Percentages and Percentage Points

Why this matters

Returns, interest rates, fees, margins, growth, tax, and inflation are all quoted as percentages. That makes the percentage the most used tool in investing and the easiest to misuse. This lesson starts from the beginning and covers a handful of common mistakes. If you're comfortable with percentages already, the sections on losses and on percentage points are the ones to read.

A percent of what?

A percentage is a fraction of something, and that something is called the base. To find a percentage change, divide the change by where you started.

Percentage change = (New value − Old value) ÷ Old value

A share that goes from $50 to $60 has risen by $10 ÷ $50 = 20%. If it then falls back from $60 to $50, it has fallen by $10 ÷ $60 = 16.7%. The $10 is the same both times. The percentage differs because the base does.

Whenever you meet a percentage, the first question is what it's a percent of. A fee of 1% of your assets and a fee of 1% of your gains are very different amounts. So are a company growing 30% from $1 million of sales and one growing 30% from $1 billion.

Losses and gains aren't symmetrical

After a loss, you have less money, so each percent you gain from there is worth fewer dollars. That means it takes a bigger percentage gain to undo a loss than the loss itself.

Loss	$1,000 becomes	Gain needed to get back to $1,000
10%	$900	11.1%
20%	$800	25%
30%	$700	42.9%
50%	$500	100%
75%	$250	300%
90%	$100	900%

To recover from a 50% loss, $500 has to become $1,000, which is a gain of $500 on a base of $500. The gap between the two columns widens quickly. Small losses are nearly symmetrical, and large ones are very hard to come back from.

This is plain arithmetic, and it's a large part of why investors care so much about avoiding big losses. Risk and Return covers what risk means for an investor.

Percentages multiply

A share rises 20% one year and falls 20% the next. It's tempting to say it's back where it began. It isn't.

Start with $100. A 20% rise takes it to $120. A 20% fall from $120 is $24, which leaves $96. You're down 4%.

The reliable way to combine percentage changes is to turn each one into a multiplier and multiply them. A 20% gain is × 1.20, and a 20% loss is × 0.80.

1.20 × 0.80 = 0.96, a loss of 4%

The same applies to gains. Three years of 10% growth isn't 30%.

1.10 × 1.10 × 1.10 = 1.331, a gain of 33.1%

The extra 3.1% is growth on earlier growth, which is the idea behind Compounding. Over a few years the difference between adding and multiplying is small. Over decades it's most of the result.

Multipliers also let you work backward. If a price is $100 after a 25% rise, the starting price was $100 ÷ 1.25 = $80. It wasn't $75. Taking 25% off the new price uses the wrong base.

Percent and percentage points

Things get slippery when the number that's changing is already a percentage. Suppose an interest rate rises from 4% to 5%. There are two honest ways to describe that, and they sound very different.

• It rose by one percentage point. That's the plain difference: 5 minus 4.
• It rose by 25%. That's the change relative to where it started: 1 ÷ 4.

Both are correct. On a $200,000 loan where only interest is paid, the yearly cost goes from $8,000 to $10,000. That's one percentage point of the loan, and 25% more interest.

The last row is the one to remember. "The risk doubled" and "the risk rose by a tenth of a percentage point" describe the same thing. Whoever is presenting the number picks the version that suits their point. When a headline gives only one, work out the other.

People who work with rates use a smaller unit that avoids the confusion. A basis point is one hundredth of a percentage point, or 0.01%. A rate moving from 4.00% to 4.25% has risen 25 basis points. A fund charging 0.05% a year charges 5 basis points. The unit removes any doubt about whether "a 1% rise" means points or percent.

Small percentages on large bases

A percentage that sounds small can be large in money, depending on what it's charged on and for how long. A fund fee is charged on everything you have invested, every year, whatever the return.

$10,000 invested for 30 years	Ending value
Growing 7% a year	$76,123
Growing 6% a year, after a fee of 1 percentage point	$57,435
Difference	$18,688

The fee is 1% of the assets each year. It's one seventh of each year's return. And after 30 years of compounding, it has taken about a quarter of the ending value. All three statements are true, and each uses a different base. The 7% return is an assumption for the arithmetic, not a forecast.

Four questions for any percentage

• Percent of what? Find the base.
• Percent or percentage points? If the thing measured is itself a rate, ask which.
• Over what period? 10% in a year and 10% over ten years are very different results.
• Before or after costs, taxes, and inflation? Later lessons take these one at a time.

Key takeaways

• A percentage change is the change divided by the starting value. Always ask what the base is.
• A loss needs a larger percentage gain to recover: 50% down needs 100% up.
• Combine percentage changes by multiplying, not adding.
• When a rate changes, percentage points give the plain difference and percent gives the relative change.
• One basis point is 0.01%, or one hundredth of a percentage point.

Check your understanding

A share falls 40%. What gain does it need to get back to where it started?

1. 40%, since a fall and a rise of the same size cancel each other out.
2. About 67%, since the gain is measured from the lower price.
3. 60%, since the share is now at 60% of the price it started from.

A fund lowers its yearly fee from 0.50% to 0.40%. Which description is accurate?

1. A cut of 0.10 percentage points, or 10 basis points, which is 20% lower.
2. A cut of 10%, since the fee has come down by 0.10 from where it began.
3. A cut of 10 percentage points, since fees are counted in whole percents.

What's next

If returns multiply, the ordinary average of a set of yearly returns can't be the right summary of them. Arithmetic and Geometric Averages explains which average to use and why the two differ.