An efficient method for simulating interest rate curves


 Luke Randall
 2 years ago
 Views:
Transcription
1 BANCO DE MÉXICO An efficient method for simulating interest rate curves Javier Márquez DiezCanedo Carlos E. Nogués Nivón Viviana Vélez Grajales February 2003
2 Summary The purpose of this paper is to present a methodology for simulating temporal interest rate structures which preserve curve shape, using a minimum number of parameters based on historical data. For this methodology we use Nelson and Siegel s parsimonious modeling, which is able to capture commonly observed multiple yield curve shapes using only four parameters. The process only requires a random sampling of occurrences of the values of these four parameters, and any other point in the curve is obtained directly using Nelson and Siegel s formula. 1 Historical series of parameters generated by these structures are obtained along with their combined probability distribution. Random series of parameters can thus be generated, which behave according to the historical distribution and interest rate curve structures for the corresponding maturities. I. Introduction The market risk of a portfolio of financial assets is estimated according to adjustments in the value of the portfolio originated by changes in risk factors. Regarding debt instruments, the most important risk factor is interest rates. Interest rates are also a risk factor when valuing other instruments like options, futures, and forwards. As for bonds, the valuation method commonly used consists of discounting future flows to the present value. In order to do so, both the flows and their payment dates must be known and the applicable discount factors as well. While flows and dates on which they are generated are specified in a contract, discount factors are the result of a market consensus in relation to the temporal structure of future interest rates applicable to the bond s valuation. Since this structure is uncertain, bond portfolio risk analysis often resorts to the Monte Carlo simulation in order to generate multiple temporal rate structures and obtain the risk profile of the portfolio and its VaR. 2 1 Nelson, C. R. and A. F. Siegel. Parsimonious Modeling and Yield Curves, Journal of Business 60 (October 1987), pp
3 This simulation is usually performed by previously setting the number of points (maturities) on the curve and undertaking a pointtopoint simulation. Besides the potential difficulty involved and the fact that it can lead to inconsistent results, it also involves a complicated code and computational overexertion. This paper examines the use of the Nelson and Siegel s yield curve generation model using a random sample of only four parameters, far fewer than the points on the curve that need to be simulated. 3 II. Spot and forward rates 4 In this section we review algebraic expressions that relate spot rates with forward rates. Since handling compound annual yields using algebraic expressions is more complicated, compound rates will be handled throughout this section. Studies of interest rates with different maturities are normally based on information from debt instruments or bonds. Theoretically speaking, in a complete and taxfree market, the price of a bond is the present value of the coupons paid plus the present value of the principal. Discount rates for different maturities used in the calculation correspond to a spot rate temporal structure; in other words, the yield structure at a given date t is represented by a graph of the spot rate for each maturity date. 5 2 Value at Risk. 3 Around thirty. 4 Svensson, Lars E. O., Estimating and Interpreting Forward Interest Rates: Sweden , Working Paper no. 4871, National Bureau of Economic Research, Spot rates at a given maturity correspond to the interest rate paid on a zerocoupon bond with that maturity. 2
4 Let r (t,t ) be the continuous compound interest rate of a zerocoupon bond bond negotiated at time t and maturing ont > t. L et m = T t be the time it takes to mature. The price at t of this zerocoupon bond with principal equal to 1 peso will denote the discount function d(t,t ). The discount function and the spot rate are therefore related as follows: r (t,t )(T t ) d(t,t ) = P(t,T ) = e Consider a bond with a nominal value of 100 pesos paying annual coupons of c pesos and maturing within m years. The present value at time t of the coupon payment that took place in year k, k=1,2,.., m, will be c d (t, t + k ), and the present value of the principal payable in year m will be 100 d (t, t + m). The price of the bond on the transaction date will therefore be: m P(t,t + m) = cd(t,t + k ) + 100d(t,t + m) k =1 Let f (t,t ',T ) be the forward rate continuously comprised of a forward contract negotiated at time t and whose underlying transaction begins at time t ' and ends at T where T > t ' > t. 3
5 r (t,t ) = u =t T t The forward rate thus relates to the spot rate as follows: f (t,t ',T ) = (T t )r (t,t ) (t ' t )r (t,t ' ) T t ' The instant forward rate is the forward rate of a contract maturing at infinity and is defined as follows: f (t,t ' ) = lim f (t,t ',T) T t ' The forward rate f (t,t ',T ) is the average of instant forward rates: T f (t,u)du f (t,t ',T ) = u =t ' T t ' Similarly, the spot rate r (t,t) ) at time T and maturity T is the average of instant forward rates: T f (t,u)du III. Nelson and Siegel s model Nelson and Siegel developed an adjustment model for interest rate temporal structures, flexible enough to represent the different shapes these curves generally adopt: Monotonous Humped Sshaped 4
6 Nelson and Siegel s parametric model assumes that the instantaneous forward rate is the solution of a second order differential equation with two equal roots. Thus, following the same notation as in the previous section, assuming that the negotiation of a forward contract is t = 0, and that the underlying transaction begins on day m, the function of the instantaneous forward rate is defined as follows: 5
7 where ( ) is the vector of parameters that determines the shape of the curve. As explained in the previous section, the spot rate r (0,m) is obtained by integrating the instantaneous forward rates from 0 to m and dividing them by m. This produces the spot rate function: [ ] To appreciate the flexibility of this model in adapting to the shapes that temporal rate structures commonly adopt, the function s coefficients are examined. Note how these coefficients denote the weight given to the short, medium, and longterm sections of the forward curve. First, when m tends towards infinity, the limit of r (m) is. The constant parameter therefore denotes the weight given to the longterm component. Similarly, denotes the weight given to the shortterm component and denotes relative mediumterm importance in the temporal rate structure. Forward curve components Long term 0.6 Short term (exp(m)) 0.4 Medium term (m*exp(m)) Maturity The chart above shows the different components of the forward curve. The longterm component is a constant that does not reach zero at the limit. The medium 6
