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| ==Nominal return==
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| Let ''P''<sub>''t''</sub> be the price of a security at time ''t'', including any cash dividends or [[interest]], and let ''P''<sub>''t'' − 1</sub> be its price at ''t'' − 1. Let ''RS''<sub>''t''</sub> be the simple rate of return on the security from ''t'' − 1 to ''t''. Then
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| : <math> 1 + RS_{t}=\frac{P_{t}}{P_{t-1}}.</math>
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| The '''continuously compounded rate of return''' or '''instantaneous rate of return''' ''RC<sub>t</sub>'' obtained during that period is
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| : <math> RC_{t}=\ln\left (\frac{P_{t}}{P_{t-1}}\right ).</math>
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| If this instantaneous return is received continuously for one period, then the initial value ''P''<sub>''t-1''</sub> will grow to <math>P_t = P_{t-1} \cdot e^{RC_t}</math> during that period. See also [[Compound interest|continuous compounding]].
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| Since this analysis did not adjust for the effects of [[inflation]] on the purchasing power of ''P''<sub>''t''</sub>, ''RS'' and ''RC'' are referred to as [[Real versus nominal value (economics)|nominal rates of return]].
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| ==Real return==
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| Let <math> \pi _ t</math> be the purchasing power of a dollar at time ''t'' (the number of bundles of consumption that can be purchased for $1). Then <math>\pi_t = 1/(PL_t)</math>, where ''PL''<sub>''t''</sub> is the price level at ''t'' (the dollar price of a bundle of consumption goods). The simple inflation rate ''IS''<sub>''t''</sub> from ''t'' –1 to ''t'' is <math>\tfrac {PL_t}{PL_{t-1}} - 1</math>. Thus, continuing the above nominal example, the final value of the investment expressed in [[Real versus nominal value (economics)|real]] terms is
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| :<math>P_t^{real} = P_t \cdot \frac{PL_{t-1}}{PL_t}.</math> | |
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| Then the continuously compounded real rate of return <math>RC^{real}</math> is
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| : <math> RC_{t}^{real}=\ln\left (\frac{P_{t}^{real}}{P_{t-1}}\right ).</math> | |
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| The continuously compounded real rate of return is just the continuously compounded nominal rate of return minus the continuously compounded inflation rate.
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| ==Source==
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| *[http://gsbwww.uchicago.edu/fac/eugene.fama/teaching/Reading%20List%20and%20Notes/Continuously%20Componded%20Returns.doc Eugene Fama Notes]
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| {{DEFAULTSORT:Continuously Compounded Nominal And Real Returns}}
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| [[Category:Applied mathematics]]
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Jayson Berryhill is how I'm called and my spouse doesn't like it at all. For a whilst I've been in Alaska but I will have to move in a yr or two. She is truly fond of caving but she doesn't have the time recently. He is an information officer.
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