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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fisher equation ： ウィキペディア英語版 Fisher equation The Fisher equation in financial mathematics and economics estimates the relationship between nominal and real interest rates inflation. It is named after Irving Fisher, who was famous for his works on the theory of interest. In finance, the Fisher equation is primarily used in YTM calculations of bonds or IRR calculations of investments. In economics, this equation is used to predict nominal and real interest rate behavior. Letting denote the real interest rate, denote the nominal interest rate, and let denote the inflation rate, the Fisher equation is: :$i\; \backslash approx\; r\; +\; \backslash pi$ This is a linear approximation, but as here, it is often written as an equality: :$i\; =\; r\; +\; \backslash pi$ The Fisher equation can be used in either ex-ante (before) or ex-post (after) analysis. Ex-post, it can be used to describe the real purchasing power of a loan: :$r\; =\; i\; -\; \backslash pi$ Rearranged into an ''expectations augmented Fisher equation'' and given a desired real rate of return and an expected rate of inflation (with superscript meaning "expected") over the period of a loan, it can be used as an ex-ante version to decide upon the nominal rate that should be charged for the loan: :$i\; =\; r\; +\; \backslash pi^e$ This equation existed before Fisher,〔http://ia700304.us.archive.org/6/items/appreciationinte00fish/appreciationinte00fish.pdf〕〔http://www.policonomics.com/irving-fisher/〕〔http://199.169.211.101/publications/research/economic_review/1983/pdf/er690301.pdf〕 but Fisher proposed a better approximation which is given below. The approximation can be derived from the exact equation: :$1\; +\; i\; =\; (1\; +\; r)(1\; +\; \backslash pi).$ ==Derivation== Although time subscripts are sometimes omitted, the intuition behind the Fisher equation is the relationship between nominal and real interest rates, through inflation, and the percentage change in the price level between two time periods. So assume someone buys a $1 bond in period while the interest rate is . If redeemed in period, , the buyer will receive dollars. But if the price level has changed between period and , then the real value of the proceeds from the bond is therefore :$(1+r\_)=\; \backslash frac$ 1 + i_t &= \left (1 + r_ \right ) \left (1 + \pi_ \right ) \\ &= 1 + r_ + \pi_ + r_ \pi_ \end Therefore :$\backslash begin$ i_t &= r_ + \pi_ + r_ \pi_ \\ &\approx r_ + \pi_ \end The last line follows from the assumption that both real interest rates and the inflation rate are fairly small, (perhaps on the order of several percent, although this depends on the application) therefore is much larger than and so can be dropped. More formally, this linear approximation is given by using two 1st order Taylor expansions, namely: :$\backslash begin$ \frac &\approx 1-x,\\ (1+x)(1+y) &\approx 1+x+y. \end Combining these yields the approximation: :$1+r=\; \backslash frac\; \backslash approx\; (1+i)(1-\backslash pi)\; \backslash approx\; 1+i-\backslash pi,$ and hence :$r\; \backslash approx\; i\; -\; \backslash pi.$ These approximations, valid only for small changes, can be replaced by equalities, valid for any size changes, if logarithmic units are used, notably centinepers, which are infinitesimally equal to percentages (approximately equal for small values); other logarithmic units differ by scale factors. 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「Fisher equation」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.