8 term component is associated with the only maturity that begins with zero (and thus it is not short term) and reaches zero (and thus it is not long term). The shortterm maturity is the only one that falls repeatedly to zero. At the same time, parameter " " determines the speed at which maturities that include it in the equation tend towards their limits. Therefore, with a small value of " ", curve approximations are better for shortterm maturities than for longterm maturities. Similarly, with large values of " ", a better adjustment is achieved in the long term than in the short term. The chart below shows the feature of variable " ". The curve adjusted with =90 represents better the curve of the original data in the short term, unlike the curve adjusted with =150, which differs more from the real curve in the short term but adjusts better to longterm data. Note how if it were not for " ", the model is linear in other parameters Adjusted data and curves Original data Adjusted with tao = 90 Adjusted with tao = Maturity in days Figure 2 IV. Methodology for adjusting temporal rate structures As described in the corresponding paragraph, simple interest rates for different maturities observed at a given point in time must be converted to continuously compound rates. A yield structure corresponding to the Nelson and Siegel s model is applied to these original data using least squares. 7
9 Although one can simply try to estimate the parameters using a group of conventional nonlinear least squares, the fact the model is linear in all parameters except for can be used with some degree of efficiency. The form used by the authors consists of iterating on the model s nonlinear parameter and successively solving linear regressions on the other parameters until convergence by maximizing R 2 is obtained. A one dimensional search procedure 6 is therefore used on the nonlinear section of the model, and efficiency of the estimation in the linear part is exploited. To facilitate this procedure, we chose the Golden Section method for the one dimensional search. 7 First, Nelson and Siegel s equation is noted as follows: [ ] where r (m) represents the continuously compounded rate at maturity m. The iterative process is as follows. Assuming there is data on spot rates, compounded at different maturities m1,m2,..., mn, and for a given value of parameter τ, parameters a, b and c are obtained as least squares solutions of the system of equations presented below in matrix form. Thus: ( ) [ ] [ ] ( [ ] ) 6 On a single variable. 7 Generally speaking, it is easier to undertake onedimensional searches than to solve the problem all at once. This also results in the need to choose good starting points for the iterative process, which is easier using the method proposed. 8
10 ( ) [ ] [ ] ( [ ] ) Thus, the system of equations simply described by X = MC is: ( ) [ ] [ ] ( [ ] ) which corresponds to a multiple linear regression problem represented by wellknown normal equations ; in other words: C * = (M T M ) 1 M T X. Next, the vector of estimated rates Xˆ is calculated for the same maturities as for the original data using the values obtained; in other words: Xˆ = MC. The sum of the squares for error and the adjusted R 2 corresponding to the Nelson and Siegel model are subsequently calculated as follows: [( ) ( )] After setting parameters a, a, b and c, is optimized using the golden section method within an interval [u,v ].With the new value of parameter the process is repeated until convergence is established at R 2. The vector of parameters (τ, β_0, β_1, β_2) is therefore obtained, which represents the structure of 9
11 dataadjusted rates. The golden method section for finding a global maximum requires the function to be optimized in order for it to be unimodal and quasiconcave. 8 Otherwise, only convergence to a local optimum can be guaranteed. When this occurs, there are normally two local optimums at the limits of the interval, and convergence to one or the other will very likely depend on the starting point. Unfortunately, the function to optimize cannot be guaranteed, which in this case is R 2 adjusted to the Nelson and Siegel model, either unimodal or quasiconcave in relation to parameter. 9 For the four cases examined in this paper (Cetes, Udibonos, Libor, and Tbill curves) mixed concave/convex patterns were found, and a general way of choosing the variability interval to carry out the search was not found. For this reason, prior to optimizing to choose the function domain, target functions were charted at several equidistant points within a broad interval. Depending on the shape of the curves and the maturities of the original data, an initial interval was chosen where the optimum value of was located. Below are the charts of the type of functions produced by the research Adjusted R squared Tao Adjusted R squared Tao Figures 3 and 4 10
12 The good or poor performance of R 2 as a function of tao has nothing to do with the algorithm used to optimize. In the first two functions, R 2 versus are unimodal and quasiconcave and their global optimum is within the interval. Since they behave well, no other precaution is required. In the subsequent one, a function that is also quasiconcave can be observed, but since its rises monotonously in, it is not unimodal within the interval. The optimum would appear to be (which makes no practical sense). The interval in this case is thus truncated once R 2 has been accurately obtained, and the corresponding limit is chosen as the optimum value of. The interval should not be overly extended, since besides from it causing numerical stability problems, the model would adjust well in the long term but at the expense of a poor shortterm adjustment. 0.5 Adjusted R squared Tao Figure 5 The chart below shows a case in which after the local optimum around = 50, the function decreases before rising indefinitely again, although it appears to never surpass the local maximum mentioned. In this case, the interval was narrowed in order for the search to converge to the aforementioned local optimum. 11
13 Adjusted R squared Tao Figure 6 Finally, curves that were not quasiconcave and with local optimums at the two interval limits were also found. In this case, the larger of the two is chosen (the one on the left). Adjusted R squared Tao Figure 7 Each case should be carefully analyzed in order to obtain the best parameter adjustment. For example, in the case of the Cetes curve, a golden section algorithm was run initially allowing to vary in interval (1,1000), but resulted in negatives; in other words, over the longterm, estimated yields had negative values while shortterm yields were excessively high. After several tests, the problem was resolved by narrowing the interval to (10,364), as 364 is the longest maturity for 12
14 which historical data are available. On the basis of this observation, the decision was taken to run the model using a longer maturity, and data was obtained for each of the remaining instruments. V. Simulation of temporal structures In order to assess the technique and illustrate it, we present the results of adjustments to the temporal structures of Cetes, Udibonos, Libor, and Tbill yield curves. The four series of the corresponding parameters were built based on daily data from January 2001 to January A histogram of frequencies per parameter was obtained for the four series of parameters, and the variancecovariance matrix was calculated for the four parameters (,,, ) in that order, and then decomposed using the Cholesky A factorization such that = ΑΑ'. The histograms of the parameters were considered as empirical probability distributions. The simulation model consists simply of applying the following formula: In this equation, denotes the vector of the simulated parameters, is the vector of middle values of the parameters, A is the Cholesky factorization of the variancecovariance matrix of the parameters, and is a vector of random samples of the corresponding empirical distributions of such parameters
