Getting the valuation formulas right when it comes to annuities

PurposeThe purpose of this paper is to establish the flow-to-equity method, the free cash flow (FCF) method, the adjusted present value method and the relationships between these methods when the FCF appears as an annuity. More specifically, we depart from the two most widely used evaluation settings. The first setting is that of Modigliani and Miller who based their analysis on a stationary FCF. The second setting is that of Miles and Ezzell who worked with an FCF that represents an autoregressive possess of first order.Design/methodology/approachInspired by recent observations in the literature concerning cash flows, discount rates and values in discounted cash flow (DCF) methods, we mathematically derive DCF valuation formulas for annuities.FindingsThe following relationships are established: (a) the correct discount rate of the tax shield when the free cash flow takes the form of a first-order autoregressive annuity, (b) the direct valuation of the tax shield from the free cash flow for a first-order autoregressive annuity, (c) the correct translation from the required return on unlevered equity to the levered equity, when the free cash flow is a stationary annuity and (d) direct calculation of the unlevered and levered firm values and the value of the tax shield for a stationary annuity.Originality/valueUntil now the complete set of formulas for the valuation of stochastic annuities by different DCF methods has not been established in the literature. These formulas are developed here. These formulas are important for practitioners and academics when it comes to the valuation of cash flows of finite lifetime.

Point process simulation of generalised inverse Gaussian processes and estimation of the Jaeger integral

In this paper novel simulation methods are provided for the generalised inverse Gaussian (GIG) Lévy process. Such processes are intractable for simulation except in certain special edge cases, since the Lévy density associated with the GIG process is expressed as an integral involving certain Bessel functions, known as the Jaeger integral in diffusive transport applications. We here show for the first time how to solve the problem indirectly, using generalised shot-noise methods to simulate the underlying point processes and constructing an auxiliary variables approach that avoids any direct calculation of the integrals involved. The resulting augmented bivariate process is still intractable and so we propose a novel thinning method based on upper bounds on the intractable integrand. Moreover, our approach leads to lower and upper bounds on the Jaeger integral itself, which may be compared with other approximation methods. The shot noise method involves a truncated infinite series of decreasing random variables, and as such is approximate, although the series are found to be rapidly convergent in most cases. We note that the GIG process is the required Brownian motion subordinator for the generalised hyperbolic (GH) Lévy process and so our simulation approach will straightforwardly extend also to the simulation of these intractable processes. Our new methods will find application in forward simulation of processes of GIG and GH type, in financial and engineering data, for example, as well as inference for states and parameters of stochastic processes driven by GIG and GH Lévy processes.

Numerical solution of the gas compression problem at a specified law of action

Выполнено численное моделирование одномерных течений политропного газа, описывающее сжатие покоящегося газа с плотностью 1 в покоящийся газ, сжатый до значения 10. Описываемое сжатие происходит без ударных волн эффективным с точки зрения энерговложения способом, так как энергия тратится только на сжатие газа, но не на его разгон Controlled thermonuclear fusion (CTF) is an almost unlimited source of energy and scientists have been studying it for several decades. This requires an efficient and stable compression of diyterium-tritium fuel to a very high density. This work addresses shockless one-dimensional (plane, cylindrical and spherical symmetry cases) “compression from rest to rest”, when gas from the initial resting state under the influence of an impenetrable piston is shocklessly transferred to a resting homogeneous state, but compressed by 10000 times. This compression is energetically most advantageous, because work is spent only on the compression, but not on the gas acceleration. Earlier [10] this problem was solved in the opposite direction of time change. In this case, a density jump occurs on the piston which was taken into account in calculations [3] at the final moment of compression. The numerical solution of this problem in the opposite direction of time variation allows calculating the trajectory of the compressing piston in the form of a set of points ( t,r ) at which the gas velocity and density are determined. In this paper, the problem of shockless “compression from rest to rest” is numerically solved in the forward direction of time change if the compressing piston trajectory is known. The compression piston moves along a monotonous trajectory away from the axis or center of symmetry. It is important, when calculating in forward direction of time change, no internal characteristics are initially entered. They, like all gas flow in the calculation area, are determined in the process of direct calculation. This indicates that the trajectory of compressing piston is the recommendation for appropriate physical experiments

