Chapter05_Mathematics of Finance_13e

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Mathematical finance

Mathematical financeMathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical or numerical models suggested by financial economics. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the fair value of derivatives of the stock (see: Valuation of options; Financial modeling). The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results.In terms of practice, mathematical finance also overlaps heavily with the field of computational finance (also known as financial engineering). Arguably, these are largely synonymous, although the latter focuses on application, while the former focuses on modeling and derivation (see: Quantitative analyst), often by help of stochastic asset models. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk- and portfolio management on the other hand. These are discussed below.Many universities around the world now offer degree and research programs in mathematical finance; see Master of Mathematical Finance.History: Q versus PThere exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other hand. One of the main differences is that they use different probabilities, namely the risk-neutral probability, denoted by "Q", and the actual probability, denoted by "P".Derivatives pricing: the Q worldFurther information: Black–ScholesThe goal of derivatives pricing is to determine the fair price of a given security in terms of more liquid securities whose price is determined by the law of supply and demand. Examples of securities being priced are plain vanilla and exotic options, convertible bonds, etc.Once a fair price has been determined, the sell-side trader can make a market on the security. Therefore, derivatives pricing is a complex "extrapolation" exercise to define the current market value of a security, which is then used by the sell-side community.Derivatives pricing: the Q worldGoal"extrapolate the present"Environmentrisk-neutral probabilityProcesses continuous-time martingalesDimension lowTools Ito calculus, PDE’sChallenges calibrationBusiness sell-sideQuantitative derivatives pricing was initiated by Louis Bachelier in The Theory of Speculation (published 1900), with the introduction of the most basic and most influential of processes, the Brownian motion, and its applications to the pricing of options. Bachelier modeled the time series of changes in the logarithm of stock prices as a randomwalk in which the short-term changes had a finite variance. This causes longer-term changes to follow a Gaussian distribution. Bachelier's work, however, was largely unknown outside academia.The theory remained dormant until Fischer Black and Myron Scholes, along with fundamental contributions by Robert C. Merton, applied the second most influential process, the geometric Brownian motion, to option pricing. For this M. Scholes and R. Merton were awarded the 1997 Nobel Memorial Prize in Economic Sciences. Black was ineligible for the prize because of his death in 1995.The next important step was the fundamental theorem of asset pricing by Harrison and Pliska (1981), according to which the suitably normalized current price Pof a security is arbitrage-free, and thus truly fair, only if there exists awith constant expected value which describes its future evolution:stochastic process Pt(1 )A process satisfying (1) is called a "martingale". A martingale does not reward risk. Thus the probability of thenormalized security price process is called "risk-neutral" and is typically denoted by the blackboard font letter " ".The relationship (1) must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time.The quants who operate in the Q world of derivatives pricing are specialists with deep knowledge of the specific products they model.Securities are priced individually, and thus the problems in the Q world are low-dimensional in nature. Calibration is one of the main challenges of the Q world: once a continuous-time parametric process has been calibrated to a set of traded securities through a relationship such as (1), a similar relationship is used to define the price of new derivatives.The main quantitative tools necessary to handle continuous-time Q-processes are Ito’s stochastic calculus and partial differential equations (PDE’s).Risk and portfolio management: the P worldRisk and portfolio management aims at modelling the probability distribution of the market prices of all the securities at a given future investment horizon.This "real" probability distribution of the market prices is typically denoted by the blackboard font letter " ", as opposed to the "risk-neutral" probability " " used in derivatives pricing.Based on the P distribution, the buy-side community takes decisions on which securities to purchase in order to improve the prospective profit-and-loss profile of their positions considered as a portfolio.Risk and portfolio management: the P worldGoal"model the future"Environment real probabilityProcesses discrete-time seriesDimension largeTools multivariate statisticsChallenges estimationBusiness buy-sideThe quantitative theory of risk and portfolio management started with the mean-variance framework of Harry Markowitz (1952), who caused a shift away from the concept of trying to identify the best individual stock forinvestment. Using a linear regression strategy to understand and quantify the risk (i.e. variance) and return (i.e. mean) of an entire portfolio of stocks, bonds, and other securities, an optimization strategy was used to choose a portfolio with largest mean return subject to acceptable levels of variance in the return. Next, breakthrough advances were made with the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT) developed by Treynor (1962), Mossin (1966), William Sharpe (1964), Lintner (1965) and Ross (1976).For their pioneering work, Markowitz and Sharpe, along with Merton Miller, shared the 1990 Nobel Memorial Prize in Economic Sciences, for the first time ever awarded for a work in finance.The portfolio-selection work of Markowitz and Sharpe introduced mathematics to the "black art" of investment management. With time, the mathematics has become more sophisticated. Thanks to Robert Merton and Paul Samuelson, one-period models were replaced by continuous time, Brownian-motion models, and the quadratic utility function implicit in mean–variance optimization was replaced by more general increasing, concave utility functions.[1] Furthermore, in more recent years the focus shifted toward estimation risk, i.e., the dangers of incorrectly assuming that advanced time series analysis alone can provide completely accurate estimates of the market parameters [2]Much effort has gone into the study of financial markets and how prices vary with time. Charles Dow, one of the founders of Dow Jones & Company and The Wall Street Journal, enunciated a set of ideas on the subject which are now called Dow Theory. This is the basis of the so-called technical analysis method of attempting to predict future changes. One of the tenets of "technical analysis" is that market trends give an indication of the future, at least in the short term. The claims of the technical analysts are disputed by many academics.CriticismOver the years, increaingly sophisticated mathematical models and derivative pricing strategies have been developed, but their credibility was damaged by the financial crisis of 2007–2010.Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by Nassim Nicholas Taleb in his book The Black Swan[3] and Paul Wilmott. Taleb claims that the prices of financial assets cannot be characterized by the simple models currently in use, rendering much of current practice at best irrelevant, and, at worst, dangerously misleading. Wilmott and Emanuel Derman published the Financial Modelers' Manifesto in January 2008[4] which addresses some of the most serious concerns.Bodies such as the Institute for New Economic Thinking are now attempting to establish more effective theories and methods.[5]In general, modeling the changes by distributions with finite variance is, increasingly, said to be inappropriate. In the 1960s it was discovered by Benoît Mandelbrot that changes in prices do not follow a Gaussian distribution, but are rather modeled better by Lévy alpha-stable distributions. The scale of change, or volatility, depends on the length of the time interval to a power a bit more than 1/2. Large changes up or down are more likely than what one would calculate using a Gaussian distribution with an estimated standard deviation.[3] See also Financial models with long-tailed distributions and volatility clustering.Mathematical finance articlesSee also Outline of finance: § Financial mathematics; § Mathematical tools; § Derivatives pricing. Mathematical tools•Asymptotic analysis•Mathematical models•Stochastic calculus•Brownian motion•Lévy process•Calculus•Monte Carlo method•Stochastic differential equations•Copulas•Numerical analysis•Stochastic volatility•Numerical partial differential equations•Crank–Nicolson method•Finite difference method •Differential equations•Real analysis•Value at risk•Expected value•Partial differential equations•Volatility•ARCH model•GARCH model•Ergodic theory•Probability•Feynman–Kac formula•Probability distributions•Binomial distribution•Log-normal distribution•Fourier transform•Quantile functions•Heat equation•Gaussian copulas•Radon–Nikodym derivative•Girsanov's theorem•Risk-neutral measure•Itô's lemma•Martingale representation theoremDerivatives pricing•The Brownian Motion Model of Financial Markets•Rational pricing assumptions •Risk neutral valuation•Arbitrage-free pricing •Forward Price Formula •Futures contract pricing •Swap Valuation •Options•Put–call parity (Arbitragerelationships for options)•Intrinsic value, Time value•Moneyness•Pricing models•Black–Scholes model•Black model•Binomial options model•Monte Carlo option model•Implied volatility, Volatility smile•SABR Volatility Model•Markov Switching Multifractal•The Greeks•Finite difference methods foroption pricing•Vanna Volga method•Trinomial tree•Garman-Kohlhagen model•Optimal stopping (Pricing ofAmerican options)•Interest rate derivatives•Black model•caps and floors•swaptions•Bond options•Short-rate models•Rendleman-Bartter model•Vasicek model•Ho-Lee model•Hull–White model•Cox–Ingersoll–Ross model•Black–Karasinski model•Black–Derman–Toy model•Kalotay–Williams–Fabozzi model•Longstaff–Schwartz model•Chen model•Forward rate-based models•LIBOR market model(Brace–Gatarek–Musiela Model, BGM)•Heath–Jarrow–Morton Model (HJM)Notes[1]Karatzas, Ioannis; Shreve, Steve (1998). Methods of Mathematical Finance. Secaucus, NJ, USA: Springer-Verlag New York, Incorporated.ISBN 9780387948393.[2]Meucci, Attilio (2005). Risk and Asset Allocation. Springer. ISBN 9783642009648.[3]Taleb, Nassim Nicholas (2007). The Black Swan: The Impact of the Highly Improbable. Random House Trade. ISBN 978-1-4000-6351-2.[4]"Financial Modelers' Manifesto" (/blogs/paul/index.cfm/2009/1/8/Financial-Modelers-Manifesto). PaulWilmott's Blog. January 8, 2009. . Retrieved June 1, 2012.[5]Gillian Tett (April 15, 2010). "Mathematicians must get out of their ivory towers" (/cms/s/0/cfb9c43a-48b7-11df-8af4-00144feab49a.html). Financial Times. .References•Harold Markowitz, Portfolio Selection, Journal of Finance, 7, 1952, pp. 77–91•William Sharpe, Investments, Prentice-Hall, 1985•Attilio Meucci, versus Q: Differences and Commonalities between the Two Areas of Quantitative Finance (http:// /abstract=1717163''P), GARP Risk Professional, February 2011, pp. 41-44Article Sources and Contributors6Article Sources and ContributorsMathematical finance Source: /w/index.php?oldid=500952080 Contributors: A.j.g.cairns, Acroterion, Ahd2007, Ahoerstemeier, Albertod4, Allemandtando, Amckern, Angelachou, Arthur Rubin, Author007, Avraham, Ayonbd2000, Baoura, Beetstra, Billolik, Brad7777, Btyner, Burakg, Burlywood, CapitalR, Celuici, Cfries, Charles Matthews, Christoff pale, Christofurio, Ciphers, Colonel Warden, Cursive, DMCer, DocendoDiscimus, DominicConnor, Drootopula, DuncanHill, Dysprosia, Edward, Elwikipedista, Eric Kvaalen, Evercat, Eweinber,FF2010, Fastfission, Feco, Financestudent, Fintor, Flowanda, Gabbe, Gary King, Gene Nygaard, Giftlite, Giganut, HGB, Halliron, Hannibal19, HappyCamper, Headbomb, Hroðulf, Hu12,Hégésippe Cormier, JBellis, JYolkowski, Jackol, Jamesfranklingresham, Jimmaths, Jmnbatista, JohnBlackburne, JonHarder, JonMcLoone, Jonhol, Jrtayloriv, Kaslanidi, Kaypoh, Kimys,Kolmogorov Complexity, Kuru, Lamro, Langostas, Looxix, MER-C, MM21, Mav, Mic, Michael Hardy, Michaltomek, Mikaey, Minesweeper, MrOllie, Msh210, Mydogategodshat,Nikossskantzos, Niuer, Nparikh, Oleg Alexandrov, Onyxxman, Optakeover, Paul A, Pcb21, PhotoBox, Pnm, Portutusd, Ppntori, Punanimal, Quantchina, Quantnet, Ralphpukei, Rasmus Faber, Rhobite, Riskbooks, Rodo82, Ronnotel, Ruud Koot, SUPER-QUANT-HERO, Sardanaphalus, Sentriclecub, Silly rabbit, SkyWalker, Smaines, Smesh, Stanislav87, SymmyS, Tassedethe, Taxman, Template namespace initialisation script, Tesscass, Tigergb, Timorrill, Timwi, Uxejn, Vabramov, Vasquezomlin, WebScientist, Willsmith, Woohookitty, Xiaobajie, YUL89YYZ, Yunli, Zfeinst, Zfr, 253 anonymous editsLicenseCreative Commons Attribution-Share Alike 3.0 Unported///licenses/by-sa/3.0/。

Mathematics of Finance

If we are investing an annuity payment, we often want to calculate what it will grow to.
FVannuity Pmt(1 k)n
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Pmt(1 k)n
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.... Pmt(1 k)0
(1 k)n 1 FV Pmt k
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Example: Assume you have $1,200 to invest and want to know how long it will take for that sum to grow to $3,000 if invested in a CD at 7%:
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Solving For k Or n (III)
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Present Value
Assume that your parents are making the same calculation for your sibling’s (who is just starting high school) first tuition payment.
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Solving For k Or n
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Why “Mathematics” Matters
Characteristics of financial securities: Invest today in exchange for future cash flow promises. Control rights in case cash flows fail to materialize. Examples: Corporations issuing stocks. Corporations and Government issuing bonds. Individuals receiving bank/finance company loan Individuals make deposits in saving accounts What are these “promises” worth? Payments occur in the future. Payments may be risky.

