it declined further still in 2009 Combined with the decline in value of other financial assets held by U.S citizens, by 2010, U.S homeowners’ wealth had declined by $14 trillion! This is a staggering result, and it affected millions of lives: people had to alter their retirement decisions, housing decisions, and other important consumption decisions Just about every other large economy in the world suffered a decline in the market value of financial assets, as a result of the global financial crisis of 2008–2009 This chapter will explain why people buy houses (other than as a place to live), why they buy other types of financial assets, and why businesses sell those financial assets in the first place The chapter will also give us insight into why financial markets and assets go through boom and bust cycles like the one described here Introduction to Financial Markets In this chapter, you will learn about: • How Businesses Raise Financial Capital • How Households Supply Financial Capital • How to Accumulate Personal Wealth When a firm needs to buy new equipment or build a new facility, it often must go to the financial market to raise funds Usually firms will add capacity during an economic expansion when profits are on the rise and consumer demand is high Business investment is one of the critical ingredients needed to sustain economic growth Even in the sluggish economy of 2009, U.S firms invested $1.4 trillion in new equipment and structures, in the hope that these investments would generate profits in the years ahead Between the end of the recession in 2009 through the second quarter 2013, profits for the S&P 500 companies grew to 9.7 % despite the weak economy, with much of that amount driven by cost cutting and reductions in input costs, according to the Wall Street Journal [link] shows corporate profits after taxes (adjusted for inventory and capital consumption) Despite the steep decline in quarterly net profit in 2008, profits have recovered and surpassed pre-Recession levels 2/4 Introduction to Financial Markets Corporate Profits After Tax (Adjusted for Inventory and Capital Consumption) Since 2000, corporate profits after tax have mostly continued to increase each year save for a substantial decrease between 2008 and 2009 (Source: http://research.stlouisfed.org/fred2) Many firms, from huge companies like General Motors to startup firms writing computer software, not have the financial resources within the firm to make all the desired investments These firms need financial capital from outside investors, and they are willing to pay interest for the opportunity to get a rate of return on the investment for that financial capital On the other side of the financial capital market, suppliers of financial capital, like households, wish to use their savings in a way that will provide a return Individuals cannot, however, take the few thousand dollars that they save in any given year, write a letter to General Motors or some other firm, and negotiate to invest their money with that firm Financial capital markets bridge this gap: that is, they find ways to take the inflow of funds from many separate suppliers of financial capital and transform it into the funds desired by demanders of financial capital Such financial markets include stocks, bonds, bank loans, and other financial investments Visit this website to read more about financial markets 3/4 Introduction to Financial Markets Our perspective then shifts to consider how these financial investments appear to suppliers of capital such as the households that are saving funds Households have a range of investment options: bank accounts, certificates of deposit, money market mutual funds, bonds, stocks, stock and bond mutual funds, housing, and even tangible assets like gold Finally, the chapter investigates two methods for becoming rich: a quick and easy method that does not work very well at all, and a slow, reliable method that can work very well indeed over a lifetime 4/4 Mathematics for Finance: An Introduction to Financial Engineering Marek Capinski Tomasz Zastawniak Springer Springer Undergraduate Mathematics Series Springer London Berlin Heidelberg New York Hong Kong Milan Paris To k yo Advisory Board P.J. Cameron Queen Mary and Westfield College M.A.J. Chaplain University of Dundee K. Erdmann Oxford University L.C.G. Rogers University of Cambridge E. Süli Oxford University J.F. Toland University of Bath Other books in this series A First Course in Discrete Mathematics I. Anderson Analytic Methods for Partial Differential Equations G. Evans, J. Blackledge, P. Yardley