[Maths Class Notes] on Extrapolation Pdf for Exam

What is the Extrapolation Method?

The process in which you estimate the value of given data beyond its range is called an extrapolation method. In other words, the extrapolation method means the process that is used to estimate a value if the current situation continues for a longer period. The Extrapolation Method is a vital component in Mathematics. It also has its branches in Statistics, Sociology, Psychology, and other fields of study too. In this section, we shall discuss what extrapolation means, the definition, extrapolation formula, and have a better understanding of the extrapolation method with the help of a few example problems. In addition to that, you should also know about interpolation. This is the process of estimating the value of the given data. 

What is Meant by the Term Extrapolation?

Extrapolation Method is a process in which you estimate value by understanding the known factors beyond a particular area. The other definition would be the data values are assumed to be points like a1, a2,…………an. It exists as statistical data and when this data is tested periodically,  it can give you the necessary information or the next data point. An easy example of extrapolation that you can observe in your everyday life can be driving a car or riding a bike. You usually tend to look further at your sight to understand the condition of the road better.

Extrapolation word has come from the word ‘extra’, meaning ‘outside’, and is a shortened form of the word interpolation. An interpolation defined as an insertion between two points. An extrapolation is an insertion outside any existing points. For example, if we know something about Monday and Tuesday, we might be able to make an extrapolation about Wednesday.

Difference between Interpolation and Extrapolation

In Interpolation, the value of the independent variable is used within certain limits to evaluate the value of dependent variable with a known independent variable. Although, if the value of a dependent variable lies outside the limits, the extrapolation method is used to evaluate the value of the independent variable.

Extrapolation in Statistics

Here you try to understand the future or you predict the future with the available data or information. With the historical data present, you can estimate the rate at which some incidents might increase or some time it might decrease. For example, the rate at which the population of the world is increasing or it can be determining the weather conditions. 

Extrapolation Method

Extrapolation method is of three types – linear, conic, and polynomial extrapolation. Given below is a brief description of these methods:

1. Linear Extrapolation

When you want to predict the value that is not too far away from the existing data, linear extrapolation will help you for any linear function. When you have been given a graph, you use this method to draw a tangent line at the last point and extend this line beyond its limits. 

2. Conic Extrapolation

This type of extrapolation helps you create a conic section with the last five endpoints of the data. When you have a para or a hyperbola, the conic section’s curve is relative to the x-axis and doesn’t curve, but in the case of an ellipse or a circle, it curves on itself. 

3. Polynomial Extrapolation

You can create a polynomial curve using all the data points given to you. This method is applied using Newton’s System of Finite Series or Lagrange Interpolation. With the associated points, you can find the required data. 

Extrapolation Formula

In a linear graph, let’s assume two endpoints (a1, b1 ) and (a2, b2 ). Here you’ll have to find the value of the point “a” that has to be extrapolated. Therefore, the extrapolation formula goes by:

b(a) = b1 + ((a – a1)/(b – b1))(b2 – b1)

Extrapolate Graph

We know that The process in which you estimate the value of given data beyond its range is called an extrapolation method. See the example below to understand the extrapolate graph. Here the unknown values are  a1, a2,  a3 and you need to find  a4. Now finding the a4 is the called extrapolation point. 

Methods of customizing Settings for the Yield Curve

In the customizing settings for the yield curve, any of the  following methods can be used:

1. Keep Continuously Interest-Bearing Zero Rate Constant

It is assumed that the continuously interest-bearing zero rate for all terms in the extrapolation area is the same as the last available continuously interest-bearing zero rate in the yield curve.

For this, the following equation is applied:

Zcc(T>Tn) = Zcc(Tn)

2. Keep Par Interest Rate Constant

It is assumed that the last available par interest rate in the yield curve, once converted into the continuous interest-bearing portrayal, is the same as the continuously interest-bearing forward interest rate in the entire extrapolation area. The system performs the following calculations:

• Calculation of the par interest rate for the last yield curve grid point based on the conditions set for the yield curve.

• Conversion of the par interest rate P into a continuously interest-bearing portrayal (m = number of interest payments per year) and calculation of the factor F, as portrayed in the following graphic:

             F = exp(mlog(1+p/100m))

            d(T>Tn) = d(Tn)F-(T-Tn) 

           Zcc(T>Tn) = -log(d(T>Tn))/T

3. Keep C
ontinuously Interest-Bearing Forward Rate Constant

The system assumes that the continuously interest-bearing forward rate on the last grid point of the yield curve is the same as the continuously interest-bearing forward interest rate in the entire extrapolation area. The following calculations are performed by the system:

            S = Zcc(Tn) – Zcc(Tn-1)/ Tn-Tn-1

           f(Tn) = sTn + Zcc(Tn)

Zcc(T>Tn) = f(Tn) + (Zcc(Tn)-f(Tn))Tn/T

Stepwise Calculation of Linear Extrapolation

The formula for Linear Extrapolation can be divided into the following steps:

1. The data is first analysed whether the data is following the trend and whether the same can be forecasted.

2. Two variables are there where one is a dependent variable, whereas the second is an independent variable.

3. The numerator of the formula has to start with the previous value of a dependent variable, and then one needs to add back the fraction of the independent variable.

4. Then, the value arrived in step 3 by a difference of immediate given dependent values is multiplied.

5. Therefore, the value of the dependent variable will give us the extrapolated value.

Solved Examples

Example 1: The two points on a straight line are (2, 6) and (4, 10). Find the value of b when a = 6 using linear extrapolation. 

Solution:

Given, 

a1 = 2

b1 = 6

a2 = 4

b2= 10

a = 6

b(a) = b1 + ((a – a1)/(a2 – a1))(b2 – b1)

Substituting the values: 

b(6) = 6 + ((6-2)/(4-2)) (10-6)

b(6) = 6 + (4/2) (4)

b(6) = 6 + (2) (4)

b(6) = 6 + 8

b(6) = 14

Example 2: The two points on a straight line are (4,8 ) and (10, 6). Find the value of b when a = 8 using linear extrapolation. 

Solution:

Given, 

a1 = 4

b1 = 2

a2 = 10

b2= 6

a = 8

b(a) = b1 + ((a – a1)/(a2 – a1))(b2 – b1)

Substituting the values: 

b(8) = 8 + ((8-4)/(10-4)) (6-2)

b(8) = 6 + (4/6) (4)

b(8) = 6 + (0.667) (4)

b(8) = 6 + 2.667

b(8) = 8.667

Example 3: The two points on a straight line are (6, 3) and (8, 9). Find the value of b when a = 12 using linear extrapolation. 

Solution:

Given, 

a1 = 6

b1 = 3

a2 = 8

b2= 9

a = 8

b(a) = b1 + ((a – a1)/(a2 – a1))(b2 – b1)

Substituting the values: 

b(12) = 3 + ((12-6)/(8-6)) (9-3)

b(12) = 3 + (6/2) (6)

b(12) = 3 + (3) (6)

b(12) = 3 + 18

b(12) = 21