scholarly journals Perturbation analysis of a nonlinear equation arising in the Schaefer-Schwartz model of interest rates

2018 ◽  
Vol 68 (3) ◽  
pp. 617-624
Author(s):  
Beáta Stehlíková
Keyword(s):  
Nonlinear Equation ◽  
Interest Rates ◽  
Closed Form ◽  
Special Functions ◽  
Analytical Formula ◽  
Elementary Functions ◽  
Bond Prices ◽  
The Difference ◽  
The Interest Rate ◽  
Infinite Sums

Abstract We deal with the interest rate model proposed by Schaefer and Schwartz, which models the long rate and the spread, defined as the difference between the short and the long rates. The approximate analytical formula for the bond prices suggested by the authors requires a computation of a certain constant, defined via a nonlinear equation and an integral of a solution to a system of ordinary differential equations. A quantity entering the nonlinear equation is expressed in a closed form, but it contains infinite sums and evaluations of special functions. In this paper we use perturbation methods to compute the constant of interest as an asymptotic serie with coefficients given in closed form and expressed using elementary functions. A quick computation of the bond prices, which our approach allows, is essential for example in calibration of the model by means of fitting the observed yields, where the theoretical bond prices need to be recalculated for every observed date and maturity, as well as every combination of parameters considered. The first step of our derivation is identification of a small parameter in the problem, since it is not immediately clear. We verify our choice by numerical experiments using the values of parameters from the literature.

scholarly journals Reforming LIBOR and Other Financial Market Benchmarks

2015 ◽  
Vol 29 (2) ◽  
pp. 191-212 ◽  
Author(s):  
Darrell Duffie ◽  
Jeremy C. Stein
Keyword(s):  
Financial Markets ◽  
Interest Rates ◽  
Banking System ◽  
Economic Role ◽  
News Reports ◽  
Legal Risks ◽  
The Difference ◽  
The Interest Rate ◽  
Orderly Transition

LIBOR is the London Interbank Offered Rate: a measure of the interest rate at which large banks can borrow from one another on an unsecured basis. LIBOR is often used as a benchmark rate— meaning that the interest rates that consumers and businesses pay on trillions of dollars in loans adjust up and down contractually based on movements in LIBOR. Investors also rely on the difference between LIBOR and various risk-free interest rates as a gauge of stress in the banking system. Benchmarks such as LIBOR therefore play a central role in modern financial markets. Thus, news reports in 2008 revealing widespread manipulation of LIBOR threatened the integrity of this benchmark and lowered trust in financial markets. We begin with a discussion of the economic role of benchmarks in reducing market frictions. We explain how manipulation occurs in practice, and illustrate how benchmark definitions and fixing methods can mitigate manipulation. We then turn to an overall policy approach for reducing the susceptibility of LIBOR to manipulation before focusing on the practical problem of how to make an orderly transition to alternative reference rates without raising undue legal risks.

scholarly journals Pengaruh Faktor Internal Spesifik Bank Terhadap Interest Rate di Perusahaan Sektor Perbankan

2019 ◽  
Vol 19 (1) ◽  
pp. 16
Author(s):  
Ilma Meidira Eprianto ◽  
Catur Rahayu Martiningtiyas
Keyword(s):  
Interest Rate ◽  
Credit Risk ◽  
Interest Rates ◽  
Design Methodology ◽  
Operational Efficiency ◽  
Internal Factors ◽  
Specific Factors ◽  
The Difference ◽  
The Interest Rate ◽  
Positive Effect

