latex ir 1 defxing/ams320/documents/handout03.pdf · example 1 (pricing a coupon bond) figure 1:...

Outline Types Measures Spot rate Bond pricing Bootstrap Forward rates FRA Duration Convexity Term structure

Interest Rates

Haipeng Xing

Department of Applied Mathematics and Statistics

Haipeng Xing, AMS320, Textbook: Hull (2009)

Interest Rates


Outline

1 Types of interest rates

2 Measuring interest rates

3 The n-year spot rate

4 Bond pricing

5 Determining treasury zero rates — the bootstrap method

6 Forward interest rates

7 Forward rate agreements

8 Duration

9 Convexity

10 Theories of the term structure of interest rates


Interest Rates


Types of interest rates — treasury rates

Treasury rates are the rates an investor earns on Treasury Bill andTreasury bonds. They are instruments used by a government toborrow in its own currency.

It is usually assumed that there is no chance that a government willdefault on an obligation denominated in its own currency.

Treasury rates are therefore considered as risk-free rates in the sensethat an investor who buys a Treasury bill or Treasury bond is certainthat interest and principal payments will be made as promised.


Interest Rates


Types of interest rates — LIBOR

LIBOR (London Interbank O↵ered Rate) is an unsecured short-termborrowing rate between banks.

LIBOR rates have traditionally been calculated each business day for10 currencies and 15 borrowing periods. The borrowing periodsrange from 1 day to 1 year.

LIBOR rates are used as reference rates for hundreds of trillions ofdollars of transactions throughout the world.

In recent years there have been suggestions that some banks mayhave manupulated their LIBOR quotes.


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Types of interest rates — the federal funds rate

In the United States, financial institutions are required to mantain acertain amount of cash with the Federal Reserve. The reserverequirement for a bank at any time depends on its outstanding assetsand liabilities. At the end of a day, some financial institutions havesurplus funds in their federal account while others have requirementfor funds. This leads to borrowing and lending overnight. In theU.S., the overnight rate is called the federal funds rate.

The weighted average of the rates in brokered transactions (withweights being determined by the size of the transaction) is termedthe e↵ective federal funds rate.

This overnight rate is monitored by the central bank, which mayintervene with its own transactions in an attempt to raise or lower it.

Both LIBOR and the federal funds rates are unsecured borrowingrates.


Interest Rates


Types of interest rates — repo rates

A repurchase agreement (repo) is an agreement where a financialinstitution that owns securities agrees to sell them for a certain priceand buy them bank in the future (usually the next day) for a slightlyhigher price.

The financial institution obains a loan in the repo. The interest ratethat the financial institution pays is referred to as the repo rate.

Repo rates are secured rates, hence a repo rate is generally slightlybelow the correpsonding fed funds rate.


Interest Rates


Measuring interest rates — simple interest

Interest: In exchange for the use of an investor’s money, banks pay afraction of the account balance back to the investor. The fractionalpayment is known as interest. The money a bank uses to payinterest is generated by investments and loans that the bank makeswith the investor’s money.

An interest rate, denoted as r, is the fraction of the invested amountused to compute the interest. It is usually expressed as a percentagepaid per year.

The initially invested amount which earns the interest, denoted asP , is called principal. The sum of the prinicpal amount and earnedinterest, denoted as A, is called the compound amount.

Simple interest: The compound amount after t interest periods(think of them as years) is

A = P (1 + r)t.


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Measuring interest rates: Examples

If the interest rate is measured with annual compounding, the bank’sstatement that the interest rate is 10% means that $100 grows to

$100⇥ 1.1 = $110

at the end of 1 year.When the interest rate is measured with semiannual compounding, itmeans that 5% is earned every 6 months, with the interest beingreinvested. In this case, $100 grows to

$100⇥ 1.052 = $110.25.

at the end of 1 year.When the interest rate is measured with quarterly compounding, thebank’s statement means that 2.5% is earned every 3 months, withthe interest being reinvested. The $100 then grows to

$100⇥ 1.0254 = $110.38.

at the end of 1 year.Haipeng Xing, AMS320, Textbook: Hull (2009)

Interest Rates




Interest Rates


Measuring interest rates — compound interest

suppose that an amount A is invested for n years at an interest rateof R per annum. If the rate is compounded once per annum, theterminal value of the investment is

A(1 +R)n.

