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Given a collection of pairs (time, cash flow) representing a project, the NPV is a function of the rate of return. The internal rate of return is a rate for which this function is zero, i.e. the internal rate of return is a solution to the equation NPV = 0 (assuming no arbitrage conditions exist).
For example, irregular internal rate of return (XIRR), a financial function, operates on a collection of cash flow values from one column, but must also apply variable period lengths from another column and an initial iterative assumption from a third, in order to return a single, summarizing value.
As another example, a two-year return of 10% converts to an annualized rate of return of 4.88% = ((1+0.1) (12/24) − 1), assuming reinvestment at the end of the first year. In other words, the geometric average return per year is 4.88%. In the cash flow example below, the dollar returns for the four years add up to $265.
MIRR is calculated as follows: = (), where n is the number of equal periods at the end of which the cash flows occur (not the number of cash flows), PV is present value (at the beginning of the first period), FV is future value (at the end of the last period).
The modified Dietz method [1] [2] [3] is a measure of the ex post (i.e. historical) performance of an investment portfolio in the presence of external flows. (External flows are movements of value such as transfers of cash, securities or other instruments in or out of the portfolio, with no equal simultaneous movement of value in the opposite direction, and which are not income from the ...
Halley's method is a numerical algorithm for solving the nonlinear equation f(x) = 0.In this case, the function f has to be a function of one real variable. The method consists of a sequence of iterations:
For this reason this particular function λ(x) is sometimes called the Cramér function. The rate function defined above in this article is a broad generalization of this notion of Cramér's, defined more abstractly on a probability space , rather than the state space of a random variable.
Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to ...