• Discuss the mechanics of swaps and the role of financial intermediaries in swap markets:
  • Interest rate swaps
  • Currency swaps
• Explain how interest rate and currency swaps transform assets and liabilities
• Discuss the comparative advantage argument as rationale for swap transactions
• Design interest rate swaps and currency swaps
• Calculate the value of interest rate and currency swaps in terms of bond prices
• Define credit (default) risk and explain how it relates to swap transactions

25556 Financial Securities (FS) – Lecture 11: Interest Rate Swaps

• Features of a plain vanilla swap
• Default risk and credit exposure
• Using swaps as a hedging instrument
• The comparative advantage argument
• The role of swap dealers and market makers
• Deriving the swap rate
• Demonstration of the use of a swap to alter cash flow profiles

• Key structural features of interest rate swaps:
  • No initial exchange of principal: Notional principal serves only as the scale for calculating interest payments.
  • Netting of interest payments: At each settlement date, only the net interest difference changes hands between counterparties.
  • Reset in advance, paid in arrears: Floating interest rates are fixed at the beginning of each period, with the resulting cash flow paid at the end of the period.
• Core economic functions:
  • Convert a liability from fixed rate to floating rate, or from floating rate to fixed rate.
  • Convert an investment (asset) from fixed rate to floating rate, or vice versa.
• Historical note: The first formal swap was executed in 1981 between IBM and the World Bank.

Microsoft agrees to receive 6-month LIBOR and pay a fixed rate of 5% p.a. semi-annually for 3 years on a notional principal of $100 million:

• Fixed cash flow paid = $\mathrm{\$}100\text{M}\times 0.05\times 1/2=\mathrm{\$}2.50\text{M}$
• Current 6-month LIBOR = 4.2%: floating cash flow received = $\mathrm{\$}100\text{M}\times 0.042\times 1/2=\mathrm{\$}2.10\text{M}$
• In 6 months, 6-month LIBOR = 4.8%: floating cash flow received = $\mathrm{\$}100\text{M}\times 0.048\times 1/2=\mathrm{\$}2.40\text{M}$

Figure: see Table 7.1, page 176 (Hull). Each floating flow is set by the LIBOR of the previous row.

• FIs act as Market Makers for swaps:
  • It is very unlikely that two counterparties with exact offsetting needs will approach the bank simultaneously.
  • Most swaps are entered into by the FI with one counterparty without an immediately offsetting position – the FI “warehouses” the swap.
  • The FI charges a bid-ask spread or commission (typically 3–4 basis points).
• The FI must actively hedge its open risk:
  • Market Risk (Interest Rate Risk): Hedged dynamically using interest rate futures (e.g. Eurodollar or bank bill futures), government bonds, and offsetting swap transactions.
  • Credit (Default) Risk: Managed by demanding collateral (credit support annex – CSA), netting agreements, and hedging in credit default swap (CDS) markets.

Suppose the FI requires a 3 basis points (0.03%) annual commission, split equally as 0.015% to each party:

• Microsoft pays the FI 5.015% (instead of 5.000%).
  Net MS borrowing cost: $(\text{LIBOR}+0.1\mathrm{\%})-\text{LIBOR}+5.015\mathrm{\%}=5.115\mathrm{\%}$ (instead of 5.10% without the FI: 0.015% worse off).
• Intel receives 4.985% from the FI (instead of 5.000%).
  Net Intel borrowing cost: $5.20\mathrm{\%}-4.985\mathrm{\%}+\text{LIBOR}=\text{LIBOR}+0.215\mathrm{\%}$ (instead of LIBOR + 0.20% without the FI: 0.015% worse off).
• FI Bank Profit: Receives 5.015% • Pays 4.985% • Net spread = $5.015\mathrm{\%}-4.985\mathrm{\%}=0.030\mathrm{\%}$ (3 bps).

The following borrowing rates are offered to two corporations:

• Notice the spread differential:
  • AAACorp pays 1.20% less in the fixed market, but only 0.70% less in the floating market.
  • AAACorp has a comparative advantage in the FIXED market.
  • BBBCorp has a comparative advantage in the FLOATING market.
• AAACorp has a fixed rate loan but wants to borrow floating.
• BBBCorp has a floating rate loan but wants to borrow fixed.
• A swap allows each party to borrow where it has a comparative advantage and transform the nature of its liability:
  • AAACorp accesses a comparatively better floating rate;
  • BBBCorp gains access to a comparatively better fixed rate.