15 In order to use this simulation process, the empirical distribution of the parameters had to be standardized by deducting their mean at each interval and dividing it between the distribution s standard deviation in order to not duplicate the effects of multiplying the vector by the matrix using Cholesky s factorization. This process can be used to generate as many temporal structure curves as required. V.1. Examples of curve and parameter distribution adjustments Below are adjustments of the temporal structure on January 28, 2002 for Cetes data. The figures show the initial data based on which Nelson and Siegel s model was adjusted and the values the adjusted curve adopts for certain maturities. The original points are charted and the curve that is formed when substituting the optimum value of the parameters. Subsequently, the proportions of the curves generated by the simulators and of the original curves for which the model was adjusted are presented by grouping them according to the most commonly observed shapes, namely: normal, inverted and mixed. s of yield structures Figure Maturity in days 10 This is achieved by generating random numbers evenly distributed between zero and one (i.e. ) and associating them with the corresponding interval of the accumulated empirical distribution of each parameter using the Monte Carlo method. 13
16 The 1 structure is known as a normal structure in which the yield is an incremental function of the maturity; in other words, the greater the maturity, the higher the yield, as uncertainty increases over time. The 2 structure is known as a mixed structure; it increases up to a certain point during the maturity and then decreases in the long term. The last type of curve is known as an inverted structure because the yield decreases throughout the maturity indicating that shortterm rates are very high and investors expect them to decrease in the future to more normal levels. Subsequently, only the most relevant results obtained for other instruments whose temporal yield structures were adjusted are shown as well as the process used to make the adjustment and the outcome of the Monte Carlo simulation. V.2. Example: Cetes V.2.1. Curve adjustment In the case of the Cetes curve, a decision was made to work with maturities generally observed in the market usually corresponding to 28, 91, 182, and 364 days. The table on the left shows the Cetes yield data at the close of January 28, The optimum parameter vector for the curve at this date obtained using the technique described in paragraph IV is: (,,, ) = ( ) 11 Data obtained from the curve file of Valmer, the price supplier. The file shows both market yields and estimated supplier prices. Note that the estimates of the supplier in the case of longterm Cetes are very similar every day of the year. 14
17 Table 1 Cetes data for 28/01/2002 Maturity Simple yield Continuous yield Cetes adjustment Maturity Simple yield Continuous yield The box on the right and the graph below show the adjustment obtained for the Nelson and Siegel model both numerically and graphically. Simple yield Cetes Maturity in days Cetes data Cetes adj. Figure 9 V.2.2. Empirical distribution of parameters 15
18 From the series of 4 parameters, and the mean vector, and variance and covariance matrix was obtained. In the case of Cetes, the following results were obtained: Matrix A resulting from Cholesky s factorization of is as follows: Histograms corresponding to each parameter were also carried out and normalized as described. The figures below show the histograms obtained for the Cetes yield curve parameters. V.2.3. Simulation process Below are graphs of both the proportions of the structures generated by the simulation and the original structures the model was adjusted to according to the yield structure type: Cetes Cetes Figures 10 and 11 16
19 Frequency Both graphs are very similar, 1 being the most frequently observed structure. There is also a graph showing only some of the structures the simulation generated. Cetes Cetes Figures 12 and 13 Shape of the original curves Shape of the curves generated by the simulation 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Curve type 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Curve type 17
20 V.3. Examples: Udibonos, Libor and Tbill V.3.1. Summary Below is a table summarizing Cholesky s Mean Vector and VarianceCovariance matrixes, derived from the parameter estimates used for their respective structure simulations. The complete results of the adjustments, histograms, and simulation process can be found in the appendices. Table 2 Cetes Udibonos Mean VarCovar
21 V.3.2. Obtaining data and estimating parameters Udibonos Data for Udibonos were obtained from the Valmer vector file. 12 This file contains the Udibonos yield observed in the market and the price supplier s estimated yield. Unlike the Valmer file from which Cetes data were obtained, in the case of Udibonos it is not known which data correspond to market transactions and which are Valmer estimates. For this reason, all maturities published were used. Maturities for daily data w were between 10 and 3700 days. 12 Valmer is a price provider. 19
22 The optimization algorithm was run allowing to vary at interval (10,3700). During the first six months of 2001, Udibonos yields for 15day and longer maturities were published. In the second half of the year, the maturities published begin as of around 200 days. In other words, the model is adjusted using data as of 200 days and is used to estimate shortterm yields. Libor Data for the Libor rate was obtained from Bloomberg. The published rates correspond to 28, 91,182, 273 and 365day maturities. Running the algorithm and allowing vary within a very large interval resulted in negative values of to for estimated longterm yields while shortterm yields took on very high values in relation to observed ones. After several tests, and based on the results found, the problem was solved by reducing the interval to (10,150). The optimum value of 100. was therefore generally below Tbill Data corresponding to Tbill yields were also obtained from Bloomberg. It is important to note that, in general terms, there is data for 91, 182, 365, 730, 1825, 3650 and days. There are very longterm yields and the optimum is high (the mean is 2857). This means that the curve adjusts better in the long term than in the short term. If adjustment is required, the short term may have to reduce the interval of parameter, or an optimization of parameters that ignores longterm data may be needed. In the beginning, the model was adjusted by letting vary in interval (1,1000), with a result in which the parameter always assumed an extreme value of 999; in other words, the interval was too short. By considering a greater interval of (500,6000), parameter generally found optimum values within the interval.
