**Future value with continuous compounding = PV × e**^{rt}

**PV = Present value**

**r = rate**

**t = time**

**e = mathematical constant**

Future value with continuous compounding helps to determine the future value of money at the present time. FV with continuous compounding inherit some underlying concepts behind the idea which leads to establishing this equation for the sake of personal use.

Before moving further, one must be able to distinguish between the time value of money, future value, and continuous compounding.

## How continuous compounding formula derived

The formula for the present value of continuous compoundingwas derived from the future value of an interest-bearing investment. Here isthe formula is written for the future value of interest-bearing account;

**Future value (FV) = PV × [1 + (i ÷ n)] ^{n × t}**

From the aboveequation if we calculate the limit of formula as ** n** approaches to infinitythen we get a more simplified version of this formula;

**FV = PV × e ^{i × t}**

Now we will discuss some basic concepts of present value, time value of money, and continuous compounding. So, that one can understand the present value of continuous compounding precisely.

**Present value:**Time value of money plays the underlying role to bring-up the present valueconcept. Present value says that if a person has two options to receive anamount of $500 today vs. $570 after 5 years. Then to make a precise decisionand make more profits he/she must have to calculate the value of $570 receivedafter 5 years at the present time.

**Time value of money:**It is basically the idea that a specific amount available today will worth morethan the same amount available after 2 years. The underlying concept for thatis inflation because prices changes over time and thus the currency devaluesover time. For example, if $500 is received today it will worth more than $500received after 2 years.

**Continuouscompounding: **Continuous compounding concept says that the compounding isconstant. While ordinary compounding has a compound basis like annual, semi-annual,monthly, or quarterly. But continuous compounding is constant and exhibits aninfinite level of compounding within a specific time.

### Future value continuous compounding example

For instance, let’s assume that Miss. Olivia wants tocalculate the balance of her investment account after 5 years from today’sdate. This account earns 6% per annum and uses continuous compounding approachand current balance in the account is $3,600.

Now putting values in the FV with continuous compoundingformula, where time ** t** is 5-years and 0.06 for the rate

**, while the presentvalue is $3,600 and the final equation is;**

*r*FV w/t continuouscompounding = $3,600 × e^{(0.06) (5)}

FV w/t continuouscompounding = $4,859.49

So, the final value by using FV with continuous compoundingis $4,859.49 after 5 years of continuous compounding.