• Method 1 – Total Borrowing Cost Comparison:
  • Without swap: AAA borrows floating (LIBOR − 0.10%) + BBB borrows fixed (5.20%)
    Total cost = $\text{LIBOR}+5.10\mathrm{\%}$.
  • With swap: AAA borrows fixed (4.00%) + BBB borrows floating (LIBOR + 0.60%)
    Total cost = $\text{LIBOR}+4.60\mathrm{\%}$.
  • Total Gain: $(\text{LIBOR}+5.10\mathrm{\%})-(\text{LIBOR}+4.60\mathrm{\%})=0.50\mathrm{\%}$ p.a.
• Method 2 – Difference of Spreads:
  • Fixed spread: $a=5.20\mathrm{\%}-4.00\mathrm{\%}=1.20\mathrm{\%}$.
  • Floating spread: $b=(\text{LIBOR}+0.60\mathrm{\%})-(\text{LIBOR}-0.10\mathrm{\%})=0.70\mathrm{\%}$.
  • Total Gain: $a-b=1.20\mathrm{\%}-0.70\mathrm{\%}=0.50\mathrm{\%}$.
• Sharing the Gain:
  • If FI intermediary requires 0.04% commission:
  • Remaining benefit = $0.50\mathrm{\%}-0.04\mathrm{\%}=0.46\mathrm{\%}$. Split equally: $$\frac{0.46\mathrm{\%}}{2}=0.23\mathrm{\%}\text{ saving for each company}$$

Both companies achieve a 0.23% saving relative to their direct borrowing markets. This is the comparative advantage of the swap (in the diagram, $L$ means LIBOR, e.g. $L-0.33\mathrm{\%}$ is LIBOR − 0.33%):

• Step 1: AAA borrows at 4.00% fixed; BBB borrows at LIBOR + 0.60% from market.
• Step 2: FI passes 4.00% to AAA (reimbursing fixed) and (LIBOR + 0.60%) to BBB.
• Step 3: AAA pays FI $(\text{LIBOR}-0.10\mathrm{\%})-0.23\mathrm{\%}=\text{LIBOR}-0.33\mathrm{\%}$.
  BBB pays FI $5.20\mathrm{\%}-0.23\mathrm{\%}=4.97\mathrm{\%}$.
• FI Profit: $(4.97\mathrm{\%}-4.00\mathrm{\%})+(\text{LIBOR}-0.33\mathrm{\%}-\text{LIBOR}-0.60\mathrm{\%})$
  $=0.97\mathrm{\%}-0.93\mathrm{\%}=0.04\mathrm{\%}$ (4 bps).

• Alternative Swap Designs:
  • The parties exchange plain LIBOR and adjust the fixed legs; the FI earns 4.37% − 4.33% = 0.04% (in the diagram, $L$ means LIBOR).
• Criticisms of the Comparative Advantage Argument:
  • Maturity Mismatch: The 4.0% and 5.2% fixed rates available to AAA and BBB are multi-year (e.g. 5-year) fixed commitments.
  • Rollover Risk: The floating rates (LIBOR − 0.10% and LIBOR + 0.60%) are 6-month borrowing facilities. The credit spread is reviewed every 6 months!
  • If BBB's credit rating deteriorates, its bank lender will increase the spread above LIBOR, so BBB does not truly lock in a fixed rate for 5 years.
  • The apparent comparative advantage reflects the lender's option to increase credit spreads or cancel floating credit lines if credit quality drops.

An interest rate swap can be valued as the difference between two bonds:

• Fixed-rate bond ($B_{\text{fix}}$): Valued in the usual way by discounting all scheduled fixed coupon and principal cash flows using the zero curve.
• Floating-rate bond ($B_{\text{fl}}$): Valued by using the fundamental property that a floating-rate bond is worth exactly par immediately after every payment date: $B_{\text{fl}}=(L+k^{\ast })e^{-r{t}_1}$, where $L$ is notional principal (here $L$ no longer means LIBOR), $k^{\ast }$ is the known upcoming floating payment set at the last reset date, and $t_1$ is time until the next payment.
• Net Swap Value:
  $$V_{\text{swap}}=B_{\text{fl}}-B_{\text{fix}}\text{(for party receiving floating, paying fixed)}$$ $$V_{\text{swap}}=B_{\text{fix}}-B_{\text{fl}}\text{(for party receiving fixed, paying floating)}$$
  Example: receiving floating and paying fixed = long a floating-rate bond, short a fixed-rate bond.
• Alternative: a portfolio of Forward Rate Agreements (FRAs) – own study (Hull, p. 187).

Quiz Question 7.2 (page 207): Receive 6-month LIBOR, pay 7% semi-annually on $100M notional. 10 months remaining. Discount rate $r=5\mathrm{\%}$ p.a. (continuous) for all maturities. Latest LIBOR was 4.6%.

