MBF1243 Derivatives

L5: Interest Rates

Types of Rates

• An interest rate in a particular situation defines the amount of

money a borrower promises to pay the lender. For any given

currency, many different types of interest rates are regularly

quoted.

• These include mortgage rates, deposit rates, prime borrowing

rates, and so on. The interest rate applicable in a situation

depends on the credit risk.

• This is the risk that there will be a default by the borrower of

funds, so that the interest and principal are not paid to the

lender as promised.

• The higher the credit risk, the higher the interest rate that is

promised by the borrower. 2

Types of Rates

• Treasury rate - are the rates an investor earns on Treasury bills

and Treasury bonds.

• These are the instruments used by a government to borrow in its

own currency.

• Japanese Treasury rates are the rates at which the Japanese

government borrows in yen; US Treasury rates are the rates at

which the US government borrows in US dollars; and so on.

• It is usually assumed that there is no chance that a government will

default on an obligation denominated in its own currency.

• Treasury rates are therefore totally risk-free rates in the sense that

an investor who buys a Treasury bill or Treasury bond is certain

that interest and principal payments will be made as promised.

3

Types of Rates

LIBOR

• LIBOR is short for London Interbank Offered Rate. It is a reference interest

rate, produced once a day by the British Bankers’ Association, and is

designed to reflect the rate of interest at which banks are prepared to make

large wholesale deposits with other banks.

• LIBOR is quoted in all major currencies for maturities up to 12 months: 1-

month LIBOR is the rate at which 1-month deposits are offered, 3-month

LIBOR is the rate at which 3-month deposits are offered, and so on.

• A deposit with a bank can be regarded as a loan to that bank. A bank must

therefore satisfy certain creditworthiness criteria in order to be able to

receive deposits from another bank at LIBOR. Typically it must have a AA

credit rating.

• A rate closely related to LIBOR is LIBID. This is the London Interbank Bid

Rate and is the rate at which banks will accept deposits from other banks.

At any specified time, there is a small spread between LIBID and LIBOR

rates (with LIBOR higher than LIBID).

4

Types of Rates • Repo Rates

• This is a contract where an investment dealer who owns securities agrees

to sell them to another company now and buy them back later at a slightly

higher price.

• The other company is providing a loan to the investment dealer. The

difference between the price at which the securities are sold and the price

at which they are repurchased is the interest it earns.

• The interest rate is referred to as the repo rate. If structured carefully, the

loan involves very little credit risk. If the borrower does not honor the

agreement, the lending company simply keeps the securities. If the lending

company does not keep to its side of the agreement, the original owner of

the securities keeps the cash.

• The most common type of repo is an overnight repo, in which the

agreement is renegotiated each day.

5

Types of Rates The Risk-Free Rate

• The ‘‘risk-free rate’’ is used extensively in the evaluation of derivatives.

• It might be thought that derivatives traders would use the interest rates implied

by Treasury bills and bonds as risk-free rates. In fact, they do not do this.

• However, there are a number of tax and regulatory issues that cause Treasury

rates to be artificially low.

• Financial institutions have traditionally used LIBOR rates as risk-free rates.

• For a AA-rated financial institution LIBOR is the short-term opportunity cost of

capital. The financial institution can borrow short-term funds at the LIBOR

quotes of other financial institutions and can lend funds to other financial

institutions at its own LIBOR quotes.

6

Measuring Interest Rates

• A statement by a bank that the interest rate on one-year deposits is 10% per

annum sounds straightforward and unambiguous. In fact, its precise meaning

depends on the way the interest rate is measured.

• If the interest rate is measured with annual compounding, the bank’s

statement that the interest rate is 10% means that $100 grows to

$100 x 1.1 = $110 at the end of 1 year.

• When the interest rate is measured with semiannual compounding,

• it means that 5% is earned every 6 months, with the interest being reinvested.

In this case $100 grows to

$100 x 1.05 x 1.05 = $110.25 at the end of 1 year.

• When the interest rate is measured with quarterly compounding, the bank’s

statement means that 2.5% is earned every 3 months, with the interest being

reinvested. The $100 then grows to

$100 x 1.0254 = $110.38 at the end of 1 year.

Table 4.1 shows the effect of increasing the compounding frequency further. 7

Measuring Interest Rates

8

Measuring Interest Rates

• The compounding frequency defines the units in which an interest rate

is measured.

• A rate expressed with one compounding frequency can be converted

into an equivalent rate with a different compounding frequency.

• For example, from Table 4.1 we see that 10.25% with annual

compounding is equivalent to 10% with semiannual compounding.

• We can think of the difference between one compounding frequency

and another to be analogous to the difference between kilometers and

miles. They are two different units of measurement.

9

Measuring Interest Rates

• To generalize our results, suppose that an amount A is invested for

n years at an interest rate of R per annum. If the rate is

compounded once per annum, the terminal value of the investment

is

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

the investment is

• When m = 1, the rate is sometimes referred to as the equivalent

annual interest rate.

10

Continuous Compounding

• The limit as the compounding frequency, m, tends to infinity is

known as continuous compounding.