S_FMA_4

Financial Mathematics for ActuariesChapter4Rates of ReturnLearning Objectives •Internal rate of return(yield rate)•One-period rate of return of a fund:time-weighted rate of return and dollar-weighted(money-weighted)rate of return•Rate of return over longer periods:geometric mean rate of return and arithmetic mean rate of return•Portfolio return and return of a short-selling strategy •Crediting interest:investment-year method and portfolio method •Inﬂation:real rate of return•Capital budgeting and project appraisal4.1Internal Rate of Return •Consider a project with initial investment C0.We assume the cash ﬂows occur at regular intervals.•The project lasts for n years and the future cashﬂows are denoted by C1,···,C n.•We adopt the convention that cash inﬂows to the project(invest-ments)are positive and cash outﬂows from the project(withdrawals) are negative.•We deﬁne the internal rate of return(IRR)(also called the yield rate)as the rate of interest such that the sum of the present values of the cashﬂows is equated to zero.•Denoting the internal rate of return by y ,we have n X j =0C j (1+y )j =0,(4.1)where j is the time at which the cash ﬂow C j occurs.•This equation can also be written as C 0=−n X j =1C j (1+y )j .(4.2)•Thus,the net present value of all future withdrawals (injections are negative withdrawals)evaluated at the IRR is equal to the initial investment.Example 4.1:A project requires an initial cash outlay of $2,000and is expected to generate $800at the end of year 1and $1,600at the endof year2,at which time the project will terminate.Calculate the IRR of the project.Solution:If we denote v=1/(1+y),we have,from(4.2)2,000=800v+1,600v2,or5=2v+4v2.Dropping the negative answer from the quadratic equation,we havev=−2+√4+4×4×52×4=0.8956.Thus,y=(1/v)−1=11.66%.Note that v<0implies y<−1,i.e.,the loss is larger than100%,which is precluded from consideration.2•There is generally no analytic solution for y in(4.1)when n>2, and numerical methods have to be used.The Excel function IRR enables us to compute the answer easily.Its usage is described as follows:Example4.2:An investor pays$5million for a5-year lease of a shopping mall.He will receive$1.2million rental income at the end of each year.Calculate the IRR of his investment.Solution:See Exhibit4.1.2•If the cashﬂows occur more frequently than once a year,such as monthly or quarterly,y computed from(4.1)is the IRR for the payment interval.•Suppose cashﬂows occur m times a year,the nominal IRR in an-nualized term is m·y,while the annual eﬀective rate of return is (1+y)m−1.Example4.3:A cash outlay of$100generates incomes of$20after4 months and8months,and$80after2years.Calculate the IRR of the investment.Solution:If we treat one month as the interest conversion period,the equation of value can be written as100=20(1+y1)4+20(1+y1)8+80(1+y1)24,where y1is the IRR on monthly interval.The nominal rate of return on monthly compounding is12y1.Alternatively,we can use the4-month interest conversion interval,and the equation of value is100=201+y4+20(1+y4)2+80(1+y4)6,where y4is the IRR on4-month interval.The nominal rate of return on4-monthly compounding is3y4.The eﬀective annual rate of return is y=(1+y1)12−1=(1+y4)3−1.The above equations of value have to be solved numerically for y1or y4. We obtain1.0406%as the IRR per month,namely,y1.The eﬀective annualized rate of return is then(1.010406)12−1=13.23%.Solving for y4with Excel,we obtain the answer4.2277%.Hence,the annualized eﬀective rate is(1.042277)3−1=13.23%,which is equal to the eﬀective rate computed using y1.2•When cashﬂows occur irregularly,we can deﬁne y as the annualized rate and express all time of occurrence of cashﬂows in years.•Suppose there are n+1cashﬂows occurring at time0,t1,···,t n, with cash amounts C0,C1,···,C n.Equation(4.2)is rewritten asC0=−nX j=1C j(1+y)t j,(4.3)which requires numerical methods for the solution of y.•We may also use the Excel function XIRR to solve for y.Unlike IRR, XIRR allows the cashﬂows to occur at irregular time intervals.The speciﬁcation of XIRR is as follows:Example4.4:A project requires an outlay of$2.35million in return for$0.8million after9months,$1million after15months and$1million after2years.What is the IRR of the project?Solution:Returns of the project occur at time(in years)0.75,1.25 and2.We solve for v numerically from the following equation using Excel Solver(see Exhibit4.2)235=80v0.75+100v1.25+100v2to obtain v=0.879,so thaty=10.879−1=13.77%,which is the eﬀective annualized rate of return of the project.Alterna-tively,we may use the Excel function XIRR as shown in Exhibit4.3.2•A project with no subsequent investment apart from the initial cap-ital is called a simple project.•For simple projects,(4.1)has a unique solution with y>−1,so that IRR is well deﬁned.Example4.5:A project requires an initial outlay of$8million,gen-erates returns of$50million1year later,and requires$50million to terminate at the end of year2.Solve for y in(4.1).Solution:We are required to solve8=50v−50v2,which has v=0.8and0.2as solutions.This implies y has multiple solu-tions of25%and400%.24.2One-Period Rate of Return•We consider methods of calculating the return of a fund over a1-period interval.The methodology adopted depends on the data available.•We start with the situation where the exact amounts of fund with-drawals and injections are known,as well as the time of their occur-rence.•Consider a1-year period with initial fund amount B0(equal to C0). Cashﬂow of amount C j occurs at time t j(in fraction of a year)for j=1,···,n,where0<t1<···<t n<1.•Note that C j are usually fund redemptions and new investments,and do not include investment incomes such as dividends and couponpayments.•Denoting the fund value before and after the transaction at time t j by B B j and B A j,respectively,we have B A j=B B j+C j for j=1,···,n.•The diﬀerence between B B j and B A j−1,i.e.,the balance before the transaction at time t j and after the transaction at time t j−1,is due to investment incomes,as well as capital gains and losses.•Let the fund balance at time1be B1,and deﬁne B B n+1=B1andB A0=B0(this notation will allow us to express the gross return as(4.4)below).Figure4.1illustrates the time diagram.•We now introduce two methods to calculate the1-year rate of re-turn:the time-weighted rate of return(TWRR)and the dollar-weighted rate of return(DWRR).•To compute the TWRR weﬁrst calculate the return over each subin-terval between the occurrences of transactions by comparing the fund balances just before the new transaction to the fund balance just af-ter the last transaction.•If we denote R j as the rate of return over the subinterval t j−1to t j,we have1+R j=B B jB j−1,for j=1,···,n+1.(4.4)•Then TWRR over the year,denoted by R T,isR T=⎡⎣n+1Y j=1(1+R j)⎤⎦−1.(4.5)•The TWRR requires data of the fund balance prior to each with-drawal or injection.In contrast,the DWRR does not require thisinformation.It only uses the information of the amounts of the withdrawals and injections,as well as their time of occurrence.•In principle,when cash of amount C j is injected(withdrawn)at time t j,there is a gain(loss)of capital of amount C j(1−t j)for the remaining period of the year.•Thus,the eﬀective capital of the fund over the1-year period,denoted by B,is given byB=B0+nX j=1C j(1−t j).•Denoting C=P n j=1C j as the net injection of cash(withdrawal if negative)over the year and I as the interest income earned over the year,we have B1=B0+I+C,so thatI=B1−B0−C.(4.6)Hence the DWRR over the1-year period,denoted by R D,isR D=IB=B1−B0−CB0+P n j=1C j(1−t j).(4.7)Example4.6:On January1,a fund was valued at100k(1k=1,000). On May1,the fund increased in value to112k and30k of new principal was injected.On November1,the fund value dropped to125k,and42k was withdrawn.At the end of the year,the fund was worth100k.Calculate the DWRR and the TWRR.Solution:As C=30−42=−12,there is a net withdrawal.From (4.6),the interest income earned over the year isI=100−100−(−12)=12.Hence,from (4.7),the DWRR isR D =12100+23×30−16×42=10.62%.The fund balance just after the injection on May 1is 112+30=142k,and its value just after the withdrawal on November 1is 125−42=83k.From (4.4),the fund-value relatives over the three subperiods are1+R 1=112100=1.120,1+R 2=125142=0.880,1+R 3=10083=1.205.Hence,from (4.5),the TWRR isR T =1.120×0.880×1.205−1=18.76%.2•TWRR compounds the returns of the fund over subperiods after purging the eﬀects of the timing and amount of cash injections and withdrawals.•As fund managers have no control over the timing of fund injection and withdrawal,the TWRR appropriately measures the performance of the fund manager.•DWRR is sensitive to the timing and amount of the cashﬂows.•If the purpose is to measure the performance of the fund,the DWRR is appropriate.•It allows superior market timing to impact the return of the fund.•For funds with frequent cash injections and withdrawals,the com-putation of the TWRR may not be feasible.The diﬃculty lies in the evaluation of the fund value B B j,which requires the fund to be constantly marked to market.•In some situations the exact timing of the cashﬂows may be diﬃcult to identify.•In this situation,we may approximate(4.7)by assuming the cash ﬂows to be evenly distributed throughout the1-year evaluation pe-riod.•Hence,we replace1−t j by its mean value of0.5so that P n j=1C j(1−t j)=0.5C,and(4.7)can be written asR D'IB0+0.5C=IB0+0.5(B1−B0−I))=I0.5(B1+B0−I).(4.8)Example4.7:For the data in Example4.6,calculate the approximate value of the DWRR using(4.8).Solution:With B0=B1=100,and I=12,the approximate R D isR D=120.5(100+100−12)=12.76%.24.3Rate of Return over Multiple Periods •We now consider the rate of return of a fund over a m-year period.•Weﬁrst consider the case where only annual data of returns are available.Suppose the annual rates of return of the fund have been computed as R1,···,R m.Note that−1≤R j<∞for all j.•The average return of the fund over the m-year period can be cal-culated as the mean of the sample values.We call this measure the arithmetic mean rate of return,denoted by R A,which is givenbyR A=1mmX j=1R j.(4.9)•An alternative is to use the geometric mean to calculate the average, called the geometric mean rate of return,denoted by R G,whichis given byR G=⎡⎣m Y j=1(1+R j)⎤⎦1m−1=[(1+R1)(1+R2)···(1+R m)]1m−1.(4.10)Example4.8:The annual rates of return of a bond fund over the last 5years are(in%)as follows:6.48.92.5−2.17.2Calculate the arithmetic mean rate of return and the geometric mean rate of return of the fund.Solution:The arithmetic mean rate of return isR A=(6.4+8.9+2.5−2.1+7.2)/5=22.9/5=4.58%,and the geometric mean rate of return isR G=(1.064×1.089×1.025×0.979×1.072)15−1=(1.246)15−1=4.50%.2 Example4.9:The annual rates of return of a stock fund over the last 8years are(in%)as follows:15.218.7−6.9−8.223.2−3.916.91.8 Calculate the arithmetic mean rate of return and the geometric mean rate of return of the fund.Solution:The arithmetic mean rate of return isR A=(15.2+18.7+···+1.8)/8=7.10%and the geometric mean rate of return isR G=(1.152×1.187×···×1.018)1−1=6.43%.2•Given any sample of data,the arithmetic mean is always larger than the geometric mean.•If the purpose is to measure the return of the fund over the holding period of m years,the geometric mean rate of return is the appro-priate measure.•The arithmetic mean rate of return describes the average perfor-mance of the fund for one year taken randomly from the sample period.•If there are more data about the history of the fund,alternative mea-sures of the performance of the fund can be used.The methodology of the time-weighted rate of return in Section4.2can be extended to beyond1period(year).•Suppose there are n subperiods,with returns denoted by R1,···, R n,over a period of m years.Then we can measure the m-year return by compounding the returns over each subperiod to form the TWRR using the formulaR T=⎡⎣n Y j=1(1+R j)⎤⎦1−1.(4.11)•We can also compute the return over a m-year period using the IRR. We extend the notations for cashﬂows in Section4.2to the m-year period.•Suppose cashﬂows of amount C j occur at time t j for j=1,···,n, where0<t1<···<t n<m.Let the fund value at time0and time m be B0and B1,respectively.We treat−B1as the last transaction, i.e.,fund withdrawal of amount B1.•The rate of return of the fund is calculated as the IRR which equates the discounted values of B0,C1,···,C n,and−B1to zero.•This is referred to as the DWRR over the m-year period.We denote it as R D,which solves the following equationB0+nX j=1C j(1+R D)t j−B1(1+R D)m=0.(4.12)•The example below concerns the returns of a bond fund.When a bond makes the periodic coupon payments,the bond values drop and the coupons are cash amounts to be withdrawn from the fund.Example4.10:A bond fund has an initial value of$20million.The fund records coupon payments in six-month periods.Coupons received from January1through June30are regarded as paid on April1.Likewise, coupons received from July1through December31are regarded as paid on October1.For the2-year period2008and2009,the fund values and coupon payments were recorded in Table4.1.Table4.1:Cashﬂows of fundTime Coupon received Fund value beforemm/dd/yy($millions)date($millions)01/01/0820.004/01/080.8022.010/01/08 1.0222.804/01/090.9721.910/01/090.8523.512/31/0925.0Calculate the TWRR and the DWRR of the fund.Solution:As the coupon payments are withdrawals from the fund(the portfolio of bonds),the fund drops in value after the coupon payments. For example,the bond value drops to22.0−0.80=21.2million on April 1,2008after the coupon payments.Thus,the TWRR is calculated asR T=∙2220×22.8022.0−0.8×21.9022.80−1.02×23.5021.90−0.97×25.023.50−0.85¸0.5−1=21.42%.To calculate the DWRR we solve R D from the following equation20=0.8(1+R D)+1.02(1+R D)+0.97(1+R D)+0.85(1+R D)+25(1+R D)=0.8v+1.02v3+0.97v5+0.85v7+25v8,where v=(1+R D)−0.25.We let(1+y)−1=v and use Excel to obtain y= 4.949%,so that the annual eﬀective rate of return is R D=(1.04949)4−1= 21.31%.24.4Portfolio Return•We now consider the return of a portfolio of assets.•Suppose a portfolio consists of N assets denoted by A1,···,A N.Let the value of asset A j in the portfolio at time0be A0j,for j= 1,···,N.•We allow A0j to be negative for some j,so that asset A j is sold short in the portfolio.•The portfolio value at time0is B0=P N j=1A0j.Let the asset values at time1be A1j,so that the portfolio value is B1=P N j=1A1j.•Denote R P as the return of the portfolio in the period from time0to time1.Thus,R P=B1−B0B0=B1B0−1.•We deﬁnew j=A0j B0,which is the proportion of the value of asset A j in the initial portfolio, so thatNX j=1w j=1,and w j<0if asset j is short sold in the portfolio.•We also denoteR j=A1j−A0jA0j=A1jA0j−1,which is the rate of return of asset j.Thus,1+R P=B1 B0=1B0N X j=1A1j=NX j=1A0j B0×A1j A0j=NX j=1w j(1+R j),which impliesR P=NX j=1w j R j,(4.13)so that the return of the portfolio is the weighted average of the returns of the individual assets.•Eq(4.13)is an identity,and applies to realized returns as well as returns as random variables.If we take the expectations of(4.13),we obtainE(R P)=NX j=1w j E(R j),(4.14)so that the expected return of the portfolio is equal to the weighted average of the expected returns of the component assets.•The variance of the portfolio return is given byVar(R P)=NX j=1w2j Var(R j)+N X h=1N X j=1|{z}h=jw h w j Cov(R h,R j).(4.15)•For example,consider a portfolio consisting of two funds,a stock fund and a bond fund,with returns denoted by R S and R B,respec-tively.Likewise,we use w S and w B to denote their weights in the portfolio.•Then,we haveE(R P)=w S E(R S)+w B E(R B),(4.16)andVar(R P)=w2S Var(R S)+w2B Var(R B)+2w S w B Cov(R S,R B),(4.17)where w S+w B=1.Example4.11:A stock fund has an expected return of0.15and variance of0.0625.A bond fund has an expected return of0.05and variance of0.0016.The correlation coeﬃcient between the two funds is −0.2.(a)What is the expected return and variance of the portfolio with80%in the stock fund and20%in the bond fund?(b)What is the expected return and variance of the portfolio with20%in the stock fund and80%in the bond fund?(c)How would you weight the two funds in your portfolio so that yourportfolio has the lowest possible variance?Solution:For(a),we use(4.16)and(4.17),with w S=0.8and w B= 0.2,to obtainE(R P)=(0.8)(0.15)+(0.2)(0.05)=13%Var(R P)=(0.8)2(0.0625)+(0.2)2(0.0016)+2(0.8)(0.2)(−0.2)q(0.0016)(0.0625) =0.03942.√0.03942=19.86%. Thus,the portfolio has a standard deviation ofFor(b),we do similar calculations,with w S=0.2and w B=0.8,toobtain E(R P)=7%,Var(R P)=0.002884and a standard deviation of √0.002884=5.37%.Hence,we observe that the portfolio with a higher weightage in stock has a higher expected return but also a higher standard deviation,i.e.,higher risk.For(c)we rewrite(4.17)asVar(R P)=w2S Var(R S)+(1−w S)2Var(R B)+2w S(1−w S)Cov(R S,R B). To minimize the variance,we diﬀerentiate Var(R P)with respect to w S to obtain2w S Var(R S)−2(1−w S)Var(R B)+2(1−2w S)Cov(R S,R B).Equating the above to zero,we solve for w S to obtainw S=Var(R B)−Cov(R S,R B)Var(R B)+Var(R S)−2Cov(R S,R B)=5.29%.The expected return of this portfolio is5.53%,its variance is0.001410, and its standard deviation is3.75%,which is lower than the standard deviation of the bond fund of4%.Hence,this portfolio dominates the bond fund,in the sense that it has a higher expected return and a lower standard deviation.Note that the fact that the above portfolio indeed minimizes the variance can be veriﬁed by examining the second-order condition.24.5Short sales•A short sale is the sale of a security that the seller does not own.It can be executed through a margin account with a brokerageﬁrm.•The seller borrows the security from the brokerageﬁrm to deliver to the buyer.•The sale is based on the belief that the security price will go down in the future so that the seller will be able to buy back the security at a lower price,thus keeping the diﬀerence in price as proﬁt.•Proceeds from the short sale are kept in the margin account,and cannot be invested to earn income.•The seller is required to place cash or securities into the margin account.The initial percentage margin m is the percentage ofthe proceeds of the short sold security that the seller must place into the margin account.•If P0is the price of the security when it is sold short,the initial deposit D is P0m.The seller will earn interest from the deposit.•At any point in time,there is a maintenance margin m∗,which is the minimum percentage of the seller’s equity in relation to the current value of the security sold short.•If the current security price is P1,the equity E is P0+D−P1and E/P1must be larger than m∗.•If E/P1falls below m∗,the seller will get a margin call from the broker instructing him to top up his margin account.Example4.12:A person sold1,000shares of a stock short at$20. If the initial margin is50%,how much should he deposit in his margin account?If the maintenance margin is30%,how high can the price go up before there is a margin call?Solution:The initial deposit is1,000×20×0.5=$10,000.At price P1,the percent of equity is10,000+1,000(20−P1),1,000P1which must be more than0.3.Thus,the maximum P1is30=$23.08.1.32•To calculate the rate of return of a short sale strategy we note that the capital is the deposit D in the margin account.The return includes the interest earned in the deposit.The seller,however, pays the dividend to the buyer if there is any dividend payout.The net rate of return will thus take account of the interest earned and the dividend paid.Example4.13:A person sells a stock short at$30.The stock pays a dividend of$1at the end of the year,after which the man covers his short position by buying the stock back at$27.The initial margin is50%and interest rate is4%.What is his rate of return over the year? Solution:The capital investment per share is$15.The gross return after one year is(30−27)−1+15×0.04=$2.6.Hence,the return overthe1-year period is2.6=17.33%.15Note that in the above calculation we have assumed that there was no margin call throughout the year.24.6Crediting Interest:Investment-Y ear Method andPortfolio Method•A fund pools the investments of individual investors.The investors may invest new money into the fund at any time.•While the fund invests the aggregate of the investments,there is an issue of how to credit interest to the individual investors’accounts.•A simple method is to credit the average return of the fund to all investors.This is called the portfolio method.•The portfolio method may not be equitable when the individual investments are made at diﬀerent times.•For example,at a time when returns to securities are going up,new investments are likely to achieve higher returns compared to old。

