Complex argument

comparison of atan and atan2 functions

The principal value of complex number argument measured in radians can be defined as:

· values in the range [0, 2π)

· values in the range (-π, π].

To compute these values one can use functions:

· atan2 with principal value in the range (-π, π]

· atan with principal value in the range (-π/2, π/2]

Term of investment

In economics or macroeconomics

In economic theory or in macroeconomics, investment is the amount purchased per unit time of goods which are not consumed but are to be used for future production (i.e. capital). Examples include railroad or factory construction. Investment in human capital includes costs of additional schooling or on-the-job training. Inventory investment is the accumulation of goods inventories; it can be positive or negative, and it can be intended or unintended. In measures of national income and output, "gross investment" (represented by the variable I) is also a component of gross domestic product (GDP), given in the formula GDP = C + I + G + NX, where C is consumption, G is government spending, and NX is net exports, given by the difference between the exports and imports, X − M. Thus investment is everything that remains of total expenditure after consumption, government spending, and net exports are subtracted (i.e. I = GDP − C − G − NX).

Non-residential fixed investment (such as new factories) and residential investment (new houses) combine with inventory investment to make up I. "Net investment" deducts depreciation from gross investment. Net fixed investment is the value of the net increase in the capital stock per year.

Fixed investment, as expenditure over a period of time ("per year"), is not capital. The time dimension of investment makes it a flow. By contrast, capital is a stock — that is, accumulated net investment to a point in time (such as December 31).

Investment is often modeled as a function of Income and Interest rates, given by the relation I = f(Y, r). An increase in income encourages higher investment, whereas a higher interest rate may discourage investment as it becomes more costly to borrow money. Even if a firm chooses to use its own funds in an investment, the interest rate represents an opportunity cost of investing those funds rather than lending out that amount of money for interest.^[1]

In finance

In finance, investment is the application of funds to hold assets over a longer term in the hope of achieving gains and/or receiving income from those assets. It generally does not include deposits with a bank or similar institution. Investment usually involves diversification of assets in order to avoid unnecessary and unproductive risk.

In contrast, dollar (or pound etc) cost averaging and market timing are phrases often used in marketing of collective investments and can be said to be associated with speculation.

Investments are often made indirectly through intermediaries, such as pension funds, banks, brokers, and insurance companies. These institutions may pool money received from a large number of individuals into funds such as investment trusts, unit trusts, SICAVs etc to make large scale investments. Each individual investor then has an indirect or direct claim on the assets purchased, subject to charges levied by the intermediary, which may be large and varied.

interest rate.

An interest rate is the rate at which interest is paid by borrowers for the use of money that they borrow from a lender. Specifically, the interest rate (I/m) is a percent of principal (P) paid at some rate (m). For example, a small company borrows capital from a bank to buy new assets for its business, and in return the lender receives interest at a predetermined interest rate for deferring the use of funds and instead lending it to the borrower. Interest rates are normally expressed as a percentage of theprincipal for a period of one year

Real vs nominal interest rates

The nominal interest rate is the amount, in percentage terms, of interest payable.

For example, suppose a household deposits $100 with a bank for 1 year and they receive interest of $10. At the end of the year their balance is $110. In this case, the nominal interest rate is 10% per annum.

The real interest rate, which measures the purchasing power of interest receipts, is calculated by adjusting the nominal rate charged to take inflation into account. (See real vs. nominal in economics.)

If inflation in the economy has been 10% in the year, then the $110 in the account at the end of the year buys the same amount as the $100 did a year ago. The real interest rate, in this case, is zero.

After the fact, the 'realized' real interest rate, which has actually occurred, is given by the Fisher equation, and is

where p = the actual inflation rate over the year. The linear approximation

is widely used.

The expected real returns on an investment, before it is made, are:

where:

= real interest rate

= nominal interest rate

= expected or projected inflation over the year

Accumulated Value

The total amount an investment currently holds, including the capital invested and the interest (gain) it has earned to date. Accumulated value is important in the insurance field because it refers to the total acquired value in cash value life insurance. It is calculated as the sum or total of the initial investment plus the interest earned to date.

Also referred to as accumulated amount or cash value.

Simple and compound interest

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