Percentages · RiskAtoms Academy

By the end you will be able to

1Switch between per cents, fractions and decimals, and find a percentage of an amount.
2Calculate a percentage change, and tell percentage points from per cent.
3Chain percentage changes by multiplying growth factors, and explain why +10% then −10% is not zero.
4Read and use basis points.

Why you need it

A share that falls 50% has to rise 100%, not 50%, to get back to where it was. Percentages look simple, but they hide traps like this.

Returns, interest rates, inflation and almost every ratio in finance are percentages. These lessons use them:

L1.1.06 How prices are formed
Price moves and spreads as a percentage of the price.
L1.1.08 Returns and risk: a first look
Returns as percentage changes, and why gains and losses are not symmetric.
L1.3.01 The time value of money
Interest rates written as percentages and used as decimals.
L1.3.07 Real vs nominal rates
Inflation and real rates, quoted in per cent and percentage points.
L1.5.04 Valuation multiples
Ratios and growth rates expressed as percentages.
L1.6.04 Repo and securities financing
Repo rates and haircuts, often quoted in basis points.
L1.8.05 An insurer’s income statement
Ratios on an insurer's income statement, in per cent.
L1.8.06 Capital, equity and leverage
Capital and leverage ratios.
L1.8.07 Key ratios for banks and insurers
Key ratios for banks and insurers, and how they change.

The idea

Per cent means per hundred

Per cent comes from the Latin per centum, “for each hundred”. So 25% means 25 out of every 100. Picture a block of 100 squares and shade some of them:

Squares shaded25

Per cent

25%

Fraction

25/100

Decimal

0.25

Three ways to write the same amount. In a formula, always use the decimal: 25% is 0.25.

To find a percentage of something, turn the percentage into a decimal and multiply. “Of” means “times”:

20\% \text{ of } 250 = 0.20 \times 250 = 50

Here is the one idea that makes the rest easy. Going up 10% means keeping all of what you had (1) and adding 10% more (0.10): multiply by 1.10. Going down 10% means keeping 90%: multiply by 0.90. The number you multiply by is the growth factor.

The rules

Rule 1: percentage change. A percentage change compares a move with where it started.

Rule 2: changes in a row multiply. Each change works on the new value, not the original one.

Rule 3: percentage points and basis points. When the thing that changes is itself a percentage, such as an interest rate, there are two different ways to describe the change.

The rules on one card

\text{\% change} = \frac{\text{new} - \text{old}}{\text{old}} \qquad \text{new} = \text{old} \times (1 + p)

\text{overall change} = (1 + p_1)(1 + p_2)\cdots(1 + p_n) - 1

1 percentage point = 100 basis points. 1 bp = 0.01% = 0.0001. To undo a change $p$ , you need $\tfrac{1}{1+p} - 1$ .

Try it

Set two moves and watch what happens to 100. Try the presets first, then look at the curve: losses need bigger gains to recover, and the gap grows fast as the loss gets larger.

Try itPercentage moves in a row

Open the percentage calculator →

Move 1−20%

Move 2+20%

100 ends up as

96.00

Overall change−4.00%

Adding the moves says0%

To undo move 1 you need+25.00%

The value after each move

The move needed to get backactually neededwhat adding suggests

Across: move 1. Up: the move that undoes it.

Worked examples

1. A share price goes from $40.00 to $46.00, then back

Step 1 of 3

1
The rise
$\dfrac{46.00 - 40.00}{40.00} = \dfrac{6.00}{40.00} = 0.15 = 15\%$

2. A policy rate rises from 4.25% to 4.50%

Step 1 of 3

1
Percentage points
$4.50\% - 4.25\% = 0.25$ percentage points.

3. A fund returns +20%, −15% and +5% over three years

Step 1 of 3

1
Turn each return into a growth factor
$1.20,\quad 0.85,\quad 1.05$

Common mistakes

Each of these is tempting. Decide what is wrong before you reveal the fix.

“A price went from 50 to 40. That is a fall of 10 ÷ 40 = 25%.”
“It fell 20% and then rose 20%, so it is back where it started.”
“Unemployment went from 4% to 5%, so it rose by 1%.”
“50 basis points is 5%.”
“Up 6%, so multiply by 6.” Or: “Up 6%, so multiply by 0.06.”

Reaction

Check what stuck

Question 1 · objective 2

A price rises from 80 to 92. What is the percentage change, in %?

%

Question 2 · objective 2

An interest rate goes from 2% to 2.5%. Which statement is correct?

AIt rose by 0.5%BIt rose by 0.5 percentage points, which is 25% of its old levelCIt rose by 25 percentage pointsDIt rose by 5 basis points

Question 3 · objective 3

An investment falls by 25%. By what percentage must it now rise to get back to where it started, in %?

%

Question 4 · objective 3

A price goes up 10% and then down 10%. What is the overall percentage change, in %?

%

Question 5 · objective 4

A spread widens from 1.20% to 1.55%. By how many basis points has it widened?

bp

Glossary

Per cent (%): Per hundred. 15% means 15 out of every 100, or 0.15 as a decimal.
Percentage change: How much a value has moved, as a share of where it started: (new − old) ÷ old.
Growth factor: The number you multiply by to apply a percentage change: 1 + p. Up 10% is × 1.10; down 10% is × 0.90.
Percentage point: The plain difference between two percentages. From 2% to 3% is a rise of 1 percentage point.
Basis point (bp): A basis point (bp) is one hundredth of a percentage point: 0.01%, or 0.0001 as a decimal.
Relative change: A change measured as a share of the starting value. A percentage change is a relative change.