经济学（Economics）
一、本课定位
| 课次 | 主题 | 能力 |
|---|---|---|
| L186 | 利率平价（IRP） | 能够运用IRP公式判断汇率是否被高估或低估，计算无套利远期汇率，并理解IRP在国际资本流动中的核心作用 |
二、我们要解决什么问题？
假设你是一位中国出口商，3个月后将收到100万美元货款。目前即期汇率为1美元=7.15元人民币，美国3个月无风险利率为4.8%（年化），中国3个月无风险利率为2.4%（年化）。你想通过远期合约锁定人民币收入，但银行报出的3个月远期汇率是1美元=7.18元人民币。这个价格是否合理？如果不合理，你应该如何套利？利率平价（IRP）正是解决“不同货币利率差异与汇率升贴水必须保持无套利均衡”这一核心问题的理论工具。
三、利率平价的基本概念
利率平价理论认为，在资本可以自由流动且不存在交易成本和资本管制的情况下，两种货币的利率差异必须被其远期汇率与即期汇率的升（贴）水完全抵消，否则就会出现无风险套利机会。
核心逻辑：高利率货币会面临远期贴水（Forward Discount），低利率货币会面临远期升水（Forward Premium），以消除套利空间。
IRP分为两种形式： - 抛补利率平价（Covered Interest Rate Parity, CIRP）：使用远期合约锁定汇率风险。 - 无抛补利率平价（Uncovered Interest Rate Parity, UIRP）：依赖对未来即期汇率的预期，不使用远期合约。
CFA一级重点考察抛补利率平价。
四、抛补利率平价的公式推导与含义
设： - $S_0$：当前即期汇率（本币/外币，例如CNY/USD） - $F_{0,T}$：T期远期汇率（本币/外币） - $r_d$：本币（国内）无风险利率（年化） - $r_f$：外币（国外）无风险利率（年化） - $T$：以年为单位的时间
抛补利率平价的核心公式为： $$F_{0,T} = S_0 \times \frac{1 + r_d \times T}{1 + r_f \times T}$$
近似公式（当利率和期限较小时）： $$\frac{F_{0,T}-S_0}{S_0} \approx (r_d - r_f) \times T$$
其中，$\frac{F-S}{S}$称为远期升贴水率（Forward Premium/Discount）。
经济含义： - 如果本国利率高于外国利率（$r_d > r_f$），则远期汇率应高于即期汇率（本币远期贬值），即本币出现远期贴水。 - 公式右侧代表“借外币、投资本币并用远期锁定”的无套利成本。
五、汇率高估与低估的判断
若市场远期汇率 $F_{market} > F_{IRP}$，则称远期外币被高估（或本币被低估），套利策略为： 1. 借入本币 2. 即期买入外币并投资于外币无风险资产 3. 同时卖出远期外币（锁定卖出价格）
反之，若 $F_{market} < F_{IRP}$，则远期外币被低估，套利方向相反。
六、连续复利形式（补充知识）
在连续复利下，IRP公式变为： $$F_{0,T} = S_0 \times e^{(r_d - r_f)T}$$
CFA一级主要使用离散复利公式，但需了解连续形式在高频交易中的应用。
完整案例演算
案例 1：计算理论远期汇率
即期汇率 $S_0 = 7.15$ CNY/USD，美国年化利率 $r_f = 5\%$，中国年化利率 $r_d = 2.5\%$，期限3个月（$T=0.25$）。
计算IRP远期汇率： $$F = 7.15 \times \frac{1 + 0.025 \times 0.25}{1 + 0.05 \times 0.25} = 7.15 \times \frac{1.00625}{1.0125} = 7.15 \times 0.993827 = 7.1064$$
结论：理论远期汇率为7.1064，低于即期，说明人民币远期升水（因为中国利率更低）。
案例 2：套利机会判断与计算
接案例1，若银行报出的3个月远期汇率为7.14（高于IRP的7.1064），判断是否存在套利机会并计算100万美元名义金额的套利利润（假设可借贷金额100万美元）。
判断：$F_{market}=7.14 > 7.1064=F_{IRP}$，外币（美元）远期被高估。
套利步骤（以100万美元规模）： 1. 今日借入人民币：$100 \times 7.15 = 715$万元人民币 2. 即期买入100万美元并投资美国3个月存款，到期获得：$100 \times (1 + 0.05 \times 0.25) = 101.25$万美元 3. 同时卖出101.25万美元的3个月远期合约，按7.14锁定，未来收到人民币：$101.25 \times 7.14 = 722.925$万元 4. 归还人民币借款本息：$715 \times (1 + 0.025 \times 0.25) = 719.46875$万元
无风险利润：$722.925 - 719.46875 = 3.45625$万元人民币
案例 3：UIRP与预期汇率
假设年化利率差 $r_d - r_f = -2.5\%$，根据UIRP，预期未来一年即期汇率变化率约为$-2.5\%$。若当前即期为7.15，则一年后预期即期汇率约为： $$E(S_1) = 7.15 \times (1 - 0.025) = 6.97125$$
这意味着市场预期人民币将升值2.5%，以抵消利率劣势。注意UIRP在现实中预测能力较弱，常因风险溢价而偏差。
易错陷阱对照
| 易错点 | 错误做法 | 正确做法 |
|---|---|---|
| 汇率标价法混淆 | 直接用USD/CNY代替CNY/USD代入公式 | 必须保持公式中S和F的标价法一致（本币/外币） |
| 利率期限不匹配 | 用年化利率直接代入而不乘T | 必须将利率调整为期间利率：$r \times T$ |
