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Every profitability ratio is one figure from the income statement divided by one from the balance sheet:

$ROE = \frac{\text{Net income}}{\text{Shareholders' equity}}, \qquad ROA = \frac{\text{Net income}}{\text{Total assets}}$

Both are usually quoted against average equity or assets — the mean of opening and closing balances — because the numerator covers a period while the denominator is a snapshot.

The DuPont identity splits ROE into three drivers by inserting revenue and assets and cancelling:

$ROE = \underbrace{\frac{NI}{\text{Rev}}}_{\text{margin}} \times \underbrace{\frac{\text{Rev}}{\text{Assets}}}_{\text{turnover}} \times \underbrace{\frac{\text{Assets}}{\text{Equity}}}_{\text{leverage}}$

Multiply the three and the middle terms cancel back to $NI/\text{Equity}$. The decomposition is useful precisely because it shows why two firms with the same ROE are not alike: one may earn it on margin, another on leverage.

Note the relation $ROE = ROA \times \text{leverage}$ falls straight out of the same cancellation.

Present value and NPV

A cash flow $C_t$ received at time $t$ is worth $C_t/(1+r)^t$ today. For a level stream, the annuity factor collapses the sum:

$PV = C\cdot\frac{1 - (1+r)^{-n}}{r}, \qquad NPV = PV - C_0$

IRR

The internal rate of return is the discount rate that sets $NPV = 0$:

$\sum_{t=1}^{n}\frac{C_t}{(1+\text{IRR})^t} - C_0 = 0$

Like a loan rate, it has no closed form and is found by iteration.

Liquidity and leverage

$\text{Current} = \frac{CA}{CL}, \qquad \text{Quick} = \frac{CA - \text{Inventory}}{CL}, \qquad D/E = \frac{\text{Total debt}}{\text{Equity}}$

The quick ratio strips out the current asset that is slowest to turn into cash.

Every one of these is arithmetic on figures you supply. What counts as debt, which equity balance to use, and what a "healthy" level looks like are accounting and industry judgements, not calculations — this page does not make them for you.

• Mixing a period figure with a point-in-time figure: a full year of net income over a year-end equity balance overstates or understates the return whenever the balance moved. Use the average.
• Comparing ROE across capital structures: leverage inflates ROE without any operating improvement. ROA is the leverage-neutral comparison.
• Forgetting the sign on the initial outlay: $NPV$ subtracts $C_0$. Adding it makes every project look brilliant.
• Discounting with a mismatched period: annual cash flows need an annual $r$; monthly flows need $r/12$ and $n$ in months.
• Ranking projects by IRR alone: IRR ignores scale, and cash-flow streams that change sign more than once can have several roots.
• Reading a ratio without its industry: a current ratio of $1.1$ is ordinary in one sector and alarming in another. The number is the easy part.