Payment

A payment is the voluntary tender of money or its equivalent or of things of value by one party (such as a person or company) to another in exchange for goods, or services provided by them, or to fulfill a legal obligation. The party making a payment is commonly called the payer, while the payee is the party receiving the payment.
Payments can be effected in a number of ways, for example:
In general, the payee is at liberty to determine what method of payment he or she will accept; though normally laws require the payer to accept the country's legal tender up to a prescribed limit. Payment is most commonly effected in the local currency of the payee, unless the parties agree otherwise. Payment in another currency involves an additional foreign exchange transaction. The payee may compromise on a debt, i.e., accept a part payment in full settlement of a debtor's obligation, or may offer a discount, E.G: For payment in cash, or for prompt payment, etc. On the other hand, the payee may impose a surcharge, for example, as a late payment fee, or for use of a certain credit card, etc.
Payments are frequently preceded by an invoice or bill, which follow the supply of goods or services, but in some industries (such as travel and hotels) it is becoming common for pre-payments to be required before the service is performed or provided. In some industries, a deposit may be required before services are performed, which acts as a part pre-payment or as security to the service provider. In some cases, progress payments are made in advance, and in some cases part payments are accepted, which do not extinguish the payer’s legal obligations. The acceptance of a payment by the payee extinguishes a debt or other obligation. A creditor cannot unreasonably refuse to accept a payment, but payment can be refused in some circumstances, for example, on a Sunday or outside banking hours. A payee is usually obligated to acknowledge payment by producing a receipt to the payer. A receipt may be an endorsement on an account as "paid in full". The giving of a guarantee or other security for a debt does not constitute a payment.
Etymology.
The root word "pay" in "payment" comes from the Latin "pacare" (to pacify), from "pax", meaning "peace". In the Middle Ages, the term began to be used more broadly, to mean "to pacify one's creditors". As the Latin word was made part of Old French "paier", it retained the meaning "appease" but gained the meaning "to pay" (as in paying a debt). The Middle English word "payen", which came from French, was also used in both ways.
Methods.
There are two types of payment methods; exchanging and provisioning. Exchanging involves the use of money, comprising banknotes and coins. Provisioning involves the transfer of money from one account to another, and involves a third party. Credit card, debit card, cheque, money transfers, and recurring cash or ACH (Automated Clearing House) disbursements are all electronic payments methods. Electronic payments technologies include magnetic stripe cards, smartcards, contactless cards, and mobile payments.
Parties involved.
A payment may involve more than two parties. For example, a pre paid card transaction usually involves four parties: the purchaser, the seller, the issuing bank and the acquiring bank. A cash payment requires at least three parties: the seller, the purchaser and the issuer of the currency. A barter payment requires a minimum of two parties: the purchaser and the seller.
Providers.
The infrastructure and electronic clearing methods are formed by the payment provider. Global credit card payment providers are Diners Club, Visa, American Express and MasterCard. Maestro and Cirrus are international debit card payment providers.
Blockchain also provides an infrastructure for payments using digital currency and potentially better security, fewer intermediaries, and lower transaction costs.
Interbank.
The "central bank CB(Ccy)" of a currency can maintain a loro account for a bank (which the bank would call a nostro account).
Direct.
A bank "P" (the "payer") with a central bank nostro can "pay directly" to another bank "R" (the "receiver") which has also a nostro with "CB(Ccy)" by instructing the central bank to make a payment of "N[Ccy]" (or an amount "N" in "Ccy").
The central bank however will only accept "P"s payment instruction if the balance "B" on P's loro account before the payment is "B≥N". The payment itself is a booking in the loro accounts of the central bank where P's loro account is debited with "N" (it will be "B-N≥0" thereafter) and simultaneously "R"s loro account is credit with "N" (its balance "C" will be "C+N≥0" thereafter).
We denote the payment symbolically with "[P→CB→R]".
Indirect.
