# Future Value with Continuous Compounding

Future value of a single sum compounded continuously can be worked out by multiplying it with e (2.718281828) raised to the power of product of applicable annual percentage rate (r) and time period (t).

Let’s say you have $1,000 deposited in an account that earns 8% per annum. If there is annual compounding, value of$1,000 after one year will be $1,080 (=$1,000 × (1 + 8%). If there is semi-annual, quarterly, monthly and daily compounding, the future value will be :

Semi-annual Compounding

$$=\text{\1,000}\times{\left(\text{1}+\frac{\text{8%}}{\text{2}}\right)}^\text{2}=\text{\1,081.60}$$

Quarterly Compounding

$$=\text{\1,000}\times{\left(\text{1}+\frac{\text{8%}}{\text{4}}\right)}^\text{4}=\text{\1,082.43}$$

Monthly Compounding

$$=\text{\1,000}\times{\left(\text{1}+\frac{\text{8%}}{\text{12}}\right)}^{\text{12}}\\=\text{\1,083.00}$$

Daily Compounding

$$=\text{\1,000}\times{\left(\text{1}+\frac{\text{8%}}{\text{365}}\right)}^{\text{365}}\\=\text{\1,083.28}$$

If you notice, the differential that results from increase in compounding frequency drops as we move to smaller and smaller time units. If interest is compounded each nanosecond, the future value will equal $1,083.287 as worked out below: $$\text{FV}\ (\text{continous compounding})\\=\text{\1,000}\times{\text{2.718281828}}^{\text{0.08}\times\text{1}}\\=\text{\1,083.287}$$ ## Formula Following is the formula for determining future value of a single sum in case of continuous compounding: $$\text{FV}\ (\text{continous compounding})=\text{PV}\times \text{e}^{\text{r}\times \text{t}}$$ Where PV is the value of the single sum at t=0, e is a constant which equals 2.718281828, r is the annual nominal percentage rate and t is the time period in years. Similarly, future value of an annuity that is subject to continuous compounding can be worked out using the following formula: $$\text{FV of Annuity}\ (\text{Continous Compounding})\\=\text{PMT}\times\frac{\text{e}^{\text{r}\times \text{t}}-\text{1}}{\text{e}^\text{r}-\text{1}}$$ Where e is 2.718281828, r is the periodic nominal interest rate (i.e. interest rate applicable to each payment period) and t is the total number of payments. ## Example Find out future value of$1,000 deposited each quarter for 3 years if interest rate is 9%.

The periodic interest rate is 2.25% (=9%/4) and applicable number of periods is 12 (=4×3).

Future value of the annuity can be worked out as follows:

$$\text{FV of Annuity}\ (\text{Continous Compounding})\\=\text{\1,000}\times\frac{{\text{2.718281828}}^{\text{0.0225}\times\text{12}}-\text{1}}{{\text{2.718281828}}^{\text{0.0225}}-\text{1}}=\text{\13,621.8}$$