第1章 Introductory Econometrics for Finance(金融计量经济学导论-东北财经大学陈磊)

1.2 The Special Characteristics of Financial Data
• 宏观经济计量分析的数据问题：
• 小样本；测量误差与数据修正
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• 金融数据的观测频率高，数据量大 • 金融数据的质量高
这些意味着可以采用更强有力的分析技术，研究结果也更可靠。
• 金融数据包含很多噪音（noisy），更难以从随机的和无关的变动中分辨出趋势和规律 • 通常不满足正态分布 • 高频数据经常包含反映市场运行方式的、但人们并不感兴趣的其它模式(pattern) ，需要在建模时加以考虑
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Types of Data
• Problems that Could be Tackled Using a Time Series Regression - How the value of a country’s stock index has varied with that country’s macroeconomic fundamentals. - How the value of a company’s stock price has varied when it announced the value of its dividend payment. - The effect on a country’s currency of an increase in its interest rate • Cross-sectional data(截面数据) are data on one or more variables collected at a single point in time, e.g. - Cross-section of stock returns on the New York Stock Exchange - A sample of bond credit ratings for UK banks

货币金融学(第十二版)英文版题库及答案chapter 13

Economics of Money, Banking, and Financial Markets, 12e (Mishkin)Chapter 13 Central Banks and the Federal Reserve System13.1 Origins of the Federal Reserve System1) The First Bank of the United StatesA) was disbanded in 1811 when its charter was not renewed.B) had its charter renewal vetoed in 1832.C) was fundamental in helping the Federal Government finance the War of 1812.D) None of the above.Answer: AQues Status: Previous EditionAACSB: Reflective Thinking2) The Second Bank of the United StatesA) was disbanded in 1811 when its charter was not renewed.B) had its charter renewal vetoed in 1832.C) is considered to be the primary cause of the bank panic of 1907.D) None of the above.Answer: BQues Status: Previous EditionAACSB: Reflective Thinking3) The public's fear of centralized power and distrust of moneyed interests led to the demise of the first two experiments in central banking, otherwise known asA) the First Bank of the United States and the Second Bank of the United States.B) the First Bank of the United States and the Central Bank of the United States.C) the First Central Bank of the United States and the Second Central Bank of the United States.D) the First Bank of North America and the Second Bank of North America.Answer: AQues Status: Previous EditionAACSB: Reflective Thinking4) The financial panic of 1907 resulted in such widespread bank failures and substantial losses to depositors that the American public finally became convinced thatA) the First Bank of the United States had failed to serve as a lender of last resort.B) the Second Bank of the United States had failed to serve as a lender of last resort.C) the Federal Reserve System had failed to serve as a lender of last resort.D) a central bank was needed to prevent future panics.Answer: DQues Status: Previous EditionAACSB: Reflective Thinking5) What makes the Federal Reserve so unique compared to other central banks around the world is itsA) centralized structure.B) decentralized structure.C) regulatory functions.D) monetary policy functions.Answer: BQues Status: Previous EditionAACSB: Reflective Thinking13.2 Structure of the Federal Reserve System1) Which of the following is NOT an entity of the Federal Reserve System?A) Federal Reserve BanksB) the Comptroller of the CurrencyC) the Board of GovernorsD) the Federal Open Market CommitteeAnswer: BQues Status: Previous EditionAACSB: Reflective Thinking2) Which of the following is an entity of the Federal Reserve System?A) the U.S. Treasury SecretaryB) the FOMCC) the Comptroller of the CurrencyD) the FDICAnswer: BQues Status: Previous EditionAACSB: Reflective Thinking3) The three largest Federal Reserve banks (New York, Chicago, and San Francisco) combined hold more than ________ percent of the assets of the Federal Reserve System.A) 25B) 33C) 50D) 67Answer: CQues Status: Previous EditionAACSB: Analytical Thinking4) The Federal Reserve Banks are ________ institutions since they are owned by the ________.A) quasi-public; private commercial banks in the district where the Reserve Bank is locatedB) public; private commercial banks in the district where the Reserve Bank is locatedC) quasi-public; Board of GovernorsD) public; Board of GovernorsAnswer: AQues Status: Previous EditionAACSB: Reflective Thinking5) Each Federal Reserve bank has nine directors. Of these ________ are appointed by the member banks and ________ are appointed by the Board of Governors.A) three; sixB) four; fiveC) five; fourD) six; threeAnswer: DQues Status: Previous EditionAACSB: Reflective Thinking6) The nine directors of the Federal Reserve Banks are split into three categories: ________ are professional bankers, ________ are leaders from industry, and ________ are to represent the public interest and are not allowed to be officers, employees, or stockholders of banks.A) 5; 2; 2B) 2; 5; 2C) 4; 2; 3D) 3; 3; 3Answer: DQues Status: Previous EditionAACSB: Reflective Thinking7) Member commercial banks have purchased stock in their district Fed banks; the dividend paid by that stock is limited by law to ________ percent annually.A) fourB) fiveC) sixD) eightAnswer: CQues Status: Previous EditionAACSB: Reflective Thinking8) The Federal Reserve Bank of ________ houses the open market desk.A) BostonB) New YorkC) ChicagoD) San FranciscoAnswer: BQues Status: Previous EditionAACSB: Reflective Thinking9) The president from which Federal Reserve Bank always has a vote in the Federal Open Market Committee?A) PhiladelphiaB) BostonC) San FranciscoD) New YorkAnswer: DQues Status: Previous EditionAACSB: Reflective Thinking10) An important function of the regional Federal Reserve Banks isA) setting reserve requirements.B) clearing checks.C) determining monetary policy.D) setting margin requirements.Answer: BQues Status: Previous EditionAACSB: Reflective Thinking11) Which of the following functions is NOT performed by any of the twelve regional Federal Reserve Banks?A) check clearingB) conducting economic researchC) setting interest rates payable on time depositsD) issuing new currencyAnswer: CQues Status: Previous EditionAACSB: Reflective Thinking12) All ________ are required to be members of the Fed.A) state chartered banksB) national banks chartered by the Office of the Comptroller of the CurrencyC) banks with assets less than $100 millionD) banks with assets less than $500 millionAnswer: BQues Status: Previous EditionAACSB: Reflective Thinking13) Of all commercial banks, about ________ belong to the Federal Reserve System.A) 10%B) one halfC) one thirdD) 90%Answer: CQues Status: Previous EditionAACSB: Reflective Thinking14) Prior to 1980, member banks left the Federal Reserve System due toA) the high cost of discount loans.B) the high cost of required reserves.C) a desire to avoid interest rate regulations.D) a desire to avoid credit controls.Answer: BQues Status: Previous EditionAACSB: Reflective Thinking15) The Fed's support of the Depository Institutions Deregulation and Monetary Control Act of 1980 stemmed in part from itsA) concern over declining Fed membership.B) belief that all banking regulations should be eliminated.C) belief that interest rate ceilings were too high.D) belief that depositors had to become more knowledgeable of banking operations. Answer: AQues Status: Previous EditionAACSB: Reflective Thinking16) Banks subject to reserve requirements set by the Federal Reserve System includeA) only nationally chartered banks.B) only banks with assets less than $100 million.C) only banks with assets less than $500 million.D) all banks whether or not they are members of the Federal Reserve System.Answer: DQues Status: Previous EditionAACSB: Reflective Thinking17) The Depository Institutions Deregulation and Monetary Control Act of 1980A) established higher reserve requirements for nonmember than for member banks.B) established higher reserve requirements for member than for nonmember banks.C) abolished reserve requirements.D) established uniform reserve requirements for all banks.Answer: DQues Status: Previous EditionAACSB: Reflective Thinking18) There are ________ members of the Board of Governors of the Federal Reserve System.A) 5B) 7C) 12D) 19Answer: BQues Status: Previous EditionAACSB: Reflective Thinking19) Members of the Board of Governors areA) chosen by the Federal Reserve Bank presidents.B) appointed by the newly elected president of the United States, as are cabinet positions.C) appointed by the president of the United States and confirmed by the Senate.D) never allowed to serve more than 7-year terms.Answer: CQues Status: Previous EditionAACSB: Reflective Thinking20) Each governor on the Board of Governors can serveA) only one nonrenewable fourteen-year term.B) one full nonrenewable fourteen-year term plus part of another term.C) only one nonrenewable eight-year term.D) one full nonrenewable eight-year term plus part of another term.Answer: BQues Status: Previous EditionAACSB: Reflective Thinking21) The Chairman of the Board of Governors is chosen from among the seven governors and serves a ________, renewable term.A) one-yearB) two-yearC) four-yearD) eight-yearAnswer: CQues Status: Previous EditionAACSB: Reflective Thinking22) While the discount rate is "established" by the regional Federal Reserve Banks, in truth, the rate is determined byA) Congress.B) the president of the United States.C) the Senate.D) the Board of Governors.Answer: DQues Status: Previous EditionAACSB: Reflective Thinking23) Which of the followings is a duty of the Board of Governors of the Federal Reserve System?A) setting margin requirements, the fraction of the purchase price of the securities that has to be paid for with cashB) setting the maximum interest rates payable on certain types of time deposits under Regulation QC) regulating credit with the approval of the president under the Credit Control Act of 1969D) All governors advise the president of the United States on economic policy.Answer: AQues Status: Previous EditionAACSB: Reflective Thinking24) Which of the followings is NOT a current duty of the Board of Governors of the Federal Reserve System?A) setting margin requirements, the fraction of the purchase price of the securities that has to be paid for with cashB) setting the maximum interest rates payable on certain types of time deposits under Regulation QC) approving the discount rate "established" by the Federal Reserve banksD) voting on the conduct of open market operationsAnswer: BQues Status: Previous EditionAACSB: Reflective Thinking25) The Federal Open Market Committee usually meets ________ times a year.A) fourB) sixC) eightD) twelveAnswer: CQues Status: Previous EditionAACSB: Reflective Thinking26) The Federal Reserve entity that makes decisions regarding the conduct of open market operations is theA) Board of Governors.B) chairman of the Board of Governors.C) Federal Open Market Committee.D) Open Market Advisory Council.Answer: CQues Status: Previous EditionAACSB: Reflective Thinking27) The Federal Open Market Committee consists of theA) five senior members of the seven-member Board of Governors.B) seven members of the Board of Governors and seven presidents of the regional Fed banks.C) seven members of the Board of Governors and five presidents of the regional Fed banks.D) twelve regional Fed bank presidents and the chairman of the Board of Governors. Answer: CQues Status: Previous EditionAACSB: Reflective Thinking28) The majority of members of the Federal Open Market Committee areA) Federal Reserve Bank presidents.B) members of the Federal Advisory Council.C) presidents of member banks.D) the seven members of the Board of Governors.Answer: DQues Status: Previous EditionAACSB: Reflective Thinking29) Each Fed bank president attends FOMC meetings; although only ________ Fed bank presidents vote on policy, all ________ provide input.A) three; tenB) five; tenC) three; twelveD) five; twelveAnswer: DQues Status: Previous EditionAACSB: Reflective Thinking30) Although reserve requirements and the discount rate are not actually set by the ________, decisions concerning these policy tools are effectively made there.A) Federal Reserve Bank of New YorkB) Board of GovernorsC) Federal Open Market CommitteeD) Federal Reserve BanksAnswer: CQues Status: Previous EditionAACSB: Reflective Thinking31) The research document given to the Federal Open Market Committee that contains information on the state of the economy in each Federal Reserve district is called theA) beige book.B) green book.C) blue book.D) black book.Answer: AQues Status: Previous EditionAACSB: Reflective Thinking32) The teal book is the Fed research document containingA) the forecast of national economic variables for the next three years.B) forecasts of the money aggregates conditional on different monetary policy stances.C) information on the state of the economy in each Federal Reserve district.D) both A and B.E) A, B and C.Answer: DQues Status: Previous EditionAACSB: Reflective Thinking33) The Federal Open Market Committee's "balance of risks" is an assessment of whether, in the future, its primary concern will beA) higher exchange rates or higher unemployment.B) higher inflation or a stronger economy.C) higher inflation or a weaker economy.D) lower inflation or a stronger economy.Answer: CQues Status: Previous EditionAACSB: Reflective Thinking34) Subject to the approval of the Board of Governors, the decision of choosing the president ofa district Federal Reserve Bank is made byA) all nine district bank directors.B) the six district bank directors elected by the member banks.C) three district bank directors who are professional bankers.D) district bank directors who are not professional bankers.E) class A and class B directors.Answer: DQues Status: Previous EditionAACSB: Ethical Understanding and Reasoning Abilities35) Why does the Federal Reserve Bank of New York play a special role within the Federal Reserve System?Answer: The New York district contains the largest banks in the country. The New York Fed supervises and examines these banks to insure their soundness and the safety of the nation's financial system. The New York Fed conducts open market operations and foreign exchange transactions for the Fed and Treasury. The New York Fed belongs to the Bank for International Settlements, so its president and the chairman of the Board of Governors represent the U.S. at the monthly meetings of the world's central banks. The New York Fed president is the only president of a regional Fed who is a permanent voting member of the FOMC.Ques Status: Previous EditionAACSB: Reflective Thinking36) Who are the voting members of the Federal Open Market Committee and why is this committee important? Where does the power lie within this committee?Answer: The FOMC determines the monetary policy of the United States through its decisions about open market operations. It also effectively determines the discount rate and reserve requirements. The seven members of the Board of Governors, the president of the New York Fed, and four of the other eleven regional bank presidents are voting members on a rotating basis. Within the FOMC, the chairman of the Board of Governors wields the power.Ques Status: Previous EditionAACSB: Reflective Thinking13.3 How Independent is the Fed?1) Instrument independence is the ability of ________ to set monetary policy ________.A) the central bank; goalsB) Congress; goalsC) Congress; instrumentsD) the central bank; instrumentsAnswer: DQues Status: Previous EditionAACSB: Reflective Thinking2) The ability of a central bank to set monetary policy instruments isA) political independence.B) goal independence.C) policy independence.D) instrument independence.Answer: DQues Status: Previous EditionAACSB: Reflective Thinking3) Goal independence is the ability of ________ to set monetary policy ________.A) the central bank; goalsB) Congress; goalsC) Congress; instrumentsD) the central bank; instrumentsAnswer: AQues Status: Previous EditionAACSB: Reflective Thinking4) The ability of a central bank to set monetary policy goals isA) political independence.B) goal independence.C) policy independence.D) instrument independence.Answer: BQues Status: Previous EditionAACSB: Reflective Thinking5) Members of Congress are able to influence monetary policy, albeit indirectly, through their ability toA) withhold appropriations from the Board of Governors.B) withhold appropriations from the Federal Open Market Committee.C) propose legislation that would force the Fed to submit budget requests to Congress, as must other government agencies.D) instruct the General Accounting Office to audit the foreign exchange market functions of the Federal Reserve.Answer: CQues Status: Previous EditionAACSB: Reflective Thinking6) Explain two concepts of central bank independence. Is the Fed politically independent? Why do economists think central bank independence is important?Answer: Instrument independence is the ability of the central bank to set its instruments, and goal independence is the ability of a central bank to set its goals. The Fed enjoys both types of independence. The Fed is largely independent of political pressure due to its earnings and the conditions of appointment of the Board of Governors and its chairman. However, some political pressure can be applied through the threat or enactment of legislation affecting the Fed. Independence is important because there is some evidence that independent central banks pursue lower rates of inflation without harming overall economic performance.Ques Status: Previous EditionAACSB: Reflective Thinking13.4 Should the Fed Be Independent?1) The case for Federal Reserve independence does NOT include the idea thatA) political pressure would impart an inflationary bias to monetary policy.B) a politically insulated Fed would be more concerned with long-run objectives and thus be a defender of a sound dollar and a stable price level.C) policy is always performed better by an elite group such as the Fed.D) a Federal Reserve under the control of Congress or the president might make the so-called political business cycle more pronounced.Answer: CQues Status: Previous EditionAACSB: Reflective Thinking2) The political business cycle refers to the phenomenon that just before elections, politicians enact ________ policies. After the elections, the bad effects of these policies (for example,________ ) have to be counteracted with ________ policies.A) expansionary; higher unemployment; contractionaryB) expansionary; a higher inflation rate; contractionaryC) contractionary; higher unemployment; expansionaryD) contractionary; a higher inflation rate; expansionaryAnswer: BQues Status: Previous EditionAACSB: Analytical Thinking3) The strongest argument for an independent Federal Reserve rests on the view that subjecting the Fed to more political pressures would impartA) an inflationary bias to monetary policy.B) a deflationary bias to monetary policy.C) a disinflationary bias to monetary policy.D) a countercyclical bias to monetary policy.Answer: AQues Status: Previous EditionAACSB: Ethical Understanding and Reasoning Abilities4) Critics of the current system of Fed independence contend thatA) the current system is undemocratic.B) voters have too much say about monetary policy.C) the president has too much control over monetary policy on a day-to-day basis.D) the Board of Governors is held responsible for policy missteps.Answer: AQues Status: Previous EditionAACSB: Diverse and Multicultural Work Environments5) Recent research indicates that inflation performance (low inflation) has been found to be best in countries withA) the most independent central banks.B) political control of monetary policy.C) money financing of budget deficits.D) a policy of always keeping interest rates low.Answer: AQues Status: Previous EditionAACSB: Reflective Thinking6) Make the case for and against an independent Federal Reserve.Answer: Case for: 1. An independent Federal Reserve can shield the economy from the political business cycle, and it will be less likely to have an inflationary bias to monetary policy. 2. Control of the money supply is too important to leave to inexperienced politicians.Case against: 1. It is undemocratic to have monetary policy be controlled by a small number of individuals that are not accountable. 2. In the past, an independent Fed has not used its freedom wisely. 3. Its independence may encourage it to pursue its own self-interest rather than the public's interest.Ques Status: Previous EditionAACSB: Ethical Understanding and Reasoning Abilities13.5 Explaining Central Bank Behavior1) The theory of bureaucratic behavior suggests that the objective of a bureaucracy is to maximizeA) the public's welfare.B) profits.C) its own welfare.D) conflict with the executive and legislative branches of government.Answer: CQues Status: Previous EditionAACSB: Reflective Thinking2) The theory of bureaucratic behavior when applied to the Fed helps to explain why the FedA) was supportive of congressional attempts to limit the central bank's autonomy.B) was so secretive about the conduct of future monetary policy.C) sought less control over banks in the 1980s.D) was willing to take on powerful groups that may threaten its autonomy.Answer: BQues Status: Previous EditionAACSB: Reflective Thinking3) What is the theory of bureaucratic behavior and how can it be used to explain the behavior of the Federal Reserve?Answer: The theory of bureaucratic behavior concludes that the main objective of any bureaucracy is to maximize its own welfare, which is related to power and prestige. This can explain why the Federal Reserve has defended its autonomy, avoids conflict with Congress and the president, and its push to gain more control over banks.Ques Status: Previous EditionAACSB: Analytical Thinking13.6 Structure and Independence of the European Central Bank1) Under the European System of Central Banks, the Executive Board is similar in structure to the ________ of the Federal Reserve System.A) Board of GovernorsB) Federal Open Market CommitteeC) Federal Reserve BanksD) Federal Advisory CouncilAnswer: AQues Status: Previous EditionAACSB: Reflective Thinking2) Under the European System of Central Banks, the Governing Council is similar in structure to the ________ of the Federal Reserve System.A) Board of GovernorsB) Federal Open Market CommitteeC) Federal Reserve BanksD) Federal Advisory CouncilAnswer: BQues Status: Previous EditionAACSB: Reflective Thinking3) Under the European System of Central Banks, the National Central Banks have the same role as the ________ of the Federal Reserve System.A) Board of GovernorsB) Federal Open Market CommitteeC) Federal Reserve BanksD) Federal Advisory CouncilAnswer: CQues Status: Previous EditionAACSB: Reflective Thinking4) Members of the Executive Board of the European System of Central Banks are appointed to ________ year, nonrenewable terms.A) fourB) eightC) tenD) fourteenAnswer: BQues Status: Previous EditionAACSB: Reflective Thinking5) Which of the following statements comparing the European System of Central Banks and the Federal Reserve System is TRUE?A) The budgets of the Federal Reserve Banks are controlled by the Board of Governors, while the National Central Banks control their own budgets and the budget of the European Central Bank.B) The European Central Bank has similar power over the National Central Banks when compared to the level of power the Board of Governors has over the Federal Reserve Banks. C) Just like the Federal Reserve System, monetary operations are centralized in the European System of Central Banks with the European Central Bank.D) None of the above.Answer: AQues Status: RevisedAACSB: Reflective Thinking6) The Governing Council usually meets ________ times a year.A) fourB) sixC) eightD) twelveAnswer: DQues Status: Previous EditionAACSB: Reflective Thinking7) In the Governing Council, the decision of what policy to implement is made byA) majority vote of the Executive Board members.B) majority vote of the heads of the National Banks.C) consensus.D) majority vote of all members of the Governing Council.Answer: CQues Status: Previous EditionAACSB: Reflective Thinking8) The central bank which is generally regarded as the most independent in the world because its charter cannot be changed by legislation is theA) Bank of England.B) Bank of Canada.C) European Central Bank.D) Bank of Japan.Answer: CQues Status: Previous EditionAACSB: Reflective Thinking9) Explain the similarities and differences between the European System of Central Banks and the Federal Reserve System.Answer: The similarities between the two are in their structure. The National Central Banks of the member countries of the Eurosystem have the same role as the Federal Reserve Banks in the Federal Reserve System. The Executive Board and the Governing Council of the Eurosystem resemble the Board of Governors and the Federal Open Market Committee of the Federal Reserve System, respectively. There are three major differences between the two. The first difference is concerning the control of the budgets. In the Fed, the Board of Governors controls the budgets of the Reserve Banks while in the Eurosystem, the National Banks control the budget of the European Central Bank. The second difference is the monetary operations of the Eurosystem are conducted by the National Banks, so they are not as centralized as the monetary operations in the Federal Reserve System.Ques Status: RevisedAACSB: Reflective Thinking13.7 Structure and Independence of Other Foreign Central Banks1) On paper, the Bank of Canada has ________ instrument independence and ________ goal independence when compared to the Federal Reserve System.A) less; lessB) less; moreC) more; lessD) more; moreAnswer: AQues Status: Previous EditionAACSB: Reflective Thinking2) The oldest central bank, having been founded in 1694, is theA) Bank of England.B) Deutsche Bundesbank.C) Bank of Japan.D) Federal Reserve System.Answer: AQues Status: Previous EditionAACSB: Reflective Thinking3) While legislation enacted in 1998 granted the Bank of Japan new powers and greater autonomy, its critics contend that its independence isA) limited by the Ministry of Finance's veto power over a portion of its budget.B) too great because it need not pursue a policy of price stability even if that is the popular will of the people.C) too great since the Ministry of Finance no longer has veto power over the bank's budget.D) limited since the Ministry of Finance can dismiss senior bank officials.Answer: AQues Status: Previous EditionAACSB: Reflective Thinking4) Regarding central bank independenceA) the Fed is more independent than the European Central Bank.B) the European Central Bank is more independent than the Fed.C) the trend in industrialized nations has been to reduce central bank independence.D) the Bank of England has the longest tradition of independence of any central bank in the world.Answer: BQues Status: Previous EditionAACSB: Reflective Thinking5) The trend in recent years is that more and more governmentsA) have been granting greater independence to their central banks.B) have been reducing the independence of their central banks to make them more accountable for poor economic performance.C) have mandated that their central banks focus on controlling inflation.D) have required their central banks to cooperate more with their Ministers of Finance. Answer: AQues Status: Previous EditionAACSB: Reflective Thinking6) Which of the following statements about central bank structure and independence is TRUE?A) In recent years, with the exception of the Bank of England and the Bank of Japan, most countries have reduced the independence of their central banks, subjecting them to greater democratic control.B) Before the Bank of England was granted greater independence, the Federal Reserve was the most independent of the world's central banks.C) Both theory and experience suggest that more independent central banks produce better monetary policy.D) While the European Central Bank is independent, it is not as independent as the Federal Reserve.Answer: CQues Status: Previous EditionAACSB: Reflective Thinking。

methods of mathematical finance 金融数学方法

methods of mathematical finance 金融数学方法There are various methods used in mathematical finance to model and analyze financial markets. Some of the key methods include:1. Probability Theory: Mathematical finance heavily relies on probability theory to model uncertain future outcomes. It allows for the calculation of expected values, standard deviations, and other statistical properties of financial variables.2. Stochastic Calculus: Stochastic calculus is used to model and analyze the dynamics of financial variables that follow stochastic processes. It provides tools to derive differential equations and solve them using techniques such as Itô's lemma and stochastic differential equations.3. Option Pricing Theory: Option pricing theory is a fundamental concept in mathematical finance. It includes models like the Black-Scholes model, which determine the fair price of financial derivatives based on underlying asset prices, volatility, time, and other factors.4. Monte Carlo Simulation: Monte Carlo simulation is a popular method used to analyze and value complex financial instruments and portfolios. It involves generating a large number of random simulations to estimate the probability distribution of potential outcomes.5. Optimization Techniques: Optimization techniques are used to find the optimal allocation strategy for investment portfolios. These techniques aim to maximize returns while consideringconstraints such as risk tolerance or liquidity requirements.6. Time Series Analysis: Time series analysis is used to model and analyze financial data that evolves over time. It involves techniques like autoregressive integrated moving average (ARIMA) models and volatility modeling to forecast future prices and evaluate risk.7. Financial Econometrics: Econometrics applies statistical methods to analyze economic and financial data. In mathematical finance, financial econometrics is used to estimate and test various models, validate assumptions, and forecast future market behavior.8. Numerical Methods: Numerical methods like finite difference methods and finite element methods are used to solve complex mathematical equations or partial differential equations that arisein option pricing and risk management.These are just a few of the many methods used in mathematical finance. The field continues to evolve, with new techniques and models constantly being developed to capture the complexities and dynamics of financial markets.。

Mathematical Techniques in Finance (4)

12
Incomplete markets without redundant basis assets (2.1) • Example 2.2 (p. 27): delta hedging
13
Hedging problem with redundant basis assets and incomplete market (2.1)
15
Least Squares Hedge – Example (2.2.1) • Example 2.4 (p. 31)
– Focus asset: b* = [1 2 3] – Basis assets: (A•1)* = [1 1 0]
2
Complete market (1.9, p.14) • If dimension of marketed subspace spanned by basis assets equals the number of scenarios we say market is complete • In a complete market any focus asset which does not require extra scenarios can be hedged perfectly
• Suppose A is invertible (square, full rank)
–Complete market, no redundant basis assets
• Perfect hedge Ax = b: • Multiply by A-1 from the left x = A-1b • Two ways to find the price of focus asset b 1) By replication: Focus asset price = S*x 2) Using a pricing kernel: Focus asset price = ψ*b

