This is the sixth theory session of the Problem Solving Test Course. I explain a shortcut for computing compound growth (1+x%)^y and define boundaries of its application. As always, a few examples follow

[TRANSCRIPT]

How large is 5 percent growth over six years?

What about 10 years?

You need to find the answers quickly in order to succeed in PST

Likely there is an approximation technique which we will be able to use to help us cut the time solving this question and this is what’s the sixth session of the problem-solving test course will be dedicated to.

And some examples will illustrate the theory.

Let us start with the following example.

You might want to pause the video now and try to solve a question on your own.

Which of the following is the closest estimate in billions of dollars of the projected revenue ABZ industry in five years in the upside case? From the text we learn that the revenue ABZ industries now is eleven point sixty four billion dollars and in the upside case the revenue growth is projected at seven percent per anum. Revenue in year zero equals eleven point sixty four billion: growth in the upside case equals 7 percent. Revenue in year 5 equals revenue in year zero times one plus the growth rate in the upside case to the power of five, because we are looking at the five year horizon. And using numbers this would look like eleven point sixty four times one plus seven percent to the power of five.

Okay.

The straightforward way of solving this question will look like this. We would need to write things down again and do the lengthy multiplication. Well of course this calculation is solvable but I think we need to spend about 15 minutes trying to tackle it.

And of course I cannot afford to do that in a real PST setting.

So another way of solving this question would be through a roundabout called compound margin. We can write out compound growth statement in terms of simple growth plus something else. This something else is called the compound margin and in this case its value will be about 5 or 6 percent. In this theory session I will explain to you how I got this number.

So the results would be roughly eleven point sixty four times one point four.

And then the calculation becomes quite easy right.

So we are ready to pick our option here.

Our choice is B.

Let us look at the mechanics of compound growth approximation.

We are in this compound growth situation which is so difficult to compute.

And we want a transition to simple growth.

The question is how do we do? Binomianl expansion is a way of representing a sum of two values in the end power via sum of their power from zero to end with certain binomial coefficients.

The expression might seem a bit over complicated at this point but now a special case, it becomes straightforward. Indeed

The first segment is just one, the second is an X and everything that follows is a small number.

Whenever the absolute value of x is small than one. so we can rewrite our combined growth expression in the following way. Compound growth equals simple growth, plus a certain small number called compound margin. To make the combined growth expression operational, we need a better understanding of how the compound margin behaves. I computed the values of the compound margin for different growth rates X and numbers of years N.

And here’s what I got.

Green cells indicates that is up to 10 percentage points and red values indicates values about 200 percentage points.

A few interesting observations.

Firstly the combined margin is always positive even when the growth rate is negative.

This is explained by the fact that in the expression for compound margin even powers of X are always positive and outweigh subsequent odd powers of x which might be negative.

Secondly when the growth rate and the number of years are small the compound margin is small to typically below 10 percentage points but the compound margins shoot for the stars when X and N become large.

So we cannot neglect when X and N are bigger than about 10.

Let’s write this properties down and see why they matter.

Compound margin greater than zero means that the compound result is always greater than the simple result.

In other words a compounded growth is higher than simple growth, while a compound decline is smaller than simple decline.

Here’s an example. One plus 5 percent to the power 7 equals 1 plus 3 5 percent through simple growth, plus compound margin. And the result is about one point four, which is greater than the result from the simple growth.

So the growth is bigger and the compound growth case bending the simple growth case.

Now let’s look at the declining examples. 1 minus 5 percent to the power of seven equals 1 minus 35 percent plus again compound margin; and the result is about zero point seven, which is again greater than the simple growth result of simple decline result of 0.65.

So the drop is smaller in the compound decline case than in this simple decline case.

Another interesting result is that when X times N is less than, or equal to 45 in absolute value the compound margin does not exceed 10 percent.

So for example 1 plus 6 percent to the power 7 equals one point forty two plus compound margin which is about 150.