The hindcast of recent tsunamis in Northern Pacific

Three earthquakes occurred in the North Pacific in 2020, causing observable tsunamis. The tsunamis were not devastating. Numerical modelling of tsunami propagation was performed to reproduce operational forecasting (retrospective analysis) of waveforms at deep-water stations. Direct calculation of tsunami using USGS finite-fault source data on GPU was carried out. The leap-frog (Arakawa staggered grid) scheme calculation over the Pacific Ocean on a regular grid with a spatial step of 0.5 arc minutes of 1440 min (1 day) tsunami propagation was performed in approximately 90 min of computer time. With use of a hybrid cluster with several GPU accelerators and proper optimization of the simulation algorithm, this time can be reduced by tens of times. Consequently, the time for estimating the transfer function will be comparable to the travel time of a tsunami to the stations, where the forecasts data is. It will make possible to forecast the shape of a tsunami at any point with a lead time enough to decide for tsunami alert at sites where a tsunami poses a real danger. The calculation results are in good agreement with the real data of deep-ocean measurements. The quality of the forecast is comparable to the quality of calculations by other methods.

Survival landscape of different tumor regression grades and pathologic complete response in rectal cancer after neoadjuvant therapy based on reconstructed individual patient data

Background Neoadjuvant therapy can lead to different tumor regression grades (TRG) in rectal cancer after neoadjuvant therapy. The purposes of this study are to investigate the relationships among TRG, pathologic complete response (pCR) and long-term survival, on the basis of reconstructed individual patient data (IPD). Methods The PubMed, Embase, Ovid and Cochrane CENTRAL databases were searched. The primary endpoint was to evaluate the survival landscape of different TRGs after neoadjuvant therapy and the secondary endpoint was to evaluate the associations between pCR and survival. IPD were reconstructed with Kaplan–Meier curves. Results The 10-year overall survival (OS) and 5-year disease-free survival (DFS) were clearly higher in the pCR group than in the non-pCR (npCR) group (80.5% vs. 48.3, 90.1% vs. 69.8%). Furthermore, the OS and DFS increased with improvement in tumor regression after neoadjuvant therapy. According to the IPD, the pCR group had longer OS (HR = 0.240, 95% CI = 0.177–0.325, p < 0.001) and DFS (HR = 0.274, 95% CI = 0.205–0.367, p < 0.001) than the npCR group. Better tumor regression was associated with better survival outcomes (p < 0.005). Direct calculation of published HR values yielded similar results. Conclusions Our results indicate a positive relationship between better tumor regressions and improved survival benefits among the npCR group and patients with rectal cancer achieving pCR had much longer OS and DFS than patients achieving npCR, presenting a survival landscape of different TRGs and pCR in rectal cancer after neoadjuvant therapy.

Synthesis of four new neuro-statistical tests for testing the hypothesis of independence of small samples of biometric data

The paper considers the analysis of small samples according to several statistical criteria to test the hypothesis of independence, since the direct calculation of the correlation coefficients using the Pearson formula gives an unacceptably high error on small biometric samples. Each of the classical statistical criteria for testing the hypothesis of independence can be replaced with an equivalent artificial neuron. Neuron training is performed based on the condition of obtaining equal probabilities of errors of the first and second kind. To improve the quality of decisions made, it is necessary to use a variety of statistical criteria, both known and new. It is necessary to form networks of artificial neurons, generalizing the number of artificial neurons that is necessary for practical use. It is shown that the classical formula for calculating the correlation coefficients can be modified with four options. This allows you to create a network of 5 artificial neurons, which is not yet able to reduce the probability of errors in comparison with the classical formula. A gain in the confidence level in the future can only be obtained when using a network of more than 23 artificial neurons, if we apply the simplest code to detect and correct errors.

Displacement prediction and optimization of a non-circular planetary gear hydraulic motor

The non-circular planetary gear hydraulic motor is a low-speed and high-torque hydraulic motor with excellent performance. It has the characteristics of a wide speed range, low weight and is widely used in various fields. Aiming to solve the problem of there being no intuitive formula for calculating the displacement of the non-circular planetary gear hydraulic motor at present, based on the analysis of the effects of structural parameters on the displacement of the motor, this paper proposes a formula for calculating the displacement of a non-circular planetary gear hydraulic motor when the pitch curve of the sun wheel is a high-order ellipse. The formula allows the direct calculation and prediction of the displacement of the motor. To improve the unit volume displacement of the hydraulic motor (which determines the power density of the motor), based on the analysis of the unit volume displacement constraints, an optimization equation is proposed by adding an optimization factor to the original equation of the pitch curve of the sun wheel. It is seen that the addition of the new optimization factor eliminates the self-interlacing of the pitch curve of inner ring gear. This elimination increases the unit volume displacement of the motor.