# FV of Annuity w/t Continuous Compounding Calculates the Total Returns From a Projects

FV of Annuity – Continuous Compounding = Cash flow × [{ert – 1} ÷ {er – 1}]

r = rate
t = time

Future value of an annuity with continuous compoundingformula helps an analyst to determine the ending balance of an annuity which iscompounded continuously. To understand the FV of an annuity with continuouscompounding one must have a good grip on underlying concepts which includescontinuous compounding and future value of the annuity.

Continuous Compounding:It is a type of compounding which is constant rather than of monthly,quarterly, semi-annually, or annual basis. It may likely to be similar to thebacterial growth which multiplies with time and never stops. A question arisesin mind that how much I can earn more with continuous compounding as opposed tosimple interest. The formula for continuous compounding is;

CC = P × ert

P = principal amount

r = rate

t = time

Future value of an annuity:The future value of the annuity is basically the ending balance of a series ofperiodic payments which is also sometimes involves compounding of interest. Futurevalue of annuity formula just only solves for the ending balance for investmentafter a given period of time.

The future value of an annuity using continuous compoundingformula employs both concepts to calculate the ending balance for an investmentor a project having continuous compounding.

## How FV of an annuity with continuous compounding formula induced?

Future value of an annuity with continuous compoundingformula basically calculates the future cash flows along with interest earnedon it. The formula for that can be shown as;

FV of annuity with continuous compounding = CF(e)r + CF(e)2r … CF(e)rt

The formula assumes that all cash flows are equal and CFindicates for cash flows. If we take er common from the aboveequation then we can rewrite it as;

FV of an annuity with cont. comp. = CF × [(1 – ert) ÷ (1 – er)]

If the above equation is multiplied with -1/-1then it will generate the equation available at the top of the page.

### FV of an annuity with continuous compounding example

Assume that if an individual is looking in to calculate theending balance of a savings account after 5 years. While he has a plan to save\$2,000 at each month and the bank offers 4% continuous compounding interest onsavings accounts.

Here, an important point to note that the example above doesnot say that the initial deposit is instant therefore ‘annuity due’ formulawould be applied instead of a simple annuity. Now putting values in the formulato get;

FV of an annuity withCont. Comp. = \$2,000 × [{e(0.0034)(60) – 1} ÷ {e(0.0034)(60) }]

FV of an annuity with Cont. Comp. = \$120,106.26

Here in the above example, we use the monthly rate (i.e.4%/12/100 = 0.0034). In fact, continuous compounding may offer much more amountcompared to the above figures.