23 V.3.3. Results obtained Since the initial histograms of adjusted parameters resulted in very regular shapes, thought was given to adjusting a known distribution to the estimated parameters, by simplifying the Montecarlo Simulation process even further. However, this is not generally the case, as when you review the histograms for instruments like LIBOR and Udibonos, shown in figures 17 and 18 and further on, the same smooth performance of the histogram was not observed. For this reason, it was decided to simulate parameters directly from their empirical distribution. Udibonos Libor As can be observed, the most problematic parameters were, and ; in other words, those associated with the short, mid and longterm components of the forward curve. 21
24 VI. Conclusions Considering the aforementioned, as is usually the case in such empirical studies certain difficulties were encountered when estimating parameters which were solved by examining the data and by making thoughtful use of the nonlinear least squares optimization algorithm. In order to validate the yield curve simulation methodology, in this paper it was only necessary to simulate 200 temporal structures to obtain distribution parameter convergence. For the purpose of calculating the VaR of a portfolio of financial debt instruments using the Monte Carlo simulation, as many as required should be simulated to guarantee that convergence is as accurate as possible. At present, with the typical portfolios of Mexican banks, this may imply simulating a number of curves greater than 10,000. This is why the Nelson and Siegel model turns out to be relevant, as it implies large savings since the curve is fully specified using four parameters. Perhaps the major flaw of the Nelson and Siegel model is that the adjustment may not be at all convincing in terms of certain maturities, especially on curves contemplating very long maturities. This is because, as seen in section three, depending on the value of parameter, the curve adjusts well in the short term or in the long term, but not both. An interesting hypothesis is to include a new term in the equation in order to solve the problem. Also, this term could provide the function with better concavity properties, thus facilitating optimization by paying a small price by increasing only one parameter. Finally, another pending issue is finding correlations between the parameters of curves associated with different instruments. This would complete the process, as although the dimension of the simulation process considerably increases, it would allow the simulation of congruent groups of curves in order to achieve a better representation of interest rate scenarios. However, this simply implies obtaining the covariance matrix across parameters and the simulation model remains the same.
25 Appendix 1 Golden section method The Golden section search is an iterative method in which the interval for searching the optimum of a variable s function is successively narrowed. In the case of maximizing R 2 to adjust the Nelson and Siegel equation on, the search was performed until an interval less than or equal to 1 was found. Below we explain the optimization algorithm for a minimization by taking [10,m] as the starting interval, where m represents the longest maturity for which data for each of the temporal structures of the yields examined could be obtained. Let [u 1,v 1 ] be the initial interval on which the search for the optimum point will be performed. The subindexes indicate the number of iterations the search is found in. k denotes the iteration counter. Thus, in this initial iteration k=1. Let 1 = u 1 + (1 )(v 1 u 1 ) and 1 = a 1 + (v 1 u 1 ) where = Regressions are calculated for = 1 and for = 1, and the following steps are repeated: 1. If v k u k < 1 stop, the optimum solution is in interval [u k,v k ]. Otherwise, if 3. R 2 > R 2,go to step2 and if R 2 R 2, go to step k k k k 2. u k +1 = k and v k +1 = v k. Also, let k +1 = k y k +1 = u k +1 + (v k +1 u k +1 ). Calculate R 2 ( k +1 ) and go to step u k +1 = u k and v k +1 = k. Also, let k +1 = k and k +1 = u k +1 + (1 )(v k +1 u k +1 ). Calculate R 4. Substitute k for k + 1 and go to step 1. 2 ( k +1 ) and go to step 4. 23
26 Levels Udibonos Appendix 2 Adjustment of yield structures 28/01/2002 Vector of parameters (,,, ) = ( ) Udibonos data for 28/01/2002 Maturity Simple yield Continuous yield Udibonos adjustment Maturity Simple yield Continuous yield Simple yield Udibonos Udibonos data Udibonos adjustment Maturity in days 24
27 Libor Vector of parameters (,,, ) = ( ) Libor data for 28/01/2002 Maturity Simple yield Continuous yield Simple yield Continuous yield Simple yield Libor Libor data Libor adjustment Maturity in days 25
28 Tbill Vector of parameters (,,, ) = ( ) Tbill data for 28/01/2002 Maturity Simple yield Continuous yield Simple yield Continuous yield Simple yield Tbill Maturity in days Tbill data Tbill adjustment 26
29 Frequency Udibonos Appendix 3 s Udibonos Udibonos Udibonos Udibonos
30 Frequency Frequency Frequency Libor Libor Libor Libor Libor
31 Tbill Tbill Tbill Frequency Frequency Tbill Frequency Frequency Tbill 29
32 Frequency Frequency Frequency Appendix 4 Simulations Udibonos Shape of original curves Shape of curves generated by simulation 50% 50% 40% 40% 30% 30% 20% 20% 10% 10% 0% % Curve type Curve type Libor Shape of original curves Shape of curves generated by 80% 70% 70% 60% 60% 50% 50% 40% 40% 30% 30% 20% 20% 10% 10% 0% % Curve type Curve type 30
33 Frequency Tbill Shape of original curves Shape of curves generated by simulation 80% 70% 60% 50% 40% 30% 70% 60% 50% 40% 30% 20% 20% 10% 10% 0% Curve type 0% Curve type 31
