翻訳と辞書 Words near each other ・ Continuous symmetry ・ Continuous test-driven development ・ Continuous testing ・ Continuous tone ・ Continuous Tone-Coded Squelch System ・ Continuous track ・ Continuous training ・ Continuous transmission mode ・ Continuous truss bridge ・ Continuous wave ・ Continuous wavelet ・ Continuous wavelet transform ・ Continuous wound infiltration ・ Continuous-flow intersection ・ Continuous-flow manufacturing ・ Continuous-repayment mortgage ・ Continuous-rod warhead ・ Continuous-time Markov chain ・ Continuous-time quantum walk ・ Continuous-time random walk ・ Continuous-time stochastic process ・ Continuous-wave radar ・ Continuously compounded nominal and real returns ・ Continuously embedded ・ Continuously variable slope delta modulation ・ Continuously variable transmission ・ Continuum ・ Continuum (American Horror Story) ・ Continuum (Art Fair) ・ Continuum (chamber ensemble) Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Continuous-repayment mortgage ： ウィキペディア英語版 Continuous-repayment mortgage Analogous to continuous compounding, a continuous annuity〔 - Entry on (continuous annuity )〕〔Mathematics Dictionary p.86〕 is an ordinary annuity in which the payment interval is narrowed indefinitely. A (theoretical) continuous repayment mortgage is a mortgage loan paid by means of a continuous annuity. Mortgages (i.e., mortgage loans) are generally settled over a period of years by a series of fixed regular payments commonly referred to as an annuity. Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. Given loan ''P''0, per period interest rate i, number of periods ''n'' and fixed per period payment ''x'', the end of term balancing equation is: ::$P\_0(1+i)^\; =\; \backslash sum\_^n\; x(1+i)^=\backslash frac$ Summation can be computed using the standard formula for summation of a geometric sequence. In a (theoretical) continuous-repayment mortgage the payment interval is narrowed indefinitely until the discrete interval process becomes continuous and the fixed interval payments become—in effect—a literal cash "flow" at a fixed annual rate. In this case, given loan ''P''0, annual interest rate ''r'', loan timespan ''T'' (years) and annual rate ''M''''a'', the infinitesimal cash flow elements ''M''''a''''δt'' accumulate continuously compounded interest from time t to the end of the loan timespan at which point the balancing equation is: ::$P\_0e^=\backslash int\backslash limits\_^\; M\_ae^\backslash ,\; dt=\backslash frac.$ Summation of the cash flow elements and accumulated interest is effected by integration as shown. It is assumed that compounding interval and payment interval are equal—i.e., compounding of interest always occurs at the same time as payment is deducted.〔Strictly speaking compounding occurs momentarily before payment is deducted so that interest is calculated on the balance as it was before deduction of the period payment.〕 Within the timespan of the loan the time continuous mortgage balance function obeys a first order linear differential equation (LDE)〔Beckwith p. 116: ''"Technically speaking, the underlying equation is known as an ordinary, linear, first order, inhomogenous, scalar differential equation with a boundary condition."''〕 and an alternative derivation thereof may be obtained by solving the LDE using the method of Laplace transforms. Application of the equation yields a number of results relevant to the financial process which it describes. Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of ''n'' fixed monthly payments amount ''x'' invested at a monthly interest rate ''i''% is: :$P\_v(n)\; =\; \backslash frac$ The formula may be re-arranged to determine the monthly payment ''x'' on a loan of amount ''P''0 taken out for a period of ''n'' months at a monthly interest rate of ''i''%: :$x\; =\; \backslash frac)^)\}$ :$N\backslash cdot\; x(N)\; =\; \backslash frac)^\}$ Note that ''N''·''x''(''N'') is simply the amount paid per year – in effect an annual repayment rate ''M''''a''. It is well established that: :$\backslash lim\_\backslash left(1+\backslash frac\backslash right)^=e^$ 〔Beckwith p.115〕〔Munem and Foulis p.273〕 Applying the same principle to the formula for annual repayment, we can determine a limiting value: :$M\_a=\backslash lim\_N\backslash cdot\; x(N)=\backslash lim\_\backslash frac)^\}=\backslash frac)^)\}$ Applying the limiting expression developed above we may write present value as a purely time dependent function: :$P\_v(t)=\backslash lim\_P\_v(N,t)=\backslash frac(1-e^)$〔See also: ps. 470–471〕 Noting that the balance due ''P''(''t'') on a loan ''t'' years after its inception is simply the present value of the contributions for the remaining period (i.e. ''T'' − ''t''), we determine: :$P(t)\; =\; \backslash frac(1\; -\; e^)=\backslash frac\{1\; -\; e^\{-rT\}\}$ 〔Beckwith: Equation (31) p. 124.〕 The graph(s) in the diagram are a comparison of balance due on a mortgage (1 million for 20 years @ ''r'' = 10%) calculated firstly according to the above time continuous model and secondly using the Excel PV function. As may be seen the curves are virtually indistinguishable – calculations effected using the model differ from those effected using the Excel PV function by a mere 0.3% (max). The data from which the graph(s) were derived can be viewed here. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Continuous-repayment mortgage」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.