《货币金融学(第十三版)》英文版教学课件mishkin_econ13e_ppt_10

The Spread of Government Deposit Insurance Throughout the World: Is This a Good Thing?
• Has government deposit insurance helped improve the performance of the financial system and prevent banking crises? The answer seems to be “no.” Research at the World Bank seems to answer “no,” since on average, the adoption of explicit government deposit insurance is associated with less banking sector stability and a higher incidence of banking crises. Furthermore, on average, deposit insurance seems to retard financial development.
Copyright © 2022, 2019, 2016 Pearson Education, Inc. All Rights Reserved
Types of Financial Regulation: Restrictions on Asset Holdings
• Attempts to restrict financial institutions from too much risk taking: – Bank regulations ▪ Promote diversification ▪ Prohibit holdings of common stock – Capital requirements ▪ Minimum leverage ratio (for banks) ▪ Basel Accord: risk-based capital requirements ▪ Regulatory arbitrage

金融书单

金融书单.txt如果中了一千万，我就去买30套房子租给别人，每天都去收一次房租。

哇咔咔~~充实骑白马的不一定是王子，可能是唐僧；带翅膀的也不一定是天使，有时候是鸟人。

Financial mathematics is a strong applied science, and its special features is that, unlike many other disciplines such as biological applications than it is more akin to the difficulty of mathematical physics, and on the other hand, It can be applied and engineering analogy, because good results to be "profitable", you may cheat a Journal, but you can not cheat the Market ... and the more unique, It requires a person is extremely Mr. miscellaneous knowledge, a very good list of important大体而言,所需要的知识分为三类Broadly speaking, what is needed is divided into three types of knowledge1.数量1. Volume2.经济金融2. Financial3.编程,这方面我比较弱,至今还算不上professional programmer3. Programming, which I quite weak, is still not a professional programmer 大致上来说,一个人需要吃透如下LEVEL的书籍:Generally, a person needs a good understanding of the books are as follows LEVEL :1.Thinking in C++ Vol 1 & 21.Thinking in C + + Vol 1 & 22.The C++ Programming Language2.The C + + Programming Language另外,还需要data structure & alogrithms的知识 好在编程高手尽多,这方面也不太需要我业余的意见,呵呵In addition, also need data structure knowledge & alogrithms "br> Fortunately, the high Programming do more hands, which I also do not need an amateur, Oh现在我列一下数量方面的书单Now I have listed what quantity of the books1.概率论1. Probability Theory很不幸的事实是,概率论基本上没有好的中文教材(1998之前,之后我就不清楚了)Unfortunately the fact that probability theory is basically no good Chinese language teaching (1998, after I do not know.)Ross的书适合本科和硕士生,胜在例子详尽Ross's book for undergraduate and master's degree students, detailed examples in victoryBillingsley的概率论和弱收敛的两本教材是非常好的入门书Billingsley probability theory and the weak convergence of two materials is a very good introductory book chung的概率论教材很严格,读起干巴巴的来会有点累,不过是真长工夫的密籍On the probability chung materials are very strict, and read dry to be a little bit tired, but efforts Managa membership of the FederatedDurrett的书很流行,不过里面的小错误很多 如果你真的想理解概率论,feller的两本书是不可不读的,可以说,从高中水平到博士以上学位的读者,都会从中获益---如果要推选概率论里面最有影响的教材,feller的书无可比拟,不过读时要一路自己算,feller书里面错误非常多,虽然都显然是笔误Durrett very popular book, but there's a lot of small mistakes "br> If you really want to understand probability theory, Feller of the two book is important reading, we can say that the high school level to doctoral degreesreaders, will benefit from -- if elected probability theory to the most influential inside the material, the book unparalleled Feller, But reading all the way to their calculation, Feller book is wrong, although obvious typoBreiman的书也是经典,概率味比chung的浓Breiman book is classic, probability flavor than the concentration chungloeve的书可以作为工具书使用Loeve the book can be used as a tool2.随机分析 黄志远的随机分析入门是一本很好的书 严加安的鞅论可以做工具书用2. Stochastic Analysis "br> HUANG Zhi-yuan of stochastic analysis portal is a very good chapter of the" br> strict installation of the martingale theory can be Tools to do with bookRoss的Inrto to probability model可以做本科生随机过程入门,例子很多Ross Inrto to the probability model can do random undergraduate introductory course , for example, many Karlin & Taylor的两本书非常适合硕士生用Karlin & Taylor, the two books are very suitable for student useresnick的几本书概率味很不错,应用性也很强Resnick of several books probability taste quite good, and also very strong applicationoksendal的书是SDE里面最简单的Oksendal book is the most simple SDE inside the Karatzas Shreve有好几本书,金融数学的博士不可不读Karatzas Shreve has several books, financial mathematics can not read Dr.Revuz Yor的连续鞅是很好的书Revuz 01:53 continuous martingale is a very good book Protter的书是严格随机分析里面最容易读的,文笔很好Protter book is strictly random analysis inside the easiest time, writes wellwilliams的书深入浅出,入门很合适Williams book for visitors, the entry is appropriate Chung Williams的书比oksendal稍微难一点,作为应用随机分析的标准教材很不错Chung Williams book than oksendal slightly difficult, Application of stochastic analysis as a standard curriculum is pretty good3-控制论3-cybernetics控制论在portfolio selection problem和risk management里面有很多的应用,optimal stopping在美式derivative非常重要 金融数学里面用的主要是随机控制,和粘性解(因为operator is often degenerate)Cybernetics in portfolio selection problem and manage risk ment that there are many applications, optimal stopping the American derivative very important "for the financial br> learn inside the main random control, and viscous solutions (because operator is often degenerate)经典的随机控制书是Randomized controlled classic book1.FLEMING and RISHEL, (1975) Deterministic and Stochastic Optimal Control.1.FLEMING and RISHEL, (1975) Optimal Deterministic and Stochastic C ontrol.2.KRYLOV, (1980) Controlled diffusion processes2.KRYLOV. (1980) Controlled diffusion processes3.BORKAR, (1989) Optimal control of diffusion processes.3.BORKAR. (1989) Optimalcontrol of diffusion processes .4.BENSOUSSAN and LIONS, (1982) Controle Impulsionnel et Inequations Variationnelles4.BENSOUSSAN and LIONS, (1982) Controle et Inequations V Impulsionnel ariationnelles粘性解的标准文献是Viscous solution is the standard literature1. Crandall, Ishii and Lions, User''s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992),1. Crandall, Ishii and Lions. User '' s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992),2.Fleming and Soner, Controlled Markov Processes and Viscosity Solutions, 1992.2.Fleming and ?zen. Controlled Markov Processes and Viscosity Sol utions, 1992.4.数值算法4. Numerical Algorithms首先,finite difference是极其常用的算法,这方面书籍很多,比如Ames的经典教材 计算矩阵: Golub and Van Loan, Matrix Computations, 1996First, the finite difference is the most commonly used algorithm, which a lot of books, For example, Ames classic textbook "br> matrix calculation : Golub and Van Loan. Matrix 1,776,240, 1,996 Kushner and Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, 1992. Kushner''s Markov chain approximation method是控制论里最有用的算法Kushner and Dupuis. Numerical Methods for Stochastic Control Prob lems in Continuous Time, 1992. Kushner '' s Markov chain approximation m ethod control theory is the most useful algorithmROGERS and TALAY, Numerical Methods in Financial Mathematics. 1997.论文集TALAY distinct and, Numerical Methods in Financial Mathematics. 1 997. PapersKloeden and Platen, Numerical Solution of Stochastic Differential Equations, 1997. 偏理论,实用性差一点Kloeden and Platen. Numerical Solution of Stochastic Differentia Equations 1, 1997. Partial theory, practical, almostGlasserman, Monte Carlo Methods in Financial Engineering, 2003这本书非常非常实用,可以说是金融数学数值算法的最新经典Glasserman, Monte Carlo Methods in Financial Engineering, 2,003 this book very useful, it can be said that the financial mathematical algorithm to the latest classic5-时间序列 当然,学习时间序列之前,统计特别是多变量统计要先学好5-time series "br> Of course, time sequence, in particular statistical multivariate statistical study firstA Guide to Econometrics: by Peter Kennedy可能是最通俗易懂的入门书 A Guide to Econometrics : by Peter Kennedy is probably the most accessible book for beginners Econometric Analysis,by William H. Greene和Time Series Analysis by James Douglas Hamilton是非常标准的教材,许多学校都在用Econometric Analysis, by William H.. Greene and Time Series Analysis by Ja mes Douglas Hamilton is a very standard materials, many schools are usingBox Jerkins的Time Series Analysis: Forecasting & Control,当之无愧的经典Box Jerkins of Time Series Analysis : Forecasting & Control, the well-deserved classics Time Series and Dynamic Models by Christian Gourieroux,Gourieroux写了许多书,但似乎他的书不如他的研究文章水准高Time Series and Dynamic Models by Christian Gou rieroux, Gourieroux wrote many books, But it seems that his books as his high standard of research articlesThe Econometrics of Financial Markets,by John Y. Campbell, Andrew W. Lo, A. Craig MacKinlay,新经典The Econometrics of Financial Markets. by John Y. Campbell, Andrew W.. Lo, A. Craig MacKinlay, the new classical现在我们来看一下经济金融方面的书单Now we look at the economic and financial aspects of the books首先要强调,金融不是经济,经济考虑的是国计民生,环球宇宙之类的大问题,而金融考虑的是money making, risk control之类的充满铜臭味的小问题First emphasize that the economy is not financial, economic considerations are the national economy, the global universe category of major problems and the financial consideration is making money. risk control such as the stench of money with little problems当然,经济背景也是需要的,比如说Of course, the economic background is necessary, for instance,Varian: Microeconomic Analysis(1992)Varian : Microeconomic Analysis (1992) Samuelson: EconomicsSamuelson : Economics如果有时间,最有价值的书大概是Keynes的general principle,If I had the time, the most valuable book about Keynes is the general principle,看的时候的感觉会跟第一次学微积分差不多See when the feeling will be the first school with calculus almost现在我们进入金融书单Now we enter the financial books1.理论金融1. Monetary TheoryMerton: Continuous time financeMerton : Continuous time financeHuang Litzenberger: Foundation for financial economicsHuang Litzenberger : Foundation for financial economicsIngersoll: Theorey of financial decision makingIngersoll : Theorey of financial decision makingRoss: Neoclassical FinanceRoss : Neoclassical FinanceRoss, Westerfield, Jaffe: Corporate FinanceRoss, Westerfield, Jaffe : Corporate FinanceDuffie: security marketDuffie : security marketDuffie: Dynamic Asset Pricing TheoryDuffie : Dynamic Asset Pricing Theory当然,金融文献浩如烟海,上面的书单是针对ASSET PRICING一块的,因为这一块最为定量化.至于做underwriting, M&A,一般不是很需要数量出身的人,至少到目前为止:)Of course, the multitude of financial literature, the above book just with a ASSET PRICING, Thisis because one of the most quantitative. As regards the question of underwriting, M & A, generally not very origin of the number needed, at least so far :)2.入门和综合类2. And integrated portal然后就要开始看一些实际的入门书了Then we should start to see some real beginners book Hull, Options, Futures and Other DerivativesHull, Options, Futures and Other DerivativesBaxter and Rennie, Financial CalculusBaxter and Rennie, Financial Calculus Shreve:Stochastic Calculus Models for Finance vol 1 & 2Shreve : Stochastic Calculus Models for Finance, vol 1 & 2Wilmott: quantitative financeWilmott : quantitative finance然后ThenBjork: Arbitrage theory in continuous timeMP : Arbitrage theory in continuous time Cvitanic, Zapatero: Introduction to the economics and mathematics of financial marketsCvitanic, Zapatero : Introduction to the economics and mathematics of financial marketsElliott, Kopp: Mathematics of Financial marketsElliott, Kopp : Mathematics of Financial marketsKaratzas Shreve: Method of math financeKaratzas Shreve : Method of math finance Musiela and Rutkowski: martingale method for financeMusiela and Rutkowski : martingale method for financeBielecki, Rutkowski: Credit Risk : Modeling , Valuation and HedgingBielecki, Rutkowski : Credit Risk : Modeling, Valuation and HedgingDuffie Singleton: Credit RiskDuffie Singleton : Credit RiskAmman: Credit risk valuationAmman : Credit risk évaluationTalebTaleb ynamic HedgingYnamic Hedging3. Fixed income3. Fixed incomeTuckman: Fixed Income Securities: Tools for Today''s Markets是入门的最佳选择Tuckman : Fixed Income Securities : Tools for Today '' s Markets portal is the best choice然后,就不得不面对Fabozzi的无数厚书乐:)Then, it had to face numerous Fabozzi thick book of music :)Fixed Income MathematicsFixed Income MathematicsFixed Income SecuritiesFixed Income SecuritiesBond Markets : Analysis and StrategiesBond Markets : Analysis and Strategies The Handbook of Fixed Income Securities,The Handbook of Fixed Income Securities. Handbook of Mortgage Backed SecuritiesHandbook of Mortgage Backed Securities Collateralized Debt Obligations: Structures and AnalysisCollateralized Debt Obligations : Structures and AnalysisInterest Rate, Term Structure, and Valuation ModelingInterest Rate, Term Structure, Valuation ModelingJessica James, Nick Webber Interest Rate Modelling: Financial Engineering,这本书乱而全Jessica James, Nick Webber Interest Rate Modeling : Financial Engineering, this book full of chaos andBrigo, Mercurio:Interest Rate Models 数学上难一些Brigo, Mercurio:Interest Rate Models mathematical some hardTavakoli: Collateralized Debt Obligations and Structured FinanceTavakoli : Collateralized Debt Obligations and Structur ed FinanceTavakoli: Credit Derivatives & Synthetic Structures: A Guide to Instruments and ApplicationsTavakoli : Credit Derivatives & Synthetic Structures : A Guide to Instruments and ApplicationsHayre: Salomon Smith Barney Guide to Mortgage-Backed and Asset-Backed SecuritiesHayre : Salomon Smith Barney Guide to Mortgage-Backed and ABS Securities4:其他类4 : OtherRebonato有几本很好的书:Rebonato several very good book :Volatility and Correlation : The Perfect Hedger and the FoxVolatility and Correlation : The Perfect Hedger and the FoxModern Pricing of Interest-Rate Derivatives : The LIBOR Market Model and BeyondModern Pricing of Interest-Rate Derivatives : The LIBOR Market Model and BeyondInterest-Rate Option Models : Understanding, Analysing and Using Models for Exotic Interest-Rate OptionsInterest-Rate Option Models : Understanding, Analyzing and Using Models for Exotic Interest -Rate OptionsSch?nbucher:Credit Derivatives Pricing Models: Model, Pricing and Implementation 写得很乱但是无可替代Sch?nbucher : Credit Derivatives Pricing Models : Model, Pricing and Implementation written into confusion but irreplaceableGENCAY: An Introduction to High-Frequency Finance第一本关于high frequency的书GENCAY : An Introduction to the first High-Frequency Finance on the high frequency bookO''Hara:Market Microstructure TheoryO '' Hara:Market Microstructure Theory Harris:Trading and Exchanges: Market Microstructure for PractitionersHarris:Trading and Exchanges : Market Microstructure for Practitioners。

finance第5章

13
Finance
School of Management
The Influence of Mandatory Saving on Voluntary Saving
Suppose that social security benefits are equal to what you would have if
n 30 i 3% PV 0 FV ? PMT 2000 Result $95,151
n 15
i 3%
PV 95,151
FV 0
PMT ?
Result $7,970
14
Finance
School of Management
Age 35 65 70 80
Salary 30,000 30,000 0 0
1
Finance
School of Management
Chapter 5 Contents
Analyze how much to save for retirement. Determine whether to defer taxes or pay them now. Determine whether to get a professional degree. Determine whether to buy or rent an apartment.
Result $11.94
6
Finance
School of Management
47 . 58 ( 30 , 000 C ) = 11 . 94 C
C = $ 23 , 982
— Permanent income
C = ∑ t t =1 (1 + i )

《货币金融学(第十三版)》英文版教学课件mishkin_econ13e_ppt_09

$90M
$80M $10M
Bank Capital
Blank
$10M
Blank
– Suppose a bank’s required reserves are 10%.
– If a bank has ample excess reserves, a deposit outflow does not necessitate changes in other parts of its balance sities
Required reserves
Loans
+$10 +$90
Checkable deposits
Blank
+$100
Blank
• Asset transformation: selling liabilities with one set of characteristics and using the proceeds to buy assets with a different set of characteristics
• The bank borrows short and lends long
Copyright © 2022, 2019, 2016 Pearson Education, Inc. All Rights Reserved
General Principles of Bank Management
• Liquidity Management • Asset Management • Liability Management • Capital Adequacy Management • Credit Risk • Interest-rate Risk

《货币金融学(第十三版)》英文版教学课件mishkin_econ13e_ppt_07

Copyright © 2022, 2019, 2016 Pearson Education, Inc. All Rights Reserved
Application: The Coronavirus Stock Market Crash of 2020
• The spread of the coronavirus in February 2020 triggered a stock market crash in which the Dow Jones Industrial Average fell from a peak of 29,551 on February 12 to 18,561 on March 20, a decline of 37%.
Copyright © 2022, 2019, 2016 Pearson Education, Inc. All Rights Reserved
Copyright © 2022, 2019, 2016 Pearson Education, Inc. All Rights Reserved
Application: Monetary Policy and Stock Prices (2 of 2)
• Furthermore, a lowering of interest rates is likely to stimulate the economy, so the growth rate in dividends, g, is likely to be somewhat higher. This rise in g also causes the denominator in Equation 5 to decrease, which also leads to a rise in stock prices.

