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Basic formula for exponential growth

How is exponential growth defined?

Before we go into the details of what exponential growth is and how is it related to exponential decay, let us take a step back and understand what exactly the logic of the term is. Every mathematical function holds a value and that value has a growth rate. Now, since we have defined all the terms required for understanding exponential growth, let us go straight to the definition.

Exponential form of growth occurs as a result of the proportional relation between the growth rate attached to the value that a mathematical function carries and the current value that the function holds. In this way, time turns out to be an exponential function. In a similar way, we can define exponential decay but the only difference between this and the former is the rate of growth that happens to be negative in case of decay. Let us consider equal intervals for discrete domains and this is something which we can also refer to as geometric decay or growth. The values of the function obtained lead to the formation of a geometric progression. Let us move towards the basic formula for exponential growth.

Formula of exponential growth

Let us pick a variable and name it “x”. Adding to that, we will consider growth rate as “r”, which can be positive as well as negative. The time “t” will go in intervals that would be discrete in nature. Now since we have all the variables in place required for the formula, let us put it all together.

xt = x0 (1+r)t

x0 is the value that x holds when the value of time is 0. Let us consider an example where the growth rate is 5% which equivalents to 0.05 starting from an integer value at a particular point of time till the second time, which will be approximately 5% more than the first time. The time variable here acts as an exponent and hence, makes the whole formula an exponential function.

Role of exponential growth in finance

Whenever you see prices rising in the market, the representation will always be graphical. If you have ever seen a graph that depicts the increasing market prices, you must have noticed that it takes the form of a curve. That curve is actually an exponential function and the cause for the curve is the compounding returns that we get on investments. Within the due course of time, compound interest can even convert a small principal amount into an amount much larger than expected.

The concept of exponential growth is extensively used for financial modelling and the concept is quite tough to understand in one go. Stock markets always follow a trend, but there are always surprises which only noted economists are able to predict. The concept of long-term averages is not exactly ideal when we make the predictions based on financial calculations. There are other methods which help in analysing portfolio values as well as expectations for a long-term period. One such method is the Monte Carlo simulation that has grown popular over the past decade. It is a defined as a broad class of computer-based algorithms that follow the random sampling method in order to get numerical results. In the world of finance, it is one of the most widely used algorithms for mathematical problems. There are three classes that this method is divided into. These are numerical integration, optimization and using the concept of probability distribution to compute exponential growth.

Examples of exponential growth

If the growth rate and size of the graph are constantly going high, then it is known as exponential growth. One of the most researched examples is compound interest. Exponential functions always take the form of the equation y=a.bx where x is a variable component that is directly proportional to the extent of exponential growth. If you are asked to draw the graph of an exponential function, remember that it can never be a straight line. The reason for the same is that if you consider a straight line, then you will notice that the rate of change associated with the line remains the same throughout the length. But in the case of exponential functions, the rate of change gradually keeps on increasing or decreases along the curve of the graph. Hence, the value of the slope will differ at every point as opposed to that of a straight line where the value and angle of curve remain the same throughout the length of the curve.

Most of the phenomena related to the real world can easily be modelled using exponential functions which show how the decay or growth happens in relation to the time function. Some of the commonly researched examples are the studies related to bacteria, populations, radioactive substances, AIDS virus, temperature, credit payments and electricity just to name a few. Any variable related to any real world scenario can be assigned a value of x and its growth, as well as decay, can be computed using a fixed percentage done at regular intervals of time. This process of computing the growth trend is known as exponential growth or decay.

Exponential growth at Algebra level

There are basically two functions at the algebraic level that we make use of in order to show the growth and decay concepts in various real-world situations. As the trend says, whenever we witness a growth that occurs at fixed percentage and takes place at regular intervals of time, these patterns can be graphically represented.

Decay and exponential growth are changes that take place at the mathematical level. The rate of change will continue to follow an increasing or decreasing trend as it is directly related to the time function. If the time function has a high value, then the rate will be increasing as soon as the value of time function increases and this is what we term as growth. On the other hand, if the value of time function decreases, then the rate of change also goes down and this is what we term as exponential decay. Also, we discussed in the previous section of the article that the rate of change, if represented on a graph, will never give us a straight line as it will be a curve that will have a different angle at every point of the slope. These exponential functions can never be straight lines.

The exponential growth mathematical model based on physical phenomena will only apply to limited regions as this form of growth is unbounded that is also considered as physically realistic. Towards the beginning, the rate of growth might be exponential, but over the course of time, the phenomena model might reach a region that might have been ignored before. Negative feedback results in decay which gradually becomes significant. This results in the formation of a logistic model of growth. There are many other assumptions related to exponential growth and some of these panders to instantaneous feedback, continuity or breakdowns.

On the whole, exponential growth happens to be an immensely interesting topic of research for economists and mathematicians.

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