34 Bibliography Bliss Robert R., Testing Term Structure Estimation Methods, Working Paper 9612a, Federal Reserve Bank of Atlanta, November Márquez DiezCanedo, Javier. Fundamentos de Teoría de Optimización, Limusa Mokhtar S Bazaraa and C. M. Shetty, Nonlinear Programming, Theory and Algorithms, John Wiley & Sons, USA,1979. J.P. Morgan and Reuters, RiskMetricsTechnical Document, New York, Fourth Edition, Nelson, C. R. and A. F. Siegel. Parsimonious Modeling and Yield Curves, Journal of Business 60 (October 1987), Svensson Lars E. O., Estimating and Interpreting Forward Interest Rates: Sweden , Working Paper No. 4871, National Bureau of Economic Research,
Analysis of Deterministic Cash Flows and the Term Structure of Interest Rates
Analysis of Deterministic Cash Flows and the Term Structure of Interest Rates Cash Flow Financial transactions and investment opportunities are described by cash flows they generate. Cash flow: payment
More informationIntroduction to Bond Valuation. Types of Bonds
Introduction to Bond Valuation (Text reference: Chapter 5 (Sections 5.15.3, Appendix)) Topics types of bonds valuation of bonds yield to maturity term structure of interest rates more about forward rates
More informationAn introduction to ValueatRisk Learning Curve September 2003
An introduction to ValueatRisk Learning Curve September 2003 ValueatRisk The introduction of ValueatRisk (VaR) as an accepted methodology for quantifying market risk is part of the evolution of risk
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationModeling VaR of Swaps. Dr Nitin Singh IIM Indore (India)
Modeling VaR of Swaps Dr Nitin Singh IIM Indore (India) nsingh@iimidr.ac.in Modeling VaR of Swaps @Risk application Palisade Corporation Overview of Presentation Swaps Interest Rate Swap Structure and
More informationSimple Linear Regression Chapter 11
Simple Linear Regression Chapter 11 Rationale Frequently decisionmaking situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related
More informationBonds historical simulation value at risk
Bonds historical simulation value at risk J. Beleza Sousa, M. L. Esquível, R. M. Gaspar, P. C. Real February 29, 2012 Abstract Bonds historical returns can not be used directly to compute VaR by historical
More informationOLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique  ie the estimator has the smallest variance
Lecture 5: Hypothesis Testing What we know now: OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique  ie the estimator has the smallest variance (if the GaussMarkov
More informationFINANCIAL AND INVESTMENT INSTRUMENTS. Lecture 6: Bonds and Debt Instruments: Valuation and Risk Management
AIMS FINANCIAL AND INVESTMENT INSTRUMENTS Lecture 6: Bonds and Debt Instruments: Valuation and Risk Management After this session you should Know how to value a bond Know the difference between the term
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement Primary
Shape, Space, and Measurement Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two and threedimensional shapes by demonstrating an understanding of:
More informationCalculating VaR. Capital Market Risk Advisors CMRA
Calculating VaR Capital Market Risk Advisors How is VAR Calculated? Sensitivity Estimate Models  use sensitivity factors such as duration to estimate the change in value of the portfolio to changes in
More informationAsymmetry and the Cost of Capital
Asymmetry and the Cost of Capital Javier García Sánchez, IAE Business School Lorenzo Preve, IAE Business School Virginia Sarria Allende, IAE Business School Abstract The expected cost of capital is a crucial
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Prealgebra Algebra Precalculus Calculus Statistics
More informationProfit Forecast Model Using Monte Carlo Simulation in Excel
Profit Forecast Model Using Monte Carlo Simulation in Excel Petru BALOGH Pompiliu GOLEA Valentin INCEU Dimitrie Cantemir Christian University Abstract Profit forecast is very important for any company.
More informationRegression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture  2 Simple Linear Regression
Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture  2 Simple Linear Regression Hi, this is my second lecture in module one and on simple
More informationMarket Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk
Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day
More informationYIELD CURVE GENERATION
1 YIELD CURVE GENERATION Dr Philip Symes Agenda 2 I. INTRODUCTION II. YIELD CURVES III. TYPES OF YIELD CURVES IV. USES OF YIELD CURVES V. YIELD TO MATURITY VI. BOND PRICING & VALUATION Introduction 3 A
More informationLecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier
Lecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier Prof. Massimo Guidolin Portfolio Management Spring 2016 Overview The
More informationVALUING FLOATING RATE BONDS (FRBS)
VALUING FLOATING RATE BONDS (FRBS) A. V. Rajwade * Valuing Floating Rate Bonds (FRBs) 1. The principal features of floating rate bonds can be summarised simply: these are bonds day Tbill, refixed every
More informationBond valuation and bond yields
RELEVANT TO ACCA QUALIFICATION PAPER P4 AND PERFORMANCE OBJECTIVES 15 AND 16 Bond valuation and bond yields Bonds and their variants such as loan notes, debentures and loan stock, are IOUs issued by governments
More informationSENSITIVITY ANALYSIS AND INFERENCE. Lecture 12
This work is licensed under a Creative Commons AttributionNonCommercialShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationPrentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1
STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason
More informationEC247 FINANCIAL INSTRUMENTS AND CAPITAL MARKETS TERM PAPER
EC247 FINANCIAL INSTRUMENTS AND CAPITAL MARKETS TERM PAPER NAME: IOANNA KOULLOUROU REG. NUMBER: 1004216 1 Term Paper Title: Explain what is meant by the term structure of interest rates. Critically evaluate
More informationCHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES. From Exploratory Factor Analysis Ledyard R Tucker and Robert C.
CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES From Exploratory Factor Analysis Ledyard R Tucker and Robert C MacCallum 1997 180 CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES In
More informationAdvanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 28 Fourier Series (Contd.) Welcome back to the lecture on Fourier
More informationDoes the Spot Curve Contain Information on Future Monetary Policy in Colombia? Por : Juan Manuel Julio Román. No. 463
Does the Spot Curve Contain Information on Future Monetary Policy in Colombia? Por : Juan Manuel Julio Román No. 463 2007 tá  Colombia  Bogotá  Colombia  Bogotá  Colombia  Bogotá  Colombia  Bogotá
More informationMargin Calculation Methodology and Derivatives and Repo Valuation Methodology
Margin Calculation Methodology and Derivatives and Repo Valuation Methodology 1 Overview This document presents the valuation formulas for interest rate derivatives and repo transactions implemented in
More informationUsing least squares Monte Carlo for capital calculation 21 November 2011
Life Conference and Exhibition 2011 Adam Koursaris, Peter Murphy Using least squares Monte Carlo for capital calculation 21 November 2011 Agenda SCR calculation Nested stochastic problem Limitations of
More informationWebbased Supplementary Materials for Bayesian Effect Estimation. Accounting for Adjustment Uncertainty by Chi Wang, Giovanni
1 Webbased Supplementary Materials for Bayesian Effect Estimation Accounting for Adjustment Uncertainty by Chi Wang, Giovanni Parmigiani, and Francesca Dominici In Web Appendix A, we provide detailed
More informationChapter 2: Systems of Linear Equations and Matrices:
At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,
More informationThis paper is not to be removed from the Examination Halls
~~FN3023 ZA d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON FN3023 ZA BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences,
More information15.433 INVESTMENTS Class 14: The Fixed Income Market Part 2: Time Varying Interest Rates and Yield Curves. Spring 2003
15.433 INVESTMENTS Class 14: The Fixed Income Market Part 2: Time Varying Interest Rates and Yield Curves Spring 2003 TimeVarying Interest Rates 12 10 TBill Rates (monthyly, %) 8 6 4 2 0 Jun85 Jun86
More informationJava Modules for Time Series Analysis
Java Modules for Time Series Analysis Agenda Clustering Nonnormal distributions Multifactor modeling Implied ratings Time series prediction 1. Clustering + Cluster 1 Synthetic Clustering + Time series
More informationRisk and Return in the Canadian Bond Market
Risk and Return in the Canadian Bond Market Beyond yield and duration. Ronald N. Kahn and Deepak Gulrajani (Reprinted with permission from The Journal of Portfolio Management ) RONALD N. KAHN is Director
More informationDURATION AND CONVEXITY
CHAPTER 5 DURATION AND CONVEXITY KEY CONCEPTS Duration Modified duration Convexity DURATION Bonds, as discussed in previous chapters of this book, are subject to the following major risks: Risk of default
More informationThe Classical Linear Regression Model
The Classical Linear Regression Model 1 September 2004 A. A brief review of some basic concepts associated with vector random variables Let y denote an n x l vector of random variables, i.e., y = (y 1,
More informationBest Credit Data Bond Analytics Calculation Methodology
Best Credit Data Bond Analytics Calculation Methodology Created by: Pierre Robert CEO and CoFounder Best Credit Data, Inc. 50 Milk Street, 17 th Floor Boston, MA 02109 Contact Information: pierre@bestcreditanalysis.com
More informationInternet Appendix for Taxes on TaxExempt Bonds
Internet Appendix for Taxes on TaxExempt Bonds Andrew Ang Columbia University and NBER Vineer Bhansali PIMCO Yuhang Xing Rice University This Version: 11 November 2008 Columbia Business School, 3022 Broadway
More informationt = 1 2 3 1. Calculate the implied interest rates and graph the term structure of interest rates. t = 1 2 3 X t = 100 100 100 t = 1 2 3
MØA 155 PROBLEM SET: Summarizing Exercise 1. Present Value [3] You are given the following prices P t today for receiving risk free payments t periods from now. t = 1 2 3 P t = 0.95 0.9 0.85 1. Calculate
More informationConcepts in Investments Risks and Returns (Relevant to PBE Paper II Management Accounting and Finance)
Concepts in Investments Risks and Returns (Relevant to PBE Paper II Management Accounting and Finance) Mr. Eric Y.W. Leung, CUHK Business School, The Chinese University of Hong Kong In PBE Paper II, students
More informationKEYWORDS Monte Carlo simulation; profit shares; profit commission; reinsurance.
THE USE OF MONTE CARLO SIMULATION OF MORTALITY EXPERIENCE IN ESTIMATING THE COST OF PROFIT SHARING ARRANGEMENTS FOR GROUP LIFE POLICIES. By LJ Rossouw and P Temple ABSTRACT This paper expands previous
More informationLecture Quantitative Finance
Lecture Quantitative Finance Spring 2011 Prof. Dr. Erich Walter Farkas Lecture 12: May 19, 2011 Chapter 8: Estimating volatility and correlations Prof. Dr. Erich Walter Farkas Quantitative Finance 11:
More informationSenior Secondary Australian Curriculum
Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero
More informationImplied Vol Constraints
Implied Vol Constraints by Peter Carr Bloomberg Initial version: Sept. 22, 2000 Current version: November 2, 2004 File reference: impvolconstrs3.tex I am solely responsible for any errors. I Introduction
More informationSpot rates, forward rates and plot of the term structure of interest rate.
