SL 内容覆盖
Simple & compound interest, annuities, loan amortization, inflation, currency conversion
HL 额外内容
Net Present Value (NPV), Internal Rate of Return (IRR), more complex financial modeling
一、复利与年金 / Compound Interest & Annuities
金融数学的起点永远是复利公式。SL 阶段你需要熟练掌握以下公式：FV = PV x (1 + r/k)^(k x n)
其中 FV 表示终值（Future Value），PV 表示现值（Present Value），r 为年利率，k 为每年复利次数，n 为年数。 FV represents the future value, PV the present value, r the annual interest rate, k the compounding frequency per year, and n the number of years. 年金 (Annuity) 是等时间间隔的一系列等额支付。SL考试中常考两类年金： 普通年金 Ordinary Annuity每期期末支付，如按揭还款
即付年金 Annuity Due每期期初支付，如租金预付
解题关键：先画时间轴（timeline），标注每笔现金流的时点和方向（流入/流出），再代入公式。许多同学丢分不是不会算，而是时点搞反了。 Key tip: Always draw a timeline first, label the timing and direction (inflow/outflow) of each cash flow, then plug into your formula. Most marks are lost not from calculation errors but from getting the timing wrong.二、贷款摊销 / Loan Amortization
贷款摊销是年金公式的核心应用。给定贷款总额 P、年利率 r、还款期数 n，你需要求解每期还款额 A。 Loan amortization is the core application of annuity formulas. Given the loan amount P, annual rate r, and number of payments n, you solve for the periodic payment A.A = P x [r/k x (1+r/k)^(kxn)] / [(1+r/k)^(kxn) - 1]
考试常见题型：计算每月还款额、计算已支付的总利息、提前还款后的余额计算。务必区分"月利率 = 年利率 / 12"和"等效月利率"两种处理方式——IB 默认使用前者（名义利率换算）。 Common exam questions: calculating monthly payments, total interest paid, and outstanding balance after early repayment. Make sure you distinguish between "monthly rate = annual rate / 12" and the equivalent monthly rate -- IB defaults to the former (nominal rate conversion).三、净现值与内部收益率 / NPV & IRR
进入 HL 部分，净现值（NPV）和内部收益率（IRR）是金融数学的核心概念。 Moving into HL territory, Net Present Value (NPV) and Internal Rate of Return (IRR) are the centerpieces of financial mathematics. 净现值 NPV 衡量一项投资的当前价值。将所有未来现金流按给定折现率折现到当前，求和后减去初始投资。NPV > 0 → 项目可行；NPV < 0 → 不划算。 NPV measures the present value of an investment. Discount all future cash flows to the present using a given discount rate, sum them up, and subtract the initial investment. NPV > 0 means the project is viable; NPV < 0 means it is not worthwhile.NPV = Σ [CF_t / (1+r)^t] - Initial Investment
内部收益率 IRR 是使 NPV 恰好等于零的折现率。它代表项目的预期年化回报率。当 IRR > 要求回报率时，项目值得投资。 IRR is the discount rate that makes NPV exactly equal to zero. It represents the project's expected annualized return. When IRR exceeds the required rate of return, the project is worth investing in. 在 IB 考试中，IRR 使用 GDC（图形计算器）求解，不需要手算迭代。你只需设置方程：NPV = 0，然后用计算器的 solver 或 finance 功能求解未知折现率。 In IB exams, IRR is solved using your GDC (Graphing Display Calculator) -- no manual iteration required. Simply set up the equation NPV = 0, then use your calculator's solver or finance function to find the unknown discount rate. 永续年金 Perpetuity: 当支付期数趋于无穷时，年金公式简化为 PV = PMT / r。这是股票估值中 Gordon Growth Model 的数学基础。虽然 IB 不直接考永续年金公式的推导，但理解这一概念对把握"货币时间价值"的极限情况很有帮助。 A perpetuity is an annuity that continues forever. The formula simplifies to PV = PMT / r. This is the mathematical foundation of the Gordon Growth Model used in stock valuation. While IB does not directly test perpetuity derivation, understanding this concept helps you grasp the limiting case of the time value of money. 折现率的选择 The Discount Rate Dilemma: NPV 分析中最具争议的环节是折现率的选择。折现率过高会低估长期项目价值，过低则可能批准不经济的项目。IB 考题中通常会直接给定折现率，但在 IA 写作中，你需要为自己的折现率选择提供合理依据——常见方法包括参考无风险利率加上风险溢价，或使用 WACC（加权平均资本成本）。 The most debated part of NPV analysis is the choice of discount rate. A rate that is too high undervalues long-term projects; a rate too low may approve uneconomic ones. IB exam questions usually provide the discount rate directly, but in your IA, you need to justify your choice -- common approaches include referencing the risk-free rate plus a risk premium, or using WACC (Weighted Average Cost of Capital). 通胀与汇率 Inflation & Currency Conversion: SL 大纲明确要求掌握通胀调整和货币兑换。名义利率与实际利率的关系遵循 Fisher 方程：1 + i = (1 + r)(1 + p)，其中 i 为名义利率、r 为实际利率、p 为通胀率。真题中常以"某国年通胀率 3%，银行存款利率 5%，求实际购买力增长"的形式出现。 The SL syllabus explicitly requires mastery of inflation adjustment and currency conversion. The relationship between nominal and real interest rates follows the Fisher equation: 1 + i = (1 + r)(1 + p), where i is the nominal rate, r the real rate, and p the inflation rate. Past papers often ask: "If a country has 3% annual inflation and bank deposits pay 5%, what is the real growth in purchasing power?"常见误区
- 把 NPV 和利润混为一谈
- 忽略时间价值直接加总现金流
- 混淆名义利率和实际利率
- IRR 计算漏掉初始投资的负号
加分技巧
- 画时间轴标注所有现金流
- 明确写出公式再代入数字
- 用 GDC 验证手动计算结果
