All Terrain Thinking

A Compendium of things I think are Important

Algebra: Calculating why it's not in your wallet

How often do you find yourself in a situation where you are concerned with change? I suspect it happens frequently. How much will your paycheck increase if the government increases payroll taxes by $5.00 a pay period? What time will we get in if we are taking a 5 hour flight that leaves at 9 AM? What will the balance be in my saving account in five years if interest accumulates at 5 percent per year? These are just a sampling of the questions that involve change and behind the answers to each one is an equation of change.

What measures of change can we have between any two dates, the original point, which we will called old (O) and the new point (N). Two popular ones would be change and rate of change, what you may have heard of as percentage change. They may sound very much the same, but they are not, and if you think they are, you are ripe to get ripped off. We saw that earlier when we looked at the two graphs of population growth which appear below.

In the first graph Africa appears to be the center of population growth while in the second Africa trails significantly behind Asia. So how do we explain these substantial differences when in each graph we are representing change? The differences become clear when we look at the two equations that generated the graphs.

Let's begin with a measure of change-what is represented in the second graph of population change. We will use the formula: Change = (New - Old). We have one equation in three variables, Change (C), New (O), and Old (O). Given that there is only one equation, we can solve for only one unknown which means that we will have to have information on the other two unknowns. If you know where you started and where you end up, you can certainly calculate the change, while if you know where you started and how much things changed, you could easily calculate the new. Finally, you could determine where you started if you knew where you ended up and how much you changed to get there.

To answer each of these questions you can use the logic of algebra to generate three equations, each one isolating the unknown variable on the left side of the = sign. The three formulations would be:

We would use the first equation to solve something like the following: What was the change in population between 1980 and 1995 if it increased from 5 to 5.4 billion. The second approach would be appropriate when we need to know our bank account balance next year if it is currently $2,000 and it will increase $200 this year. Finally, if we needed to know the balance in a checking account at the beginning of the period and knew that it is now $1,200 and it had grown $200 during the period, we would use the third specification.

A second popular measure of change is percentage change, a concept that has caused trouble for many students over the years. When we are measuring percentage change (PC) we will use the following formula

Once again we have one equation with three unknowns (PC, N, and O) so we will once again have three formulations. In addition to the formula for percentage change above, we can rewrite the equation as follows:

Questions that would be best answered by the three specifications would be:

What was the growth rate in sales if they went from $120 to $150 million? (1a)

What would the world's population be in ten years if it increases by 5 percent from its current level of 5 billion? (2a)

What must we put in the bank today if we want to have $20,000 in the bank next year if we expect the interest rate to be 6 percent? (3a)

If we call the old value the present value (PV) and the new value the future value (FV), the equation that describes the mathematical structure of the relationship between these four variables will be the formula for compound growth that you will find in your economics and finance courses.

• PV = Present Value ( value at beginning of the period)
• FV = Future Value ( value at the end of the period)
• g = Average growth rate per period
• T = Number of time periods

As you would expect by now, we can think of this as an equation with 4 unknowns. You can use it to solve for any one of the variables when you have information on the other three variables. To solve for the unknown all that you need to do is rewrite the equation so that the unknown is on the left hand side of the = sign. To make it easy for you I have done the algebra to help you solve the three most common problems.

Computation of Future Value: givens=initial value, growth rate, time period

Computation of Present Value: givens=future value, growth rate, time period

Computation of Average Growth Rate: givens-future and present value, time

Questions that would be best answered by the three specifications would be:

What was the growth rate in sales if they went from $120 to $150 million in 5 years? (3b)

What would the world's population be in ten years if it increases by 5 percent from its current level of 5 billion? (1b)

What must we put in the bank today if we want to have $20,000 in the bank in 10 years if we expect the interest rate to be 6 percent? (2b)

Below you will find a few examples of where one could use this formula. You can also check out the bond pricing example-one of the most important applications of present value. This example should help you better understand the negative relationship between bond prices and interest rates.

There is one final measure of change which we should briefly mention. Consider the problem of estimating change in a variable that is actually the product of two or more other variables for example, let's look at estimating the change in revenue. We know from our introductory economics course, or our own experiences, that revenue is calculated simply by multiplying the price times quantity [ R = P * Q]. In this situation where we have this multiplicative relationship, the equation for change in revenue (DR) in terms of change in price (DP) and change in quantity (DQ) would be price times the change in quantity plus price times the change in quantity.

This formula allows us to decompose any change into its component parts, a valuable tool when we attempt to answer the questions concerning the decline in growth and the link between productivity growth and wage growth. With a little bit of algebra we could also transform this equation into a decomposition of percentage change in revenue %D R

This formula allows us to decompose any change into its component parts, a valuable tool which we will use often.

To see how we might use the compound growth equation, let's look at three simple questions and their answers before we return to a few of the Future Questions from our introduction.

Q1. What will the population of India be in the year 2020 if the population in 1985 was estimated to be 751 million and the growth rate is expected to remain at 2.5% a year for the entire time period?

A1: This is a future value problem. The initial value is 751, the growth rate is 2.5% (.0251), and the time horizon is 35 years.

Q2. How much should you pay for a piece of paper that guarantees you $1000 three years from now if you expect the interest rate to be 8% per year for this three year period? Stated somewhat differently, how much would I have to invest today to receive $1,000 in three years

A2: This is a present value problem. The end value is $1,000, the time horizon is 3 years, and the growth rate of money is 8%.

Q3. The average weekly wage rate for workers in the US increased from $102 per week in 1970 to $389 a week in 1989. What was the average yearly rate of increase?

A3. This is an average yearly growth rate problem. The initial value is 102, the end value is 389, and the time horizon is 19.

If you feel confident about your mastery of the compounding formula, you are ready to look at a few of those opening questions. More specifically, you should check out the problems concerned with calculating the value of a college education, the future costs of a college degree, and the value of a time-share unit.

• Some important financial MATH
• What is the return on my stock dividend?
• How will the interest rate affect the price of my bond?
• How do exchange rates affect my return on stock?

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