| 套利方向记反 | 看到高利率货币就认为其远期应升值 | 高利率货币远期必然贴水（F > S when quoted as domestic/foreign） |
| 忽略复利基数 | 用$(1+r_d)/(1+r_f)$时把分子分母写反 | 分子为本币利率，分母为外币利率 |
| 把UIRP当CIRP计算 | 用预期汇率代替远期汇率计算套利 | CIRP使用可观察的远期汇率，UIRP使用预期汇率 |
关键公式 / 关系速记
- 核心公式：$F_{0,T} = S_0 \times \frac{1 + r_d \times T}{1 + r_f \times T}$
- 远期升贴水率：$\frac{F-S}{S} \approx r_d - r_f$（以年化表示时需乘T）
- 近似关系：远期贴水率 ≈ 利率差（高利率货币贴水）
- 连续复利形式：$F = S_0 \times e^{(r_d-r_f)T}$
- 套利无风险条件：$F_{market} = F_{IRP}$ 时套利利润为0
练习题（含计算与情景）
Q1. 根据抛补利率平价，若本国利率高于外国利率，则：
A. 本币远期升水
B. 本币远期贴水
C. 远期汇率等于即期汇率
D. 无法判断
Q2. 即期汇率为1.25 USD/EUR，欧元区利率4%，美国利率2%，6个月远期汇率理论值最接近：
A. 1.2375
B. 1.2626
C. 1.2150
D. 1.2500
Q3. 若市场远期汇率高于IRP计算的远期汇率，则最可能的套利策略是：
A. 借外币、投资本币、买入远期外币
B. 借本币、投资外币、卖出远期外币
C. 借本币、投资本币、卖出远期本币
D. 不存在套利机会
Q4. 连续复利下，若$r_d=0.03$，$r_f=0.01$，$T=1$，$S_0=6.5$，则$F$约为：
A. 6.630
B. 6.371
C. 6.532
D. 6.469
Q5. 以下哪项不是利率平价成立的前提假设？
A. 资本自由流动
B. 无交易成本
C. 投资者风险中性
D. 存在资本管制
Q6. 当前即期汇率7.20 CNY/USD，中国3个月利率1.5%（年化），美国3个月利率2.8%（年化）。根据IRP，3个月远期汇率应为：
A. 7.176
B. 7.224
C. 7.183
D. 7.217
Q7. 在UIRP中，预期汇率变化率约等于：
A. 本币利率减外币利率
B. 远期升水率
C. 通胀率差异
D. 实际利率差异
Q8. 若抛补利率平价被打破且市场远期汇率过低，最直接的后果是：
A. 套利者会买入远期外币，推动远期汇率上升
B. 套利者会卖出远期外币，推动远期汇率下降
C. 即期汇率立即调整
D. 央行必须干预外汇市场
答案与详解
| 题号 | 答案 | 详解 |
|---|---|---|
| Q1 | B | 高利率货币（本币利率高）在远期必须贴水才能抵消利率优势，防止套利 |
| Q2 | A | $F=1.25 \times \frac{1+0.02\times0.5}{1+0.04\times0.5}=1.25\times\frac{1.01}{1.02}\approx1.2377$，选A |
| Q3 | B | $F_{market}>F_{IRP}$时，外币远期被高估，应借本币买外币投资并卖出远期外币锁定高价 |
| Q4 | A | $F=6.5\times e^{(0.03-0.01)\times1}=6.5\times e^{0.02}\approx6.5\times1.0202=6.631$ |
| Q5 | D | 利率平价成立的前提是资本自由流动、无交易成本、无资本管制，D是反面 |
| Q6 | A | $F=7.20\times\frac{1+0.015\times0.25}{1+0.028\times0.25}=7.20\times\frac{1.00375}{1.007}\approx7.176$ |
| Q7 | A | UIRP认为预期本币升值率≈外国利率-本国利率，即$r_f - r_d$，等价于本币利率减外币利率的相反数，但选项A最接近标准表述 |
| Q8 | A | 远期汇率过低意味着远期外币被低估，套利者会买入远期外币（卖出远期本币），推动远期汇率上升直至均衡 |
本节要点速记
- 抛补利率平价是无套利均衡条件，高利率货币必然远期贴水
- 核心公式：$F = S \times \frac{1+r_d T}{1+r_f T}$，分子分母千万不能写反
- 判断套利方向关键在于比较$F_{market}$与$F_{IRP}$的大小
- 汇率标价法必须与公式中“本币/外币”定义一致
- UIRP使用预期汇率，现实预测偏差大，CFA重点仍在CIRP
- 掌握套利全流程计算（借、换、投、锁、还）是得分关键
Economics
I. Lesson Focus
This lesson explains the Interest Rate Parity (IRP) condition that links interest rate differentials between two countries to the forward premium or discount in their exchange rates. Candidates must master the covered interest rate parity (CIRP) formula, determine whether a quoted forward rate is mispriced, execute arbitrage steps, and calculate risk-free profits. The material also introduces uncovered interest rate parity (UIRP) for conceptual contrast. Mastery of IRP is essential for understanding international capital flows, currency valuation, and derivative pricing in the CFA curriculum.
