Interest Rates

Outline1. Types of 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

9. Theories of the term structure of interest rates

10. Summary

Types of Rates(1/3)(1) Treasury Rates( 公債利率 )

A government to borrow in its own currency.

An investor earns on Treasury bills and Treasury bonds.

It is usually assumed “risk-free rates”.

American Treasury Rates

Types of Rates(2/3)(2) LIBOR( 倫敦銀行間放款利率 )

The London Interbank Offered Rate is a daily reference rate based on the interest rates at which banks borrow funds from other banks in the London wholesale money market .

LIBOR is short term .

Derivatives traders regard LIBOR rates as a better indication of the “true” risk-free rate than Treasury rates .

Large banks also quote LIBID rates ( London Interbank Bid Rate ) and is the rate at which they will accept deposits from other banks.

LIBOR

Types of Rates(3/3)(3) Repo Rates( 附買回利率 )

A Repurchase agreement (also known as a repo or Sale and Repurchase Agreement) allows a borrower to use a financial security as collateral for a cash loan at a fixed rate of interest.

The difference between the price is the interest it earns. We called the interest rate is repo rate.

The most common type of repo is an overnight repo.

Measuring interest rates(1/4) Suppose that an amount A is invested for n years at an interest rate

of R per annum. If the rate is compounded m times per annum, the terminal value

of the investment is A (1+ R/m)mn(1)

Compound times Future value (R=10%)

Every year (m=1) 110.00

Half year (m=2) 110.25

Season (m=4) 110.38

Monthly (m=12) 110.47

Weekly (m=51) 110.51

Example1. A pension fund manager invests $10 million in a debt

obligation that promises to pay 7.3% per year for four years. What is the future value of the $10 million?

To determine the future value of any sum of money invested today, we can use the future value equation, which is:

mn)m

R (1A

4.66$13,255,58)1

0.073 (110,000,000 4

Measuring interest rates(2/4) With continuous compounding, it can be shown that

an amount A invested for n years at rate R grows

to

AeRn (2)

Compounding a sum of money at a continuously compounded rate R for n years involves multiplying it by eRn. Discounting it at a continuously compounded rate R for n years involves multiplying e-Rn.

Which of the following amounts is closest to the end value of invest-ing $3000 for ¾ years at s continuously compounded rate of 12%?

32833000 4

3%12

e

Measuring interest rates(3/4) Suppose that Rc is a rate of interest with continuous

compounding and Rm is the equivalent rate with compounding m times per annum. From the results in equations (1) and (2), we have

eRc= ( 1+ Rm/m )mn

Rc = m ln(1+ Rm/m)

Rm=m (eRc/m - 1 )

(3)

(4)

These equations can be used to convert a rate with a compounding frequency of m times per annum to a continuously compounded rate and vice versa.

Measuring interest rates(4/4)

(1 )

(1 )

ln(1 )

ln(1 )(3)

c

c

R nmnm

R n mnm

mc

mc

RA Ae

mR

em

RR n mn

mR

R mm

ln(1 )

ln(1 )

ln(1 )

1

( 1)(4)

c m

c

c

mc

c m

R R

m m

Rmm

R

mm

or

RR n mn

mR n R

mn m

e e

Re

m

R m e

Example(1) Consider an interest rate is quoted as 10% per annum with semiannual

compounding . From equation (3) with m=2 and Rm=0.1 ,the equivalent rate with continuous compounding is

2ln (1+ 0.1/2) = 0.09758 Or 9.758% per annum

(2)Suppose that a lender quotes the interest rate on loan as 8% per annum with continuous compounding , and that interest is actually paid quarterly . From equation (1) with m=4 and Rc=0.08 , the equivalent rate with quarterly compounding is

4(e0.08/4 -1)=0.0808

Or 8.08 per annum. This means that on a $1,000 loan ,interest payments of $20.20 would be required each quarter

Zero rates

Definition:

1. The n-year zero-coupon interest rate is the rate of interest earned on an investment that starts today and lasts for n years. All the interest and principal is realized at the end of n years.

2. Sometimes also referred to as the n-year spot rate.

Bond Pricing(1/4)

Suppose that a 2-year Treasury bond with a principal of $100 provides coupons at the rate of 6% per annum semiannually.

Maturity(years)

Zero Rate(% cont comp)

0.5 5.0

1.0 5.8

1.5 6.4

2.0 6.8

Bond Pricing(2/4)

(1)Theoretical price of the bond ( 理論價格 )

3e-0.05*0.5 ＋ 3e-0.058*1.0 ＋ 3e-0.064*1.5 ＋ 103e-0.068*2.0 ＝ 98.39

Maturity (years)

Zero Rate (% cont comp)

0.5 5.0

1.0 5.8

1.5 6.4

2.0 6.8

Continuous

Compounding

Ae-Rn

Bond Pricing(3/4)

(2) Bond Yield (債券報酬率 )

A bond’s yield is the single discount rate that, when applied to all cash flows, gives a bond price equal to its market price. Assumed: y is the yield on the bond.