北美精算师大纲

基本教育阶段(6门课程)：课程1：精算科学的数学基础说明：这门课程的目的是为了培养关于一些基础数学工具的知识，形成从数量角度评估风险的能力，特别是应用这些工具来解决精算科学中的问题。

并且假设学员在学习这门课程之前已经熟练掌握了微积分、概率论的有关内容及风险管理的基本知识。

主要内容及概念：微积分、概率论、风险管理（包括损失频率、损失金额、自留额、免赔额、共同保险和风险保费）课程2：利息理论、经济学和金融学说明：这门课程包括利息理论，中级微观经济学和宏观经济学，金融学基础。

在学习这门课程之前要求具有微积分和概率论的基础知识。

主要内容及概念：利息理论，微观经济学，宏观经济学，金融学基础课程3：随机事件的精算模型说明：通过这门课程的学习，培养学员关于随机事件的精算模型的基础知识及这些模型在保险和金融风险中的应用。

在学习这门课程之前要求熟练掌握微积分、概率论和数理统计的相关内容。

建议学员在通过课程1和课程2后学习这门课程。

主要内容及概念：保险和其它金融随机事件，生存模型，人口数据分析，定量分析随机事件的金融影响课程4：精算建模方法说明：该课程初步介绍了建立模型的基础知识和用于建模的重要的精算和统计方法。

在学习这门课程之前要求熟练掌握微积分、线性代数、概率论和数理统计的相关内容。

主要内容及概念：模型－模型的定义－为何及如何使用模型－模型的利弊－确定性的和随机性的模型－模型选择－输入和输出分析－敏感性检验－研究结果的检验和反馈方法－回归分析－预测－风险理论－信度理论课程5－精算原理应用说明：这门课程提供了产品设计，风险分类，定价/费率拟定/建立保险基金，营销，分配，管理和估价的学习。

覆盖的范围包括金融保障计划,职工福利计划，事故抚恤计划，政府社会保险和养老计划及一些新兴的应用领域如产品责任，担保的评估，环境的维护成本和制造业的应用。

该课程的学习材料综合了各种计划和覆盖范围以展示精算原理在各研究领域中应用的一致性和差异性。

会计学推荐书目

50、Derivative Securities﹡ Robert Jarrow & Stuart Turnbull
注：﹡标明难度较高的读物，Prentice Hall 表示美国的一家出版社，其余为书的编著者。
该书目由中国人民大学财政金融学院院长陈雨露教授和应用金融系主任瞿强副教授提供。
上海财大金融学推荐书目
4、Macroeconomic Theory﹡ Sargent T.J.
5、Dynamic Macroeconomics﹡ Sargent T.J.
6、Handbook of Macroeconomics (Volume1A 1B 1C) J.B. Taylor & M. Woodford
10、《华尔街追梦实录》
英文名：Monkey Business: Swinging Through the Wall Street Jungle
约翰·罗尔夫（John Rolfe）彼德·特鲁珀（Peter Troob）著，机械工业出版社2001年5月出版
推荐理由：两位刚从美国商学院毕业的MBA学生，用幽默的笔触，回顾了他们从走进华尔街到成为“标准的投资银行家”的一路摸爬滚打的历程。对于志在从Analyst、Associate起步，进军投资银行业的MBA学生来说，此书的参考价值不言而喻.
二、金融学基础类
16、Foundation for Financial Economics Huang C.F. & Litzenberger R.H.
17、Principles of Financial Economics S.F. Leroy & Jan Werner
18、Handbooks of Monetary Economics(1 and 2) Benjamin M. Friedman& Frank H.Hahn

数理金融初步英文版第三版课程设计

数理金融初步英文版第三版课程设计Course OverviewThis course provides an introduction to the field of mathematical finance. It covers fundamental concepts and methods from probability theory, statistics, calculus, and linear algebra, and their applications to finance and economics. Topics include financial derivatives, risk management, portfolio optimization, and asset pricing.Throughout the course, students will gn a solid understanding of the mathematical foundations of finance. They will also learn how to apply mathematical techniques to real-world problems in finance and investment.Course ObjectivesUpon completion of this course, students will be able to:•Understand the fundamental concepts and methods of mathematical finance•Use probability theory, statistics, calculus, and linear algebra to analyze financial problems•Appreciate the mathematical models that underlie the pricing and hedging of financial derivatives•Expln the principles of risk management and portfolio optimization•Apply mathematical techniques to real-world problems in finance and investmentCourse StructureThe course will consist of 12 lectures and 12 problem sets. Each lecture will be 90 minutes long and will cover a specific topic in mathematical finance. The lectures will be conducted in English.The problem sets will provide students with the opportunity to apply the concepts and methods learned in class to practical problems. The problem sets will be assigned at the end of each lecture and will be due one week later. Students will be required to submit their solutions online.In addition to the lectures and problem sets, there will be a final exam. The exam will cover all the topics discussed in the course.Course Topics1.Introduction to Mathematical Finance2.Probability Theory and Stochastic Calculus3.Discrete-Time Models and Derivatives4.Continuous-Time Models and Derivatives5.Black-Scholes Model6.Options Trading Strategies7.Exotic Options and Structured Products8.Risk Management9.Portfolio Optimization10.Asset Pricing Models11.Empirical Finance12.Summary and ReviewAssessmentThe final grade for the course will be determined as follows: •Problem sets (40%)•Final exam (60%)To pass the course, students must obtn a grade of at least 60%. In addition, attendance is mandatory for all lectures.Recommended TextbookHull, J. C. (2014). Options, futures, and other derivatives (9th ed.). Prentice Hall.ConclusionThis course provides a solid foundation in the mathematics of finance. It is ideal for students who wish to pursue careers in finance, banking, investment, or risk management. The course assumes a basic knowledge of calculus, linear algebra, and probability theory. Students who successfully complete the course will be well-equipped to applytheir mathematical skills to real-world financial problems.。

《货币金融学(第十三版)》英文版教学课件mishkin_econ13e_ppt_23

Copyright © 2022, 2019, 2016 Pearson Education, Inc. All Rights Reserved
The Bottom Line: The Relationship Between Stabilizing Inflation and Stabilizing Economic Activity
Copyright © 2022, 2019, 2016 Pearson Education, Inc. All Rights Reserved
Learning Objectives (1 of 2)
23.1 Illustrate and explain the policy choices that monetary policy makers face under the conditions of aggregate demand shocks, temporary supply shocks, and permanent supply shocks. 23.2 Identify the lags in the policy process and summarize why they weaken the case for an activist policy approach. 23.3 Explain why monetary policy makers can target any inflation rate in the long run but cannot target aggregate output in the long run.
• Nonactivists believe government action is unnecessary to eliminate unemployment.

金融数学或者经济数学方向的书单

金融数学或者经济数学方向的书单=。

=+这些书单里的书俺都下了电子书...想催眠的想打发时间的想提高英语阅读能力的可以问我要...改天我自个整理个我下的经济学的电子书单...不想老是在后面再附上自己的推荐...****************************************************** 1.Futures,Options and other derivatives--by John Hull.这本书不用多说了，买就是了。

不管是找工作还是senior quant 都会用到。

John Hull也是非常厉害的，各个方面都有开创性的成果。

现在Toronto Uni.2.Arbitrage theory in continuous time--by Tomas Bjork 这本书非常适合数学/物理背景的人读，注重数学理论的培养。

本来我觉得也没什么，但是被公司老板大加赞扬后就改变看法了。

Bjork现在瑞典SSE。

3.Financial Calculus--Martin Baxter&Rennie非常薄但是elegant的一本书，1996年，算是比较早了，但是和Hull的那本书齐名。

也是聪聪的first book。

作者1现在野村证券伦敦（nomura)，作者2在美林伦敦(ml)，都是fixed income。

4.Financial calculus for finance II--ShreveShreve的新书，非常elegant，非常仔细，数学完备，适合数学背景，但是比较厚，对于入门来说还是3好。

作者现在CMU纽约。

教授。

顶尖人物。

5。

Martingale methods in Financial modelling--Musiela& Rutkovski作者现在BNP(巴黎银行）和华沙理工？都是顶尖人物。

数学背景1.Brownian motion and stochastic calculus--Shreve& Karasatz如果想在这一行发paper或者搞研究的话，或者读phd,这是必须的。

Mathematics_for_Finance

Mathematics for Finance: An Introduction to Financial EngineeringAmerican Mathematical Monthly, The, Dec 2004 by Protter, PhilipMathematical finance (or financial engineering, as it is often known) is a young subject for mathematics, but is highly popular with students. No doubt the allure of being connected to vast sums of money is a part of the attraction. Yet it is a difficult subject, requiring a broad array of knowledge of subjects that are traditionally considered hard to learn.Forty years ago, options and what are now called "financial derivatives" were little known. Options were traded on the Chicago Board Options Exchange (CBOE), primarily for commodities such as pork bellies, orange juice, coffee, and precious metals. Let us take a minute to describe a situation where an option is useful. Imagine a small Indiana farmer raising pigs. The price is high now, but his pigs are only 80% grown. He can market them now and make a handsome profit, or he can wait until they are fully grown and make a larger profit if the price stays up, but end up making significantly less money if the price falls. He could solve the problem by buying a forward contract, locking in a prearranged price and thus a sure profit, but what if the price rises further? Then he will kick himself for having locked in the price. An option, on the other hand, gives him the right, but not the obligation, to sell his pigs at the prearranged price, thus guaranteeing him the nice profit but not excluding a potentially bigger one.The prices of options such as the one just described were set by the market: supply and demand. In the United States there is a fervent belief that the market knows best and, if left alone, will arrive at a fair and just price. There are many unspoken hypotheses involved with this belief, and in the case of commodities, several were violated. It will suffice to point out that small farmers were buying options sold by large banks and companies. In the early 1970s, Black, Scholes, and Merton showed that by using the ItO stochastic calculus and a simple model describing the dynamics of the price of a risky asset, one could arrive at a fair price for an option. They did this using a key idea: if one sells the option for $x, there is a hedging strategy by which one can use that $x to trade in the commodity over time until the option is due and end up with exactly what is owed to the option purchaser at the settlement time. There is no risk at all, except the implicit risk that the model for the dynamic price of the commodity is wrong. Therefore, if the market price is larger than $x, one can charge the market price and match the option and have money left over. If the market price is less than $x, one can buy the options and make money in reverse. It turned out that the market is often wrong, but the breakthrough of Black and Scholes went largely ignored by the financial players. This gradually changed, largely through the efforts of a few visionary people at Wells Fargo Bank, who worked not so much with commodities as with the (then) new concepts of portfolio insurance and index funds (see [2]).The option I described has the result of removing the risk for the pig farmer. For a (usually rather small) fee, he can buy what amounts to an insurance policy on the price of pork bellies. This is known as a transfer of risk: the option seller assumes the risk the farmer is not willing to assume, just as an insurance company assumes (for a fee) the financial risk of one's house burning down. This example also shows the utility of such insurance, since now the farmer will not slaughter the pigs before they are fully grown, and society as a whole will benefit (assuming that people eat pork). Once the methodology for pricing this transfer of risk became widely known, the concept spread widely. It has transformed modern business and arguably helped to create the financial boom years of the 1990s. One can now insure against currency fluctuations, dangerous drops in stock prices in one's portfolio, and all manner of (often fairly esoteric) business operations by this form of risk transfer. Options are also widely used for less virtuous goals, such as helping companies and executives avoid paying taxes, and of course for what amounts, simply, to gambling。

清华大学外国教材中心提供

Options, futures, & other derivatives/期权期货和其他衍生品/John C. Hull/4th ed (eprinted ed.)
Tsinghua uni英文版教材
本科生和研究生的教材
F830.9 FH91f
Fundamentals of futures and options markets/期货和期权市场基本原理/John C. Hull./4th ed. (reprinted ed.)
Tsinghua University Press/c2001
本科生会计学课程的教材
F270 FM961 2001
Guide to managerial communication :effective business writing and speaking/管理沟通指南:有效商务写作与交谈/by Mary Munter./5th ed. (reprinted ed.)
为美国商学院研究生国际市场营销课程应用最普遍的一本案例教材
F713.3 FK87m
Marketing management/营销管理/Philip Kotler/Millennium ed (reprinted ed.)
Tsinghua university Press/c2001
清华MBA核心课程英文版教材
Prentice Hall/c1994.
Prentice Hall international series in industrial and systems engineering
本书是专门为工业、产品生产、制造和系统工程专业或相关专业的高年级本科生和低年级的研究生编写的
F273 FH471 2001