If the payer "P" maintains a loro account for another bank "X", "P" can act as a payment agent for "X":
"X" instructs "P" to pay to "R"; then "P" instructs "CB(Ccy)" to pay on behalf of "X": ["X→P→CB→R"].
In this example the first half of the payment is "indirect" and the second one (from "CB(Ccy)" to "R") is still "direct". If "X" would pay via "P" to "R" and finally to another party "Y", the payment would be 'fully indirect': ["X→P→CB→R→Y"].
In practice, not every payment agent might have a direct nostro with the central bank, thus rather weird payment constellations can exist, which are hard to describe.
Directness coefficient "(p,q)".
If we define (p,q) where "p" is number of parties on the sender side and "q" on the receiver side, we can classify the directness of payments.
"Sn,Rm" Description.
We denote with "Sn" the senders and with "Rm" the receivers in a payment process, such that
"R0=S0=CBCCY" is the central bank of the currency which has 'no' distance ("n=m=0") to the payment process,
"m,n=1": the "direct" nostro agents of the sender / receiver: 
S1 is the "direct payer"; R1 is the "direct receiver" (or nostro agent)
"m,n=2": the "indirect nostro agents" of the sender / receiver: 
S2 is the "indirect payer"; R2 is the "indirect receiver"
"m,n=3": the "sender / payer bank" : 
S3 is the "sender bank"; R3 is the "receiver bank"
"m,n=4": the "sender / payer": 
S4 is the "sender (payer)"; R4 is the "receiver (payee)"
Here is an (p=q=4) example:
 [sender→sender bank → indirect agent of sender bank → direct agent of sender bank → central bank → direct agent of receiver bank → indirect agent of receiver bank → receiver bank → receiver(payee)]
or
 [S4 → S3 → S2 → S1 → S0=R0 → R1 → R2 → R3 → R4]
Global market.
In 2005, an estimated $40 trillion globally passed through some type of payment system. Roughly $12 trillion of that was transacted through various credit cards, mostly the 21,000 member banks of Visa and MasterCard. Processing payments, including the extending of credit, produced close to $500 billion in revenue. In 2012, roughly $377 trillion passed through noncash payment systems. This led to total account and transaction revenues close to $524 billion.
Debit cards.
In the U.S., debit cards are the fastest growing payment technology. In 2001, debit cards accounted for 9 percent of all purchase transactions, and this is expected to double to 18.82 percent in 2011.
Cheques.
Historically, cheques have been one of the primary means of payment for purchasing goods and services, though its share in the payment mix is falling worldwide. In 2001, in the United States, cheques accounted for 25% of the U.S.-based payment mix; and in 2006, this was projected to fall to 17%.
In the United States, a cheque as a form of payment can legally be refused for any reason (or no reason). A payment by cheque is not a "payment" until the cheque has been cashed (i.e., deposited) and cleared by the banking system.
Timing.
The timing of a payment has legal implication in some situations. For tax purposes, for example, the timing of a payment may determine whether it qualifies as a deduction in a taxpayer's calculation of taxable income in one year or the next.
For U.S. tax purposes, cash payments generally are taken to occur at the time of payment. Payment may also occur when a person transfers property or performs a service to the payee in satisfaction of an obligation. A payment by check is normally deemed to occur when the check is delivered, as long as the check is honored on presentation by the payee. This rule also generally applies where the check is not presented to the bank until the next taxable year, and even though the payer could stop payment on the check in the meantime. Postdated checks, however, are not considered payment when delivered. Generally, payments by credit card take effect at the point of the sale and not when a payer is billed by the credit card company or when the payer pays the credit card company's bill. A business that reports on an accrual basis, would report income in the year of sale though payment may be received in a subsequent year.
Payment of most fees to government agencies by cheque, if permitted, usually takes effect after a set number of days for clearance or until the cheque is actually cleared. Payments by credit card, if permitted, and cash payments take immediate effect. Normally, no other forms of payment are permitted or accepted.