<p><strong>Abstrak</strong></p><p><strong>Tujuan</strong> - Penelitian ini bertujuan untuk mengetahui pengaruh faktor spesifik internal bank terhadap <em>interest rate</em>.</p><p><strong>Desain/Metodologi/Pendekatan</strong>  - Regresi data panel berganda yang digunakan  untuk mengukur pengaruh faktor spesifik internal bank seperti <em>liquidity</em>, <em>operational efficiency</em>, <em>credit risk</em>, <em>capitalization</em>, dan <em>lending out ratio</em> terhadap interest rate</p><p><strong>Hasil</strong> – Penelitian ini menemukan bahwa <em>efficiency</em> dan <em>credit</em> <em>risk</em> memiliki pengaruh positif yang signifikan terhadap <em>interest rate </em>sedangkan <em>liquidity</em>, <em>capitalization</em> dan <em>lending out ratio </em>tidak berpengaruh terhadap <em>interest rate</em>.</p><p><strong>Keterbatasan/Nilai </strong>– Pengukuran <em>interest rate</em> tidak menggunakan suku bunga sbi tetapi perhitungan selisih antara suku bunga pinjaman dan suku bunga deposito.</p><p><strong> </strong></p><p><strong>Abstract</strong></p><p><strong>Proposed</strong> - This study aims to determine the effect of bank's specific internal factors on interest rates.</p><p><strong>Design/Methodology/Approach</strong>  - Mutiple panel data was used to analyse bank internal specific factors, namely liquidity, operational efficiency, credit risk, capitalization, and lending out ratio to the interest rate.</p><p><strong>Result</strong>  – The results of this study indicate that efficiency and credit risk have a significant positive effect on interest rates while liquidity but capitalization and lending out ratio do not affect the interest rate</p><p><strong>Novelty/Value</strong> - Interest rate measurement does not use the SBI interest rate but calculates the difference between the loan interest rate and the deposit rate.</p>

Interest Rate Pass-Through: Empirical Results for the Euro Area

2005 ◽  
Vol 6 (1) ◽  
pp. 37-78 ◽  
Author(s):  
Gabe J. de Bondt
Keyword(s):  
Interest Rate ◽  
Interest Rates ◽  
Euro Area ◽  
Vector Autoregressive ◽  
Empirical Results ◽  
Marginal Costs ◽  
Lending Rates ◽  
The Difference ◽  
Pass Through ◽  
The Interest Rate

Abstract This paper empirically examines the interest rate pass-through at the euro area level. The focus is on the pass-through of official interest rates, approximated by the overnight interest rate, to longer-term market interest rates, which, in turn, are a proxy for the marginal costs for banks to attract deposits or grant loans, and therefore passed through to retail bank interest rates. Empirical results, on the basis of a (vector) error-correction and vector autoregressive model, suggest that the pass-through of official interest to market interest rates is complete for money market interest rates up to three months, but not for market interest rates with longer maturities. Furthermore, the immediate pass-through of changes in market interest rates to bank deposit and lending rates is found to be at most 50%, whereas the final pass-through is typically found to be close to 100%, in particular for lending rates. Empirical results for a sub-sample starting in January 1999 show qualitatively similar findings and are supportive of a quicker interest rate pass-through since the introduction of the euro. It is shown that the difference between the adjustment speed of bank deposit and lending rates (typically around one versus three months since the common monetary policy) can to a large extent significantly be explained by credit risk considerations.

scholarly journals How to handle negative interest rates in a CIR framework

SeMA Journal ◽  
2021 ◽  
Author(s):  
Marco Di Francesco ◽  
Kevin Kamm
Keyword(s):  
Interest Rate ◽  
Interest Rates ◽  
Yield Curve ◽  
Analytical Formula ◽  
Market Prices ◽  
Cir Model ◽  
Term Structures ◽  
Negative Interest Rates ◽  
The Difference ◽  
Mean Variance

AbstractIn this paper, we propose a new model to address the problem of negative interest rates that preserves the analytical tractability of the original Cox–Ingersoll–Ross (CIR) model without introducing a shift to the market interest rates, because it is defined as the difference of two independent CIR processes. The strength of our model lies within the fact that it is very simple and can be calibrated to the market zero yield curve using an analytical formula. We run several numerical experiments at two different dates, once with a partially sub-zero interest rate and once with a fully negative interest rate. In both cases, we obtain good results in the sense that the model reproduces the market term structures very well. We then simulate the model using the Euler–Maruyama scheme and examine the mean, variance and distribution of the model. The latter agrees with the skewness and fat tail seen in the original CIR model. In addition, we compare the model’s zero coupon prices with market prices at different future points in time. Finally, we test the market consistency of the model by evaluating swaptions with different tenors and maturities.

scholarly journals The Integral Mittag-Leffler, Whittaker and Wright Functions

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3255
Author(s):  
Alexander Apelblat ◽  
Juan Luis González-Santander
Keyword(s):  
Closed Form ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Laplace Transforms ◽  
Graphical Form ◽  
Generalized Hypergeometric Functions ◽  
Infinite Sums ◽  
First Time ◽  
Mathematical Literature

Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in closed-form. In most reported cases, these new integral functions are expressed as generalized hypergeometric functions but also in terms of elementary and special functions. The behavior of some of the new integral functions is presented in graphical form. By using the MATHEMATICA program to obtain infinite sums that define the Mittag-Leffler, Whittaker, and Wright functions and also their corresponding integral functions, these functions and many new Laplace transforms of them are also reported in the Appendices for integral and fractional values of parameters.