Applied Geometry for Computer Graphics and CAD D. Marsh Basic Linear Algebra, Second Edition T.S. Blyth and E.F. Robertson Basic Stochastic Processes Z. Brze´zniak and T. Zastawniak Elementary Differential Geometry A. Pressley Elementary Number Theory G.A. Jones and J.M. Jones Elements of Abstract Analysis M. Ó Searcóid Elements of Logic via Numbers and Sets D.L. Johnson Essential Mathematical Biology N.F. Britton Fields, Flows and Waves: An Introduction to Continuum Models D.F. Parker Further Linear Algebra T.S. Blyth and E.F. Robertson Geometry R. Fenn Groups, Rings and Fields D.A.R. Wallace Hyperbolic Geometry J.W. Anderson Information and Coding Theory G.A. Jones and J.M. Jones Introduction to Laplace Transforms and Fourier Series P. P. G . D y k e Introduction to Ring Theory P. M . C o h n Introductory Mathematics: Algebra and Analysis G. Smith Linear Functional Analysis B.P. Rynne and M.A. Youngson Matrix Groups: An Introduction to Lie Group Theory A. Baker Measure, Integral and Probability M. Capi´nski and E. Kopp Multivariate Calculus and Geometry S. Dineen Numerical Methods for Partial Differential Equations G. Evans, J. Blackledge, P. Yardley Probability Models J. Haigh Real Analysis J.M. Howie Sets, Logic and Categories P. C a m e r o n Special Relativity N.M.J. Woodhouse Symmetries D.L. Johnson Topics in Group Theory G. Smith and O. Tabachnikova Topologies and Uniformities I.M. James Vector Calculus P.C. Matthews Marek Capi´nski and Tomasz Zastawniak Mathematics for Finance An Introduction to Financial Engineering With 75 Figures 1 Springer Marek Capi´nski Nowy Sa  cz School of Business–National Louis University, 33-300 Nowy Sa  cz, ul. Zielona 27, Poland Tomasz Zastawniak Department of Mathematics, University of Hull, Cottingham Road, Kingston upon Hull, HU6 7RX, UK Cover illustration elements reproduced by kind permission of: Aptech Systems, Inc., Publishers of the GAUSS Mathematical and Statistical System, 23804 S.E. Kent-Kangley Road, Maple Valley, WA 98038, USA. Tel: (206) 432 - 7855 Fax (206) 432 - 7832 email: info@aptech.com URL: www.aptech.com. American Statistical Association: Chance Vol 8 No 1, 1995 article by KS and KW Heiner ‘Tree Rings of the Northern Shawangunks’ page 32 fig 2. Springer-Verlag: Mathematica in Education and Research Vol 4 Issue 3 1995 article by Roman E Maeder, Beatrice Amrhein and Oliver Gloor ‘Illustrated Mathematics: Visualization of Mathematical Objects’page 9 fig 11, originally published as a CD ROM ‘Illustrated Mathematics’ by TELOS: ISBN 0-387-14222-3, German edition by Birkhauser: ISBN 3-7643-5100-4. Mathematica in Education and Research Vol 4 Issue 3 1995 article by Richard J Gaylord and Kazume Nishidate ‘Traffic Engineering with Cellular Automata’ page 35 fig 2. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Michael Trott ‘The Implicitization of a Trefoil Knot’ page 14. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Lee de Cola ‘Coins, Trees, Bars and Bells: Simulation of the Binomial Process’ page 19 fig 3. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Richard Gaylord and Kazume Nishidate ‘Contagious Spreading’ page 33 fig 1. Mathematica in Education and Research Vol 5 [...]... formally referred to as: Primary markets (issuer -to- investor transactions with investment banks as intermediaries in the securities markets, and banks, insurance companies, and others in the loan markets) Secondary markets (investor -to- investor transactions with broker-dealers and exchanges as intermediaries in the securities markets, and mostly banks in the loan markets) Secondary markets play a critical... corporate finance staff, like that of a loan banker, is to evaluate the issuing company’s business, its financial condition and to prepare a valuation analysis for the offered security As we stated before, financial markets for securities are organized into two segments defined by the parties to a securities transaction: Primary markets Secondary markets The Purpose and Structure of Financial Markets 19... 