高等数学

Annals of Mathematics,163(2006),1–35Finite and inﬁnite arithmetic progressionsin sumsetsBy E.Szemer´e di and V.H.Vu*AbstractWe prove that if A is a subset of at least cn1/2elements of{1,...,n}, where c is a suﬃciently large constant,then the collection of subset sums of A contains an arithmetic progression of length n.As an application,we conﬁrma long standing conjecture of Erd˝o s and Folkman on complete sequences.1.IntroductionFor a(ﬁnite or inﬁnite)set A of positive integers,S A denotes the collection ofﬁnite subset sums of AS A=x∈Bx|B⊂A,|B|<∞.Two closely related notions are that of lA and l∗A:lA denotes the set of numbers which can be represented as a sum of l elements of A and l∗A denotes the set of numbers which can be represented as a sum of l diﬀerent elements of A,respectively.(If l>|A|,then l∗A is the empty set.)It is clear thatS A=∪∞l=1l∗A.One of the fundamental problems in additive number theory is to estimate the length of the longest arithmetic progression in S A,l A and l∗A,respectively.The purpose of this paper is multi-fold.We shall prove a sharp result concerning the length of the longest arithmetic progression in S A.Via the proof,we would like to introduce a new method which can be used to handle many other problems.Finally,the result has an interesting application,as we can use it to settle a forty-year old conjecture of Erd˝o s and Folkman concerning complete sequences.*Research supported in part by NSF grant DMS-0200357,by an NSF CAREER Grant and by an A.Sloan Fellowship.2 E.SZEMER´EDI AND V.H.VUTheorem1.1.There is a positive constant c such that the following holds. For any positive integer n,if A is a subset of[n]with at least cn1/2elements, then S A contains an arithmetic progression of length n.Here and later[n]denotes the set of positive integers between1and n.The proof Theorem1.1introduces a new and useful method to prove the existence of long arithmetic progressions in sumsets.Our method relies on inverse and geometrical arguments,rather than on Fourier analysis like most papers on this topic.This method opens a way to attack problems which previously have seemed very hard.Let us,for instance,address the problem of estimating the length of the longest arithmetic progression in lA (where A is a subset of[n]),as a function of l,n and|A|.In special cases sharp results have been obtained,thanks to the works of several researchers, including Bourgain,Freiman,Halberstam,Ruzsa and S´a rk¨o zy[2],[6],[8],[17]. Our method,combined with additional arguments,allows us to derive a sharp bound for this length for a wide range of l and|A|.For instance,we can obtain a sharp bound whenever l=nαand|A|=nβ,whereαandβare arbitrary positive constants at most1.Details will appear in a subsequent paper[19].An even harder problem is to estimate the length of the longest arithmetic progression in l∗A.The distinction that the summands must be diﬀerent fre-quently poses a great challenge.(A representative example is Erd˝o s-Heilbronn vs Cauchy-Danveport[15].)On the other hand,one of our arguments(the tiling technique discussed in§5)seems to provide an eﬀective tool to overcome this challenge.Although there are still many details to be veriﬁed,we believe that with this tool,we could handle l∗A as successfully as lA.As a conse-quence,one can prove a sharp bound for the length of the longest arithmetic progression in S A even when the cardinality of A is much smaller than n1/2,ex-tending Theorem1.1.Our method also works for multi-sets(where an element may appear many times).A result concerning multi-sets will be mentioned in Section7.Let us now make a few comments on the content of Theorem1.1.The bound in this theorem is sharp up to the constant factor c.In fact,it is sharp from two diﬀerent points of view.First,it is clear that if A is the interval[cn1/2],then the length of the longest arithmetic progression in S A is O(n).Second,and more interesting,there is a positive constantαsuch that the following holds:For all suﬃciently large n there is a set A⊂[n]with cardinalityαn1/2such that the longest arithmetic progression in S A has length O(n3/4).We provide a concrete construction at the end of Section5.We next discuss an application of Theorem1.1.We can use this theorem to conﬁrm a well-known and long standing conjecture of Erd˝o s,dating back to1962.In fact,the study of Theorem1.1was partially motivated by this conjecture.FINITE AND INFINITE ARITHMETIC PROGRESSIONS IN SUMSETS 3An inﬁnite set A is complete if S A contains every suﬃciently large positive integer.The notion of complete sequences was introduced by Erd˝o s in the early sixties and has since then been studied extensively by various researchers (see §6of [5]or §4.3of [15]for surveys).The central question concerning complete sequences is to ﬁnd suﬃcient conditions for completeness.In 1962,Erd˝o s [4]made the following conjectureConjecture 1.2.There is a constant c such that the following holds.Any increasing sequence A ={a 1<a 2<a 3<...}satisfying(a)A (n )≥cn 1/2(b)S A contains an element of every inﬁnite arithmetic progression ,is complete.Here and later A (n )denotes the number of elements of A not exceeding n .The bound on A (n )is best possible,up to the constant factor c ,as shown by Cassels [3](see also below for a simple construction).The second assumption (b)is about modularity and is necessary as shown by the example of the sequence of even numbers.So Erd˝o s’s conjecture basically says that a sequence is complete if it is suﬃciently dense and satisﬁes a trivially necessary modular condition.Erd˝o s [4]proved that the statement of the conjecture holds if one replaces (a)by a stronger condition that A (n )≥cn (√1)/2.A few years later,in 1966,Folkman [9]improved Erd˝o s’result by showing that A (n )≥cn 1/2+εis suﬃcient,for any positive constant ε.The ﬁrst and simpler step in Folkman’s proof is to show that any sequence satisfying (b)can be partitioned into two subsequences with the same density,one of which still satisﬁes (b).In the next and critical step,Folkman shows that if A is a sequence with density at least n 1/2+εthen S A contains an inﬁnite arithmetic progression.His result follows immediately from these two steps.In the following we say that A is subcomplete if S A contains an inﬁnite arithmetic progression.Folkman’s proof,quite naturally,led him to the following conjecture,which (if true)would imply Conjecture 1.2.Conjecture 1.3.There is a constant c such that the following holds.Any increasing sequence A ={a 1<a 2<a 3<...}satisfying A (n )≥cn 1/2is subcomplete.Here is an example which shows that the density n 1/2is best possible (up to a constant factor)in both conjectures.Let m be a large integer divisible by 8(say,104)and A be the sequence consisting of the union of the intervals[m 2i /4,m 2i /2](i =0,1,2...).It is clear that this sequence has density Ω(n 1/2)and satisﬁes (b).On the other hand,the diﬀerence between m 2i /4and the4 E.SZEMER´EDI AND V.H.VUsum of all elements preceding it tends to inﬁnity as i tends to inﬁnity.Thus S A cannot contain an inﬁnite arithmetic progression.(The constants1/4and 1/2might be improved to slightly increase the density of A.)Folkman’s result has further been strengthened recently by Hegyv´a ri[11] and L uczak and Schoen[13],who(independently)reduced the density n1/2+εto cn1/2log1/2n,using a result of Freiman and S´a rk¨o zy(see§7).Together with Conjecture1.3,Folkman also made a conjecture about nondecreasing sequences(where the same number may appear many times).We address this conjecture in the concluding remarks(§7).An elementary application of Theorem1.1helps us to conﬁrm Conjecture 1.3.Conjecture1.2follows immediately via Folkman’s partition argument.In fact,as we shall point out in Section7,the statement we need in order to conﬁrm Conjecture1.3is weaker than Theorem1.1.Corollary1.4.There is a positive constant c such that the following holds.Any increasing sequence of density at least cn1/2is subcomplete.Corollary1.5.There is a positive constant c such that the following holds.Any increasing sequence A={a1<a2<a3<...}satisfying(a)A(n)≥cn1/2(b)S A contains an element of every inﬁnite arithmetic progression,is complete.Let us conclude this section with a remark regarding notation.Through the paper,we assume that n is suﬃciently large,whenever needed.The asymp-totic notation is used under the assumption that n tends to inﬁnity.Greek lettersε,γ,δetc.denote positive constants,which are usually small(much smaller than1).Lower case letters d,h,g,l,m,n,s denote positive integers. In most cases,we use d,h and g to denote constant positive integers.The logarithms have base two,if not otherwise speciﬁed.For the sake of a better presentation,we omit unnecessaryﬂoors and ceilings.For a positive integer m,[m]denotes the set of positive integers in the interval from1to m,namely, [m]={1,2,...,m}.The notion of sumsets is central in the proofs.If A and B are two sets of integers,A+B denotes the set of integers which can be represented as a sum of one element from A and one element from B:A+B={a+b|a∈A,b∈B}. We write2A for A+A;in general,lA=(l−1)A+A.A graph G consists of a(ﬁnite)vertex set V and an edge set E,where an element of E(an edge)is a(unordered)pair(a,b),where a=b∈V.The degree of a vertex a is the number of edges containing a.A subset I of V(G)is called an independent set if I does not contain any edge.A graph is bipartiteFINITE AND INFINITE ARITHMETIC PROGRESSIONS IN SUMSETS 5if its vertex set can be partitioned into two sets V 1and V 2such that every edge has one end point in V 1and one end point in V 2(V 1and V 2are referred to as the color classes of V ).2.Main lemmas and ideasLet us start by presenting a few lemmas.After the reader gets him-self/herself acquainted with these lemmas,we shall describe our approach to the main theorem (Theorem 1.1).As mentioned earlier,our method relies on inverse arguments and so we shall make frequent use of Freiman type inverse theorems.In order to state these theorems,we ﬁrst need to deﬁne generalized arithmetic progressions.A generalized arithmetic progression of rank d is a subset Q of Z of the form {a + d i =1x i a i |0≤x i ≤n i };the product d i =1n i is its volume,which we denote by Vol(Q ).The a i ’s are the diﬀerences of Q .In fact,as two diﬀerent generalized arithmetic progressions might represent the same set,we always consider generalized arithmetic progressions together with their structures.Let A ={a + d i =1x i a i |0≤x i ≤n i }and B ={b + d i =1x i a i |0≤x i ≤m i }be two generalized arithmetic progressions with the same set of diﬀerences.Then their sum A +B is the generalized arithmetic progression {(a +b )+ d i =0z i a i |0≤z i ≤n i +m i }.Freiman’s famous inverse theorem asserts that if |A +A |≤c |A |,where c is a constant,then A is a dense subset of a generalized arithmetic progression of constant rank.In fact,the statement still holds in a slightly more general situation,when one considers A +B instead of A +A ,as shown by Ruzsa [16],who gave a very nice proof which is quite diﬀerent from the original proof of Freiman.The following result is a simple consequence of Freimain’s theorem and Pl¨u nnecke’s theorem (see [18,Th.2.1],for a proof).The book [14]of Nathanson contains a detailed discussion on both Pl¨u nnecke’s and Ruzsa’s results.Theorem 2.1.For every positive constant c there is a positive integer d and a positive constant k such that the following holds.If A and B are two subsets of Z with the same cardinality and |A +B |≤c |A |,then A +B is a subset of a generalized arithmetic progression P of rank d with volume at most k |A |.In the case A =B ,it has turned out that P has only log 2c essential dimensions.The following is a direct corollary of Theorem 1.3from a paper of Bilu [1].One can also see that it is a direct consequence of Freiman’s cube lemma and Freiman’s homomorphism theorem [7].Theorem 2.2.For any positive constant c ≥2there are positive con-stants δand c such that the following holds.If A ⊂Z satisﬁes |A |≥c 26 E.SZEMER´EDI AND V.H.VUand|2A|≤c|A|,then there is a generalized arithmetic progression P of rank log2c such that Vol(P)≤c |A|and|P∩A|≥δ|A|≥δcVol(P).Next,we take a closer look at generalized arithmetic progressions of rank2. The following two lemmas show that under certain circumstances,a generalized arithmetic progression P of rank2contains a long arithmetic progression whose length is proportional to the cardinality of P.Lemma2.3.Let P={x1a1+x2a2|0≤x i≤l i}be a generalized arithmetic progression of rank2where l i≥5a i>0for i=1,2.Then P contains anarithmetic progression of length35|P|and diﬀerence gcd(a1,a2).This lemma was proved in an earlier paper[18];we sketch the proof for the sake of completeness.Proof of Lemma2.3.We shall prove that P contains an arithmeticprogression of length35gcd(a1,a2)(l1a1+l2a2)and diﬀerence gcd(a1,a2).A simpleargument shows that35gcd(a1,a2)(l1a1+l2a2)≥35|P|.It suﬃces to consider the case when a1and a2are co-prime.In this case weshall actually show that P contains an interval of length35(l1a1+l2a2).In the following we identify P with the cube Q={(x1,x2)|0≤x i≤l i}of integer points in Z2together with the canonical mapf:Z2→Z:f((x1,x2))=x1a1+x2a2.The desired progression will be provided by a walk in this cube,following a speciﬁc rule.Once the walk terminates,its two endpoints will be far apart, showing that the progression has large length.As a1and a2are co-prime,there are positive integers l 1,l 1,l 2and l 2such that l 1,l 1<a2,l 2,l 2<a1andl 1a1−l 2a2=l 2a2−l 1a1=1.(1)We show that P contains the interval[15(l1a1+l2a2),45(l1a1+l2a2)].Letu1and u2denote the vectors(l 1,−l 2)and(−l 1,l 2),respectively.Set v0=(l1/5,l2/5).We construct a sequence v0,v1,...,such that f(v j+1)=f(v j)+1as follows.Once v j is constructed,set v j+1=v j+u i given that one canﬁnd1≤i≤2such that v j+u i∈Q(if both i satisfy this condition then choose any of them).If there is no such i,then stop.Let v t=(y t,z t)be the lastpoint of this sequence.As neither v t+u1nor v t+u2belong to Q,both of thefollowing two conditions(∗)and(∗∗)must hold:(∗)y t+l 1>l1or z t−l 2≤0.