📚 Pre-U CIE Business: Formulas & Theorems Quick Reference Guide | Pre-U CIE 商务：公式定理速查手册
This quick-reference handbook collects the essential formulas, ratios and theorems you need for the CIE Pre-U Business examination. Each section provides the core calculations, explains their significance and shows the underlying logic in a paired bilingual format. Use it as a last-minute revision tool or as a structured checkpoint while working through past papers.
本速查手册汇集了 CIE Pre-U 商务考试必备的公式、比率和定理。每个章节都以中英双语配对的形式提供核心计算、解释其含义并展示背后的逻辑。可将其用作考前最后冲刺的复习工具，或作为刷题过程中的结构化知识清单。
1. Break-even Analysis | 盈亏平衡分析
The break-even point is the level of output where total revenue equals total costs, so the business makes neither profit nor loss. Break-even output (units) = Total Fixed Costs ÷ (Selling Price per unit – Variable Cost per unit). The denominator is the contribution per unit: Contribution per unit = Selling price – Variable cost per unit. Break-even revenue can also be found using the contribution to sales ratio: Break-even revenue = Fixed Costs ÷ (Contribution per unit ÷ Selling price).
盈亏平衡点是指总收入等于总成本的产量水平，企业既不盈利也不亏损。盈亏平衡产量（单位）= 总固定成本 ÷ (单位售价 – 单位变动成本)。分母是单位贡献：单位贡献 = 售价 – 单位变动成本。盈亏平衡收入也可用贡献销售比求得：盈亏平衡收入 = 固定成本 ÷ (单位贡献 ÷ 售价)。
The margin of safety shows how far actual output can fall before the business reaches break-even. Margin of safety (units) = Actual output – Break-even output. For a target level of profit, the required output is Output for target profit = (Fixed Costs + Target Profit) ÷ Contribution per unit.
安全边际表示实际产出在降至盈亏平衡点之前还能下降多少。安全边际（单位）= 实际产量 – 盈亏平衡产量。对于目标利润，所需产量为 目标利润产量 = (固定成本 + 目标利润) ÷ 单位贡献。
2. Costs, Revenue and Contribution | 成本、收入与贡献
Total Costs = Fixed Costs + Variable Costs. Variable costs change directly with the level of output, whereas fixed costs remain constant within a relevant range. Total Revenue = Selling Price × Quantity sold. Gross profit is calculated as Gross Profit = Sales Revenue – Cost of Sales, while Net Profit = Gross Profit – Expenses.
总成本 = 固定成本 + 变动成本。变动成本随产量直接变化，而固定成本在相关范围内保持不变。总收入 = 售价 × 销售量。毛利润的计算为 毛利 = 销售收入 – 销售成本，而 净利润 = 毛利 – 费用。
Contribution is the amount each unit sold contributes towards covering fixed costs and then generating profit. Contribution = Selling Price – Variable Cost per unit. Total contribution can be expressed as Total Contribution = Contribution per unit × Quantity sold or Total Revenue – Total Variable Costs.
贡献是每售出一单位产品用于弥补固定成本并随后产生利润的金额。贡献 = 售价 – 单位变动成本。总贡献可表示为 总贡献 = 单位贡献 × 销售量 或 总收入 – 总变动成本。
Break-even analysis relies on contribution thinking: once fixed costs are covered, further sales add directly to profit. Keep in mind the distinction between financial accounting profit formats and contribution-based internal reports.
盈亏平衡分析依赖贡献思维：一旦固定成本被覆盖，额外的销售将直接增加利润。请记住财务会计利润格式与基于贡献的内部报告之间的区别。
3. Profitability Ratios | 盈利能力比率
Profitability ratios measure how effectively a business converts sales or capital into profits. Gross Profit Margin = (Gross Profit ÷ Revenue) × 100%. It reflects efficiency in production and pricing. Operating Profit Margin = (Operating Profit ÷ Revenue) × 100%, which accounts for overheads. Net Profit Margin = (Net Profit ÷ Revenue) × 100% shows the residual return after all expenses.
盈利能力比率衡量企业将销售或资本转化为利润的效率。毛利率 = (毛利 ÷ 收入) × 100%，反映生产与定价效率。营业利润率 = (营业利润 ÷ 收入) × 100%，考虑了间接费用。净利润率 = (净利润 ÷ 收入) × 100%，显示扣除所有费用后的剩余回报。
The most comprehensive profitability measure is Return on Capital Employed (ROCE) = (Operating Profit ÷ Capital Employed) × 100%. Capital employed can be calculated as Total Assets – Current Liabilities or Equity + Non‑current Liabilities. A higher ROCE means management is using long‑term funds more effectively.
