Interest RatesInterest RatesFinance (Derivative Securities) 312

Tuesday, 8 August 2006

Readings: Chapter 4

Types of RatesTypes of Rates

Treasury Rates• Short-term government securities

LIBOR• London Interbank Offer Rate• Rate applicable to wholesale deposits

between banksRepo Rates

• Repurchase agreements

Measuring RatesMeasuring Rates

Compounding frequency is unit of measurement

Increased frequency leads to continuous compounding• $100 grows to $100eRT when invested at a

continuously compounded rate R for time T• $100 received at time T discounts to $100e–RT

at time zero when continuously compounded discount rate is R

Conversion FormulaConversion Formula

Rc : continuously compounded rate

Rm: same rate with compounding m times per year

R m

R

m

R m e

cm

mR mc

ln

/

1

1

Bond PricingBond Pricing

Relies on interest rates on zero-coupon bonds (zero rates)• Interest is realised

only at maturity date

Maturity(years)

Zero Rate(% cont comp)

0.5 5.0

1.0 5.8

1.5 6.4

2.0 6.8

3 3 3

103 98 39

0 05 0 5 0 058 1 0 0 064 1 5

0 068 2 0

e e e

e

. . . . . .

. . .

To calculate the price of a two year coupon bond paying 6% semi-annually:

Bond YieldBond Yield

Single interest rate that discounts remaining CFs to equal the price

Using the previous example, solve the following equation for y:

y = 0.0676 or 6.76%

3 3 3 103 98 390 5 1 0 1 5 2 0e e e ey y y y . . . . .

Par YieldPar Yield

Coupon rate that equates a bond’s price to its face value

Using previous example:

g)compoundin s.a. (with get to 876

1002

100

222

0.2068.0

5.1064.00.1058.05.005.0

.c=

ec

ec

ec

ec

Par YieldPar Yield

If: m = no. of coupon payments per year

P = present value of $1 received at maturity

A = present value of an annuity of $1 on each coupon date

then:

cP m

A

( )100 100

Calculating Zero RatesCalculating Zero Rates

Bond Time to Annual Bond

Principal Maturity Coupon Price

(dollars) (years) (dollars) (dollars)

100 0.25 0 97.5

100 0.50 0 94.9

100 1.00 0 90.0

100 1.50 8 96.0

100 2.00 12 101.6

Bootstrap MethodBootstrap Method

2.5 can be earned on 97.5 after three months

3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding, and 10.127% with continuous compounding

Similarly the 6-month and 1-year rates are 10.469% and 10.536% with continuous compounding

Bootstrap MethodBootstrap Method

To calculate 1.5-year rate, solve:

to get R = 0.10681 or 10.681%

Similarly the two-year rate is 10.808%

9610444 5.10.110536.05.010469.0 Reee

Zero CurveZero Curve

Zero Rate (%)

Maturity (yrs)

10.127

10.469 10.536

10.681

10.808

9

10

11

12

0 0.5 1 1.5 2 2.5

Forward RatesForward Rates

Future zero rates implied by the current term structure

Zero Rate for Forward Rate

an n -year Investment for n th Year

Year (n ) (% per annum) (% per annum)

1 10.0

2 10.5 11.0

3 10.8 11.4

4 11.0 11.6

5 11.1 11.5

Calculating Forward Calculating Forward RatesRates

Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded

The forward rate for the period between times T1 and T2 is:

R T R T

T T2 2 1 1

2 1

Slope of Yield CurveSlope of Yield Curve

For an upward sloping yield curve:

Fwd Rate > Zero Rate > Par Yield

For a downward sloping yield curve:

Par Yield > Zero Rate > Fwd Rate

Forward Rate Forward Rate AgreementsAgreements

Agreement that a fixed rate will apply to a certain principal during a specified future time period

Equivalent to agreement where interest at a predetermined rate, RK , is exchanged for interest at the market rate

Can be valued by assuming that the forward interest rate will be realised

Forward Rate Forward Rate AgreementsAgreements

Let:• RK = interest rate agreed to in FRA

• RF = forward LIBOR rate for period T1 to T2 calculated today

• RM = actual LIBOR rate for period T1 to T2 observed at T1

• L = principal underlying the contract

Forward Rate Forward Rate AgreementsAgreements

If X lends to Y under the FRA, then:• Cashflow to X at T2 = L(RK – RM)(T2 – T1)• Cashflow to Y at T2 = L(RM – RK)(T2 – T1)

Since FRAs are settled at T1, payoffs must be discounted at [1 + RM (T2 – T1)]

Value of FRA is the payoff, based on forward rates, discounted at R2T2

• ValueX = L(RK – RF)(T2 – T1)e–R2T2

• ValueY = L(RF – RK)(T2 – T1)e–R2T2

Theories of Term Theories of Term StructureStructure

Expectations Theory: forward rates equal expected future zero rates

Market Segmentation: short, medium and long rates determined independently of each other

Liquidity Preference Theory: forward rates higher than expected future zero rates