DETERMINAN INFLASI DI INDONESIA PERIODE 1980-2015

2016 ◽  
Vol 21 (1) ◽  
pp. 1-7
Author(s):  
Risna Risna
Keyword(s):  
Interest Rate ◽  
Interest Rates ◽  
Money Supply ◽  
Government Spending ◽  
Positive Impact ◽  
Secondary Data ◽  
Quantitative Approach ◽  
State Spending ◽  
Factors Affecting ◽  
The Interest Rate

This study aims to determine the effect of government spending, the money supply, the interest rate of Bank Indonesia against inflation.This study uses secondary data. Secondary data were obtained directly from the Central Bureau of Statistics and Bank Indonesia. It can be said that there are factors affecting inflationas government spending, money supply, and interest rates BI. The reseach uses a quantitative approach to methods of e-views in the data. The results of analysis of three variables show that state spending significantand positive impact on inflationin Indonesia, the money supply significantand negative to inflationin Indonesia, BI rate a significantand positive impact on inflation in Indonesia

The Demand For Money in Pakistan: Reply (Notes and Comments)

1975 ◽  
Vol 14 (3) ◽  
pp. 370-375
Author(s):  
M. A. Akhtar
Keyword(s):  
Interest Rates ◽  
Money Demand ◽  
Strong Support ◽  
Physical Capital ◽  
Careful Examination ◽  
Demand For Money ◽  
Weak Support ◽  
Complementarity Hypothesis ◽  
The Difference ◽  
The One

I am grateful to Abe, Fry, Min, Vongvipanond, and Yu (hereafter re¬ferred to as AFMVY) [1] for obliging me to reconsider my article [2] on the demand for money in Pakistan. Upon careful examination, I find that the AFMVY results are, in parts, misleading and that, on the whole, they add very little to those provided in my study. Nevertheless, the present exercise as well as the one by AFMVY is useful in that it furnishes us with an opportunity to view some of the fundamental problems involved in an empi¬rical analysis of the demand for money function in Pakistan. Based on their elaborate critique, AFMVY reformulate the two hypo¬theses—the substitution hypothesis and the complementarity hypothesis— underlying my study and provide us with some alternative estimates of the demand for money in Pakistan. Briefly their results, like those in my study, indicate that income and interest rates are important in deter¬mining the demand for money. However, unlike my results, they also suggest that the price variable is a highly significant determinant of the money demand function. Furthermore, while I found only a weak support for the complementarity between money demand and physical capital, the results obtained by AFMVY appear to yield a strong support for that rela¬tionship.1 The difference in results is only a natural consequence of alter¬native specifications of the theory and, therefore, I propose to devote most of this reply to the criticisms raised by AFMVY and the resulting reformulation of the two mypotheses.

scholarly journals Risk Management for Bonds with Embedded Options

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 790
Author(s):  
Antonio Díaz ◽  
Marta Tolentino
Keyword(s):  
Risk Management ◽  
Interest Rate ◽  
Interest Rates ◽  
Interest Rate Risk ◽  
Interest Rate Models ◽  
Management Measures ◽  
Interest Rate Risk Management ◽  
Embedded Options ◽  
The Interest Rate ◽  
Interest Rate Change

This paper examines the behavior of the interest rate risk management measures for bonds with embedded options and studies factors it depends on. The contingent option exercise implies that both the pricing and the risk management of bonds requires modelling future interest rates. We use the Ho and Lee (HL) and Black, Derman, and Toy (BDT) consistent interest rate models. In addition, specific interest rate measures that consider the contingent cash-flow structure of these coupon-bearing bonds must be computed. In our empirical analysis, we obtained evidence that effective duration and effective convexity depend primarily on the level of the forward interest rate and volatility. In addition, the higher the interest rate change and the lower the volatility, the greater the differences in pricing of these bonds when using the HL or BDT models.

