Amortization

Amortization (or amortisation) is the process of decreasing, or accounting for, an amount over a period. The word comes from Middle English amortisen to kill, alienate in mortmain, from Anglo-French amorteser, alteration of amortir, from Vulgar Latin admortire "to kill", from Latin ad- and mort-, "death".

Applications of amortization

Amortization (business)

In business, amortisation refers to spreading payments over multiple periods. The term is used for two separate processes: amortization of loans and assets. It also refers to allocating the cost of an intangible asset over a period of time.

Amortization of loans

In lending, amortization is the distribution of payment into multiple cash flow installments, as determined by an amortization schedule. Unlike other repayment models, each repayment installment consists of both principal and interest. Amortization is chiefly used in loan repayments (a common example being a mortgage loan) and in sinking funds. Payments are divided into equal amounts for the duration of the loan, making it the simplest repayment model. A greater amount of the payment is applied to interest at the beginning of the amortization schedule, while more money is applied to principal at the end. Commonly it is known as EMI or Equated Monthly Installment.

or, equivalently,

where: P is the principal amount borrowed, A is the periodic amortization payment, r is the periodic interest rate divided by 100 (nominal annual interest rate also divided by 12 in case of monthly installments), and n is the total number of payments (for a 30-year loan with monthly payments n = 30 × 12 = 360).

Amortized analysis

In computer science, amortized analysis is a method for analyzing a given algorithm's time complexity, or how much of a resource, especially time or memory in the context of computer programs, it takes to execute. The motivation for amortized analysis is that looking at the worst-case run time per operation can be too pessimistic.

While certain operations for a given algorithm may have a significant cost in resources, other operations may not be as costly. Amortized analysis considers both the costly and less costly operations together over the whole series of operations of the algorithm. This may include accounting for different types of input, length of the input, and other factors that affect its performance.

History

Amortized analysis initially emerged from a method called aggregate analysis, which is now subsumed by amortized analysis. However, the technique was first formally introduced by Robert Tarjan in his 1985 paper Amortized Computational Complexity, which addressed the need for more useful form of analysis than the common probabilistic methods used. Amortization was initially used for very specific types of algorithms, particularly those involving binary trees and union operations. However, it is now ubiquitous and comes into play when analyzing many other algorithms as well.