- 答案保留合理精度（通常 2 dp）
四、GDC 计算器策略 / Calculator Strategies
IB AI 考试全卷允许使用 GDC。金融数学模块尤其依赖计算器的 TVM（Time Value of Money）求解器和 Finance 应用。以 TI-Nspire 为例： The entire IB AI exam allows GDC use. The financial mathematics module relies heavily on the TVM (Time Value of Money) solver and Finance application. Using TI-Nspire as an example: N总支付期数
I%
年利率（百分比）
PV现值（贷款为正）
PMT每期支付额
FV终值
P/Y每年支付次数
熟练使用 TVM 求解器可以节省大量计算时间。但要记住：计算器给出的是数值答案，Paper 1 和 Paper 2 的解答过程仍然需要展示你的推理步骤。 Mastering the TVM solver saves significant calculation time. But remember: the calculator gives you a numeric answer -- your Paper 1 and Paper 2 solutions still need to show your reasoning steps.五、考点梳理与应考策略 / Exam Focus & Strategy
Paper 1 (短答题 Short-Response): 通常包含基础复利计算、单笔年金求解。要求写出公式、代入、答案三步。时间充裕但不要在某道题上陷入过度思考。 Typically includes basic compound interest calculations and single annuity problems. You are expected to show formula, substitution, and answer. Time is generous but don't overthink any single question. Paper 2 (长答题 Extended-Response): 金融数学常与函数、数列、统计知识交叉出题。可能出现"比较两种贷款方案"或"分析投资项目可行性"的综合题目。 Financial mathematics often intersects with functions, sequences, and statistics. You might see integrated questions like "compare two loan options" or "analyze a project's feasibility." Internal Assessment (IA): 金融数学是 IA 的热门选题方向。你可以分析信用卡还款方案、对比不同按揭策略、或用 NPV/IRR 评估某家公司的投资决策。数据来源丰富，数学模型清晰，评分标准容易命中高分档。 Financial mathematics is a popular IA topic. You could analyze credit card repayment options, compare mortgage strategies, or use NPV/IRR to evaluate a company's investment decisions. Data sources are abundant, models are clear, and the marking criteria are easy to hit at the top bands.六、复习清单 / Revision Checklist
☐ Simple interest & compound interest formulas ☐ Ordinary annuity vs annuity due ☐ Loan amortization & repayment schedules ☐ Inflation & currency conversion (real vs nominal) ☐ NPV calculation & investment decision rules ☐ IRR via GDC solver (HL only) ☐ TVM solver proficiency on your GDC ☐ Timeline drawing for cash flow visualization七、真题示例 / Worked Example
Question: Sarah borrows $200,000 at 4.5% p.a. compounded monthly. The loan is to be repaid over 25 years with equal monthly payments. Calculate: (a) the monthly payment; (b) the total interest paid.
Solution (a):
r = 0.045, k = 12, n = 25, PV = 200000, FV = 0
Monthly rate = 0.045/12 = 0.00375
Number of payments = 12 x 25 = 300
PMT = 200000 x 0.00375 x (1.00375)^300 / ((1.00375)^300 - 1)
PMT ≈ $1,111.69
Solution (b):
Total paid = 300 x $1,111.69 = $333,507.00
Total interest = $333,507.00 - $200,000.00 = $133,507.00
八、NPV/IRR 真题示例 / NPV & IRR Worked Example
Question (HL): A company considers a project requiring an initial investment of $50,000. Expected net cash inflows are: Year 1: $15,000; Year 2: $20,000; Year 3: $25,000; Year 4: $15,000. The company's required rate of return is 8%. (a) Calculate the NPV. (b) Determine whether the project should be accepted. (c) Using your GDC, find the IRR.
Solution (a):
NPV = -50000 + 15000/1.08 + 20000/1.08^2 + 25000/1.08^3 + 15000/1.08^4
NPV = -50000 + 13888.89 + 17146.78 + 19845.28 + 11025.45
NPV = -50000 + 61906.40
NPV = $11,906.40
Solution (b): Since NPV > 0 ($11,906.40), the project should be accepted. It adds value to the company beyond the required 8% return.
Solution (c): Using GDC finance solver: set NPV = 0 with the above cash flows, solve for I%. The IRR is approximately 17.2%. Since 17.2% > 8% (required return), this confirms the accept decision. Note: when cash flows change sign more than once, there may be multiple IRRs -- always check with your GDC and use NPV as the primary decision criterion when in doubt.
考试警示 Exam Warning: 很多学生在计算 NPV 时忘记将初始投资设为负值。如果你算出的 NPV 异常大（比如少了一个负号导致 NPV = $111,906），回头检查第一年的现金流符号。IB 评分标准中，"正确识别现金流方向"通常占 1-2 分。 Many students forget to set the initial investment as a negative value. If your NPV looks suspiciously large (e.g., missing a minus sign giving NPV = $111,906), go back and check the sign of your Year 0 cash flow. In IB marking schemes, "correctly identifying cash flow direction" is typically worth 1-2 marks. 金融数学是 IB 数学中最"有用"的模块之一——你今天学的复利和贷款计算，明天就能用来算自己的按揭。IB 不只是一场考试，它是你理解世界运作方式的工具箱。 Financial mathematics is one of the most "practical" modules in IB Math -- the compound interest and loan calculations you learn today can be used tomorrow to figure out your own mortgage. IB is not just an exam; it is a toolkit for understanding how the world works.
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