Savings Calculator

This projection combines two classical time-value formulas: the future value of a lump sum, P times (1+r) raised to n, and the future value of an ordinary annuity, M times ((1+r) raised to n, minus 1) divided by r. The second term is a geometric series - each monthly deposit compounds for a different number of periods, and summing that series collapses to the closed form shown. Here r is the periodic rate, the annual rate divided by 12, and n is the number of months, so a 6 percent nominal annual rate becomes 0.5 percent per month across 120 months for a ten-year horizon. Deposits are assumed to arrive at the end of each month, making this an ordinary annuity; annuity-due timing, with deposits at the start of the period, would multiply the deposit term by (1+r). Two limitations dominate. First, the model holds the rate fixed for the whole term, which is unrealistic for savings accounts whose rates float with central bank policy - US high-yield accounts paid well under 1 percent through much of 2021 and above 4 percent in 2023 and 2024. Second, the output is nominal. At 3 percent inflation a balance growing at 4 percent gains only about 1 percent in real purchasing power. Interest in a taxable US account is ordinary income, so a saver in the 22 percent bracket keeps roughly 3.1 percent of a 4 percent yield. Compare accounts on APY rather than the stated nominal rate, because APY already folds in the compounding frequency.

Compound interest is far older than the notation used for it. Babylonian clay tablets from roughly 2000 BCE pose problems asking how long a sum lent at 20 percent takes to double, and medieval Italian merchant manuals worked the same questions in tabular form. The first systematic English treatment was Richard Witt in Arithmeticall Questions of 1613, which published compound interest and annuity tables computed to high precision and stated present-value and future-value relationships in essentially modern terms. Later in that century Edmond Halley and others formalized annuity valuation using mortality data, joining interest arithmetic to actuarial practice. Consumer-facing standardization is much more recent. Because banks could quote nominal rates under different compounding conventions, an advertised 5 percent could mean materially different returns from one institution to the next. In the United States the Truth in Savings Act of 1991, implemented through Regulation DD, required depository institutions to disclose a uniform annual percentage yield computed on a 365-day basis, making accounts directly comparable for the first time. Deposit safety has a parallel history: the Federal Deposit Insurance Corporation was created by the Banking Act of 1933 after widespread bank failures, and its coverage limit has been raised repeatedly, reaching 250,000 dollars per depositor per insured bank per ownership category.