The current worth of a future sum of money or stream of cash flowsgiven a specified rate of return.
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Determining the appropriate discount rate is the key to valuing future cash flows properly, whether they be earnings or obligations. An annuity is a series of equal payments or receipts that occur at evenly spaced intervals. Leases and rental payments are examples. The payments or receipts occur at the end of each period for an ordinary annuity while they occur at the beginning of each period for an annuity due. The value of an asset or cash at a specified date in the future, based on the value of that asset in the present.
The future value of a stream of payments annuityassuming the payments are invested at a given rate of interest.
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There are several basic equations that represent the equalities listed above. The solutions may be found using in most cases the formulas, a financial calculator or a spreadsheet. In the case of the standard annuity formula, there is no closed-form algebraic solution for the interest rate although financial calculators and spreadsheet programs can readily determine solutions through rapid trial and error algorithms.
These equations are frequently combined for particular uses. For example, bonds can be readily priced using these equations. WeekCalendar epochMonthLunisolar calendarand Computus Equinox seen from the astronomic calendar of Pizzo Vento at Fondachelli FantinaSicily The course of the sun and the moon are the most salient natural, regularly recurring events useful for timekeepingthus in pre-modern societies worldwide lunation and the year were most commonly used as time units.
Nevertheless, the Roman calendar contained remnants of a very ancient pre-Etruscan month solar year. This was mostly based on observation, but there may have been early attempts to model the pattern of intercalation algorithmically, as evidenced in the fragmentary 2nd-century Coligny calendar. The Julian calendar was no longer dependent on the observation of the new moon but simply followed an algorithm of introducing a leap day every four years.
This created a dissociation of the calendar month from the lunation. The Islamic calendar is based on the prohibition of intercalation nasi' by Muhammadin Islamic tradition dated to a sermon held on 9 Dhu al-Hijjah AH 10 Julian date: This resulted in an observation-based lunar calendar that shifts relative to the seasons of the solar year.
Calendar reform The first calendar reform of the early modern era was the Gregorian calendarintroduced in based on the observation of a long-term shift between the Julian calendar and the solar year. There have been a number of modern proposals for reform of the calendar, such as the World CalendarInternational Fixed CalendarHolocene calendarand, recently, the Hanke-Henry Permanent Calendar.
Such ideas are mooted from time to time but have failed to gain traction because of the loss of continuity, massive upheaval in implementation, and religious objections. Calendar systems[ edit ] A full calendar system has a different calendar date for every day. Thus the week cycle is by itself not a full calendar system; neither is a system to name the days within a year without a system for identifying the years. The simplest calendar system just counts time periods from a reference date.
This applies for the Julian day or Unix Time. Virtually the only possible variation is using a different reference date, in particular, one less distant in the past to make the numbers smaller. Computations in these systems are just a matter of addition and subtraction. Other calendars have one or multiple larger units of time. Calendars that contain one level of cycles: Sun and MoonSchedel's Nuremberg Chronicle, Lunar calendars are synchronized to the motion of the Moon lunar phases ; an example is the Islamic calendar.
Solar calendars are based on perceived seasonal changes synchronized to the apparent motion of the Sun; an example is the Persian calendar. Lunisolar calendars are based on a combination of both solar and lunar reckonings; examples include the traditional calendar of Chinathe Hindu calendar in India, and the Hebrew calendar.
The week cycle is an example of one that is not synchronized to any external phenomenon although it may have been derived from lunar phases, beginning anew every month. Very commonly a calendar includes more than one type of cycle, or has both cyclic and non-cyclic elements.
Most calendars incorporate more complex cycles. For example, the vast majority of them track years, months, weeks and days.
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The seven-day week is practically universal, though its use varies. It has run uninterrupted for millennia. Solar calendar Solar calendars assign a date to each solar day. A day may consist of the period between sunrise and sunsetwith a following period of nightor it may be a period between successive events such as two sunsets.
The length of the interval between two such successive events may be allowed to vary slightly during the year, or it may be averaged into a mean solar day. Other types of calendar may also use a solar day.
Lunar calendar Not all calendars use the solar year as a unit. A lunar calendar is one in which days are numbered within each lunar phase cycle.
Because the length of the lunar month is not an even fraction of the length of the tropical yeara purely lunar calendar quickly drifts against the seasons, which do not vary much near the equator. It does, however, stay constant with respect to other phenomena, notably tides.
An example is the Islamic calendar. Alexander Marshack, in a controversial reading,  believed that marks on a bone baton c. Other marked bones may also represent lunar calendars.
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Similarly, Michael Rappenglueck believes that marks on a 15,year-old cave painting represent a lunar calendar. An example is the Hebrew calendar which uses a year cycle. DecadeCenturyand Millennium Nearly all calendar systems group consecutive days into "months" and also into "years". In a solar calendar a year approximates Earth's tropical year that is, the time it takes for a complete cycle of seasonstraditionally used to facilitate the planning of agricultural activities.
In a lunar calendar, the month approximates the cycle of the moon phase. Consecutive days may be grouped into other periods such as the week.
Because the number of days in the tropical year is not a whole number, a solar calendar must have a different number of days in different years. This may be handled, for example, by adding an extra day in leap years.
The same applies to months in a lunar calendar and also the number of months in a year in a lunisolar calendar. This is generally known as intercalation.
Even if a calendar is solar, but not lunar, the year cannot be divided entirely into months that never vary in length. Cultures may define other units of time, such as the week, for the purpose of scheduling regular activities that do not easily coincide with months or years.
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Many cultures use different baselines for their calendars' starting years. For example, the year in Japan is based on the reign of the current emperor: Other calendar types[ edit ] Arithmetic and astronomical calendars[ edit ] Calendar of the QahalAn astronomical calendar is based on ongoing observation; examples are the religious Islamic calendar and the old religious Jewish calendar in the time of the Second Temple.
Such a calendar is also referred to as an observation-based calendar. The advantage of such a calendar is that it is perfectly and perpetually accurate. The disadvantage is that working out when a particular date would occur is difficult. An arithmetic calendar is one that is based on a strict set of rules; an example is the current Jewish calendar.
Such a calendar is also referred to as a rule-based calendar. The advantage of such a calendar is the ease of calculating when a particular date occurs.