If the rate is compounded m times per annum, the terminal value ofthe investment is

A⇣1 +

R

m

⌘mn


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Measuring interest rates: Compound interest

When interest is allowed to earn interest itself, an investment is saidto earn compound interest. In this case, part of the interest is paidto the investor more than once a year.

Let the number of compounding periods per year be n. If the annualinterest rate is r, then the interest rate per compounding period isr/n and the formula for compound interest is

A = P⇣1 +

r

n

⌘nt.


Interest Rates


Measuring interest rates: Compound interest

When the number of compunding periods, n, increases such that theprocedures of depositing and withdrawing are almost instantaneous,the compound amount A becomes

A = limn!1

P⇣1 +

r

n

⌘nt.

Using the natural logarithm, we obtain a formula for continuouslycompounded interest,

A = Pert.


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Example 4.1 When the number of compunding periods, n, increasessuch that the procedures of depositing and withdrawing are almostinstantaneous, the compound amount A becomes

A = limn!1

P⇣1 +

r

n

⌘nt.

Using the natural logarithm, we obtain a formula for continuouslycompounded interest,

A = Pert.


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Measuring interest rates: Transformation

Suppose that Rc is a rate of interest with continuous compoundingand Rm is the equivalent rate with compounding m times perannum. An amount A invested for n years at two rates Rc and Rm

grow to

AeRcn = A⇣1 +

Rm

m

⌘mn.

This means that

Rc = m ln⇣1 +

Rm

m

⌘, (1)

Rm = m(eRc/m � 1). (2)


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Example 4.1 Consider an interest rate that is quoted as 10% perannum with semiannual compounding. From equation (1) withm = 2 and Rm = 0.1, the equivalent rate with continuouscompounding is

2 ln⇣1 +

0.1

2

⌘= 0.09758.

or 9.758% per annum.

Example 4.2 Suppose that a lender quotes the interest rate on loansas 8% per annum with continuous compounding, and that interest isactually paid quarterly. From (2) with m = 4 and Rc = 0.08, theequivalent rate with quarterly compounding is

4⇥ (e0.08/4 � 1) = 0.0808

or 8.08% per annum.


Interest Rates


The n-year spot rate

The n-year zero-coupon interest rate is the rate of interest earned onan investment that starts today and lasts for n years.

All the interest and principal is realized at the end of n years, andthere are no intermediate payments.

The n-year zero-coupon interest rate is also referrred as n-year spotrate, the n-year zero rate.


Interest Rates


Bond pricing

Most bonds pay coupons to the holders periodically. The bond’sprincipal (i.e. par value or face value) is paid at the end of its life.The price of a bond can be calculated as the present value of all thecash flows that will be received by the owner of the bond.

Example 1 (Pricing a coupon bond)

Figure 1: Treasury zero raets (Hull,2014; Table 4.2).

Consider a 2-year Treasurybond with a principal of $100provides coupons at the rate of6% per annum semiannually. TheTreasury zero rates are measuredwith continuous compounding,as in the table. Thenthe theoretical price of the bond is

$3e�0.05⇥0.5 + $3e�0.058⇥1.0 + $3e�0.064⇥1.5 + $103e�0.068⇥2.0 = $98.39


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Bond yield

The bond yield is the discount rate that makes the present value ofthe cash flows on the bond equal to the market price of the bond.

Example 2 (Calculating the bond yield)

Suppose that the theoretical price of the bond we have been discussing,$98.39, is also its market price. Let y be the yield on the bond, it mustsatisfy that

3e�0.5y + 3e�1.0y + 3e�1.5y + 103e�2.0y = 98.39.

Solving this equation yields that y = 6.76%.


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Par yield

The par yield for a certain bond maturity is the coupon rate thatcauses the bond price to equal its par value (or the principal value).

Example 3 (Calculating the par yield)

Suppose that the coupon on a 2-year bond in our example is c per annum(or c

2 per 6 months). Using the table in Example 4, the value of the bondis equal to its par value of 100 when

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.