II. The Problem
A Chinese exporter will receive USD 1 million in three months. The spot rate is CNY 7.15 per USD, the annualized Chinese risk-free rate is 2.4%, and the annualized U.S. risk-free rate is 4.8%. A bank quotes a three-month forward rate of CNY 7.18 per USD. Is this forward rate fair? If not, how can an arbitrageur lock in a risk-free profit? Interest Rate Parity provides the exact no-arbitrage forward rate that must prevail when capital moves freely, solving the problem of whether interest rate differences are fairly compensated by expected or locked-in currency movements.
III. Core Concepts of Interest Rate Parity
Interest Rate Parity states that, in the absence of transaction costs and capital controls, the difference in risk-free interest rates between two countries must be exactly offset by the forward premium or discount between their currencies. Otherwise, risk-free arbitrage opportunities would exist and be instantly exploited until parity is restored.
There are two versions: - Covered Interest Rate Parity (CIRP) uses forward contracts to eliminate exchange-rate risk. - Uncovered Interest Rate Parity (UIRP) relies on the expected future spot rate instead of a forward contract.
CFA Level I focuses primarily on CIRP because it is observable and directly testable through arbitrage.
Economic intuition: The currency with the higher interest rate must trade at a forward discount (F > S when quoted as domestic per foreign) to eliminate the incentive to borrow the low-rate currency, convert, invest in the high-rate currency, and hedge with forwards.
IV. Derivation and Interpretation of the CIRP Formula
Define: - $S_0$: current spot exchange rate (domestic currency per unit of foreign currency) - $F_{0,T}$: forward exchange rate for delivery in T years - $r_d$: domestic risk-free interest rate (annualized) - $r_f$: foreign risk-free interest rate (annualized) - $T$: time to maturity expressed in years
The exact covered interest rate parity formula is: $$F_{0,T} = S_0 \times \frac{1 + r_d \times T}{1 + r_f \times T}$$
When rates and time periods are small, the approximation is: $$\frac{F_{0,T} - S_0}{S_0} \approx (r_d - r_f) \times T$$
The left side is the forward premium (if positive) or discount (if negative). The formula ensures that borrowing in one currency, converting at spot, investing in the other currency, and selling the foreign currency forward produces the same return as investing domestically.