A N A L Y T I C A L F I N A N C E I I BILLS, NOTES AND BONDS MARKETS: Spot rates, forward rates and plot of the term structure of interest rate FOTSING ARMAND HAMADOU H TAKOETA FRED 1 INTRODUCTION: 3 11
More informationCOGNITIVE TUTOR ALGEBRA
COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,
More informationMathematical Programming
1 The Addin constructs models that can be solved using the Solver Addin or one of the solution addins provided in the collection. When the Math Programming addin is installed, several new command lines
More informationThe Term Structure of Interest Rates, Spot Rates, and Yield to Maturity
Chapter 5 How to Value Bonds and Stocks 5A1 Appendix 5A The Term Structure of Interest Rates, Spot Rates, and Yield to Maturity In the main body of this chapter, we have assumed that the interest rate
More informationEfficient Curve Fitting Techniques
15/11/11 Life Conference and Exhibition 11 Stuart Carroll, Christopher Hursey Efficient Curve Fitting Techniques  November 1 The Actuarial Profession www.actuaries.org.uk Agenda Background Outline of
More informationPrincipal Components Analysis (PCA)
Principal Components Analysis (PCA) Janette Walde janette.walde@uibk.ac.at Department of Statistics University of Innsbruck Outline I Introduction Idea of PCA Principle of the Method Decomposing an Association
More informationC(t) (1 + y) 4. t=1. For the 4 year bond considered above, assume that the price today is 900$. The yield to maturity will then be the y that solves
Economics 7344, Spring 2013 Bent E. Sørensen INTEREST RATE THEORY We will cover fixed income securities. The major categories of longterm fixed income securities are federal government bonds, corporate
More informationNOTES ON THE BANK OF ENGLAND UK YIELD CURVES
NOTES ON THE BANK OF ENGLAND UK YIELD CURVES The MacroFinancial Analysis Division of the Bank of England estimates yield curves for the United Kingdom on a daily basis. They are of three kinds. One set
More informationEconomic Feasibility Studies
Economic Feasibility Studies ١ Introduction Every long term decision the firm makes is a capital budgeting decision whenever it changes the company s cash flows. The difficulty with making these decisions
More informationAppendix E: Graphing Data
You will often make scatter diagrams and line graphs to illustrate the data that you collect. Scatter diagrams are often used to show the relationship between two variables. For example, in an absorbance
More informationThe Effects of Start Prices on the Performance of the Certainty Equivalent Pricing Policy
BMI Paper The Effects of Start Prices on the Performance of the Certainty Equivalent Pricing Policy Faculty of Sciences VU University Amsterdam De Boelelaan 1081 1081 HV Amsterdam Netherlands Author: R.D.R.
More informationNotes for Lecture 3 (February 14)
INTEREST RATES: The analysis of interest rates over time is complicated because rates are different for different maturities. Interest rate for borrowing money for the next 5 years is ambiguous, because
More informationCapital Allocation and Bank Management Based on the Quantification of Credit Risk
Capital Allocation and Bank Management Based on the Quantification of Credit Risk Kenji Nishiguchi, Hiroshi Kawai, and Takanori Sazaki 1. THE NEED FOR QUANTIFICATION OF CREDIT RISK Liberalization and deregulation
More informationProblems and Solutions
Problems and Solutions CHAPTER Problems. Problems on onds Exercise. On /04/0, consider a fixedcoupon bond whose features are the following: face value: $,000 coupon rate: 8% coupon frequency: semiannual
More informationAlgebra 1 Course Information
Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationFundamentals of Operations Research. Prof. G. Srinivasan. Department of Management Studies. Indian Institute of Technology, Madras. Lecture No.
Fundamentals of Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture No. # 22 Inventory Models  Discount Models, Constrained Inventory
More informationBond valuation. Present value of a bond = present value of interest payments + present value of maturity value
Bond valuation A reading prepared by Pamela Peterson Drake O U T L I N E 1. Valuation of longterm debt securities 2. Issues 3. Summary 1. Valuation of longterm debt securities Debt securities are obligations
More informationPrinciple Component Analysis and Partial Least Squares: Two Dimension Reduction Techniques for Regression
Principle Component Analysis and Partial Least Squares: Two Dimension Reduction Techniques for Regression Saikat Maitra and Jun Yan Abstract: Dimension reduction is one of the major tasks for multivariate
More informationIntroduction. example of a AA curve appears at the end of this presentation.
1 Introduction The High Quality Market (HQM) Corporate Bond Yield Curve for the Pension Protection Act (PPA) uses a methodology developed at Treasury to construct yield curves from extended regressions
More informationVALUEatRISK (VaR) COMPUTATIONS under VARIOUS VaR MODELS and STRESS TESTING
1 VALUEatRISK (VaR) COMPUTATIONS under VARIOUS VaR MODELS and STRESS TESTING Suat TEKER* and Barış AKÇAY** *Department of Accounting and Finance, Istanbul Technical University, Maçka 80680 İstanbul.
More informationFIN 472 FixedIncome Securities Forward Rates
FIN 472 FixedIncome Securities Forward Rates Professor Robert B.H. Hauswald Kogod School of Business, AU InterestRate Forwards Review of yield curve analysis Forwards yet another use of yield curve forward
More information, plus the present value of the $1,000 received in 15 years, which is 1, 000(1 + i) 30. Hence the present value of the bond is = 1000 ;
2 Bond Prices A bond is a security which offers semiannual* interest payments, at a rate r, for a fixed period of time, followed by a return of capital Suppose you purchase a $,000 utility bond, freshly
More informationHedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies
Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative
More informationOptimal linearquadratic control
Optimal linearquadratic control Martin Ellison 1 Motivation The lectures so far have described a general method  value function iterations  for solving dynamic programming problems. However, one problem
More informationChapter 9. The Valuation of Common Stock. 1.The Expected Return (Copied from Unit02, slide 36)
Readings Chapters 9 and 10 Chapter 9. The Valuation of Common Stock 1. The investor s expected return 2. Valuation as the Present Value (PV) of dividends and the growth of dividends 3. The investor s required
More informationQuantitative Methods for Finance
Quantitative Methods for Finance Module 1: The Time Value of Money 1 Learning how to interpret interest rates as required rates of return, discount rates, or opportunity costs. 2 Learning how to explain
More informationINTRODUCTORY STATISTICS
INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore
More informationLecture 14 Option pricing in the oneperiod binomial model.
