You buy a product for $25.00. The store has a sale and drops the price by 30%. Then they raise the price again by 30%, so now the product costs$23.43. What happened? Well, it’s a typical case of misleading percentages. The store applied the price increase to the discounted price.

## Percentage calculations

What is 4% of 10? 20 is what percentage of 30?

Questions like these can be answered by the following equations. We use $$p$$ to denote the percentage amount and $$n$$ to denote the number we’re comparing to.

The percentage of a number with respect to another number is found with

$y = pn$

Remembering to convert the percentage into decimal, we can implement it in Python with

def nfind(p, n):
return (p / 100) * n

What is 4% of 10?

nfind(4, 10)
## 0.4

To find what percentage of a number a given number is, you can use

$p = \frac{n}{y}$

def pfind(n, y):
return n / y

What percent of 60 is 12?

pfind(12, 60)
## 0.2

There are more calculations and ways to pose percentage related questions; percentages can be confusing and tricky to grasp despite their widespread use in business settings.

## Account balance example

Now, let’s pose that we have a porfolio account with a balance of $1000 on a given day. Let’s also suppose that the portfolio loses 10% of its value after a week. That amounts to$100, meaning that the balance at the end of the week is $900. nfind(10, 1000) ## 100.0 The percentage change equation is $c = \frac{V_2 - V_1}{|V_1|} \times 100$ Where $$V_1$$ and $$V_2$$ are respectively the first and second values we’re comparing, and $$c$$ is the percentage change. def pchange(v1, v2): return round(((v2 - v1) / abs(v1)) * 100, 2) Interestingly, the change from$1000 to $900 is 10% but going from$900 to \$1000 is a change of 11%.

pchange(1000, 900)
## -10.0
pchange(900, 1000)
## 11.11

This mathematical artifact tells us that it takes more work to recuperate a loss. However, the amount of effort necessary to recover larger losses increases (almost) exponentially as the losses increases.

In general, to calculate the gains necessary to restore losses, you can use

$r = \frac{1}{(1 - p)} - 1$ where $$r$$ is the percentage needed to break even, and $$p$$ is the percentage loss.

def pneeded(p):
return round((1 / (1 - (p / 100))) - 1, 2)
pneeded(10)
## 0.11

When investments lose ground, they must make up more ground, percentage-wise, just to get back to even.

These numbers lead to some interesting points like:

• it’s better to cut your losses short (ideally at less than 10%)
• don’t fall prey to sunk cost fallacy reasoning
• active investing can be a better strategy than “set it and forget it”
• it’s not a bad idea to buy trending stocks, even at higher prices
• the time a stock spends recovering its value is opportunity cost
losses = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75]
gain = [pneeded(x) * 100 for x in losses]
import matplotlib.pyplot as plt
from matplotlib.ticker import PercentFormatter

fig, ax = plt.subplots()
ax.plot(losses, gain)
ax.yaxis.set_major_formatter(PercentFormatter())
ax.xaxis.set_major_formatter(PercentFormatter())
ax.set_title("Great losses require greater gains")
ax.set_xlabel("portfolio loss")
ax.set_ylabel("needed gain to recover losses")
ax