Free lesson · Math & notation

Percentages, basis points and units

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Start with the idea

Write the units beside every input before touching a calculator. A unit conversion changes how a value is expressed, not its economic size. This habit prevents common hundred-fold and ten-thousand-fold errors.

Symbols, units & horizon

Π (pi): profit in currency
V₀: positive starting capital in the same currency
R: decimal return
R%: return expressed in percent
Rbp: return in basis points
Subscript 0: beginning of the period

When and why to use this

Use these conversions in every position, return, cost and risk calculation in the course.

Start every calculation by writing what each number measures. A return is a fraction of starting capital; a dollar profit is an amount of currency. You can add two dollar profits but cannot add a dollar profit directly to a percentage return.

R = \frac{Π}{V_{0}}, R_{%} = 100 R, R_{b p} = 10,000 R

Algebra and arithmetic

Convert profit into return and basis points

Divide profit Π by starting value V₀ to obtain a dimensionless fraction. Multiply by 100 to express that fraction in percent or by 10,000 to express it in basis points.
To invert, $R = \frac{R_{%}}{100} = \frac{R_{b p}}{10,000}$ and $Π = R V_{0}$ .

Work it by hand

$25 profit on $10,000 gives .0025=0.25%=25 bp.

Here R is a decimal return, Π (capital Greek pi) is profit and V₀ is starting capital. The subscript 0 means “at the beginning”; it is a label, not multiplication. A percent is one part in a hundred; a basis point is one part in ten thousand.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def return_units(profit, starting_capital):
    """Currency inputs; returns decimal, percent, and basis points."""
    if starting_capital <= 0:
        raise ValueError("Starting capital must be positive")
    r = profit / starting_capital
    return {"decimal": r, "percent": 100*r, "bp": 10000*r}

print(return_units(25, 10000))

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