最全面的盈利能力指标是 已用资本回报率 (ROCE) = (营业利润 ÷ 已用资本) × 100%。已用资本可计算为 总资产 – 流动负债 或 股东权益 + 非流动负债。ROCE 越高，说明管理层对长期资金的使用越高效。
4. Liquidity Ratios | 流动性比率
Liquidity ratios indicate the business’s ability to meet short‑term obligations as they fall due. Current Ratio = Current Assets ÷ Current Liabilities. A ratio of 1.5:1 to 2:1 is often considered healthy, though it varies by industry. Acid‑Test (Quick) Ratio = (Current Assets – Inventories) ÷ Current Liabilities. Inventory is excluded because it may not be readily convertible into cash.
流动性比率反映企业偿还到期短期债务的能力。流动比率 = 流动资产 ÷ 流动负债。通常认为 1.5:1 至 2:1 较为健康，但会因行业而异。酸性测试（速动）比率 = (流动资产 – 存货) ÷ 流动负债。排除存货是因为其不一定能迅速变现。
An acid‑test ratio well below 1 suggests potential cash‑flow problems, while an excessively high current ratio could indicate inefficient use of working capital. Analysts should interpret liquidity ratios alongside the cash‑flow statement.
速动比率远低于 1 表明可能存在现金流问题，而过高的流动比率可能意味着营运资金使用效率低下。分析人员应结合现金流量表解读流动性比率。
5. Efficiency Ratios | 效率比率
Efficiency ratios assess how well a firm manages its assets and liabilities day to day. Inventory Turnover (cost of sales basis) = Cost of Sales ÷ Average Inventory. A higher turnover suggests stock is moving quickly. Inventory Days = (Average Inventory ÷ Cost of Sales) × 365, showing how many days inventory is held on average.
效率比率评估企业日常管理资产与负债的水平。存货周转率（基于销售成本）= 销售成本 ÷ 平均存货。周转次数越高，说明存货流动越快。存货周转天数 = (平均存货 ÷ 销售成本) × 365，显示持有存货的平均天数。
Trade Receivables Days = (Trade Receivables ÷ Credit Sales) × 365. This measures how long it takes to collect debts; a rising figure might signal poor credit control. Trade Payables Days = (Trade Payables ÷ Credit Purchases) × 365 shows how long the business takes to pay its suppliers.
应收帐款周转天数 = (应收帐款 ÷ 赊销收入) × 365，衡量收回欠款所需的时间；数字上升可能表示信贷控制不佳。应付帐款周转天数 = (应付帐款 ÷ 赊购金额) × 365，显示企业支付供应商所需的时间。
Asset Turnover = Revenue ÷ Total Assets (or Revenue ÷ Capital Employed). This indicates how much revenue is generated per currency unit of total assets, capturing overall resource efficiency.
资产周转率 = 收入 ÷ 总资产（或 收入 ÷ 已用资本），反映每单位总资产创造了多少收入，衡量整体资源利用效率。
6. Gearing and Capital Structure | 杠杆与资本结构
Gearing shows the proportion of a firm’s capital that comes from interest‑bearing debt. Gearing Ratio = (Non‑current Liabilities ÷ Capital Employed) × 100%. A gearing ratio above 50% is often considered high, though capital‑intensive industries may operate with higher levels. An alternative measure is the Debt‑to‑Equity Ratio = Non‑current Liabilities ÷ Total Equity.
杠杆比率显示企业资本中来源于付息债务的比例。杠杆比率 = (非流动负债 ÷ 已用资本) × 100%。比率超过 50% 通常被视为高杠杆，但资本密集型行业可能维持更高水平。另一种衡量方式是 负债权益比 = 非流动负债 ÷ 总权益。
Interest Cover = Operating Profit ÷ Interest Payable. This ratio measures how easily the business can meet its interest obligations from current earnings. A low interest cover (below 1.5–2) raises concerns about financial risk.
利息保障倍数 = 营业利润 ÷ 应付利息，衡量企业从当期盈利中支付利息费用的难易程度。利息保障倍数偏低（低于 1.5–2）会引发对财务风险的担忧。
High gearing can amplify returns to shareholders when profits are strong, but it also magnifies losses during downturns. Analysts should review loan covenants and repayment schedules alongside these ratios.
高杠杆能在利润强劲时放大股东的回报，但在市场下行时也会放大亏损。分析人员应结合贷款条款和还款计划来审查这些比率。
7. Investment Appraisal | 投资评估
Investment appraisal techniques help managers decide whether capital projects should proceed. Payback Period is the time taken for cash inflows to recover the initial investment. For even cash flows: Payback = Initial Investment ÷ Annual Net Cash Flow. For uneven flows, cumulative cash is summed until payback is reached.
投资评估方法帮助管理者决定是否应推进资本项目。回收期是指现金流入收回初始投资所需的时间。当现金流均匀时：回收期 = 初始投资 ÷ 年度净现金流量。现金流不均匀时，需累计现金直至收回投资。
Average Rate of Return (ARR) = (Average Annual Profit ÷ Average Investment) × 100%. Average investment is typically (Initial Cost + Scrap Value) ÷ 2 or simply Initial Investment ÷ 2 if no scrap value. ARR indicates the percentage return on the average book value of assets employed.