60.15 60.10 60.05 Stock 60.00 price 59.95 S 59.90 59.85 59.80 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 4 An Arbitrage Guide to Financial Markets A forward contract on XYZ SA’s stock can be viewed as a subset of this rectangle Suppose we enter into a contract today to purchase the stock 1 year from today for ¼ 60 We intend to hold the stock for 1 year after that The forward can be viewed as c... ventures Institutional traders do not want to take primary risks by speculating on markets to go up or down; instead, they hedge the primary risks by simultaneously buying and selling or borrowing and lending in spot, forward, and option markets They leave themselves 16 An Arbitrage Guide to Financial Markets exposed only to secondary ‘‘spread’’ risks Well-managed financial institutions are compensated... Purpose and Structure of Financial Markets 17 specifically desire these vehicles as they facilitate their day -to- day transactions and often offer security of government insurance against the bank’s insolvency For example, in the U.S the Federal Deposit Insurance Corporation (FDIC) guarantees all deposits up to $100,000 per customer per bank The bank’s customers do not want to invest directly in the bank’s... and skews Interest-rate options, caps, and floors Options on bond prices Caps and floors Relationship to FRAs and swaps An application Swaptions Options to cancel Relationship to forward swaps Exotic options Periodic caps Constant maturity options (CMT or CMS) Digitals and ranges Quantos Option Arbitrage 10.1 Cash-and-carry static arbitrage Borrowing against the box Index arbitrage with options Warrant... participants: individuals, pension and mutual funds, banks, governments, insurance companies, industrial corporations, stock exchanges, over-the-counter dealer networks, and others All these agents can at different times serve as demanders and suppliers of funds, and as transfer facilitators Economic theorists Mathematics for Finance: An Introduction to Financial Engineering Marek Capinski Tomasz Zastawniak Springer Springer Undergraduate Mathematics Series Springer London Berlin Heidelberg New York Hong Kong Milan Paris To k yo Advisory Board P.J. Cameron Queen Mary and Westfield College M.A.J. Chaplain University of Dundee K. Erdmann Oxford University L.C.G. Rogers University of Cambridge E. Süli Oxford University J.F. Toland University of Bath Other books in this series A First Course in Discrete Mathematics I. Anderson Analytic Methods for Partial Differential Equations G. Evans, J. Blackledge, P. Yardley Applied Geometry for Computer Graphics and CAD D. Marsh Basic Linear Algebra, Second Edition T.S. Blyth and E.F. Robertson Basic Stochastic Processes Z. Brze´zniak and T. Zastawniak Elementary Differential Geometry A. Pressley Elementary Number Theory G.A. Jones and J.M. Jones Elements of Abstract Analysis M. Ó Searcóid Elements of Logic via Numbers and Sets D.L. Johnson Essential Mathematical Biology N.F. Britton Fields, Flows and Waves: An Introduction to Continuum Models D.F. Parker Further Linear Algebra T.S. Blyth and E.F. Robertson Geometry R. Fenn Groups, Rings and Fields D.A.R. Wallace Hyperbolic Geometry J.W. Anderson Information and Coding Theory G.A. Jones and J.M. Jones Introduction to Laplace Transforms and Fourier Series P. P. G . D y k e Introduction to Ring Theory P. M . C o h n Introductory Mathematics: Algebra and Analysis G. Smith Linear Functional Analysis B.P. Rynne and M.A. Youngson Matrix Groups: An Introduction to Lie Group Theory A. Baker Measure, Integral and Probability M. Capi´nski and E. Kopp Multivariate Calculus and Geometry S. Dineen Numerical Methods for Partial Differential Equations G. Evans, J. Blackledge, P. Yardley Probability Models J. Haigh Real Analysis J.M. Howie Sets, Logic and Categories P. C a m e r o n Special Relativity N.M.J. Woodhouse Symmetries D.L. Johnson Topics in Group Theory G. Smith and O. Tabachnikova Topologies and Uniformities I.M. James Vector Calculus P.C. Matthews Marek Capi´nski and Tomasz Zastawniak Mathematics for Finance An Introduction to Financial Engineering With 75 Figures 1 Springer Marek Capi´nski Nowy Sa  cz School of Business–National Louis University, 33-300 Nowy Sa  cz, ul. Zielona 27, Poland Tomasz Zastawniak Department of Mathematics, University of Hull, Cottingham Road, Kingston upon Hull, HU6 7RX, UK Cover illustration elements reproduced by kind permission of: Aptech Systems, Inc., Publishers of the GAUSS Mathematical and Statistical System, 23804 S.E. Kent-Kangley Road, Maple Valley, WA 98038, USA. Tel: (206) 432 - 7855 Fax (206) 432 - 7832 email: info@aptech.com URL: www.aptech.com. American Statistical Association: Chance Vol 8 No 1, 1995 article by KS and KW Heiner ‘Tree Rings of the Northern Shawangunks’ page 32 fig 2. Springer-Verlag: Mathematica in Education and Research Vol 4 Issue 3 1995 article by Roman E Maeder, Beatrice Amrhein and Oliver Gloor ‘Illustrated Mathematics: Visualization of Mathematical Objects’page 9 fig 11, originally published as a CD ROM ‘Illustrated Mathematics’ by TELOS: ISBN 0-387-14222-3, German edition by Birkhauser: ISBN 3-7643-5100-4. Mathematica in Education and Research Vol 4 Issue 3 1995 article by Richard J Gaylord and Kazume Nishidate ‘Traffic Engineering with Cellular Automata’ page 35 fig 2. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Michael Trott ‘The Implicitization of a Trefoil Knot’ page 14. 23.5 FTCS and BTCS 241 x t 0 T L Fig. 23.2. Finite difference grid {jh, ik} N x , N t j=0,i=0 . Points are spaced at a distance of h apart in the x-direction and k apart in the t-direction. A simple method for the heat equation (23.2) involves approximating the time derivative ∂/∂t by the scaled forward difference in time, k −1  t , and the second order space derivative ∂ 2 /∂x 2 by the scaled second order central difference in space, h −2 δ 2 x . This gives the equation k −1  t U i j − h −2 δ 2 x U i j = 0, which may be expanded as U i+1 j − U i j k − U i j+1 − 2U i j + U i j−1 h 2 = 0. A more revealing re-write is U i+1 j = νU i j+1 + (1 − 2ν)U i j + νU i j−1 , (23.7) where ν := k/h 2 is known as the mesh ratio. Suppose that all approximate solution values at time level i, {U i j } N x j=0 , are known. Now note that U i+1 0 = a((i + 1)k) and U i+1 N x = b((i + 1)k) are given by the boundary conditions (23.4). Equation (23.7) then gives a formula for comput- ing all other approximate values at time level i + 1, that is, {U i+1 j } N x −1 j=1 . Since we 242 Finite difference methods Fig. 23.3. Stencil for FTCS. Solid circles indicate the location of values that must be known in order to obtain the value located at the open circle. are supplied with the time-zero values, U 0 j = g( jh) from (23.3), this means that the complete set of approximations {U i j } N x , N t j=0,i=0 can be computed by stepping for- ward in time. The method defined by (23.7) is known as FTCS, which stands for forward difference in time, central difference in space. Figure 23.3 illustrates the stencil for FTCS. Here, the solid circles indicate the location of values U i j−1 , U i j and U i j+1 that must be known in order to obtain the value U i+1 j located at the open circle. We may collect all the interior values at time level i into a vector, U i :=         U i 1 U i 2 . . . . . . U i N x −1         ∈ R N x −1 . (23.8) Exercise 23.3 then asks you to confirm that FTCS may be written U i+1 = FU i + p i , for 0 ≤ i ≤ N t − 1, (23.9) with U 0 =         g(h) g(2h) . . . . . . g((N x − 1)h)         ∈ R N x −1 , 23.5 FTCS and BTCS 243 where the matrix F has the form F =             1 − 2νν 0 0 ν 1 − 2νν 0 . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . 1 − 2νν 0 0 ν 1 − 2ν             ∈ R (N x −1)×(N x −1) , and the vector p i has the form p i =           νa(ik) 0 . . . . . . 0 νb(ik)           ∈ R N x −1 . Here, FU i denotes a matrix–vector product. Computational example Figure 23.4 illustrates a numerical solution produced by FTCS on the problem of Figure 23.1, with T = 3. We chose N x = 14 and N t = 199, so h = π/14 ≈ 0.22 and k = 3/199 ≈ 0.015, giving ν ≈ 0.3. The numerical solution appears to match the exact solution, shown in Figure 23.1. Computing the worst-case grid error, max 0≤j ≤N x ,0≤i≤N t |U i j − u( jh, ik)|, pro- duced 0.0012, which confirms the close agreement. As can be seen from Figure 23.4, we used a grid where k is much smaller than h –wedivided the x- axis into only 15 points, compared with 200 points on the t-axis. In Figure 23.5 we show what happens if we try to correct this imbalance. Here, we reduced N t to 94, so k ≈ 0.032 and ν ≈ 0.63. We see that the numerical solution has de- veloped oscillations that render it useless as an approximation to u(x, t).Taking smaller values of N t , that is, larger timesteps k,leads to more dramatic oscilla- tions. In Section 23.7 we develop some theory that explains this behaviour. We finish this section by deriving an alternative method that is more computationally expensive, but does not suffer from the type of instability seen in Figure 23.5. ♦ Replacing the forward difference in time in FTCS by a backward difference gives k −1 ∇ ... it into the funds desired by demanders of financial capital Such financial markets include stocks, bonds, bank loans, and other financial investments Visit this website to read more about financial. .. website to read more about financial markets 3/4 Introduction to Financial Markets Our perspective then shifts to consider how these financial investments appear to suppliers of capital such as the... chapter will also give us insight into why financial markets and assets go through boom and bust cycles like the one described here Introduction to Financial Markets In this chapter, you will learn