• Payment Dates: Next payment in 4 months ($4/12$); final payment in 10 months ($10/12$).
• Step 1: Value Fixed-Rate Bond ($B_{\text{fix}}$):
  • Semi-annual fixed coupon: $\mathrm{\$}100\text{M}\times 7\mathrm{\%}\times 6/12=\mathrm{\$}3.5\text{M}$.
  • $B_{\text{fix}}=3.5e^{-0.05\times 4/12}+(100+3.5)e^{-0.05\times 10/12}=3.442+99.276=\mathrm{\$}102.718\text{M}$.
• Step 2: Value Floating-Rate Bond ($B_{\text{fl}}$):
  • Floating payment at month 4 was set at 4.6%: $\mathrm{\$}100\text{M}\times 4.6\mathrm{\%}\times 6/12=\mathrm{\$}2.3\text{M}$.
  • At month 4 (after payment), the bond resets to par ($100M).
  • $B_{\text{fl}}=(100+2.3)e^{-0.05\times 4/12}=102.3\times 0.98347=\mathrm{\$}100.609\text{M}$.
• Step 3: Value of the Swap (Receiving Floating, Paying Fixed): $$V_{\text{swap}}=B_{\text{fl}}-B_{\text{fix}}=\mathrm{\$}100.609\text{M}-\mathrm{\$}102.718\text{M}=-\mathrm{\$}2.109\text{M}$$
• Note that: the swap rate is 7% – a $100 million bond with a semi-annual coupon of 7% p.a. sells for par, so the fixed leg is worth par at the swap rate.

Core Principle: You never need to forecast future floating cash flows! Because floating coupon resets to prevailing market rates, the entire floating leg from month 1 to month 13 collapses into par ($10M) at the first reset date.

• Key features of currency swaps:
  • Principal is exchanged at the beginning (at spot exchange rate) and re-exchanged at maturity at the same initial rate.
  • All interest payments are exchanged in full: Because payments are in different currencies, they cannot be netted.
  • Recall that in an interest rate swap, principal is not exchanged and interest payments are netted.
• It is used to:
  • Convert a liability (investment) from one currency to another.
  • Example: IBM has agreed to pay 11% on a sterling principal of £10 million and receive 8% on a USD principal of $15 million every year for 5 years:

The following borrowing rates are offered to General Motors (US) and Qantas (Australia):

• Notice the spread differential:
  • GM has an absolute advantage in both markets, but its advantage is much larger in USD (2.00% vs 0.40%).
  • GM has a comparative advantage in USD.
  • Qantas has a comparative advantage in AUD (its borrowing disadvantage is only 0.40% in AUD vs 2.00% in USD).
• GM wants AUD borrowing, while Qantas wants USD borrowing. A currency swap achieves mutual interest savings!

• Total Gain Calculation:
  • Without swap: GM borrows AUD directly at 7.60% + Qantas borrows USD directly at 7.00%
    Total borrowing cost = $7.60\mathrm{\%}+7.00\mathrm{\%}=14.60\mathrm{\%}$.
  • With swap: GM borrows USD at 5.00% + Qantas borrows AUD at 8.00%
    Total borrowing cost = $5.00\mathrm{\%}+8.00\mathrm{\%}=13.00\mathrm{\%}$.
  • Total Gain: $14.60\mathrm{\%}-13.00\mathrm{\%}=1.60\mathrm{\%}$.
  • Alternative way:
    • Benefit to GM of accessing Qantas's AUD rate: $b=8\mathrm{\%}-7.60\mathrm{\%}=0.40\mathrm{\%}$.
    • Benefit to Qantas of accessing GM's USD rate: $a=7\mathrm{\%}-5\mathrm{\%}=2\mathrm{\%}$.
    • Total gain from the swap: $a-b=2\mathrm{\%}-0.40\mathrm{\%}=1.60\mathrm{\%}$.
• Intermediary Commission & Net Benefit:
  • Suppose the FI charges 0.20% commission:
  • Net gain to be shared = $1.60\mathrm{\%}-0.20\mathrm{\%}=1.40\mathrm{\%}$.
  • Divided equally: $$\frac{1.40\mathrm{\%}}{2}=0.70\mathrm{\%}\text{ saving for each company}$$
  • GM target borrowing cost: $7.60\mathrm{\%}-0.70\mathrm{\%}=6.90\mathrm{\%}$ in AUD.
  • Qantas target borrowing cost: $7.00\mathrm{\%}-0.70\mathrm{\%}=6.30\mathrm{\%}$ in USD.

Structure of the swap with a Financial Intermediary taking 0.20% spread:

• Step 1: GM borrows USD at 5.00%; Qantas borrows AUD at 8.00% from domestic lenders.
• Step 2: GM receives USD 5.00% from FI (offsetting its USD loan); Qantas receives AUD 8.00% from FI (offsetting its AUD loan).
• Step 3: GM pays FI AUD $7.60\mathrm{\%}-0.70\mathrm{\%}=6.90\mathrm{\%}$; Qantas pays FI USD $7.00\mathrm{\%}-0.70\mathrm{\%}=6.30\mathrm{\%}$.
• FI Profit: Receives USD 6.3% • Pays USD 5.0% (+1.3% USD) • Receives AUD 6.9% • Pays AUD 8.0% (−1.1% AUD). Net gain = $1.30\mathrm{\%}-1.10\mathrm{\%}=0.20\mathrm{\%}$.