(∗∗)y t−l 1≤0or z t+l 2>l2.FINITE AND INFINITE ARITHMETIC PROGRESSIONS IN SUMSETS7 Since l 1<a2≤l1/2,y t+l 1>l1and y t−l 1≤0cannot occur simul-taneously.The same holds for z t−l 2≤0and z t+l 2>l2.Moreover,since f(v j)is increasing and y0=l1/5≥a2>l 1and z0=l2/5≥a1>l 2,we can conclude that z t−l 2≤0and y t−l 1≤0cannot occur simultaneously,either. Thus,the only possibility left is y t+l 1>l1and z t+l 2>l2.This implies that y t>l1−l 1≥45l1and z t>l2−l 1≥45l2.Thusf(v t)>45(l1a1+l2a2),(2)concluding the proof.Lemma2.4.If U⊂[m]is a generalized arithmetic progression of rank2 and l|U|≥20m,where both m and|U|are suﬃciently large,then lU contains an arithmetic progression of length m.Proof of Lemma2.4.Assume that U={a+x1a1+x2a2|0≤x i≤u i}.We can assume that u1,u2>10(if u1is small,then it is easy to check that lU contains a long arithmetic progression,where U ={a+x2a2|0≤x2≤u2}). Now let us considerlU={la+x1a1+x2a2|0≤x i≤lu i}.(3)By the assumption l|U|≥20m,we have l(u1+1)(u2+1)≥20m.As u1,u2≥10,it follows that lu1u2≥10m.On the other hand,U is a subset of [m]so the diﬀerence of any two elements of U has absolute value at most m. It follows that u1a1≤m.This impliesu1a1≤m≤lu1u2/10.So it follows that10a1≤lu2.Similarly10a2≤lu1.Thus lU satisﬁes the as-sumption of Lemma2.3and this lemma implies that lU contains an arithmetic progression of length at least3 5|lU|≥352m>m,concluding the proof.In the inequality35|lU|≥352m we used the fact that|lU|≥2m.This fact follows immediately(and with room to spare)from the assumption l|U|≥20m and the well-known fact that|A+B|≥|A|+|B|, unless both A and B are arithmetic progressions of the same diﬀerence.(We leave the easy proof as an exercise.)Despite its simplicity,Lemma2.4plays an important role in our proof. It shows that in order to obtain a long arithmetic progression,it suﬃces to obtain a large multiple of a generalized arithmetic progression of rank2.As the reader will see,generalized arithmetic progressions of rank2are actually the main objects of study in this paper.8 E.SZEMER´EDI AND V.H.VUThe next lemma asserts that by adding several subsets of positive densityof a certain generalized arithmetic progression of constant rank,one canﬁllan entire generalized arithmetic progression of the same rank and comparablecardinality.This is one of our main technical tools and we shall refer to it asthe“ﬁlling”lemma.Lemma2.5.For any positive constantγand positive integer d,there is a positive constantγ and a positive integer g such that the following holds.IfX1,...,X g are subsets of a generalized arithmetic progression P of rank d and |X i|≥γVol(P)then X1+···+X g contains a generalized arithmetic progres-sion Q of rank d and cardinality at leastγ Vol(P).Moreover,the distances ofQ are multiples of the distances of P.Remark.The conditions of this lemma imply that the ratio between thecardinality and the volume of P is bounded from below by a positive con-stant.The quantities Vol(P),|P|,Vol(Q),|Q|,|X i|’s diﬀer from each other by constant factors only.Let us now give a sketchy description of our plan.In view of Lemma2.4,it suﬃces to show that S A contains a(suﬃciently large)multiple of a(suﬃciently large)generalized arithmetic progression of rank2.We shall carryout this task in two steps.Theﬁrst step is to produce one relatively largegeneralized arithmetic progression.In the second step,we put many copies ofthis generalized arithmetic progression together to obtain a large multiple of it.This multiple will be suﬃciently large so that we can invoke Lemma2.4.Thesetwo steps are not independent,as both of them rely on the following structuralproperty of A:Either S A contains an arithmetic progression of length n(andwe are done),or a large portion of A is trapped in a small generalized arithmeticprogression of rank2.This is the content of the main structural lemma of ourproof.Lemma2.6.There are positive constantsβ1andβ2such that the follow-ing holds.For any positive integer n,if A is a subset of[n]with at least n1/2elements then either S A contains an arithmetic progression of length n,or thereis a subset A of A such that|A |≥β1|A|and A is contained in generalized arithmetic progression W of rank2with volume at most n1/2logβ2n.The reader might feel that the above description of our plan is somewhatvague.However,at this stage,that is the best we could do without involvingtoo much technicality.The plan will be updated gradually and become moreand more concrete as our proof evolves.There are two technical ingredients of the proof which deserve mentioning.Theﬁrst is what we call a tree argument.This argument,in spirit,works asfollows.Assume that we want to add several sets A1,...,A m.We shall addFINITE AND INFINITE ARITHMETIC PROGRESSIONS IN SUMSETS9 them in a special way following an algorithm which assigns sets to the vertices of a tree.A set of any vertex contains the sum of the sets of its children.If the set at the root of the tree is not too large,then there is a level where the sizes of the sets do not increase(compared to the sizes of their children)too much.Thus,we can apply Freiman’s inverse theorems at this level to deduce useful information.The creative part of this argument is to come up with a proper algorithm which suits our need.The second important ingredient is the so-called tiling argument,which helps us to create a large generalized arithmetic progression by tiling many small generalized arithmetic progressions together.(In fact,it would be more precise to call it wasteful tiling as the small generalized arithmetic progressions may overlap.)This technique will be discussed in detail in Section5.The rest of the paper is organized as follows.In the next section,we prove Lemma2.5.In Section4,we prove Lemma2.6.Both of these proofs make use of the tree argument mentioned above,but in diﬀerent ways.The proof of Theorem1.1comes in Section5,which contains the tiling argument.In Section6,we prove the Erd˝o s-Folkman conjectures.Theﬁnal section,Section 7,is devoted to concluding remarks.3.Proof of Lemma2.5We shall need the following lemma which is a corollary of a result of Lev and Smelianski(Theorem6of[12]).This lemma is relatively easy and the reader might want to consider it an exercise.Lemma3.1.The following holds for all suﬃciently large m.If A and B are two sets of integers of cardinality m and|A+B|≤2.1m,then A is a subset of an arithmetic progression of length1.1m.We also need the following two simple lemmas.Lemma3.2.For any positive constantεthere is a positive integer h0such that the following holds.If h≥h0and A1,...,A h are arithmetic progressions of length at leastεn of an interval I of length n,then there is a number h ≥.09ε2h and an arithmetic progression B of length.9εn such that at least hamong the A i’s contain B.Proof of Lemma3.2.Consider the following bipartite graph.Theﬁrst color class consists of A1,...,A h.The other color class consists of the arith-metic progressions of length.9εn in I.Since the diﬀerence of an arithmetic progression of length.9εn in I is at most1/(.9ε),the second color class has at most n/(.9ε)vertices.Moreover,an arithmetic progression of lengthεn contains at least.1εn arithmetic progression of length.9εn.Thus,each vertex in theﬁrst class has degree at least.1εn and so the number of edges is at least10 E.SZEMER´EDI AND V.H.VU.1εnh.It follows that there is a vertex in the second color class with degreeat least.1εnhn/(.9ε)=.09ε2h.The progression corresponding to this vertex satisﬁesthe claim of the lemma.Lemma3.3.Let B be an interval of cardinality n and B be a subset of B containing at least.8n elements.Then B +B contains an interval of length 1.2n+2.Proof of Lemma3.3.Without loss of generality we can assume that B= [n].If an integer m can be represented as a sum of two elements in B in more than.2n ways(we do not count permutations)than m∈B +B .To conclude,notice that every m in the interval[.4n+1,1.6n−1]has more than .2n representations.To prove Lemma2.5,we use induction on d.The harder part of the proof is to handle the base case d=1.To handle this case we apply the tree method mentioned in the introduction.Without loss of generality we can assume that g is a power of4,|X i|=n1 and0∈X i for all1≤i≤g.Let m be the cardinality of P;we can also assume that P is the interval[m].Set X1i=X i for i=1,...,g and g1=g.Here is the description of the algorithm we would like to study.The algorithm.At the t th step,the input is a sequence X t1,...,X t gt ofsets of the same cardinality n t where g t is an even number.Choose a pair 1≤i<j≤g t which maximizes|X t i+X t j|(if there are many such pairs choose an arbitrary one).Denote the sum X t i+X t j by X 1.Remove i and j from the index set and repeat the operation to obtain X 2and so on.After g t/2operations we obtain a set sequence X 1,...,Xg t/2which has decreasingcardinalities.Deﬁne g t+1=g t/4.Consider the sequence X 1,...,X gt+1andtruncate all but the last set so that all of them have the same cardinality(which is|X gt+1|).The truncated sets will be named X t+11,...,X t+1g t+1and theyform the input of the next step.The algorithm halts when the input sequencehas only one element.A simple calculation shows that g t=14t−1g1for allpossible t’s.Notice that X t gt is a subset of2t P and thus n t=|X t gt|is at most2t m.Onthe other hand,n1=|X1g1|/m≥γ.So,for some t≤log1.051γ,n t+1≤2.1n t.By the description of the algorithm, there are g t/2sets among the X t i such that every pair of them have cardinality at most n t+1≤2.1n t.To simplify the notations,call these sets Y1,...,Y h.Wehave thath=g t/2≥14tg1.So,by increasing g1we can assume that h is suﬃciently large,whenever needed.We have that|Y i|=n t and|Y i+Y j|≤2.1n t for all1≤i<j≤h.We are now in position to invoke an inverse statement and at this stage all we need is Lemma3.1(which is much simpler than Freiman’s general theorem).By this lemma,every Y i is a subset of an arithmetic progression A i of length at most 1.1n t.Moreover,A i is a subset of2t P.Also observe that by the deﬁnition of t,n t/|2t P|≥γ.We can extend the A i’s obtained prior to Lemma3.2so that each of them has length exactly1.1n t.By Lemma3.2,provided that g t is suﬃciently large, there are A i and A j such that B=A i∩A j is an arithmetic progression of length at least n t.Now consider Y i and Y j which are subsets of A i and A j, respectively.Since Y i and Y j both have n t elements,B =Y i∩Y j∩B has at least.8n t elements.The set B +B is a subset of Y i+Y j,which,in turn,is a subset of X1+···+X g(recall that we assume0∈X i for every i).This and Lemma3.3 complete the proof for the base case d=1.Now assume that the hypothesis holds for all d≤r;we are going to prove it for d=r+1.This proof uses a combinatorial counting argument and is independent of the previous proof.In particular,we do not need the tree method here.Consider a generalized arithmetic progression P of rank r+1and its canonical decomposition P=P1+P2,where P1is an arithmetic progression and P2is a generalized arithmetic progression of rank r(P1is theﬁrst“edge”of P).For every x∈P2,denote by P i1(x)the set of those elements y of P1 where x+y∈X i.We say that x is i-normal if P i1(x)has density at leastγ/2 in P1.Since|X i|≥γVol(P),the set N i of i-normal elements has density at leastγ/2in P2,for all possible i.Let g=g g where g and g are large constants satisfying g g 1/γ. Partition X1,...,X g into g groups with cardinality g each.Consider the ﬁrst group.Without loss of generality,we can assume that its members are X1,...,X g and also that|N1|=···=|N g |=γ|P2|/2.Order the elements in each N i increasingly.For each1≤k≤|N1|,let x k1,...,x k g be the k th elements in N1,...,N g ,respectively.Consider the sets P1(x k i),P1(x k2),...,P1(x k g ).Given that g is suﬃciently large,we can apply the statement for the base case d=1 to obtain an arithmetic progression A k of lengthγ1|P1|,for some positive con-stantγ1depending onγ.Each of the A k,k=1,2,...,|N1|,is a subset of g P1 which has length g |P1|(to be exact,the length of g P1is g |P1|+O(1);but since the error term O(1)plays no role,we omit it here and later to simplify the presentation),so the density of each A k in g P1isγ1/g .Applying Lemma 3.2with n=g |P1|andε=γ1/g ,a.09(γ1/g )2fraction of the A k’s contain the same arithmetic progression B of length.9γ1|P1|.Without loss of generality, we can assume A1,...,A L,whereL=.09(γ1/g )2|N1|=.09(γ1/g )2γ|P2|/2,all contain B.Let Y1be the collection of the sums x k=x k1+···+x k g , 1≤k≤L.By the ordering,all x k’s are diﬀerent so|Y1|=L and thus Y1has densityL/g |P2|=.09(γ1/g )2(γ/2g )in g P2.Moreover,the set Y1+B1is a subset of X1+···+X g .Next,by considering the second group,we obtain Y2+B2and so on. Now we focus on the sets Y1+B1,...,Y g +B g .Each B j is an arithmetic progression in g P1with density.9γ1|P1|/g |P1|=.9γ1/g .By Lemma3.2,at least a.09(.9γ1/g )2≥.07(γ1/g )2fraction of the B j’s contain the same arithmetic progression C of length.9(.9γ1|P1|)≥.8γ1|P1|.Without loss of generality,we can assume that B1,...,B g contain C,whereg =.08(γ1/g )2g .By setting g suﬃciently large compared to g ,we can assume that g is suﬃciently large.Now we are in position to conclude the proof.As Y1,...,Y g have density at least.09(γ1/g )2(γ/2g )in g P2,for a suﬃciently large g ,Y1+···+Y g contains a generalized arithmetic progression D of rank r of constant density in g (g P2),due to the induction hypothesis.The set C+D is a generalized arithmetic progression of rank r+1with positive constant density in g (g P). On the other hand,this generalized arithmetic progression is a subset of(Y1+ C)+···+(Y g +C).As we assumed0∈X i for1≤i≤g,the sum(Y1+ C)+···+(Y g +C)is a subset of X1+···+X g,completing the proof.4.Proof of Lemma2.6This proof is relatively long and we break it into several parts.In the ﬁrst subsection,we present two lemmas.The next subsection contains the description of an algorithm(again we use the tree method),which is somewhat more involved than the one used in the proof of Lemma2.5.In the third subsection,we analyze this algorithm and construct the desired sets A and W. The fourth andﬁnal subsection is devoted to the veriﬁcation of a technical statement which we need in order to show that W has the properties claimed by the lemma.4.1.Two simple lemmas.Theﬁrst lemma is a simple result from graph theory.。

