# Compound interest

A capital is placed at **compound interest** when the interests of each period are capitalized to increase it gradually and bear interests in turn. It is a notion antagonistic to that of simple interests , where the interests are not reinvested to become in turn interest – bearing.

## Calculation of compound interest

To calculate compound interest annually , use a geometric sequence whose formula is:

- {\ displaystyle V_ {f} = V_ {i} \ cdot (1+ \ rho) ^ {a} \,},

or {\ displaystyle V_ {f}} is the final value, {\ displaystyle V_ {i}} the initial value, {\ displaystyle \ rho} the interest rate over a period, and {\ displaystyle a}the number of periods (years, semesters, quarters, etc.). The habit is to express the interest rate in percentage , so we will write 2% for{\ displaystyle \ rho = 0,02}.

For example, by placing 10 units of any currency at a rate of 2% per year for 5 years, we obtain:

{\ displaystyle 10 \ cdot (1 + 2/100) ^ {5} = 11.04} units.

After 10 years, the total will be 12.19 units in this currency; after a century, 72.45 units.

This amount {\ displaystyle V_ {f}} is also that which is due by a borrower after {\ displaystyle a} years, at the interest rate {\ displaystyle \ rho} (if he has not refunded anything in the meantime).

Interests can also be compounded on {\ displaystyle n} fractions of a year, for example 12 months, even if the rate {\ displaystyle \ rho}remains expressed per year. An interest equal to{\ displaystyle \ rho / 12}is then paid at the end of each month. The final value after{\ displaystyle a} years is then given by

- {\ displaystyle V_ {f} = V_ {i} \ cdot (1 + \ rho / n) ^ {na} \,}.

One can also compose the interest on quarters or days. To compare the different periods of composition, calculate the effective rate over one year:

- {\ displaystyle 1 – {\ frac {V_ {f}} {V_ {i}}} = 1- \ left (1 + \ rho / n \ right) ^ {n}}.

For the same rate {\ displaystyle \ rho}the shorter the compounding period, the higher the effective rate.

But it is interesting to note that the effective rate converges towards a well-defined value when the year is divided into infinitely infinite periods of composition, that is to say when {\ displaystyle n}tends towards infinity. Indeed, it can be shown that:

- {\ displaystyle \ lim _ {n \ to + \ infty} (1 + \ rho / n) ^ {n} = e ^ {rho}.

This formula is used to calculate so-called compound interest continuously.

## Final value

- {\ displaystyle V_ {f} = V_ {i} (1+ \ rho) ^ {a} \,}

This formula gives the future value {\ displaystyle V_ {f}} an investment {\ displaystyle V_ {i}} with an increase at an interest rate of {\ displaystyle \ rho} while {\ displaystyle a} periods.

## Initial value

- {\ displaystyle V_ {i} = {\ frac {V_ {f}} {\ left (1 + \ rho \ right) ^ {a}}} \,}

This formula gives the initial value {\ displaystyle V_ {i}} (or present value) needed to obtain some future value {\ displaystyle V_ {f}} if the interest rate of {\ displaystyle \ rho} is capitalized during {\ displaystyle a} periods.

## Interest rate

- {\ displaystyle \ rho = \ left ({\ frac {V_ {f}} {V_ {i}}} \ right) ^ {\ left ({\ frac {1} {a}} \ right)} – 1}

This formula gives the compound interest rate{\ displaystyle \ rho} got if an initial investment {\ displaystyle V_ {i}} gives a final value {\ displaystyle V_ {f}} after {\ displaystyle a} periods of growth.

## Periods needed

- {\ displaystyle a = {\ frac {\ ln {\ frac {V_ {f}} {V_ {i}}}} {\ ln {(1 + {\ rho})}}}}

This formula gives the number of periods {\ displaystyle a} needed to get a final value {\ displaystyle V_ {f}} from an initial investment {\ displaystyle V_ {i}} if the interest rate is {\ displaystyle \ rho} ({\ displaystyle \ ln}denotes the natural logarithm function ).