Swaps: swap pricing with single curve pricing/Euribor discounting

Question 1

What is an interest rate swap?

Answer 1

A swap is an agreement between two counterparties to exchange cash flows in the future.
The agreement defines the dates when the cash flows are to be paid by each counterpart and the way in which the cashflows must be calculated.

Question 2

What are the most common swap types?

Answer 2

1. Interest rate swaps
2. Currency swaps
3. Equity swaps

Question 3

What are the interest rates involved in an interest rate swap?

Answer 3

1. A fixed interest rate paid by one fixed rate payer (floating rate receiver)
2. A floating rate paid by one floating rate payer (fixed rate receiver)

Question 4

What are the components defining the cashflows in an interest rate swap?

Answer 4

1. Two different interest rates
2. Notional capital: the amount on which the formula to calculate the cash-flow is applied (= size of the contract)
3. The frequency (e.g. semiannual, annual, quarterly) with which the payments between the two parties occur
4. The tenor/maturity of the swap

Question 5

What is a “plain vanilla” swap?

Answer 5

A “plain vanilla” swap is a fixed-to-floating interest rate swap where the two counterparties periodically exchange the difference between a floating and a fixed rate on the notional capital.

The cashflow is the difference between the payments that the two parties shuold make. For each period the floating rate is observed at the beginning of the period while the actual payment occurs at the end of the period.

Question 6

What is the formula for cacluating the cash flow of an interest rate swap?

Answer 6

For fixed rate payer:
(floating rate (annualized) - fixed rate (annualized)) * notional * frequency (0.5 for semiannual etc) = cash flow
For floating rate payer:
(fixed rate (annualized) - floating rate (annualized)) * notional * frequency (0.5 for semiannual etc) = cash flow

Question 7

What is the long leg of a swap?

Answer 7

Long leg is the stream of payments to be received by the party in a swap.

Question 8

What is the short leg of a swap?

Answer 8

Short leg is the stream of payments to be made by the party in a swap.

Question 9

What are the two types of swaps by frequency of payments?

Answer 9

1. Generic swaps (both legs have the same frequency of payments)
2. Non-generic swap (different frequency)

Question 10

What is the formula underlying the price calculation of interest rate swaps?

Answer 10

The price is the rate which makes PV(F) = PV(V), where F is the fixed leg and V is the variable (floating) leg.

Question 11

What equation does the fair swap rate f satisfy?

Answer 11

Σⁿ_t=1 (f * N)/(1+i_t)^t = Σⁿ_t=1 (E(v_t) * N)/(1+i_t)^t
Where:
f = fixed rate
E(v) = expected floating rates
i_0,t = zero coupon rate
t = time
N = notional

Question 12

What is the formula for estimating forward rates to be used in interest rate swap pricing?

Answer 12

(1 + i₂)² = (1 + i₁)¹ * (1 + h(1,2))¹

Where i₂ = 2-year spot zero coupon rate
i₁ = 1-year spot zero coupon rate
h(1,2) = forward rate between year 1 and 2
Same can be done for semester rates

Question 13

What is the formula for f (fixed rate of an interest rate swap used as the price)?

Answer 13

f = Σⁿ_t=1 (h(t-1, t)/(1+i_t)^t) / Σⁿ_t=1 1/(1+i_t)^t

Using discount factor (S(t) = 1/(1+i_t)^t:
f = (1 - S(n))/Σⁿ_t=1S(t)

Alternatively:
f = (Σⁿ_t=1h(t-1,t) * S(t))/Σⁿ_t=1S(t)

Single curve in this case implies using LIBOR/EURIBOR for both the numerator and denomaninator

Question 14

Why was a second curve added for pricing interest rate swaps?

Answer 14

LIBOR/EURIBOR after the 2008 financial crisi was no longer perceived as safe, as such the risk-free rate now is the overnight rate.

Question 15

What is the formula for calculating the forward rate from the discount factors?

Answer 15

(S(t-1)/S(t))-1