Lecture: 14 Course: M339D/M389D  Intro to Financial Math Page: 1 of 9 University of Texas at Austin Lecture 14 Option pricing in the oneperiod binomial model. 14.1. Introduction. Recall the oneperiod
More informationCHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
CHAPTER : THE TERM STRUCTURE OF INTEREST RATES CHAPTER : THE TERM STRUCTURE OF INTEREST RATES PROBLEM SETS.. In general, the forward rate can be viewed as the sum of the market s expectation of the future
More informationUsing simulation to calculate the NPV of a project
Using simulation to calculate the NPV of a project Marius Holtan Onward Inc. 5/31/2002 Monte Carlo simulation is fast becoming the technology of choice for evaluating and analyzing assets, be it pure financial
More informationSimple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the
More informationRegression III: Advanced Methods
Lecture 16: Generalized Additive Models Regression III: Advanced Methods Bill Jacoby Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Goals of the Lecture Introduce Additive Models
More informationCHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES 1. Expectations hypothesis. The yields on longterm bonds are geometric averages of present and expected future short rates. An upward sloping curve is
More informationLetter to the Student... 5 Letter to the Family... 6 Correlation of Mississippi Competencies and Objectives to Coach Lessons... 7 Pretest...
Table of Contents Letter to the Student........................................... 5 Letter to the Family............................................. 6 Correlation of Mississippi Competencies and Objectives
More information1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number
1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x  x) B. x 3 x C. 3x  x D. x  3x 2) Write the following as an algebraic expression
More informationMath 1111 Journal Entries Unit I (Sections , )
Math 1111 Journal Entries Unit I (Sections 1.11.2, 1.41.6) Name Respond to each item, giving sufficient detail. You may handwrite your responses with neat penmanship. Your portfolio should be a collection
More informationTimeVarying Rates of Return, Bonds, Yield Curves
1/1 TimeVarying Rates of Return, Bonds, Yield Curves (Welch, Chapter 05) Ivo Welch UCLA Anderson School, Corporate Finance, Winter 2014 January 13, 2015 Did you bring your calculator? Did you read these
More informationLeast Squares Estimation
Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN13: 9780470860809 ISBN10: 0470860804 Editors Brian S Everitt & David
More informationHolding Period Return. Return, Risk, and Risk Aversion. Percentage Return or Dollar Return? An Example. Percentage Return or Dollar Return? 10% or 10?
Return, Risk, and Risk Aversion Holding Period Return Ending Price  Beginning Price + Intermediate Income Return = Beginning Price R P t+ t+ = Pt + Dt P t An Example You bought IBM stock at $40 last month.
More informationSTATISTICA Formula Guide: Logistic Regression. Table of Contents
: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 SigmaRestricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary
More informationFE670 Algorithmic Trading Strategies. Stevens Institute of Technology
FE670 Algorithmic Trading Strategies Lecture 6. Portfolio Optimization: Basic Theory and Practice Steve Yang Stevens Institute of Technology 10/03/2013 Outline 1 MeanVariance Analysis: Overview 2 Classical
More informationExcess Implied Return (EIR) 1. Dan Gode and James Ohlson
Excess Implied Return (EIR) 1 1. Overview EXCESS IMPLIED RETURN (EIR) Dan Gode and James Ohlson Investors often start their analysis using screens that provide preliminary indicators of mispriced stocks.
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationModels of Risk and Return
Models of Risk and Return Aswath Damodaran Aswath Damodaran 1 First Principles Invest in projects that yield a return greater than the minimum acceptable hurdle rate. The hurdle rate should be higher for
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationLecture 12/13 Bond Pricing and the Term Structure of Interest Rates
1 Lecture 1/13 Bond Pricing and the Term Structure of Interest Rates Alexander K. Koch Department of Economics, Royal Holloway, University of London January 14 and 1, 008 In addition to learning the material
More informationIntroduction to Risk, Return and the Historical Record
Introduction to Risk, Return and the Historical Record Rates of return Investors pay attention to the rate at which their fund have grown during the period The holding period returns (HDR) measure the
More informationCHAPTER 16: MANAGING BOND PORTFOLIOS
CHAPTER 16: MANAGING BOND PORTFOLIOS PROBLEM SETS 1. While it is true that shortterm rates are more volatile than longterm rates, the longer duration of the longerterm bonds makes their prices and their
More informationThe Measurement of Currency Risk: Comparison of Two Turkish Firms in the Turkish Leather Industry
The Measurement of Currency Risk: Comparison of Two Turkish Firms in the Turkish Leather Industry Kaya Tokmakçıoğlu 1 Abstract Due to the increase in globalization and international trade, there has emerged
More informationECONOMICS 422 MIDTERM EXAM 1 R. W. Parks Autumn (30) Pandora lives in a two period Fisherian world. Her utility function for 2
NAME: ECONOMICS 422 MIDTERM EXAM 1 R. W. Parks Autumn 1994 Answer all questions on the examination sheets. Weights are given in parentheses. In general you should try to show your work. If you only present
More informationDisclaimer: This technical note is not intended to substitute its original version in Spanish for any legal purpose. It is intended solely for
Disclaimer: This technical note is not intended to substitute its original version in Spanish for any legal purpose. It is intended solely for guidance and didactic use. Technical Description of Bonos
More informationAppendix A: Sampling Methods
Appendix A: Sampling Methods What is Sampling? Sampling is used in an @RISK simulation to generate possible values from probability distribution functions. These sets of possible values are then used to
More informationChapter 1 Introduction. 1.1 Introduction
Chapter 1 Introduction 1.1 Introduction 1 1.2 What Is a Monte Carlo Study? 2 1.2.1 Simulating the Rolling of Two Dice 2 1.3 Why Is Monte Carlo Simulation Often Necessary? 4 1.4 What Are Some Typical Situations
More informationHB 5 College Preparatory Math Content Framework
Target Students: This course is appropriate for 12 th grade students whose performance on measures outlined in TEC 28.014 indicates that the student is not ready to perform entrylevel college coursework
More information2014 Assessment Report. Mathematics and Statistics. Calculus Level 3 Statistics Level 3
National Certificate of Educational Achievement 2014 Assessment Report Mathematics and Statistics Calculus Level 3 Statistics Level 3 91577 Apply the algebra of complex numbers in solving problems 91578
More information