Closed-Form Correlation of Buildings Energy Use With Key Design Parameters Calibrated Using a Genetic Algorithm

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
Raed I. Bourisli ◽  
Adnan A. AlAnzi
Keyword(s):  
Genetic Algorithm ◽  
Closed Form ◽  
Energy Use ◽  
Building Design ◽  
Office Buildings ◽  
Design Parameters ◽  
Wide Range ◽  
Variable Function ◽  
Real Coded Genetic Algorithm ◽  
The Difference

This work aims at developing a closed-form correlation between key building design variables and its energy use. The results can be utilized during the initial design stages to assess the different building shapes and designs according to their expected energy use. Prototypical, 20-floor office buildings were used. The relative compactness, footprint area, projection factor, and window-to-wall ratio were changed and the resulting buildings performances were simulated. In total, 729 different office buildings were developed and simulated in order to provide the training cases for optimizing the correlation’s coefficients. Simulations were done using the VisualDOE TM software with a Typical Meteorological Year data file, Kuwait City, Kuwait. A real-coded genetic algorithm (GA) was used to optimize the coefficients of a proposed function that relates the energy use of a building to its four key parameters. The figure of merit was the difference in the ratio of the annual energy use of a building normalized by that of a reference building. The objective was to minimize the difference between the simulated results and the four-variable function trying to predict them. Results show that the real-coded GA was able to come up with a function that estimates the thermal performance of a proposed design with an accuracy of around 96%, based on the number of buildings tested. The goodness of fit, roughly represented by R2, ranged from 0.950 to 0.994. In terms of the effects of the various parameters, the area was found to have the smallest role among the design parameters. It was also found that the accuracy of the function suffers the most when high window-to-wall ratios are combined with low projection factors. In such cases, the energy use develops a potential optimum compactness. The proposed function (and methodology) will be a great tool for designers to inexpensively explore a wide range of alternatives and assess them in terms of their energy use efficiency. It will also be of great use to municipality officials and building codes authors.

Unification proposals for fatigue crack propagation laws

2017 ◽  
Vol 13 (2) ◽  
pp. 262-283 ◽  
Author(s):  
Vladimir Kobelev
Keyword(s):  
Differential Equation ◽  
Crack Propagation ◽  
Crack Length ◽  
Closed Form ◽  
Crack Growth ◽  
Stress Ratio ◽  
Elementary Functions ◽  
Content Type ◽  
Number Of Cycles ◽  
Cycles To Failure

Purpose The purpose of this paper is to propose the new dependences of cycles to failure for a given initial crack length upon the stress amplitude in the linear fracture approach. The anticipated unified propagation function describes the infinitesimal crack-length growths per increasing number of load cycles, supposing that the load ratio remains constant over the load history. Two unification functions with different number of fitting parameters are proposed. On one hand, the closed-form analytical solutions facilitate the universal fitting of the constants of the fatigue law over all stages of fatigue. On the other hand, the closed-form solution eases the application of the fatigue law, because the solution of nonlinear differential equation turns out to be dispensable. The main advantage of the proposed functions is the possibility of having closed-form analytical solutions for the unified crack growth law. Moreover, the mean stress dependence is the immediate consequence of the proposed law. The corresponding formulas for crack length over the number of cycles are derived. Design/methodology/approach In this paper, the method of representation of crack propagation functions through appropriate elementary functions is employed. The choice of the elementary functions is motivated by the phenomenological data and covers a broad region of possible parameters. With the introduced crack propagation functions, differential equations describing the crack propagation are solved rigorously. Findings The resulting closed-form solutions allow the evaluation of crack propagation histories on one hand, and the effects of stress ratio on crack propagation on the other hand. The explicit formulas for crack length over the number of cycles are derived. Research limitations/implications In this paper, linear fracture mechanics approach is assumed. Practical implications Shortening of evaluation time for fatigue crack growth. Simplification of the computer codes due to the elimination of solution of differential equation. Standardization of experiments for crack growth. Originality/value This paper introduces the closed-form analytical expression for crack length over number of cycles. The new function that expresses the damage growth per cycle is also introduced. This function allows closed-form analytical solution for crack length. The solution expresses the number of cycles to failure as the function of the initial size of the crack and eliminates the solution of the nonlinear ordinary differential equation of the first order. The different common expressions, which account for the influence of the stress ratio, are immediately applicable.

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