Denotational semantics of the Actor model: Difference between revisions
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==Nominal return== | |||
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 | |||
: <math> 1 + RS_{t}=\frac{P_{t}}{P_{t-1}}.</math> | |||
The '''continuously compounded rate of return''' or '''instantaneous rate of return''' ''RC<sub>t</sub>'' obtained during that period is | |||
: <math> RC_{t}=\ln\left (\frac{P_{t}}{P_{t-1}}\right ).</math> | |||
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]]. | |||
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]]. | |||
==Real return== | |||
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 | |||
:<math>P_t^{real} = P_t \cdot \frac{PL_{t-1}}{PL_t}.</math> | |||
Then the continuously compounded real rate of return <math>RC^{real}</math> is | |||
: <math> RC_{t}^{real}=\ln\left (\frac{P_{t}^{real}}{P_{t-1}}\right ).</math> | |||
The continuously compounded real rate of return is just the continuously compounded nominal rate of return minus the continuously compounded inflation rate. | |||
==Source== | |||
*[http://gsbwww.uchicago.edu/fac/eugene.fama/teaching/Reading%20List%20and%20Notes/Continuously%20Componded%20Returns.doc Eugene Fama Notes] | |||
{{DEFAULTSORT:Continuously Compounded Nominal And Real Returns}} | |||
[[Category:Applied mathematics]] | |||
Revision as of 18:35, 4 June 2013
Nominal return
Let Pt be the price of a security at time t, including any cash dividends or interest, and let Pt − 1 be its price at t − 1. Let RSt be the simple rate of return on the security from t − 1 to t. Then
The continuously compounded rate of return or instantaneous rate of return RCt obtained during that period is
If this instantaneous return is received continuously for one period, then the initial value Pt-1 will grow to during that period. See also continuous compounding.
Since this analysis did not adjust for the effects of inflation on the purchasing power of Pt, RS and RC are referred to as nominal rates of return.
Real return
Let be the purchasing power of a dollar at time t (the number of bundles of consumption that can be purchased for $1). Then , where PLt is the price level at t (the dollar price of a bundle of consumption goods). The simple inflation rate ISt from t –1 to t is . Thus, continuing the above nominal example, the final value of the investment expressed in real terms is
Then the continuously compounded real rate of return is
The continuously compounded real rate of return is just the continuously compounded nominal rate of return minus the continuously compounded inflation rate.