Module 8 · Economics

Exchange Rate Calculations

EN: Cross rates, forward premiums/discounts, triangular arbitrage.
VN: Tỷ giá chéo, premium/discount kỳ hạn, arbitrage tam giác.

1. Cross-Rate Calculation Core

About: When two currencies aren't directly quoted, derive cross rate by chaining through a common currency (usually USD). Watch quote convention to avoid inverting.Tóm tắt: Tính tỷ giá chéo qua tiền chung (thường USD). Cẩn thận quy ước để không đảo ngược.

\[ \frac{A}{C} = \frac{A}{B} \times \frac{B}{C} \]

Practice problem

USD/EUR = 1.10, USD/GBP = 1.30. Compute EUR/GBP.

Show solution

EUR/GBP = (USD/GBP) / (USD/EUR) = 1.30/1.10

≈ 1.1818

2. Forward Premium / Discount Core

About: Forward FX rate derived from spot + interest-rate differential (covered IRP). Currency with higher rate trades at forward DISCOUNT. Set by no-arbitrage, not forecasts.Tóm tắt: Tỷ giá kỳ hạn = spot + chênh lệch lãi suất (IRP). Đồng có lãi suất cao → discount kỳ hạn.

\[ F = S \cdot \frac{1 + r_{P} \cdot (T/360)}{1 + r_{B} \cdot (T/360)} \]

Components

F, S Forward and spot rate (P/B convention).
r_P, r_B Price-currency and base-currency interest rates.
T Days to delivery (use 360 day-count for money-market quotes).

F > S: base currency at forward premium (low-rate currency). F < S: forward discount.

Practice problem

USD/EUR spot = 1.10. 90-day USD rate = 5%, EUR rate = 3%. 90-day forward USD/EUR?

Show solution

F = 1.10 × [1 + 0.05(90/360)] / [1 + 0.03(90/360)]

= 1.10 × 1.0125 / 1.0075

≈ 1.1055

3. Mark-to-Market a Forward Core

About: Value of an open forward at any point = PV of difference between current forward and original locked-in rate, scaled by notional. Critical for risk management and accounting.Tóm tắt: Giá trị forward đang mở = PV của chênh lệch giữa forward hiện tại và rate đã khóa, × notional.

\[ V_{f} = \frac{(F_{t} - F_{0}) \cdot \text{Notional}}{1 + r_{P} \cdot (T_{\text{remain}}/360)} \]

F_t = forward rate today on a contract with the same delivery date as the original; T_remain = days left to delivery.

Practice problem

Original 6-month USD/EUR forward locked at 1.10 on €1M notional. Today, 3 months remain; current 3-month forward = 1.12. USD 3-month rate = 4%. Compute MTM value to the long.

Show solution

Profit per EUR (long): 1.12 − 1.10 = 0.02 USD

Total: €1M × 0.02 = $20,000

Discount 3 months at 4% (annual): / [1 + 0.04(90/360)] = /1.01

MTM ≈ $19,802 (favorable to long)

4. Triangular Arbitrage Core

About: Profitable cycle through three currencies when implied cross-rate ≠ quoted cross-rate. Risk-free if executable instantly. Algorithmic trading closes these gaps in milliseconds.Tóm tắt: Vòng arbitrage qua 3 tiền tệ khi cross-rate ngầm ≠ cross-rate báo. Risk-free nếu thực hiện ngay.

Procedure

1. Compute the implied cross rate from quoted pair.
2. Compare to dealer's actual cross-rate quote.
3. If they differ, buy through cheap path, sell through expensive path → riskless profit.