So the compound margin here is about 8 percent less than 10.

However in the case 1 plus 6 percent to the power of seven the result will be different. And the compound margin here is about 19 percent which is greater than 10 percent. And here is a more extreme example.

1 plus 6 percent to the power 20.

So in this case the compound margin is 101 percent which is greater than 10 percent, is much greater actually.

So we see that when X and N and are small then the compound margin is small but when X and N together gets quite large, compound margin growing as well. And it can grow to huge numbers way above the simple growth.

Let us apply this new knowledge to the official practice question. As always you might want to pause the video now and attempt the question on your own and then check. By the way we’ve done this question in our first practice session and I will immediately cross out C and D, just because they’re not very interesting. The interesting options here are A and B, because now we will be able to check them.

So we need to find the valid conclusion based on the data in table 1. Option A refenue for other products was more than 20 million five years ago.

The products now it’s fifteen point one million and the decline was about 7 percent per anum . Let’s say that X is the revenue for other products five years ago.

Then the following would hold true. This expression is a translation of the option A from English into math language.

Recall that the compound margin is always positive but in this case because seven times five is less than 45, I would say that the compound margin here would be quite small, probably about 2 percent.

So our results will be roughly X times 1 minus 33 percent equals fifteen point one or two thirds of X equals fifteen point one.

But in this case obviously X is larger than 14 instead of 15 because it’s easier to compute over 2 times 3. Or 21 and 21 is greater than 20.

So that means that option A is actually true. In a real PST setting, once we know that option A is true we would stop solving this question and continue to other questions right?

But here for the sake of instruction let’s consider option B as well, because here too compound margin who will help us deal with it.

Hot dog revenue was more than 350 million five years ago.

So again here we translate English into math. And again let’s say that X is the revenue all beef hot dogs five years ago. We know that the contribution margin here is greater than zero, but it should be quite small.

I think it will be about 1 to 2 percent.

So our results will be approximately 366.7 over 1.23, which is less than three sixty six point seven or 1.2.

So that means that option B is false and we can cross it out.

Now here’s another interesting example from an official PST practice test. Again, paused the video and try to solve the question on your own.

Assuming housing starts declined at the constant rate which of the following is the closest estimate of the percentage drop in the number of first quarter housing starts in three years ago and this year.

So the value for three years ago was this number – two million 150000. This year, the value is here.

The value is about 500000.

So let’s assume that X is the desired anual percentage drop.

Now let’s translate the question from English into math language.

Two million one hundred fifty thousand times one minus x percent to the power of three, because three years of decline equals 500000.

Let’s simplify this a bit.

Y minus X percent to the power of 3 is roughly one quarter.

Let’s test some options now.

Option A But this is not possible because the compound margin is always greater than zero, and from here it should be negative.

So we get rid of A because it’s wrong.

Let’s test option B in a similar way.

Well obviously B does not work either right?

Because from this expression we need the compound margin to be zero and we know that the compound margin is always positive.

So we get rid of option B as well.

Option C. So option C might actually work, because here we see that the compound margin is positive and also it’s quite big, but that’s okay because the decline rate is quite big as well.

Let’s test option D instead. Let’s take 50 percent instead of 55 percent and then the result after one year would be 50 percent of the start.

The results after 2 years would be 25 percent and the result after 3 years would be twelve point five percent which is obviously less than zero point twenty five.

That means the D is a huge decline is just too large for us and we get rid of D as well.

So the only plausible option here is C and C is our answer.

Let us synthesize what we have learned today. We express compound growth by a simple growth plus a certain number called contribution margin. Contribution margin has two important properties.

One.

It is always positive. Two. If X and and are sufficiently small that means not exceeding 45 in absolute value,

The combined margin is also small not exceeding 10 percentage points.

That’s it for today.

If you have any questions as always feel free to leave them in the comments below.

I would be happy to answer. Thanks for watching and until next time.