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2 3 5
$457.31
Since NPV <0, business venture is not profitable if one considers the time value of money. It would be better to invest the $20,000 in a bank paying 7%, since the venture is equivalent to investing only $20,000 -$457.31= $19,542.69
2
3 5
$10,000
8000 6000
Assume an interest rate of 7% compounded annually and find the net present value of the cash flows.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.2 Present Value
P S1 r
n
Example 1 – Present Value
Find the present value of $1000 due after three years if the interest rate is 9% compounded monthly. Solution: S=1000, r =0.09/12, n =3(12) For interest rate, r 0.09 / 12 0.0075. 36 Principle value is P 10001.0075 $764.15.
INTRODUCTORY MATHEMATICAL ANALYSIS
For Business, Economics, and the Life and Social Sciences
Chapter 5 Mathematics of Finance
2011 Pearson Education, Inc.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.2 Present Value
Net Present Value
Net Present Value NPV Sum of present values - Initial investment
• To solve problems with interest is compounded continuously.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance
Chapter Outline
5.1) Compound Interest 5.2) Present Value 5.3) Interest Compounded Continuously
Chapter 5: Mathematics of Finance
ቤተ መጻሕፍቲ ባይዱ
Chapter Objectives
• To solve interest problems which require logarithms. • To solve problems involving the time value of money.
900 6001.015
n
1.015n 1.5 n ln1.015 ln 1.5
n ln 1.015 ln 1.5 ln 1.5 n 27.233 ln 1.015
It will take
27 .233 4
1 6.8083 6 years,9 2 months .
2011 Pearson Education, Inc.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance
5.1 Compound Interest
• Compound amount S at the end of n interest periods at the periodic rate of r is as
S P1 r
n
Example 1 – Compound Interest
Suppose that $500 amounted to $588.38 in a savings account after three years. If interest was compounded semiannually, find the nominal rate of interest, compounded semiannually, that was earned by the money.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.1 Compound Interest
Example 2 – Compound Interest
How long will it take for $600 to amount to $900 at an annual rate of 6% compounded quarterly? Solution: The periodic rate is r = 0.06/4 = 0.015.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance
5.2 Present Value
• P that must be invested at r for n interest periods so that the present value, S is given by
Example 2 – Net Present Value You can invest $20,000 in a business that guarantees you cash flows at the end of years 2, 3, and 5 as indicated in the table. Year Cash Flow
Example 5 – Net Present Value
Solution: Substracting the initial investment from the sum of the present values of the cash flows gives
NPV 10,0001.07 80001.07 60001.07 20,000
5001 r 588.38 1 r 6 588.38 500
6
588.38 1 r 500
6
r 6
588.38 1 0.0275 500
The semiannual rate was 2.75%, so the nominal rate was 5.5 % compounded semiannually.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.1 Compound Interest
Example 1 – Compound Interest
Solution: Let r be the semiannual rate. There are 2 × 3 = 6 interest periods.