Solve this equation gives c = 6.87 (with semiannual compounding).


Interest Rates


Determining treasury zero rates — the bootstrapmethod

One way to determine Treasury zero rates is from Treasury bills andcoupon-bearing bonds. The most popular approach is known as thebootstrap method.

Example 4 (Determining treasury zero rates)

To illustrate the bootstrap method, we use the data on the right (Hull,2014; Table 4.3) to compute the treasury zero rates.


Interest Rates


Example: Determining treasury zero rates

Because the first three bonds pay no coupons, the zero ratescorresponding to the maturities of these bonds can easily becalculated.

97.5 = 100e�R1⇥0.25 =) R1 = 10.127%

94.9 = 100e�R2⇥0.5 =) R2 = 10.469%

90 = 100e�R3⇥1.0 =) R3 = 10.536%

The fourth bond lasts 1.5 years, with payments $4, $4, and $104 atthe end of year 0.5, 1.0, and 1.5, respectively.

4e�R2⇥0.5 + 4e�R3⇥1.0 + 104e�R4⇥1.5 = 96 =) R4 = 10.681%

The 2-year zero rate can be calculated from the 6-month, 1-year,and 1.5-year zero rates. That is,

6e�R2⇥0.5 + 6e�R3⇥1.0 + 6e�R4⇥1.5 + 106e�R5⇥2.0 = 101.6

=) R5 = 10.808%


Interest Rates


Example: Determining treasury zero rates

A chart showing the zero rate as a function of maturity is known asthe zero curve.It is usually assume that (1) the zero curve is linear between thepoints determined using the bootstrap method, and (2) the zerocurve is horizontal prior to the first point and horizontal beyond thelast point.

Figure 2: Zero rates given by the bootstrap method (Hull, 2014; Table 4.4 &Figure 4.1).


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Forward interest rates

Forward interest rates are the future rates of interest implied bycurrent zero rates for periods of time in the future.

Suppose that the zero rates for time periods T1 and T2 are R1 andR2 with both rates continuously compounded. The forward rate RF

for the period between times T1 and T2 is

RF =R2T2 �R1T1

T2 � T1= R2 + (R2 �R1)

T1

T2 � T1. (3)

This formula is only approximately true when rates are not expressedwith continuous compounding.

Letting T2 ! T1 = T , we obtain the instantaneous forward rate fora maturity T , which is the forward rate that applies for a very shorttime period starting at T . It is

RF = R+ T@R

@T.


Interest Rates


Forward interest rates

RF = R2 + (R2 �R1)T1

T2 � T1.

Remark 1

If the zero curve is upward sloping between T1 and T2 so thatR2 > R1, then RF > R2.

If the zero curve is downward sloping between T1 and T2 so thatR2 < R1, then RF < R2.


Interest Rates


Forward rate agreements (FRAs)

If a large investor thinks the rates in the future will be di↵erent fromtoday’s forward rates, one strategy is to enter into a contract knownas a forward rate agreement to fix the rate in the future.

A FRA is an over-the-counter agreement that a certain rate willapply to a certain principal during a certain future time period.

The usual assumption underlying the contract is that the borrowingor lending would normally be done at LIBOR.

If the agreed fixed rate is greater than the actual LIBOR rate for theperiod, the borrower pays the lender the di↵erence between the twoapplied to the principal. If the reverse is true, the lender pays theborrower the di↵erence applied to the principal.

The present value of the payment is made at the beginning of thespecified period.


Interest Rates


Forward LIBOR I

Recall that the rate at which banks borrow or lend to each other iscalled LIBOR. On the current day t, LIBOR rates for periods ↵ = 1(12-month), 0.5 (6-month), 0.25 (3-month) etc are published.

We denote the LIBOR rate at current time t for period [t, t+ ↵] byLt[t, t+ ↵]. The LIBOR rate LT [T, T + ↵] for a future date T > t isa random variable.

Banks can deposit (or borrow) L at time t and receive (or pay back)L(1 + ↵Lt[t, t+ ↵]) at time t+ ↵.

All interest is paid at the maturity or term of the deposit, and thereis no interim compounding. Most interest rate derivatives typicallyreference 3-month or 6-month LIBOR.