平均报酬率 (ARR) = (年均利润 ÷ 平均投资) × 100%。平均投资通常为 (初始成本 + 残值) ÷ 2，或若没有残值则为 初始投资 ÷ 2。ARR 表示对所占用资产平均账面价值的百分比回报。
Net Present Value (NPV) discounts future cash flows to their present value using a discount factor 1 ÷ (1 + r)ⁿ, where r is the discount rate and n the year number. NPV = Σ (Net Cash Flow ÷ (1 + r)ⁿ) – Initial Cost. A positive NPV indicates the project adds value.
净现值 (NPV) 使用贴现因子 1 ÷ (1 + r)ⁿ 将未来现金流折现至当前价值，其中 r 为贴现率，n 为年份。 NPV = Σ (净现金流量 ÷ (1 + r)ⁿ) – 初始成本。正 NPV 表示项目能增加价值。
Internal Rate of Return (IRR) is the discount rate at which NPV = 0. It can be approximated using linear interpolation: IRR = L + [NPV at L ÷ (NPV at L – NPV at H)] × (H – L), where L is the lower discount rate and H the higher rate giving a negative NPV.
内含报酬率 (IRR) 是使 NPV 等于零的贴现率。可用线性插值法近似：IRR = L + [L 下的 NPV ÷ (L 下的 NPV – H 下的 NPV)] × (H – L)，其中 L 为较低的贴现率，H 为使 NPV 为负的较高贴现率。
8. Elasticity and Demand | 弹性与需求
Price elasticity of demand measures responsiveness of quantity demanded to a price change. PED = % change in quantity demanded ÷ % change in price. If |PED| > 1, demand is price elastic; if |PED| < 1, it is inelastic; if |PED| = 1, demand is unit elastic. The total revenue rule states: if demand is elastic, a price reduction raises total revenue; if demand is inelastic, a price increase raises total revenue.
需求价格弹性衡量需求量对价格变动的反应程度。PED = 需求量变动百分比 ÷ 价格变动百分比。如果 |PED| > 1，需求富有弹性；|PED| < 1 则缺乏弹性；|PED| = 1 为单元弹性。总收入法则指出：需求富有弹性时降价增收，缺乏弹性时涨价增收。
Income Elasticity of Demand (YED) = % change in quantity demanded ÷ % change in income. Positive YED indicates a normal good; a negative YED signals an inferior good. Goods with YED > 1 are luxury goods.
需求收入弹性 (YED) = 需求量变动百分比 ÷ 收入变动百分比。YED 为正当属正常商品；YED 为负则属低档商品。YED > 1 的商品为奢侈品。
Cross Elasticity of Demand (XED) = % change in quantity demanded of Good A ÷ % change in price of Good B. A positive XED suggests substitute goods; a negative XED points to complements.
需求交叉弹性 (XED) = A 商品需求量变动百分比 ÷ B 商品价格变动百分比。XED 为正当暗示替代品关系；为负则表明是互补品。
9. Shareholder and Market‑Value Ratios | 股东与市场价值比率
These ratios connect accounting figures with market perceptions. Earnings Per Share (EPS) = (Net Profit after Tax – Preference Dividends) ÷ Number of Ordinary Shares Issued. EPS indicates the profit attributable to each ordinary share.
这些比率将会计数据与市场认知联系起来。每股收益 (EPS) = (税后净利润 – 优先股股利) ÷ 已发行普通股股数。EPS 表示每股普通股可享有的利润。
Price/Earnings (P/E) Ratio = Market Price per Share ÷ Earnings per Share. A high P/E suggests investors expect future earnings growth; a low P/E may indicate undervaluation or limited growth prospects.
市盈率 (P/E) = 每股市价 ÷ 每股收益。高 P/E 暗示投资者预期未来盈利增长；低 P/E 可能表示估值偏低或增长前景有限。
Dividend Yield = (Dividend per Share ÷ Market Price per Share) × 100%. It shows the cash return shareholders receive relative to the share price. Dividend Cover = (Net Profit after Tax – Preference Dividends) ÷ Total Ordinary Dividends Paid. This ratio reveals how many times the dividend could be paid out of earnings, indicating dividend safety.
股息率 = (每股股息 ÷ 市价) × 100%，显示股东相对股价获得的现金回报。股息保障倍数 = (税后净利润 – 优先股股利) ÷ 已付普通股股利总额。该比率揭示了利润覆盖股息的倍数，反映股息安全性。
10. Labour and Operational Performance | 劳动力与运营绩效
Operational metrics often carry equal weight in Pre‑U case analysis. Labour Productivity = Output per period ÷ Number of employees (or output ÷ labour hours). Higher productivity lowers unit labour costs and can improve competitiveness.
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