What is the difference between ratio change vs percent change ? (finance question)

Question

I had a question which asked us to calculate the percentage decrease in operating income between years 2012-2013 and wanted us to predict the decrease rate for the next year. When I calculated the %change, I used the typical formula of (2013 value/2012 value)-1 *100. However, the feedback in the textbook had us only calculate the "ratio of operating income which is 2013 value/2012 value" without subtracting 1. Why is this? Now this may seem like a basic question, but it totally changed the final answer.

Answer 1

This is an important question. I will try to explain the idea behind the formula that "subtracts $1$".

Before thinking about percentage decrease let's get percentage increase cleared up. I'll use a numerical example.

If the old value is $50$ and the new value is $60$ you can compute the absolute change $60-50 = 10$ and then find out what fraction that is of $50$. It's $$ \frac{10}{50} = 0.2 . $$

Fractions -- that is, numbers between $0$ and $1$ - are uncomfortable things for most people, so the convention is to multiply them by $100$ and call them percents (etymology: "per" is "out of", "cent" is "$100$". The percent sign can be read literally as "divide by $100$", so $$ 0.2 = 20/100 = 20\%. $$

But subtracting and then dividing is clumsy. A neater way to make the computation is to find $$ \frac{60}{50} = 1.2. $$ Then you subtract $1$ to get the percent change. The "subtract $1$" comes from the observation that $$ 60 = 1.2 \times 50 = (1 + 0.2) \times 50 = 1 \times 50 + 0.2 \times 50, $$ the original amount plus the change.

I call this the $1+$ trick. It's useful for many things - for example, to find the result of a $20\%$ increase followed by a $40\%$ increase you calculate $$ 1.2 \times 1.4 = 1.64, $$ which is a $64\%$ increase - more than $20\% + 40\%$.

The trick is good for calculating backwards too. If the new value is $60$ and the percentage increase was $20\%$ then the original value must have been $60/1.2 = 50$. That's not the same as finding $20\%$ of $60$ and subtracting it.

When something doubles that's a $100\%$ increase and the trick says that's multiplying by $1+1=2$, which makes sense.

Now for decreases. If the operating value went from (say) $60$ to $50$ you compute $50/60 = 0.83$. Since $0.83 = 1 - 0.17$ that's a $17\%$ decrease. Another similar decrease the next year would correspond to a two year calculation of $0.83 \times 0.83 = 0.69$ (I rounded to two places) or a $31\%$ decrease. Note that's less than twice $17\%$.

Answer 2

The ratio of $V'$ to $V$ is $V'/V.$ The change from $V$ to $V'$ is $V'-V.$ The proportionate change in $V,$ from $V$ to $V',$ is $\frac {V'}{V}-1=\frac {V'-V}{V}.$ The percent change in $V,$ from $V$ to $V'$, is $(\frac {V'}{V}-1)(100).$

The execrable phrase "times more than" became popular because of its widespread use by broadcasters and other journalists who do not understand any of this.

Answer 3

So, Could this be the difference between a ratio and a percent change? A ratio is just a comparison of any two numbers, or in this case 2013 and 2012 values, by division. When you subtract 1 and multiply by 100 you are turning this "Ratio" into a percent comparison.