Interest Rates


Forward LIBOR II

Note that a FRA is a forward contract to exchange two cashflows.Specifically, the buyer of the FRA with maturity T and delivery priceor fixed rate K argrees at t( T ) to pay ↵K and receive↵LT (T, T + ↵) at time T + ↵. Thus the payo↵ of the FRA is↵(LT (T, T + ↵)�K) at time T + ↵.

Theorem 1

The forward LIBOR rate, denoted by Lt(T, T + ↵), is the value of Ksuch that the FRA has zero value at time t T , which is given by

Lt(T, T + ↵) =Z(t, T )� Z(t, T + ↵)

↵Z(t, T + ↵),

where Z(t, T ) is the price of the T -maturity zero coupon bound (ZCB)at time t.


Interest Rates


Forward LIBOR III

Proof. Consider a portfolio consisting of long 1 ZCB maturing at T andshort (1 + ↵K) ZCBs at T + ↵.At time T , we place the 1 from the T -maturity bond in a deposit withinterest rate LT (T, T + ↵). Then at T + ↵, the portfolio has value

(1 + ↵LT (T, T + ↵))� (1 + ↵K) = ↵(LT (T, T + ↵)�K),

the payo↵ of a FRA. Therefore, the value of the FRA at time t is

VK(t, T ) = Z(t, T )� (1 + ↵K)Z(t, T + ↵).

The forward LIBOR rate Lt(T, T + ↵) is the value of K such that theFRA has zero value. Hence,

Lt(T, T + ↵) =Z(t, T )� Z(t, T + ↵)

↵Z(t, T + ↵),


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Example 5 (Hull, 2014; Example 4.3)

Suppose that a company enters into an FRA that is designed to ensure itwill receive a fixed rate of 4% on a principal of $100 million for a 3-monthperiod starting in 3 years. The FRA is an exchange where LIBOR is paidand 4% is received for the 3-month period. If 3-month LIBOR proves tobe 4.5% for the 3-month period, the cash flow to the lender will be

$100, 000, 000⇥ (0.04� 0.045)⇥ 0.25 = �$125, 000

at the 3.25-year point. This is equivalent to a cash flow of

�125, 000�(1 + 0.045⇥ 0.25) = �$123, 609

at the 3-year point.


Interest Rates



Consider an FRA where company X agrees to lend money to company Yfor the period of time between T1 and T2. Define

RK : The fixed rate of interest agreed to in the FRA

RF : The forward LIBOR rate for the period between times T1 andT2, calculated today (i.e., Lt(T1, T2))

RM : The actual LIBOR interest rate observed in the market at timeT1 for the period between times T1 and T2.

L: The principal underlying the contract

Assume that the rates RK , RF , and RM are all measured with acompounding frequency reflecting the length of the period to which theyapply. For company X, the payo↵ at time T1 is

A :=L(RK �RM )(T2 � T1)

1 +RM (T2 � T1).

For company Y, the payo↵ at time T1 is �A.Haipeng Xing, AMS320, Textbook: Hull (2009)

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Values of an FRA

An FRA is worth zero when the fixed rate RK equals the forwardrate RF . — When an FRA is first entered into, RK is set equal tothe current value of RF , so that the value of the contract to eachside is 0.

As time passes, interest rates change, hence the vlaue is no longer 0.

The market value of a derivative at a particular time is referred to asits mark-to-market (MTM) value.

For an FRA that RK will be received on a principal of L betweentimes T1 and T2, its value today is

VFRA = L(RK �RF )(T2 � T1)e�R2T2 .

For an FRA that RF will be paid on a principal of L between timesT1 and T2, its value today is

VFRA = L(RF �RK)(T2 � T1)e�R2T2 .


Interest Rates


Interest rate swaps (IRS) I

A swap is an agreement between two counterparties to exchange aseries of cashflows at agreed dates. Cashflows are calculated on anotional amount, which we typically take to be 1. A swap has astart date T0, maturity Tn, and payment dates Ti, i = 1, . . . , n.

In a standard or vanilla swap, we typically have Ti+1 = Ti + ