derviatives 7: Pricing and Valuation of Interest Rates and Other Swaps

What is a swap contract?

agreement between two parties to exchange a series of future cash flows.

What is a forward contract?

an agreement to exchange a single cash flow or value at a future date.

Key difference between a swap and a forward?

• swap involves multiple cash flow exchanges over time

• forward involves only one exchange at maturity.

What is a Forward Rate Agreement (FRA)?

An FRA is a forward contract on an interest rate with a single settlement at the start of an interest period.

FRA vs swap settlement timing difference?

• FRA settles once at the beginning of the period

• swaps settle periodically at the end of each period.

What is implied forward rate (IFR)?

future interest rate implied by current spot (zero) rates for a specific future period.

• Each implied forward rate equals the FRA fixed rate for that period.

Swap vs series of FRAs?

A swap can be viewed as a series of FRAs with different forward rates for each period.

What is the par swap rate?

The fixed rate that makes the present value of fixed swap payments equal to floating payments

• equates the PV of all future expected floating cash flows to the PV of fixed cash flows.

• equivalent to an internal rate of return on forward rates.

Par swap rate interpretation?

breakeven fixed rate that makes an investor indifferent between paying fixed or receiving fixed.

Why Swaps Are Preferred Over FRAs

• One contract covers multiple periods.

• Lower administrative burden.

• Better cash flow matching.

• Higher liquidity in markets.

• FRAs are mainly used by intermediaries for short-term hedging.

1. find discount rates $z_i$

• bootstrapping the bond prices

• solve for first year zero rate $z_1$ : $PV_{}=\frac{FV}{1_{}+z_1}$

• use $z_1$ to find $z_2$ and so on: $PV=\frac{PMT}{1+z_1}+\frac{PMT+FV}{\left(1+z_2\right)^2}$

2. solve for IFR for each period

• (1 + z_A)^A × (1 + IFR_A,B–A)^B–A = (1 + z_B)^B.

3. Compare fixed and floating-

• $$\sum_{}^{}\text{PV(floating)}=\sum_{}^{}\text{PV(fixed)}$$

• $$\sum_{}^{}\frac{IFR}{\left(1+z_{i}\right)^{i}}=\sum_{}^{}\frac{s_{i}}{\left(1+z_{i}\right)^{i}}$$

how to go from floating rate to fixed rate?

fixed rate = Swap Fixed Rate+Loan Spread

how to go from fixed rate to floating rate?

Floating Cost = Floating Benchmark + Spread Adjustments

what happens when rates increase/decrease for

1. long fixed

2. short fixed

3. long floating

4. short floating

5. receive fixed swap

6. pay fixed swap

Position type	Rates ↑	Rates ↓
Long fixed	Lose	Gain
Short fixed	Gain	Lose
Long floating	Gain	Lose
Short floating	Lose	Gain
Receive fixed swap	Lose	Gain
Pay fixed swap	Gain	Lose

What is the difference between swap price and swap value?

• Swap price = fixed swap rate agreed at inception.

• Swap value = current MTM (mark-to-market) value of the swap after inception.

why does swap value change and what does it reflect?

• Passage of time

• Interest rate changes

Swap value reflects:

Current settlement value + PV of remaining future settlements

Formula for periodic settlement value for fixed-rate payer?

$$\text{Periodic settlement value}=(MRR-s_{N})\cdot\text{notional amount}\cdot\text{period}$$

MRR = “spot” price

s_N = forward price

Fixed-rate payer benefits/loses when what happens?

If MRR > fixed rate: Fixed payer gains.

If MRR < fixed rate: Fixed payer loses.

MTM gain/loss condition for fixed-rate payer?

gain: PV(Floating received)>PV(Fixed paid)

loss: PV(Floating received)<PV(Fixed paid)

Bond Interpretation of Swaps

1. pay-fixed swap

2. receive-fixed swap

pay-fixed swap = Short fixed-rate bond + long floating-rate note (FRN).

receive fixed swap = long fixed-rate bond + short FRN.