LidaNo.pptx

INTEREST

RATE

NO Lida

Content

s

1. Types of Interest Rates 2. Measuring Interest Rates 3. Zero Rates

4. Bond Pricing

5. Determining Treasury Zero Rates 6. Forward Rates

7. Forward Rate Agreements 8. Duration

1. Types of

Interest Rates

• Interest rate is a cost of borrowing.

• Many different types of interest rates include:

 Treasury rates: Interest rates earn on treasury bills/bonds

2. Measuring

Interest Rates

Simple Interest Vs. Compound Interest

Year

Simple Interest Compound Interest

Starting

Balance + Interest =

Ending Balance

Starting

Balance + Interest =

Ending Balance

1 100 + 10 = 110 100 + 10 = 110

2 110 + 10 = 120 110 + 11 = 121

3 120 + 10 = 130 121 + 12.1 = 133.1

4 130 + 10 = 140 133.1 + 13.3 = 146.4

 Simple Interest : Interest is paid only on initial Investment  Compound Interest : Interest earns on interest

2. Measuring Interest

Rates (Cont.)

Continuous Compound Interest

 In the limit as we compound more and more frequently we obtain

continuously compounded interest rates

 $100 grows to $100eRTwhen invested at a continuously

compounded rate R for time T

 $100 received at time T discounts to $100e-RTat time zero when

2. Measuring Interest

Rates (Cont.)

Continuous Compound Interest





R m R

m

R m e

c m

m R mc

   

 

 

 

ln

/

1

1

Where:

- R_c : Continuously compounded rate

- R_m : Same rate with compounding m times per year

3. 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 T

• Example: Treasury Zero Rates are quoted as below:

Maturity

(years) (% continuously compounded) Zero Rate

0.5 5.0

1.0 5.8

1.5 6.4

4. Bond Pricing

• The theoretical price of a bond can be calculated discounted all the

future cash flow received by the owner of the bond.

• Example: a 2-year Treasury bond with a principle of $100 provides

coupons at the rate of 6% per annum semiannually.

Bond Price =

4. Bond Pricing

(Cont.)

 The bond yield is the 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 (continuously compounded) is given by solving

to get y=0.0676 or 6.76%.

 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

Therefore, par yield is 6.87% (Semiannually compounding)

100

2

100

2

2

2























e

c

e

c

e

c

e

c

4. Bond Pricing

(Cont.)

Par Yield (Cont.)

• In general if m is the number of coupon payments per year, d is

the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date

or

• In the above example:

- m= 2,

-

A

m

d

c



(

100



100

)

d

m

C

A

100

100





87284

.

0





e

d

70027

.

3

0











e

e

e

e

5. Determining Treasury

Zero Rates

Bond Time to Annual Bond Cash 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 Method

E.g. with the below data, we want to determine continuously compound interest rates correspond to each bond maturity

Zero Rate

Continuous Compound (%)

5. Determining Treasury

Zero Rates (Cont.)

Bootstrap Method (Cont.)

 The 3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly

compound which equal to 10.127% with continuous compounding

 Similarly the 6 month and 1 year rates are 10.469% and 10.536%

with continuous compounding

 To calculate the 1.5 year rate we solve

to get R = 0.10681 or 10.681%

96

104

4

6. Forward Rates

• The forward rate is the future zero rate implied by today’s

term structure of interest rates

• Example: Calculating forward rates:

n-year Forward Rate

zero rate 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

 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

or

 The instantaneous forward rate for a maturity T is the forward rate that

applies for a very short time period starting at T. It is

6. Forward Rates (Cont.)

1 2 1 1 2 2

T

T

T

R

T

R

R







R T R

T    1 2 1 1 2

2

(

)

7. Forward Rates

Agreement

• A forward rate agreement (FRA) is an agreement that a certain rate

will apply to a certain principal during a certain future time period

• Suppose:

: the interest rate agreed to in the FRA

: the actual LIBOR interest rate in the market at time T1 for the period between T1 and T2

: the principle underlying the contract

The cash flow to the leader at time T2 is:

The cash flow to the leader at time T1 is: k

R

R

)

)(

(

R

R

T

T

L





) (

1

) )(

( _{2 1}

T T R T T R R

L _{k M}

 

 

7. Forward Rates

Agreement (Cont.)

• Example: A company enter into FRA to receive a fixed rate of 4%

(continuously compounding per annum) on a principle of $100 million for a 3-month period starting in 3 years.

If the 3-month LIBOR proves to be 4.5% for the 3-month period, the company’s cash flow at the 3.25-year point will be:

1,000,000 x (0.04 – 0.045) x 0.25 = - $125,000

The cash flow at 3-year point is the discount of the above amount which is:

Valuation

 Value of FRA where a fixed rate R_K will be received on a principal L

between times T₁ and T₂ is

 Value of FRA where a fixed rate is paid is

- R_F is the forward rate for the period - R₂ is the zero rate for maturity T₂

 Example: Consider an FRA which receives a rate of 6% annual

compounding on a principle of $100 millions between year 1 and year 2 which has spot interest rate of 4%. Future interest rate between year 1 and 2 is 5.127% annual compounding. The value of FRA is

2 2

) )(

(R_K R_F T₂ T₁ e R T

L   

2 2

) )(

(R_F R_K T₂ T₁ e R T

L   

800 , 805 $ ) 05127 . 0 06 . 0 ( 000 , 000 ,

1   e0.042 

8. Duration

 Duration of a bond is a measure of how long on average the holder of

the bond has to wait before receiving cash payments.

 Duration of a bond is defined as:

Where: C_i : the cash flow at time t_i B : the price of the bond

y : the bond yield (continuously compounded)



















t

c

e

B

D

yt i

n

i

i

1

B

8. Duration (Cont.)

 Example: Calculate the duration of a 3-year bond with 10% coupon

rate. Interest rate is 5% per annum.

Year Ct PV(Ct) at 5.0%

Proportion of Total Value

[PV(C_t)/V]

Proportion of Total Value Time

1 100 95.24 0.084 0.084 2 100 90.7 0.08 0.16 3 1100 950.22 0.836 2.509

V = 1136.16 1 Duration= 2.753 years

100 100 1100

1 3

2.753

 When the yield y is expressed with compounding m times per

year

 The expression

is referred to as the “modified duration”

8. Duration (Cont.)

m

y

y

BD

B











1

D

y m

9. Theories of the Term

Structure of Interest Rates

A number of theories have been developed to explain the behavior of yield curve:

 Expectations Theory: Forward rates equal expected future zero rates

 Market Segmentation: Short, medium and long rates determined

independently of each other. Interest rate is determined by supply and demand in the market.

 _{Liquidity Preference Theory: Forward rates should always be higher}

9. Theories of the Term Structure

of Interest Rates (Cont.)

Liquidity Preference Theory

 Suppose that the outlook for rates is flat and you have been offered

the following choices

 Which would you choose as a depositor? Which for your mortgage?

Maturity Deposit rate Mortgage rate

1 year 3% 6%

Liquidity Preference Theory (Cont.)

 To match the maturities of borrowers and lenders a bank has to

increase long rates above expected future short rates

 In our example the bank might offer

Maturity Deposit rate Mortgage rate

1 year 3% 6%

5 year 4% 7%