If the domestic rate exceeds the foreign rate ($r_d > r_f$), the domestic currency must depreciate in the forward market (F > S) to offset the interest advantage.
V. Identifying Overvaluation or Undervaluation and Arbitrage
Compare the market forward rate $F_{market}$ with the model-implied rate $F_{IRP}$: - If $F_{market} > F_{IRP}$, the foreign currency is overvalued in the forward market. Arbitrage: borrow domestic currency, buy foreign currency spot, invest in foreign risk-free asset, and sell the foreign currency forward at the inflated rate. - If $F_{market} < F_{IRP}$, the foreign currency is undervalued. Reverse the trades.
The arbitrage process forces $F_{market}$ back to $F_{IRP}$.
VI. Continuous Compounding Version
Under continuous compounding the formula becomes: $$F_{0,T} = S_0 \times e^{(r_d - r_f)T}$$
While CFA Level I primarily tests the discrete version, candidates should recognize the continuous form appears in more advanced fixed-income and derivatives topics.
Worked Cases
Case 1: Calculating the Theoretical Forward Rate
Spot rate $S_0 = 7.15$ CNY/USD, Chinese annualized rate $r_d = 2.5\%$, U.S. annualized rate $r_f = 5\%$, $T = 0.25$ (three months).
Theoretical forward rate: $$F = 7.15 \times \frac{1 + 0.025 \times 0.25}{1 + 0.05 \times 0.25} = 7.15 \times \frac{1.00625}{1.0125} = 7.15 \times 0.993827 \approx 7.1064$$
The forward rate is below spot, indicating the lower-yielding renminbi trades at a forward premium, consistent with IRP.
Case 2: Detecting Arbitrage and Computing Profit
Using the data from Case 1, suppose the bank quotes a three-month forward at 7.14 (higher than the IRP value of 7.1064). Determine the arbitrage direction and risk-free profit on a USD 100 million notional amount.
Diagnosis: $F_{market} = 7.14 > 7.1064 = F_{IRP}$, so the USD is overvalued forward.
Arbitrage steps (USD 100 million scale): 1. Borrow CNY 715 million today ($100m × 7.15). 2. Convert to USD 100 million at spot and invest at the U.S. rate; in three months receive USD 101.25 million ($100m × 1.0125). 3. Simultaneously sell USD 101.25 million forward at 7.14, locking in CNY 722.925 million. 4. Repay the CNY loan: CNY 715m × 1.00625 = CNY 719.46875 million.
Risk-free profit: CNY 722.925m − CNY 719.46875m = CNY 3.45625 million.
This profit would be competed away until the forward rate falls to 7.1064.
Case 3: Uncovered Interest Rate Parity and Expected Spot Rate
Assume the interest differential $r_d − r_f = −2.5\%$ per year. According to UIRP, the expected percentage change in the spot rate over the next year is approximately −2.5%. With current spot = 7.15, the expected spot rate one year later is: $$E(S_1) = 7.15 \times (1 − 0.025) = 6.97125$$
This implies the market expects the renminbi to appreciate by 2.5% to offset China’s lower interest rate. In practice, UIRP is a poor predictor because of risk premiums and behavioral biases; CFA emphasizes that CIRP is enforced by observable arbitrage.
