Fractions, percents and basis points

Idea: Learn the anchors

Most fractions are small multiples of a few anchors.

• Exact: $\frac{1}{4}=0.25$, $\frac{1}{5}=0.2$, $\frac{1}{8}=0.125$
• Repeating: $\frac{1}{3}\approx 0.333$, $\frac{1}{6}\approx 0.167$, $\frac{1}{7}\approx 0.143$
• Scale them: $\frac{5}{8}=5\times 0.125=0.625$, $\frac{7}{8}=1-0.125=0.875$

Idea: Build from 10%

Any percent of a price starts from $10\%$.

• $15\%$ of $240$: $24+12=36$
• $12.5\%$ is an eighth
• Swap percent and base: $8\%\times 25=25\%\times 8=2$

Example: Successive cuts multiply

A price of $240$ loses $15\%$, then $10\%$. What is the new price?

1. Each move acts on the latest value: $0.85\times 0.9=0.765$
2. $240\times 0.765=183.6$

Formula: A change has a base

$\text{change}=\frac{\text{new}-\text{old}}{\text{old}}$

Measured against the old value, so the way back is a different percent.

Common mistake: Undoing by adding back

Adding $10\%$ back to $63$ gives $63\times 1.1=69.3$, not the old $70$.

Instead Divide by the multiplier: $\frac{63}{0.9}=70$.

Quick check: Back from a loss

A position falls $20\%$. What gain, in percent, brings it back?

Formula: Basis points

$1bp=0.01\%=\frac{1}{10000}$

A basis point is a hundredth of a percent, so $100$ of them make $1\%$.

Example: A fee in basis points

A charge of $15$ basis points on $2000000$. What does it cost?

1. Multiply by the basis points, divide by $10000$
2. $2000000\times \frac{15}{10000}=3000$

Idea: Points or percent?

A rate from $4\%$ to $5\%$ rises by $100$ basis points and by $25\%$ of itself.

• $3.65\%-3.40\%=0.25\%$: that is $25$ basis points
• Relative change: $\frac{5-4}{4}=25\%$
• Say out loud which value is the base