Swaps: Interest Rate and Currency Swaps for Actuaries

Interest Rate Swaps: Structure and Mechanics

In a plain vanilla interest rate swap, two parties exchange interest payments on a notional principal: one pays a fixed rate and receives a floating rate, and the other does the opposite. No principal is exchanged. The notional principal is used only to calculate the interest payments. The fixed rate (swap rate) is set so the swap has zero value at inception.

On each settlement date, the two parties calculate their respective interest payments. Rather than exchanging the full amounts, they typically net the payments: only the difference changes hands. If the fixed rate payment exceeds the floating rate payment, the fixed-rate payer sends the net amount to the floating-rate payer, and vice versa.

Interest rate swaps are used for hedging and speculation. A company with floating-rate debt that wants fixed-rate exposure can enter a swap to pay fixed and receive floating. The floating leg offsets the floating-rate debt payments, effectively converting the debt to fixed rate.

Computing the Swap Rate

The swap rate R is the fixed rate that makes the swap have zero value at inception. It satisfies: R * sum(v^t for t = 1 to n) = sum(f_t * v^t for t = 1 to n), where f_t are the forward rates and v^t = 1/(1+s_t)^t are discount factors based on spot rates. Solving for R:

R = sum(f_t * v^t) / sum(v^t) = (1 - v^n) / sum(v^t for t = 1 to n).

The second formula follows because the floating leg of a swap has the same value as a floating-rate bond, which is worth par at inception. Therefore, the present value of floating payments equals 1 - v^n (in units of the notional).

Example: Given spot rates s_1 = 3%, s_2 = 4%, s_3 = 5%, compute the 3-year swap rate. The discount factors are v^1 = 1/1.03 = 0.9709, v^2 = 1/(1.04)^2 = 0.9246, v^3 = 1/(1.05)^3 = 0.8638. The swap rate is R = (1 - 0.8638) / (0.9709 + 0.9246 + 0.8638) = 0.1362 / 2.7593 = 0.04937 = 4.937%.

Valuing an Existing Swap

After inception, as market rates change, the swap acquires positive value for one party and negative value for the other. The value of the fixed-rate payer's position is the present value of remaining floating payments minus the present value of remaining fixed payments, all calculated at the current term structure.

An interest rate swap can be decomposed in two equivalent ways. First, as a portfolio of forward rate agreements (FRAs): each settlement period is a separate FRA comparing the fixed rate to the forward rate for that period. Second, as the difference between a floating-rate bond (received) and a fixed-rate bond (paid). Since a floating-rate bond is worth par just after each reset, its value is straightforward to compute.

Example: Suppose the 3-year swap above was initiated with R = 4.937%, and after 1 year the new spot rates are s_1 = 3.5%, s_2 = 4.5%. The remaining fixed payments are 4.937% for years 2 and 3. The value to the fixed payer is: (new floating bond value) - (PV of remaining fixed payments) - (PV of remaining notional). Compute using the new discount factors.

Currency Swaps

A currency swap exchanges principal and interest payments in two different currencies. Unlike interest rate swaps, the principal amounts are exchanged at inception and at maturity. The exchange rate at inception is the current spot rate. This principal exchange is necessary because the two parties have obligations in different currencies.

For example, a U.S. company that needs euros and a European company that needs dollars might enter a currency swap. At inception, they exchange principals at the spot rate. During the swap, they exchange interest payments in their respective currencies. At maturity, they re-exchange the original principals at the same exchange rate as inception (not the prevailing rate).

On Exam FM, currency swaps combine interest rate calculations with exchange rate conversions. The key principle is the same as for interest rate swaps: the swap is structured so its initial value is zero, which requires computing present values in each currency using the respective interest rates.

Swap Applications in Risk Management

Swaps are versatile tools for managing financial risk. Common applications include converting floating-rate debt to fixed-rate (or vice versa), managing currency exposure, and exploiting comparative advantages in borrowing. Two companies may benefit from a swap if one has a relative advantage in the fixed-rate market and the other in the floating-rate market.

On Exam FM, comparative advantage problems are structured as follows: Company A can borrow fixed at 6% or floating at LIBOR + 1%, while Company B can borrow fixed at 8% or floating at LIBOR + 2%. The fixed-rate differential (2%) exceeds the floating-rate differential (1%), so a swap can benefit both parties. The total savings is the difference of the differentials: 2% - 1% = 1%, which is split between the two parties.

Study Tips and Common Pitfalls

The most common mistake in swap rate calculations is using the wrong discount factors. Each discount factor must use the spot rate for that specific maturity, not a single flat rate. Another frequent error is forgetting that the swap rate formula R = (1 - v^n) / sum(v^t) assumes payments at times 1, 2, ..., n. If the swap settles more frequently (for example, quarterly), adjust the formula accordingly.

For currency swaps, remember that principals are exchanged (unlike interest rate swaps). Also note that the re-exchange at maturity uses the original spot rate, not the rate at maturity. Practice at least 3 swap rate calculations from spot rates, 2 swap valuation problems, and 2 comparative advantage problems to cover the main Exam FM question types.