3e-y*0.5 ＋ 3e-y*1.0 ＋ 3e-y*1.5 ＋ 103e-y*2.0 ＝ 98.39

y ＝ 6.76% (trial and error )

Bond Pricing(4/4)

(3) Par Yield (面額報酬率 )

The par yield for a certain bond maturity is the coupon rate that causes the bond price to equal its par value. Suppose that the coupon on a 2-year bond in our example is c per annum.

c/2e-0.05*0.5 ＋ c/2e-0.058*1.0 ＋ c/2e-0.064*1.5 ＋ (100+c/2)e-0.068*2.0 ＝100

c ＝ 6.87

The 2-year par yield is 6.87% per annum with semiannual compounding.

Determining Treasury zero rates(1/5)

The most popular approach is known as the bootstrap method( 拔靴法 or 導引法 ).

Bond Time to Annual Bond CashPrincipal 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

＊ Half the stated coupon is assumed to be paid every 6 months.

Determining Treasury zero rates(2/5)

The bootstrap method

The 3-month bond provides a return of 2.5 in 3 months on an initial investment of 97.5.

With quarterly compounding, the 3-month zero rate is (4×2.5)/97.5 ＝10.256 ％ (per annum).

The rate is expressed with continuous compounding, it becomes 4 ln （ 1 ＋ 0.10256/4 ）＝ 0.10127

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

•Six months •(2x5.1)/94.9=10.748%

2 ln （ 1 ＋ 0.10748/2 ）＝ 0.10469

One year

ln （ 1 ＋ 10/90 ）＝ 0.10536

Determining Treasury zero rates(3/5)

The bootstrap method To calculate the 1.5 year zero rate, the payments are as follows:

6 months : ＄ 4 ; 1 year: ＄ 4 ; 1.5 years : ＄ 104

Suppose the 1.5-year zero rate is denoted by R.

R ＝ 10.681 ％

Similarly the two-year rate is 10.808 ％

4e-0.10469*0.5 ＋ 4e-0.10536*1.0 ＋ 104e-y*1.5 ＝ 96

e-y*1.5 =0.85196

R=-ln(0.85196)/1.5 = 0.10681 6e-0.10469*0.5 ＋ 6e-0.10536*1.0 ＋ 6e-0.10681*1.5 ＋ 106e-R*2.0 ＝101.6 e-R*2.0 =0.80561R=-ln(0.80561)/2.0 = 0.10808

Determining Treasury zero rates(4/5)

Zero Curve Calculated from the Data (Figure 1)

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

Determining Treasury zero rates(5/5)

A common assumption is that the zero curve is linear between the points determined using the bootstrap method.

The zero curve is horizontal prior to the first point and horizontal beyond the last point.

By using longer maturity bonds, the zero curve would be more accurately determined beyond 2 years.

Forward rates(1/5)

Definition: the rates of interest implied by current zero rates for periods of time in the future.

Forward rates(2/5)Calculation of Forward Rates (Table 5)

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

Forward rates(3/5)Calculation of Forward Rates (Table 5)

The 4 ％ per annum rate for 2 years mean that, in return for an investment of ＄ 100 today, the receive

100e0.04*2 ＝＄ 108.33 .

Suppose that ＄ 100 is invested. A rate of 3 ％ for the first year and 5 ％ for the second year gives at the end of the second year.

100e0.03*1e0.05*1 ＝＄ 108.33

When interest rates are continuously compounded and in successive time periods are combined, the overall equivalent rate is simply the average rate during the whole period.

Forward rates(4/5)

The forward rate for year 3 is the rate of interest that is implied by a 4 ％ per annum 2-year zero rate and a 4.6 ％ per annum 3-year zero rate.

If R1and R2 are the zero rates for maturities T1 and T2 ,respectively, and RF is the forward interest rate for the period of time between T1 and T2 ,then

Formula of Forward Rates

12

1122

TT

TRTRRF

(5)

12

1122 )(

TT

TRRRRF

(6)

Example

Find the forward rate 3f4

Rf= R4 + (R4-R3) x T3 / (T4-T3)

= 5% + 0.4% x 3

= 6.2%

(1+ 0f4 )4 = (1+ 0f3)3 x (1+ 3f4 )

(1+5%)4 = (1+4.6%)3 x (1+ 3f4 )

= 0.0620920 0.062≒

Forward rate agreements(1/4)

Definition: FRA is an over-the-counter agreement that a certain interest rate will apply to either borrowing or lending a certain principal during a specified future period of time.

Forward rate agreements(2/4)

Rk: The rate of interest agreed to in the FRA

RF: The forward LIBOR interest rate for the period between times T1 and T2 calculated today.

RM: The actual LIBOR interest rate observed in the market at times T1 for the period between times T1 and T2

L: The principal underlying the contract.

