# continuous compounding

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## Continuous compounding

The process of accumulating the time value of money forward in time on a continuous, or instantaneous, basis. Interest is earned constantly, and at each instant, the interest that accrues immediately begins earning interest on itself.

## Continuous Compounding

Describing interest that accumulates on a constant basis. That is, if a loan has continuous compounding interest, the interest accumulates all the time, which means that the interest added to the loan balance also begins earning interest on itself. In the short and medium term, this has almost the same yield as daily compounding interest but over the long term, the continuous compounding interest can earn much more. Albert Einstein was once reported as saying that there is no more powerful force in the universe than continuous compounding interest. See also: Compounding.

## continuous compounding

Compounding of interest using the shortest possible interval of time. Although continuous compounding sounds impressive, in practice it results in virtually the same effective yield as daily compounding.
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Expressing the interest rate and the convenience yield as continuously compounded spot rates, [y.sub.t]([tau]) and [[psi].sub.t]([tau]), and rearranging, we can restate this equality as:
Simple, compound and continuous interest rate models are defined as below (Hussain 2013): Simple, compound, and continuously compounded interest formulae: The formula to calculate simple interest is (1) In this formula, I is the actual amount of interest you earn or pay, P is the principal of the investment or loan (the amount of money you invest or borrow), r is the annual interest rate, and t is the time you invest the money for or the time for the loan.
The methodology employed involved the performance of a step-wise regression analysis of future macroeconomic growth, as proxied by GDP growth (continuously compounded one period hence), against the lagged returns of the four risk factors.
We describe one other rate of return, the continuously compounded rate of return, denoted [delta].
There are two equations used to account for the effects of inflation: one if the nominal rate of return is continuously compounded and one if the nominal rate is not continuously compounded.
It should therefore be verified at any point in time, whatever returns are annually or continuously compounded.
It usually is measured as the standard deviation of expected continuously compounded rates of return on the stock.
Let y(t, n), without the symbol for infinity, denote continuously compounded yields.
PVC slugs, continuously compounded on the kneader, are hot-cut into chips that are directly conveyed to the first nip of the calender for development into credit card stock, swimming pool liners, and packaging.
The continuously compounded rate of return, [Alpha], on the assets will satisfy [e.sup.[Alpha]] = [Theta] [e.sup.[r.sub.e]] + (1 - [Theta]) [e.sup.r], where [r.sub.e] and r are the continuously compounded rates of return on tax-exempt and taxable bonds, respectively.
C = the value of the call option, S = the current stock price, [d.sup.1] = (log(S/X) + [r.sub.f]t + [[sigma].sup.2]t/2)/[sigma] [square root of t,] [d.sub.2] = (log(S/X) + [r.sub.f]t - [[sigma].sup.2]t/2)/[sigma] [square root of t,] N(d) + cumulative normal probability density function, X = exercise price associated with the option, t = time to expiration of the option, [[sigma].sup.2] = annualized variance of the continuously compounded returns on the stock, [r.sub.f] = continuously compounded return on a riskless investment with a maturity of t.
Daily continuously compounded returns of the DJIA are taken over the period 01/02/1930 to 06/01/2006.

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