• With continuous compounding, it can be shown that an amount A

invested for n years at rate R grows to

• where e = 2.71828. The exponential function, ex, is built into most

calculators, so the computation of the expression in equation (4.2)

presents no problems.

• In the example in Table 4.1, A = 100, n = 1, and R = 0.1, so that the

value to which A grows with continuous compounding is

11

Conversion Formulas

Define

Rc : continuously compounded rate

Rm: same rate with compounding m times per

year

12

R mR

m

R m e

cm

mR mc

ln

/

1

1

Examples

10% with semiannual compounding is equivalent to

2ln(1.05)=9.758% with continuous compounding

8% with continuous compounding is equivalent to

4(e0.08/4 -1)=8.08% with quarterly compounding

Rates used in option pricing are nearly always

expressed with continuous compounding

13

Zero Rates

• 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.

• There are no intermediate payments. The n-year zero-

coupon interest rate is sometimes also referred to as the n-

year spot rate, the n-year zero rate, or just the n-year zero.

• Suppose a 5-year zero rate with continuous compounding is

quoted as 5% per annum. This means that $100, if invested

for 5 years, grows to

14

Bond Pricing

• 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.

• Suppose that the theoretical price of the bond we have been

considering, $98.39, is also its market value (i.e., the market’s price of

the bond is in exact agreement with the data in Table 4.2).

• If y is the yield on the bond, expressed with continuous compounding, it

must be true that

15

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

. . . . . .

. . .

• This equation can be solved using an iterative (‘‘trial and error’’) procedure to

give y = 6:76%.

Table 4.2

16

Par Yield • The par yield for a certain bond maturity is the coupon rate that

causes the bond price to equal its par value. (The par value is

the same as the principal value.)

Usually the bond is assumed to provide semiannual coupons.

• Suppose that the coupon on a 2-year bond in our example is c

per annum (or 1/2c per 6 months).

• Using the zero rates in Table 4.2, the value of the bond is equal

to its par value of 100 when

• In our example we solve

17

• This equation can be solved in a straightforward way to give c = 6.87.

• The 2-year par yield is therefore 6.87% per annum. This has semiannual

compounding because payments are assumed to be made every 6 months.

With continuous compounding, the rate is 6.75% per annum.

Par Yield continued

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

(in our example, m = 2, d = 0.87284, and A =

3.70027)

18

A

mdc

)( 100100

Forward Rates

19

• Forward interest rates are the rates of interest implied by current zero rates for

periods of time in the future.

• To illustrate how they are calculated, we suppose that LIBOR zero rates are

as shown in the second column of Table 4.5.

• LIBOR zero rates are calculated in a similar way to the Treasury zero rates

calculated in the previous section.

• The rates are assumed to be continuously compounded.

• Thus, the 3% per annum rate for 1 year means that, in return for an

investment of $100 today, an amount

100e0.03 x1 = $103.05 is received in 1 year;

• the 4% per annum rate for 2 years means that, in return for an investment of

$100 today, an amount 100e 0:04 x2 = $108.33 is received in 2 years; and so

on.

Forward Rates

20

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 for the period between times T1 and

T2 is

• This formula is only approximately true when rates

are not expressed with continuous compounding

21

R T R T

T T

2 2 1 1

2 1

Application of the Formula

22

Year (n) Zero rate for n-year

investment (% per annum)

Forward rate for nth

year (% per annum)

1 3.0

2 4.0 5.0

3 4.6 5.8

4 5.0 6.2

5 5.5 6.5

Instantaneous Forward Rate

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

where R is the T-year rate

23

R TR

T

Upward vs Downward Sloping

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

24

Forward Rate Agreement

25

A forward rate agreement (FRA) is an OTC

agreement that a certain rate will apply to a

certain principal during a certain future time

period

Forward Rate Agreement: Key Results

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

LIBOR interest rate, RF , is certain to be realized

This means that the value of an FRA is the present

value of the difference between the interest that would

be paid at interest at rate RF and the interest that would

be paid at rate RK

26

Valuation Formulas

If the period to which an FRA applies lasts

from T1 to T2, we assume that RF and RK are

expressed with a compounding frequency

corresponding to the length of the period

between T1 and T2

With an interest rate of RK, the interest cash

flow is RK (T2 –T1) at time T2

With an interest rate of RF, the interest cash

flow is RF(T2 –T1) at time T2

27

Valuation Formulas continued

When the rate RK will be received on a principal of L

the value of the FRA is the present value of

received at time T2

When the rate RK will be received on a principal of L

the value of the FRA is the present value of

received at time T2

28

))(( 12 TTRR FK

))(( 12 TTRR KF

Example

An FRA entered into some time ago ensures

that a company will receive 4% (s.a.) on $100

million for six months starting in 1 year

Forward LIBOR for the period is 5% (s.a.)

The 1.5 year rate is 4.5% with continuous

compounding

The value of the FRA (in $ millions) is

29

467050050040100 510450 ..)..( .. e

Example continued

If the six-month interest rate in one year turns

out to be 5.5% (s.a.) there will be a payoff (in

$ millions) of

in 1.5 years

The transaction might be settled at the one-

year point for an equivalent payoff of

30

750500550040100 ..)..(

730002751

750.

