Basic Compound Growth
To model growh in a system that doubles every time unit and that starts with a value of 1, one could use:
growth = 2^t where t is how many time units to continue growth.
The above function can be transformed in to a more generic function
growth = initialValue * (1 + returnRate)^t
If we have 2 marbles, and for 3 days, we double the amount of marbles we have
- initalValue = 2
- returnRate = 1
- t = 3
- growth = 16
If we have 2 marbles, and for 3 days, we gain 50% more marbles
- Everything same as above except returnRate is now 0.5
Compounded n timers per time unit
In the above model, interest/growth is compounded at fixed timer intervals. It is possible to compound interest multiple times within a time interval.
For example, if we start with $100 and gain 100% interest every year, we could instead gain 50% interest every half year, or 25% interest every 4th of a year and so on. This will result in larger gains than just simple compound growth.
The equation for this is
growth = initialValue * (1 + returnRate/n)^(n*t) where n is how many times to split the time interval to.
Continuous growth e
Given the fallowing:
- An inital value of 1
- 100% return rate (return rate = 1)
- per ever 1 time unit (t = 1)
- Compounded n timers per time unit
As you increase the number of times to siplit the time unit (i.e. increase n), the output values will get closer and closer to the constant e.
In other words, e is the result of compounding 100% on smaller and smaller time intervals.
If we start with $1 and gain 100% interest every year. for 1 year We could divide the year into 1000 parts (n = 1000) and the final result would be about e.
growth = 1 * (1+1/1000)^(1000 * 1)
growth = (1/1000)^1000
If n were 1 million, or 1 trillion, the final result would still be about e.
- Compound Growth: Applying an amount if interest, based on the current amount, at set time intervals
- Continuous Growth: Applying an amount of interest, based on the current amount, at the smallest time intervals possible i.e. continuously.