Cash accumulation equation

Cash accumulation equation

The cash accumulation equation is an equation which calculates how much money will be in a bank account, at any point in time. The account pays interest, and is being fed a steady trickle of money.

Compound interest

We will approach the development of this equation by first considering the simpler case, that of just placing a lump sum in an account and then making no additions to the sum. With the usual notation, namely

y,!= the current sum (dollars)
P,!= principal (dollars)
i,!= force of interest (per year)
t,!= time (years)
the equation is
:y = Pe^{it},!eqn(1)
and so the sum of money grows exponentially. Differentiating this we derive
:frac{dy}{dt} = iPe^{it}eqn(2)
and eliminating Pe^{it},! between eqns (1) and (2) yields
:dy = iy,dteqn(3)
That is to say, eqn (1) is a solution of eqn (3).

Cash infeed

Having achieved this we are ready to start feeding money into the account, at a rate of F,! dollars/year. This is effected by making a small change to eqn (3) as follows

:dy = iy,dt + F,dt

and accordingly we need to solve the equation

: t = intfrac{dy}{iy+F}

From a table of integrals, the solution is

:t = frac{1}{i}ln(iy + F) + k

where k,! is the constant of integration. The initial sum deposited was P,! so we know one point on the curve :

:(t,y) = (0,P),!

and making this substitution we find that

:k = -frac{1}{i}ln(iP + F)

Using this expression for k,!, and recalling that

:ln(a) - ln(b) = lnleft(frac{a}{b} ight)

gives us the solution :

:it = lnleft(frac{iy + F}{iP + F} ight)

This is the neatest form of the cash accumulation equation, as we are calling it, but it not the most useful form. Using the exponential instead of the logarithmic function, the equation can be written out like this :

:y = Pe^{it} + frac{F}{i}(e^{it}-1)mbox{ , }i e 0eqn(4)

First special case

From this new perspective, eqn (1) is just a special case of eqn (4) - namely with F = 0,.

Second special case

For completeness we will consider the case i = 0,!, and specifically the expression:frac{e^{it}-1}{i}mbox{ , }i = 0,One way of evaluating this is to write out the Maclaurin expansion:e^{it} = 1 + it + frac{(it)^2}{2!} + cdotsAt a glance we can subtract 1,! from this series and divide by i,!, to find out that:frac{e^{it}-1}{i} = tmbox{ , }i = 0,With this result the cash accumulation equation now reads:y = P + Ftmbox{ , }i = 0,Thus the cash sum just increases linearly, as expected, if no interest is being paid.

Third special case

The only other special case to mention is F = -iP,!. Upon making this substitution, eqn (4) becomes simply:y = P,Evidently F,! is negative, and money is being withdrawn rather than deposited. Specifically, the interest is being withdrawn as fast as it is being earned.

An alternative interpretation of this special case is that P,! is negative - the account is overdrawn - and money is being fed in at a rate which just meets the interest charges. A force of interest value is always positive.

External links

* [http://uk.geocities.com/christopher_heys/cae.pdf an extended version of this article]
* [http://formularium.org/?go=61 try out the equation with you own values]


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Interest — For other uses, see Interest (disambiguation). Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money,[1] or money earned… …   Wikipedia

  • Coupon leverage — Coupon leverage, or leverage factor, is the amount by which a reference rate is multiplied to determine the floating interest rate payable by an inverse floater.[1] Some debt instruments leverage the particular effects of interest rate changes,… …   Wikipedia

  • ECONOMIC AFFAIRS — THE PRE MANDATE (LATE OTTOMAN) PERIOD Geography and Borders In September 1923 a new political entity was formally recognized by the international community. Palestine, or Ereẓ Israel as Jews have continued to refer to it for 2,000 years,… …   Encyclopedia of Judaism

  • Strategy dynamics — The word ‘dynamics’ appears frequently in discussions and writing about strategy, and is used in two distinct, though equally important senses.The dynamics of strategy and performance concerns the ‘content’ of strategy – initiatives, choices,… …   Wikipedia

  • India — /in dee euh/, n. 1. Hindi, Bharat. a republic in S Asia: a union comprising 25 states and 7 union territories; formerly a British colony; gained independence Aug. 15, 1947; became a republic within the Commonwealth of Nations Jan. 26, 1950.… …   Universalium

  • IS/LM model — The IS curve moves to the right, causing higher interest rates (i) and expansion in the real economy (real GDP, or Y). The IS/LM model (Investment Saving/Liquidity preference Money supply) is a macroeconomic tool that demonstrates the… …   Wikipedia

  • literature — /lit euhr euh cheuhr, choor , li treuh /, n. 1. writings in which expression and form, in connection with ideas of permanent and universal interest, are characteristic or essential features, as poetry, novels, history, biography, and essays. 2.… …   Universalium

  • computer — computerlike, adj. /keuhm pyooh teuhr/, n. 1. Also called processor. an electronic device designed to accept data, perform prescribed mathematical and logical operations at high speed, and display the results of these operations. Cf. analog… …   Universalium

  • Statement of retained earnings — Accountancy Key concepts Accountant · Accounting period · Bookkeeping · Cash and accrual basis · Cash flow management · Chart of accounts  …   Wikipedia

  • economic development — Process whereby simple, low income national economies are transformed into modern industrial economies. Theories of economic development the evolution of poor countries dependent on agriculture or resource extraction into prosperous countries… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”