📖 汇率基础：即期与远期

Economics

I. Lesson Focus

This lesson explains the fundamental mechanics of spot and forward foreign exchange markets. Candidates must master quotation conventions (direct vs. indirect), bid-ask spreads, calculation of forward premiums and discounts, the covered interest rate parity condition, no-arbitrage forward pricing, and the identification of covered interest arbitrage opportunities. These concepts form the foundation for later topics on currency risk management and international parity conditions.

II. The Problem

A Chinese exporter will receive USD 1 million in three months. The current spot rate is USD/CNY = 7.15. If the renminbi appreciates, the exporter will receive fewer CNY. How should a three-month forward contract be priced so that neither party can earn a riskless profit? This lesson systematically develops the definitions of spot and forward rates, quotation conventions, premium/discount calculations, the interest rate parity formula, and arbitrage mechanisms so candidates can solve such problems accurately on the exam.

III. Basic Concepts and Quotation Conventions

An exchange rate is the relative price of two currencies, expressed as the price of one currency in terms of another.

Direct quotation: Expresses the price of foreign currency in domestic currency units. In China, USD/CNY = 7.15 means 1 USD costs 7.15 CNY. China and most emerging markets use direct quotes.

Indirect quotation: Expresses the price of domestic currency in foreign currency units. GBP/USD = 1.28 means 1 GBP buys 1.28 USD. The United States and United Kingdom commonly use indirect quotes.

Base currency and price (quote) currency: - In USD/CNY = 7.15, USD is the base currency and CNY is the price currency. - An increase in the quoted rate means the base currency appreciates (or the price currency depreciates).

Bid-ask quotes: - Bid: The price at which the dealer is willing to buy the base currency (client sells base currency). - Ask (Offer): The price at which the dealer is willing to sell the base currency (client buys base currency). - Spread = Ask – Bid, reflecting liquidity and transaction costs.

Example: A bank quotes USD/CNY = 7.1480 / 7.1520. - Client selling USD (buying CNY) receives the bid rate 7.1480. - Client buying USD (selling CNY) pays the ask rate 7.1520.

IV. Spot Exchange Rates

The spot rate is the exchange rate for immediate delivery (usually T+2 business days), denoted $S_{f/d}$ where $f$ is the foreign currency and $d$ is the domestic currency.

CFA exams frequently require conversion between “foreign currency per domestic currency” and “domestic currency per foreign currency” forms.

Appreciation and depreciation: - If USD/CNY moves from 7.15 to 7.25, the renminbi has depreciated (more CNY needed to buy one USD) and the dollar has appreciated. - Percentage change: $\frac{S_1 - S_0}{S_0} \times 100\%$.

V. Forward Rates and Forward Premium/Discount

A forward rate is the exchange rate agreed today for settlement on a future date, denoted $F_{f/d,t}$ where $t$ is time to maturity in years.

Forward premium: Occurs when $F > S$; the base currency is more expensive in the forward market. Forward discount: Occurs when $F < S$.

Annualized forward premium/discount: $$ \text{Forward Premium (Discount)} = \frac{F_{f/d} - S_{f/d}}{S_{f/d}} \times \frac{360}{t} \times 100\% $$ where $t$ is days to maturity. CFA uses a 360-day year convention for money-market calculations.

When using direct quotes (domestic per foreign), a positive value indicates the foreign currency is at a forward premium (domestic currency at a forward discount).

VI. No-Arbitrage Forward Rate — Covered Interest Rate Parity (IRP)

The forward rate is determined by the no-arbitrage condition between the spot FX market and the money markets in the two currencies.

Covered Interest Rate Parity formula: $$ F_{f/d} = S_{f/d} \times \frac{1 + i_f \times (t/360)}{1 + i_d \times (t/360)} $$ where $i_f$ = foreign risk-free interest rate, $i_d$ = domestic risk-free interest rate, and $t$ = days to maturity.

When the domestic interest rate exceeds the foreign rate ($i_d > i_f$), the forward rate $F < S$, meaning the foreign (base) currency trades at a forward premium.

Approximation (valid when rates and tenor are small): $$ \frac{F - S}{S} \approx (i_d - i_f) \times \frac{t}{360} $$

VII. Covered Interest Arbitrage Mechanism

If the quoted forward rate deviates from the IRP-implied rate, riskless arbitrage exists: - When actual $F >$ theoretical $F$: Borrow foreign currency, convert to domestic at spot, invest in domestic money market, and sell domestic currency forward to repay the foreign loan. - When actual $F <$ theoretical $F$: Borrow domestic currency, convert to foreign at spot, invest in foreign money market, and buy foreign currency forward to repay the domestic loan.

Arbitrage activity quickly eliminates the mispricing, restoring parity.

