An Economic Analysis of Interest Rate Swaps

Member:

R94723046 賴又慈

R94723043 廖品荃

R94723052 陳佩忻

Agenda

Introduction The basic interest rate swap Many uses of interest rate swap The valuation of swaps

I. Introduction

Introduction Based on the principle of comparative advan

tage A useful tool of active liability management a

nd for hedging against interest rate risk.

History background Of IRS

Since late 1970’s — Significant increase in both the levels and t

he volatilities of market interest rates. Influence: Business firms and financial institutions fac

e higher interest-rate risks.

History background Of IRS (cont’d)

Since the early 1980’s — One of the most popular vehicles utilized to

hedge against interest rate risk. Reason:

Simple technique and easy to execute

II. The basic irs

The basic interest rate swap

Definition:

An agreement between two parties to exchange a series of interest rate payment.

Calculation of CF:

The cash flow is calculated at a fixed or floating rate on the “notional principal.”

Ex. fixed-rate payer v.s. floating-rate payer Voluntary market transactions

Reasons for IRS

Both parties obtain economic benefits

National and international money

and capital market imperfections

comparative advantages among

different borrowers in these

markets

Reasons for IRS (cont’d)

Gaps in Balance Sheet

U.S. financial institutions Assets － fixed rate of interest ＆ long maturities (mortgage and consumer installment loans)

Liabilities-short maturities (money market deposit accounts and variable-rate certificate

s of deposit) v.s. European financial institutionsAssets with relatively short maturities and liabilities with relativ

ely long maturities

Gaps in Balance Sheet (cont’d)

differences in inter-firm asset/liability composition represent opposite kinds of gaps in balance sheet.

interest rate swaps provide an economic mechanism whereby both financial institutions can benefit from a reduction in their respective balance-sheep gaps and a decrease of exposure to interest-rate risk.

III. Uses of IRS

Uses of IRS

Gap management Lower fixed-rate cost Restructure debt mix Manage basis risk

What is gap management? Duration gap: If there exists a duration gap, it means the organization

will benefit or suffer if interest rate changes

In order to prevent from loss of net value DA× A ＝ DD× D ＋ DE× E （ dollar duration ）

In order to sustain the debt ratio

DA ＝ DD

A. Using IRS in gap management

A. Using IRS in gap management (cont’d)

Sallie Mae (student loan marketing association )

floating rate student loans medium-term fixed rate cost have to shorten the duration of its

liabilities

Sallie Mae Financial institutionFloating rate

Fixed rate

B. Using IRS to Lower Fixed-Rate Cost

A useful tool for lowering a company’s cost of long-term fixed interest rate borrowing, especially for a company with low credit rating.

The quality spread is narrower in the floating-rate market than in the fixed-rate market.

B. Using IRS to Lower Fixed-Rate Cost (cont’d)

Example Floating

RateFixed Rate

Aaa Corp. T-Bill+0.25% 11.5%

Baa Corp. T-Bill+0.5% 13%

Quality Speard

0.25% 1.5%

B. Using IRS to Lower Fixed-Rate Cost (cont’d)

B. Using IRS to Lower Fixed-Rate Cost (cont’d)

Net Cost Original Cost

Cost Saving

Aaa Corp. T-Bill- 0.5%

T-Bill + 0.25%

0.75%

Baa Corp.

12.5% 13% 0.5%

C. Using IRS to Restructure Debt Mix

A useful tool for active liability management.

For example, a firm with too many high-coupon fixed-rate liabilities may:

(1) Refinancing

- Drawbacks: high cost / restrictions

(2) IRS

C. Using IRS to Restructure Debt Mix (cont’d)

A company has a $50 million of noncallable debt with a fixed-rate of 14%

What will happen if the company enter into the following IRS agreement ?

- Pay floating rate: prime rate + 50bp

- Receive fixed rate: 13%

C. Using IRS to Restructure Debt Mix (cont’d)

December 1, 1985

Prime rate : 9.5% Floating rate: 10%

The result of IRS

Financing cost:

Fixed rate of 14% Net floating rate of 11%

- An economic gain of 3%

D. Using IRS to Manage Basis Risk

A useful tool for manage the basis risk in the B/S.

The B/S of a bank:

- Asset yield: LIBOR + 0.75%

- Liability cost: T-Bill – 0.25%

D. Using IRS to Manage Basis Risk (cont’d)

D. Using IRS to Manage Basis Risk (cont’d)

After into an IRS, the bank has:

- Asset yield: LIBOR + 0.75%

- Liability cost: LIBOR - 0.75%

(1) Transforming T-Bill-rate-based

liability into LIBOR-based one

(2) Locking in a positive spread of

150bp

IV. The Valuation Of IRS

B. Simple Comparison of Alternative Swap

The choice between two interest rate swaps with different floating-rate indices should be based on the same economic criterion.

The economic decision rules for choosing between alternative swaps are the minimization of the present value of interest costs, or equivalently, the one that results in the largest value.

B. Simple Comparison of Alternative Swap

Compare the partial ex post performance of two swaps completed in August, 1982.

－ 10-year Treasury Bond rate +0.75% against 6-month LIBOR rate － 10-year Treasury Bond rate -0.80% against 6-month T-Bill rate

The floating rate difference was 158 basis points and the fixed rate difference was 155 basis point. So they were comparable.