Notation Define

Forward rate agreements(3/4)

If X company could earn RM from the LIBOR loan ,The FRA means that it will earn RK. The extra interest rate (which may be negative) that is earns as a result of entering into the FRA is RK-RM. The interest rate set at time T1 and paid at time T2. The extra interest rate therefore leads to a cash flow to company at time T2 of

))(( 12 TTRRLC KM

Cash flow formula

Similarly there is a cash flow to the Y company at time T2 of

))(( 12 TTRRLC MK

Example Suppose that a company enters into an FRA that specifies it will

receive a fixed rate of 4% on a principal of $1 -million for a 3-month period starting in 3 years . If 3-month LIBOR proves to be 4.5% for the 3-month period the cash flow to the lender will be

250,1$25.0)045.004.0(000,000,1 at the 3.25-year point. This is equivalent to a cash flow of

09.236,1$25.0045.01

250,1

at the 3-year point . The cash flow to the party on the opposite side of the transaction will be +$1,250 at the 3.25-year point or +1,236.09 at the 3-year point (All the example interest rates in this example are expressed with quarterly compounding . )

Forward rate agreements(4/4)

Value of FRA where a fixed rate RK will be received on a principal L between times T1 and T2 is

Value of FRA where a fixed rate is paid is

RF is the forward rate for the period and R2 is the zero rate for maturity T2

Valuation Formulas

22))(( 12TR

FK eTTRRLV

22))(( 12TR

KF eTTRRLV

Example Suppose the LIBOR zero and forward rates are as in Table 4.5 .

Consider an FRA where we will receive a rate of 6% ,measured with annual compounding , on a principal of $1-millon between the end of year 1 and the end of year 2. In this case , the forward rate is 5% with continuous compounding or 5.127% with annual compounding . The value of the FRA is

058,8$)05127.006.0(000,000,1 204.0 e

We see that an FRA can be valued if we

1.Caculate the payoff on the assumption that forward rates are realized ,that is on the assumption that RM=RF . (RM= The actual LIBOR interest rate RF =The forward LIBOR interest )

2.Discount this payoff at the risk-free rate .

Duration(1/3)

Definition: A measure of how long on average the holder of the bond has to wait before receiving cash payment.

A zero-coupon bond that lasts n years has a duration of n years.

A coupon-bearing bond lasting n years has a duration of less than n years.

Duration(2/3)

• Duration of a bond that provides cash flow c i at time t i is

where B is its price and y is its yield (continuously compounded) This leads to

B

ectD

iyti

n

ii

1

yDB

B

(12)

n

i

yti

ieCB1

Example For the bond in table 4.6 , the bond price B is 94.213 and the duration D is

2.653 ,so that equation give

y

yB

95.249

653.2213.94

When yield on the bond increase by 10 basis point (=0.1%) , it follows that

.The duration relationship predicts that

= -0.250 ,so that the bond price goes down to 94.213-0.250=93.963 . How accurate is this ? When the bond yield increase by 10 basis point to 12.1% , the bond is

001.0y 001.095.249 B

963.931055

55550.3121.05.2121.0

0.2121.05.1121.00.1121.05.0121.0

ee

eeee

Which is (to three decimal place) the same as that predicted by the duration relationship .

Duration(3/3)

Modified Duration The preceding analysis is based on the assumption

that y is expressed with continuous compounding. If y is expressed with a compounding frequency of m times per year, then

my

yBDB

/1

A variable D*, defined by

my

DD

/1*

Example The bond in table 4.6 has a price of 94.213 and a duration of 2.653. The

yield , express with semiannual compounding is 12.3673%. The modified duration , D* is given by

4985.22/123673.01

653.2*

D

y

yB

39.235

4985.2213.94

When the yield (semiannually compound) on the bond increase by 10 basis point (=0.1%) , we have The duration relationship predicts thatWe expect to be -235.39 x 0.001= -0.235 , so that the bond price goes down to 94.213-0.235= 93.978. How accurate is this ? When the bond yield increase by 10 basis point to 12.4673% , an exact calculation similar to that previous example shows that the bond price becomes 93.978. This shows that the modified duration calculation gives good accuracy .

001.0yB

Theories of the term structure of interest rates

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.

Summary(1/2) Treasury rates are the rates paid by a government on

borrowings in its own currency.

LIBOR rates are short-term lending rates offered by banks in the interbank market.

The n-year zero or spot rate is the applicable to an investment lasting for n years when all of the return is realized at the end.

Forward rates are the rates applicable to future periods of time implied by today’s zero rates.

Summary(2/2) FRA is an over-the-counter agreement that a certain

interest rate will apply to either borrowing or lending a certain principal at LIBOR during a specified future period of time.

Duration measures the sensitivity of the value of a bond portfolio to a small parallel shift in the zero-coupon yield curve.

Liquidity preference theory can be used to explain the interest rate term structures that are observed in practice.