Worked Cases

Case 1: Forward Premium/Discount Calculation

Spot USD/CNY = 7.0800, 90-day forward = 7.1200. Calculate the annualized forward premium on the USD.

Solution: $$ \text{Premium} = \frac{7.1200 - 7.0800}{7.0800} \times \frac{360}{90} \times 100\% = 0.00565 \times 4 \times 100\% = 2.26\% $$ The USD is at an annualized forward premium of 2.26% (CNY at a 2.26% forward discount).

Case 2: No-Arbitrage Forward Rate

Spot EUR/USD = 1.0850 (indirect quote from U.S. perspective). U.S. one-year rate = 3.0%, eurozone one-year rate = 1.2%. Calculate the one-year theoretical forward rate.

Solution: $$ F = 1.0850 \times \frac{1 + 0.012}{1 + 0.030} = 1.0850 \times \frac{1.012}{1.030} = 1.0850 \times 0.9825 \approx 1.0660 $$ The euro is at a forward discount ($F < S$), consistent with the higher U.S. interest rate.

Case 3: Identifying Covered Arbitrage Opportunity

Spot GBP/USD = 1.2500, UK rate = 4%, U.S. rate = 2%, actual 180-day forward = 1.2650. Compute the theoretical forward and determine whether arbitrage exists.

Solution: $$ F = 1.2500 \times \frac{1 + 0.04 \times (180/360)}{1 + 0.02 \times (180/360)} = 1.2500 \times \frac{1.02}{1.01} = 1.2500 \times 1.0099 \approx 1.2624 $$ Actual forward (1.2650) > theoretical forward (1.2624). Arbitrage: borrow GBP, convert to USD at spot, invest in USD, and sell USD forward (buy GBP forward) to repay the GBP loan.

Traps

Key Formulas

• Annualized forward premium/discount: $\frac{F-S}{S} \times \frac{360}{t} \times 100\%$
• Exact no-arbitrage forward rate: $F = S \times \frac{1 + i_f(t/360)}{1 + i_d(t/360)}$
• Approximate IRP: $\frac{F-S}{S} \approx (i_d - i_f) \times (t/360)$
• Percentage change in spot rate: $\frac{S_1 - S_0}{S_0}$
• Base currency appreciation ⇔ increase in direct quote (domestic per foreign)

Practice Questions

Q1. If the spot USD/CNY rate rises from 6.95 to 7.05, then:
A. The renminbi has appreciated
B. The USD has depreciated
C. The renminbi has depreciated by approximately 1.44%
D. The USD has depreciated by approximately 1.44%

Q2. A bank quotes EUR/USD = 1.0820/1.0825. A client wishing to buy euros must pay:
A. 1.0820
B. 1.0825
C. Mid-rate 1.08225
D. 1.0820 (the bid)

Q3. Spot GBP/USD = 1.3000, 90-day forward = 1.2920. The GBP is trading at a forward:
A. Premium
B. Discount
C. Parity
D. Cannot be determined

Q4. Spot USD/JPY = 110.00, Japanese one-year rate = 0.1%, U.S. one-year rate = 3.0%. According to IRP, the one-year forward USD/JPY is closest to:
A. 106.81
B. 113.19
C. 110.00
D. 109.78

Q5. The main reason CFA uses 360 days to annualize forward premiums is:
A. Greater precision
B. Consistency with money-market day-count conventions
C. Only the yen uses 360 days
D. To avoid leap-year effects

Q6. If the actual forward rate is higher than the IRP-implied rate, the arbitrageur should:
A. Borrow domestic, invest foreign, sell foreign forward
B. Borrow foreign, invest domestic, sell domestic forward
C. Borrow and invest in the domestic currency only
D. Do nothing; no arbitrage exists

Q7. An exporter will receive EUR 5 million in three months. Spot EUR/CNY = 7.85, three-month forward = 7.92. Using the forward contract, the exporter can lock in CNY proceeds closest to:
A. CNY 39,250,000
B. CNY 39,600,000
C. CNY 39,000,000
D. Cannot be calculated

Q8. Under direct quotation (domestic per foreign), a forward discount on the quoted rate is most likely when:
A. Domestic interest rate > foreign interest rate
B. Domestic interest rate < foreign interest rate
C. Interest rates are equal
D. Inflation differentials dominate

Answers

Takeaways

• Under direct quotation, a rising exchange rate means foreign-currency appreciation and domestic-currency depreciation.
• Forward rates are determined by covered interest rate parity; the interest-rate differential is the primary driver.
• The exact IRP formula $F = S \times \frac{1+i_f(t/360)}{1+i_d(t/360)}$ is the most important relationship.
• Lower interest rate currency trades at a forward premium.
• Always use the 360-day convention and respect bid-ask spreads in calculations.
• Any deviation between quoted and theoretical forward rates creates a covered interest arbitrage opportunity that forces prices back to equilibrium.