The comparison of LIBOR and T-Bill

The valuation Procedure

The valuation of IRS

In a fair transaction principle, the value of IRS is initially worth zero.

The valuation of CRS

The valuation of IRS IRS can be viewed as buying and selling the bond with

different interest rate at the same time.

Bfix ： Value of fixed-rate bond underlying the swap　 Bfl ： Value of floating-rate bond underlying the swap 　　 ti ： Time when ith papements are exchanged(1≦i n)≦　 L ： National principal in swap agreement　 ri ： LIBOR zero rate for a maturity ti

k： Fixed payment made on each payment date

k* ： Floating payment made on a next date

BBV fixflswap

nnii trn

i

tr

fixLekeB

1

11* tr

flekLB

The valuation of IRS (cont’d) Example:

1.Pay six-month LIBOR and receive 8% per annum (with semiannual compounding)

2. A national principal of $100 million.

3.The Swap has a remaining life of 1.25 years.

4.LIBOR rates with continuous compounding for 3-month, 9-month, and 15-month maturities are 10%, 10.5%,and 11%, respectively.

5.The 6-month LIBOR rate at the last payment date was 10.2%.

What is the value of IRS for FI?

The valuation of IRS (cont’d)

(unit: million) K = 4

　 K* = 5.1

　 Bfix = 4e-0.1×0.25 + 4e-0.105×0.75 + 104e-0.11×1.25

= 98.24

　 Bfl = (100 + 5.1)e-0.1×0.25 = 102.51

Vswap = 98.24 - 102.51 = -4.27

The valuation of IRS (cont’d) IRS also can be seen as a portfolio of forw

ard-rate agreements.V swap = sum of the value of all forward contracts

The procedure is as followed:1.Caculate forward rates for each of the LIBOR rates will eq

ual the forward rates.

2.Caculate swap cash flow on the assumption that the LIBOR rates will equal the forward rates.

3.Set the swap value equal to the present value of these cash flows.

The valuation of IRS (cont’d)First CF exchanging

Interest for six months at a rates of 8% per annum will be exchanged for interest at a rate of 10.2% per annum for six months. The value of the exchange to FI is:0.5 × 100 × (0.08－ 0.102)e-0.1×0.25 = -1.07

Second CF exchanging

Calculate the forward rate corresponding to the period between 3 and 9 months.(0.105 × 0.75 - 0.10 × 0.25)/0.5 = 0.107510.75%: per annum with continuous compounding;11.044%: per annum with semiannual compounding.

The value of the FRA corresponding to the exchange in 9 months is：0.5 × 100 × (0.08－ 0.11044)e-0.105×0.75 = -1.41

The valuation of IRS (cont’d)

Third CF exchanging

Calculate the forward rate corresponding to the period between 9 and 15 months.(0.11 × 1.25 - 0.105 × 0.75)/0.5 = 0.117511.75%: per annum with continuous compounding 12.102%: per annum with semiannual compounding

The value of the FRA corresponding to the exchange in 15 months is ：0.5 × 100 × (0.08－ 0.12102)e-0.11×1.25 = -1.79

The total value

of swap

The total value of swap-1.07 - 1.41 - 1.79 = -4.27

Supplement: Valuation of currency swaps

Decomposing a currency swap into:

(1) Two bonds

(2) A series of forward contracts

Decomposition into bonds

The value in U.S. dollars of a swap where dollars are received and a foreign currency is paid:

Vswap = BD - S0*BF

BD : the value of the U.S. dollar bond

BF : the value of the foreign bond

S0 : the current spot exchange rate ($/F)

Decomposition into bonds (cont’d)

Example

A company enters into a 3-yr currency swap where :

(1) Receive 5% in yen / ￥ 1,200 million

(2) Pay 8% in dollars / $10 million

(3) Japanese rate: 4% / U.S. rate: 9%

(4) Current exchange rate: ￥ 110 = $1

What’s the value of the swap for the company?

Decomposition into bonds (cont’d)

BD = 0.8e-0.09*1 + 0.8e-0.09*2 + 10.8e-0.09*3

= 9.644 million dollars BF = 60e-0.04*1 + 60e-0.04*2 + 1,260e-0.04*3

= 1,230.55 million yen

The value of the swap

= 1,230.55/110 – 9.644

= $ 1.543 million

Decomposition into forwards

Each of the exchanges can be view as a forward contract.

The value of a forward contract

= the forward price of the underlying

asset is realized.

Decomposition into forwards (cont’d)

Followed the above example:

S0 = 1/110 = $0.009091 / yen

F1 = 0.009091e(0.09-0.04)*1 = 0.009557

F2 = 0.009091e(0.09-0.04)*2 = 0.010047

F3 = 0.009091e(0.09-0.04)*3 = 0.010562

Decomposition into forwards (cont’d)

The values of the forward contracts in 1yr, 2yr and 3yr are:

(60*0.009557-0.8)e-0.09*1 = -0.2071

(60*0.010047-0.8)e-0.09*2 = -0.1647

(1260*0.010562-10.8)e-0.09*3 = 1.9147

Total value of the swap

= 1.9147-0.2071-0.1647 = $1.543 million

Conclusion

IRS transactions are based upon a simple economic principle of comparative advantage.

Using swaps, financial managers can readily transform the economic characteristics of their liabilities.

IRS are more effective than financial futures and options in hedging against the interest rate risk for horizons beyond two or three years.