Both parties have improved their borrowing rate by 0.70%. This is the comparative advantage of the currency swap.

One of the companies wishes to bear foreign exchange risk:
Suppose Qantas agrees to pay 1% p.a. in AUD and 5.30% p.a. in USD (splitting its payments across both currencies):

• By restructuring the currency obligations, Qantas absorbs partial FX risk while obtaining customized financing.
• Qantas pays 1% p.a. in AUD and 5.30% p.a. in USD. The FI nets +USD 0.30% (5.30% − 5.00%) and −AUD 0.10% (6.90% − 7.00%): a small residual currency mismatch, with a total margin of 0.20%.

Core Principle: In currency swaps, principal MUST be exchanged at both initiation and maturity at the original spot rate. Credit risk in currency swaps is substantially higher than in interest rate swaps because of full principal exchange.

Like interest rate swaps, currency swaps can be valued as:

• The difference between the values of two bonds.
• A portfolio of forward rate agreements (FRAs), i.e. forward foreign exchange contracts.

Valuation in Terms of Bond Prices:

• The current spot exchange rate $S_0$ is required (defined as domestic currency per unit of foreign currency).
• Let $B_D$ be the value of the domestic currency bond (in domestic currency).
• Let $B_F$ be the value of the foreign currency bond (in foreign currency).
• Paying foreign currency $F$ and receiving domestic currency $D$ = short a foreign currency bond, long a domestic currency bond:
  $$V_{\text{swap}}=B_D-S_0{B}_F$$
• For a financial institution receiving foreign currency $F$ and paying domestic currency $D$:
  $$V_{\text{swap}}=S_0{B}_F-B_D$$

Quiz Question 7.5 (page 207): Swap pays 10% on £20M, receives 6% on $30M annually. 15 months remaining. Discount rates: 4% in $, 7% in £ (annual compounding, all maturities). Exchange rate: 1 £ = $1.85.

• Payment Dates: Next payment in 3 months ($3/12=0.25$ yr); final payment in 15 months ($15/12=1.25$ yr).
• Step 1: Value Dollar Bond ($B_D$):
  • Annual coupon = $\mathrm{\$}30\text{M}\times 6\mathrm{\%}=\mathrm{\$}1.8\text{M}$. Final flow = $\mathrm{\$}31.8\text{M}$.
  • $B_D=\frac{1.8}{{1.04}^{0.25}}+\frac{31.8}{{1.04}^{1.25}}=1.782+30.279=\mathrm{\$}32.061\text{M}$.
• Step 2: Value Sterling Bond ($B_F$):
  • Annual coupon = $\pounds 20\text{M}\times 10\mathrm{\%}=\pounds 2.0\text{M}$. Final flow = $\pounds 22.0\text{M}$.
  • $B_F=\frac{2.0}{{1.07}^{0.25}}+\frac{22.0}{{1.07}^{1.25}}=1.966+20.216=\pounds 22.182\text{M}$.
• Step 3: Value of the Swap:
  • To party paying £ and receiving $: $$V_{\text{swap}}=B_D-S_0{B}_F=\mathrm{\$}32.061\text{M}-(1.85\times \pounds 22.182\text{M})$$ $$=\mathrm{\$}32.061\text{M}-\mathrm{\$}41.037\text{M}=-\mathrm{\$}8.976\text{M}$$
  • To party paying $ and receiving £: $$V_{\text{swap}}=+\mathrm{\$}8.976\text{M}$$

Core Principle: Value each bond in its own currency first using its domestic risk-free rate, then translate into the common valuation currency using today's spot rate $S_0$. Negative value indicates the FI's obligations in high-yielding AUD exceed its USD receipts.

• Credit risk arises from the possibility of default by a counterparty:
  • At initiation, a swap is worth zero ($V_0=0$), similar to a forward contract.
  • At a future time its value may be either positive or negative: interest rates and currency rates fluctuate, so the swap value becomes positive to one party and negative to the other.
• Asymmetric Credit Exposure for Financial Institutions:
  • An FI is exposed to credit risk only when the swap value is positive to the FI.
  • If the swap value to the FI is negative, the counterparty will not default because the contract is an asset to them: it will sell the contract to profit from its positive value (or bankruptcy administrators will preserve/sell it).
• Default Losses on Swaps vs Loans:
  • Potential losses from defaults on swaps are much smaller than losses on loans of the same notional principal, because principals are not lent out (in IRS).
  • At default, currency swaps generate significantly greater losses than interest rate swaps because principal amounts are exchanged and exchange rates can experience massive cumulative shifts.