.

.

Duration

• The duration of a bond, as its name implies, is a measure of how long on

average the holder of the bond has to wait before receiving cash payments.

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

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

because the holder receives some of the cash payments prior to year n.

• Suppose that a bond provides the holder with cash flows ci at time ti (1 < i <n).

• The bond price B and bond yield y (continuously compounded) are related by

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

31

B

ectD

iyt

in

i

i

1

Key Duration Relationship

Duration is important because it leads to the

following key relationship between the

change in the yield on the bond and the

change in its price

32

yDB

B

Key Duration Relationship continued

When the yield y is expressed with

compounding m times per year

The expression

is referred to as the “modified duration”

33

my

yBDB

1

D

y m1

Bond Portfolios

The duration for a bond portfolio is the weighted

average duration of the bonds in the portfolio with

weights proportional to prices

The key duration relationship for a bond portfolio

describes the effect of small parallel shifts in the yield

curve

What exposures remain if duration of a portfolio of

assets equals the duration of a portfolio of liabilities?

34

Convexity

The convexity, C, of a bond is defined as

This leads to a more accurate relationship

When used for bond portfolios it allows larger shifts in

the yield curve to be considered, but the shifts still

have to be parallel

35

B

etc

y

B

BC

n

i

ytii

i

1

2

2

21

22

1yCyD

B

B

Theories of the Term Structure Page 96-98

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

36

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?

Options, Futures, and Other Derivatives 9th Edition,

Copyright © John C. Hull 2014 37

Maturity Deposit rate Mortgage rate

1 year 3% 6%

5 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

38

Maturity Deposit rate Mortgage rate

1 year 3% 6%

5 year 4% 7%

Duration

39

• The duration of a bond, as its name implies, is a measure of how long on

average the holder of the bond has to wait before receiving cash

payments.

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

• However, a coupon-bearing bond lasting n years has a duration of less

than n years, because the holder receives some of the cash payments

prior to year n. The bond price B and bond yield y (continuously

compounded) are related by

40

Duration

• Consider a 3-year 10% coupon bond with a face value of $100.

• Suppose that the yield on the bond is 12% per annum with continuous compounding.

• Coupon payments of $5 are made every 6 months.

• Table 4.6 shows the calculations necessary to determine the bond's duration.

• The present values of the bond's cash flows, using the yield as the discount rate, are

shown in column 3 (e.g., the present value of the first cash flow is 5e-0.12x0.5 = 4.709).

• The sum of the numbers in column 3 gives the bond's price as 94.213. The weights are

calculated by dividing the numbers in column 3 by 94.213.

• The sum of the numbers in column 5 gives the duration as 2.653 years.

• Small changes in interest rates are often measured in basis points. As mentioned earlier,

a basis point is 0.01 % per annum.

41

Duration

42

Duration • Consider a 3-year 10% coupon bond with a face value of $100.

• Suppose that the yield on the bond is 12% per annum with continuous compounding.

• This means that y = 0.12.

• Coupon payments of $5 are made every 6 months.

• Table 4.6 shows the calculations necessary to determine the bond's duration.

• The present values of the bond's cash flows, using the yield as the discount rate, are

shown in column 3 (e.g., the present value of the first cash flow is 5e-O.12xO.5 =

4.709).

• The sum of the numbers in column 3 gives the bond's price as 94.213. The weights

are calculated by dividing the numbers in column 3 by 94.213.

• The sum of the numbers in column 5 gives the duration as 2.653 years.

• Small changes in interest rates are often measured in basis points. As mentioned

earlier, a basis point is 0.01 % per annum.

43

Duration of Bond Portfolio

The duration, D, of a bond portfolio can be defined as a weighted average

of the durations of the individual bonds in the portfolio, with the weights

being proportional to the bond prices.

It is important to realize that, when duration is used for bond portfolios,

there is an implicit assumption that the yields of all bonds will change by

approximately the same amount.

By choosing a portfolio so that the duration of assets equals the duration of

liabilities (i.e., the net duration is zero), a financial institution eliminates its

exposure to small parallel shifts in the yield curve. It is still exposed to

shifts that are either large or nonparallel.

44

Convexity

• The duration relationship applies only to small changes in yields.

• This is illustrated in Figure 4.2, which shows the relationship between

the percentage change in value and change in yield for two bond

portfolios having the same duration.

• The gradients of the two curves are the same at the origin.

• This means that both bond portfolios change in value by the same

percentage for small yield changes. For large yield changes, the

portfolios behave differently.

• Portfolio X has more curvature in its relationship with yields than

portfolio Y.

• A factor known as convexity measures this curvature.

45

Convexity

46

Convexity

• The convexity of a bond portfolio tends to be greatest when the portfolio

provides payments evenly over a long period of time.

• It is least when the payments are concentrated around one particular

point in time.

• By choosing a portfolio of assets and liabilities with a net duration of zero

and a net convexity of zero, a financial institution can make itself immune

to relatively large parallel shifts in the zero curve. However, it is still

exposed to nonparallel shifts.