4.1

Interest Rates

Chapter 4

4.2

Types of Rates

Treasury rates LIBOR rates: London Interbank

Offered Rate (Large banks willing to lend to other large banks)

Repo rates: Repurchase Agreement; sell with commitment to buy back (Selling company borrows, buying company lends)

4.3

Interest Rates: Compounding Period

Default assumption of textbook: interest rates quoted assuming continuous compounding

The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers, i.e. can convert

4.4

Continuous Compounding(Page 83)

In the limit as we compound more and more frequently we obtain continuously compounded interest rates

$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 the continuously compounded discount rate is R

4.5

Conversion Formulas

Define

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

4.6

Zero Rates

A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time TDefault assumption: interest rates quoted assuming continuous compounding. But FRA contracts are quoted assuming compounding period is length of FRA period.

4.7

Example: Use zero (or spot) rates to price a bond

Maturity (years)

Zero Rate (% cont. comp.)

0.5 5.0

1.0 5.8

1.5 6.4

2.0 6.8

4.8

Bond Pricing

To calculate the cash price of a bond discount each cash flow at the appropriate zero rate

The theoretical price of a two-year bond providing a 6% coupon semiannually (default assumption: semiannual payment of coupons) is

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

. . . . . .

. . .

4.9

Bond Yield The bond yield is the (single) discount rate

that makes the present value of the cash flows on the bond equal to the market price of the bond

Suppose that the market price of the bond in our example equals its theoretical price of 98.39

The bond yield is given by solving

to get 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 . . . . .

4.10

Par Yield The par yield for a certain maturity is the

coupon rate that causes the bond price to equal its face value.

In our example we solve

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

When is par yield used?

Investment bankers set the coupon rate on a new bond issue to equal the par yield. Result: new bonds initially trade at par.

Invoked in interest rate swaps: the swap rate is the par yield. (To be discussed in the swaps chapter.)

4.12

Par Yield (bond cash flows split into annuity and lump sum components)

In general if m is the number of coupon payments per year, P is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date

cP m

A

( )100 100

4.13

Sample Data Bootstrap Method Table 4.3, p. 86

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

4.14

The Bootstrap Method

First line: an amount 2.5 can be earned on 97.5 during 3 months, i.e. future value of 97.5 is 100

This is 10.127% with continuous compounding: 97.5 e^(.25R)= 100 implies R = 10.127%

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

4.15

The Bootstrap Method continued

To calculate the 1.5 year rate we 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

4.16

Zero Curve Calculated from the Data

9

10

11

12

0 0.5 1 1.5 2 2.5

Zero Rate (%)

Maturity (yrs)

10.127

10.469 10.536

10.681

10.808

4.17

Forward Rates (assumption: continuously compounded)

The forward rate is the future zero (or spot) rate implied by today’s zero or spot curve of interest rates.

The forward rate is inferred (from observed zero rates) now but pertains to a future time period.

Observed: R1 R2 ; Inferred: F

R2 T2 = R1 T1 + F(T2 – T1 )

4.18

Calculation of Forward RatesTable 4.5, page 89

Zero Rate for Forward Rate

an n -year Investment for n th Year

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

1 3.0

2 4.0 5.0

3 4.6 5.8

4 5.0 6.2

5 5.3 6.5

4.19

Formula for Forward Rates

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

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

12

1122

TT

TRTRF

Nexus: F, R1 , and R2

R2 is a weighted average of (“is between”) R1

and F R2T2= R1T1 + F (T1 – T2 ) Yield is a weighted average of (“is between”)

R1 and R2

4.21

Upward vs Downward SlopingZero (or Spot) Curve

For upward sloping zero curve, R2> R1 :Fwd Rate or F > R2 > Yield > R1

For downward sloping zero curve, R2< R1 :Fwd Rate or F < R2 < Yield < R1

4.22

Forward Rate Agreement, FRA

FRA is an agreement that a certain rate will apply to a certain principal during a certain future time period

FRA (interest) rate or RK is quoted assuming a compounding period equal to the FRA period.

FRA buyer pays the FRA rate, hedges a liability FRA seller receives the FRA rate, hedges a

deposit

4.23

Forward Rate Agreementcontinued

An FRA is equivalent to an agreement where interest at a predetermined rate, RK is exchanged for interest at the market rate

An FRA can be valued by assuming that the forward interest rate is certain to be realized, i.e. employ the prevailing forward rate to value a forward contract.

FRA at inception

V of FRA = zero Generic requirement: At inception the

value of a forward contract is zero Contractual rate specified RK = adjusted F F is adjusted so that its compounding

period equals that of the FRA period; must shift from continuous compounding to, say quarterly or semi-annual compounding

4.25

FRA seller: post-inception valuation

A company had agreed that it would receive 4% on $100 million for 3 months starting in 3 years

The forward rate for the period between 3 and 3.25 years is 3%, assuming quarterly compound.

3.25-year zero rate is 2% assuming CC The value now of the contract to the company is

+$234,267 or $250,000 discounted from time 3.25 years to now: 100M(4%-3%).25

e^[-2%(3.25)]

4.26

FRA seller continued: Settlement at start of forward period (always)

Suppose rate proves to be 4.5% (quarterly compounding)

The payoff is –$125,000 at the 3.25 year point; contract assumes value conveyance at end of forward period: 100M(4%-4.5%).25 = - 0.125M

This is equivalent to a payoff of –$123,609 at the 3-year point: -0.125M{1/[1 +(4.5%/4)]}= -123,609

Settlement occurs at start of forward period: FRA seller pays $123,609 to FRA buyer

FRA problems (Test Bank)

Determine FRA contractual rate: 4.3; 4.6 Valuation post-inception: 4.8 Determine settlement amount: 4.10. Note:

Settlement is always assumed to occur at the start of the FRA period in this course. This is true of the overwhelming majority of cases in financial markets.

4.28

Theories of the Zero (Spot) CurvePage 87

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

Management of Net Interest Income (Table 4.6, page 94)

Suppose that the market’s best guess is that future short term rates will equal today’s rates

What would happen if a bank posted the following rates?

Depositors prefer short-term; borrowers prefer long-term How can the bank manage its risks? Raise 5-year

deposit/mortgage rates to mitigate asset/liability mismatch

4.29

Maturity (yrs) Deposit Rate Mortgage Rate

1 3% 6%

5 3% 6%