Traps
| Common Mistake | Wrong Approach | Correct Approach |
|---|---|---|
| Quotation convention error | Using USD/CNY directly in a CNY/USD formula | Keep S and F consistently quoted as domestic per foreign unit |
| Maturity mismatch | Plugging full-year rates without multiplying by T | Always convert to period rates: multiply annual rates by T (in years) |
| Reversing arbitrage direction | Assuming high-interest currency should appreciate forward | High-interest currency must be at a forward discount (F > S) |
| Inverting numerator and denominator | Writing $(1+r_f)/(1+r_d)$ | Numerator is always the domestic interest factor; denominator is foreign |
| Confusing UIRP with CIRP | Using expected spot rate to calculate cash arbitrage profit | CIRP uses observable forward contracts; UIRP uses unobservable expectations |
| Ignoring compounding base | Treating rates as continuously compounded when they are discrete | Match the formula to the given compounding convention |
Key Formulas
- Exact CIRP: $F_{0,T} = S_0 \times \frac{1 + r_d \times T}{1 + r_f \times T}$
- Forward premium/discount approximation: $\frac{F-S}{S} \approx (r_d - r_f) \times T$
- Continuous compounding: $F_{0,T} = S_0 \times e^{(r_d - r_f)T}$
- No-arbitrage condition: Profit = 0 when $F_{market} = F_{IRP}$
- UIRP expectation: $E(\%\Delta S) \approx r_d - r_f$
Practice Questions
Q1. According to covered interest rate parity, if the domestic interest rate is higher than the foreign interest rate, the domestic currency will:
A. Trade at a forward premium
B. Trade at a forward discount
C. Have the same forward and spot rate
D. Cannot be determined
Q2. The spot rate is 1.25 USD/EUR, the eurozone interest rate is 4%, and the U.S. interest rate is 2%. The theoretical six-month forward rate is closest to:
A. 1.2375
B. 1.2626
C. 1.2150
D. 1.2500
Q3. If the market forward rate is higher than the rate implied by IRP, the most likely arbitrage strategy is to:
A. Borrow foreign currency, invest domestically, and buy foreign currency forward
B. Borrow domestic currency, invest in foreign currency, and sell foreign currency forward
C. Borrow domestic currency, invest domestically, and sell domestic currency forward
D. Conclude no arbitrage exists
Q4. Using continuous compounding, if $r_d = 0.03$, $r_f = 0.01$, $T = 1$, and $S_0 = 6.5$, $F$ is closest to:
A. 6.630
B. 6.371
C. 6.532
D. 6.469
Q5. Which of the following is NOT an assumption required for interest rate parity to hold?
A. Free capital mobility
B. No transaction costs
C. Risk-neutral investors
D. Presence of capital controls
Q6. The spot rate is 7.20 CNY/USD, the Chinese three-month annualized rate is 1.5%, and the U.S. three-month annualized rate is 2.8%. The three-month forward rate implied by IRP is closest to:
A. 7.176
B. 7.224
C. 7.183
D. 7.217
Q7. Under uncovered interest rate parity, the expected percentage change in the exchange rate is approximately equal to:
A. The domestic interest rate minus the foreign interest rate
B. The forward premium
C. The inflation differential
D. The real interest rate differential
Q8. If covered interest rate parity is violated and the market forward rate is too low, arbitrageurs will most likely:
A. Buy the foreign currency forward, pushing the forward rate upward
B. Sell the foreign currency forward, pushing the forward rate downward
C. Immediately adjust the spot rate
D. Require central bank intervention
Answers
| Question | Answer | Explanation |
|---|---|---|
| Q1 | B | When the domestic rate is higher, the domestic currency must trade at a forward discount to offset the interest advantage and prevent arbitrage. |
| Q2 | A | $F = 1.25 \times \frac{1 + 0.02 \times 0.5}{1 + 0.04 \times 0.5} = 1.25 \times \frac{1.01}{1.02} \approx 1.2377$. Choice A is correct. |
| Q3 | B | $F_{market} > F_{IRP}$ implies the foreign currency is expensive forward; borrow domestic, buy and invest foreign, sell foreign forward at the high locked-in rate. |
| Q4 | A | $F = 6.5 \times e^{(0.03-0.01)\times1} = 6.5 \times e^{0.02} \approx 6.5 \times 1.0202 = 6.631$. |
| Q5 | D | Capital controls prevent parity from holding; free mobility, zero transaction costs, and no barriers are required. |
| Q6 | A | $F = 7.20 \times \frac{1 + 0.015 \times 0.25}{1 + 0.028 \times 0.25} = 7.20 \times \frac{1.00375}{1.007} \approx 7.176$. |
| Q7 | A | UIRP states $E(\%\Delta S) \approx r_d - r_f$; the currency with the higher interest rate is expected to depreciate. |
| Q8 | A | A forward rate that is too low means the foreign currency is cheap forward; arbitrageurs buy it forward, bidding the forward price up until parity is restored. |
Takeaways
- Covered IRP enforces the no-arbitrage forward rate: $F = S \times (1 + r_d T)/(1 + r_f T)$.
- A higher domestic interest rate must be accompanied by forward depreciation of the domestic currency.
- Always compare the actual forward quote with the IRP-implied rate to detect arbitrage direction.
- Quotation convention and correct period adjustment of interest rates ($r \times T$) are frequent traps.
- UIRP links expected spot change to the interest differential but is less reliable in practice.
- The complete arbitrage sequence (borrow → convert → invest → hedge → repay) must be calculated precisely for numerical questions.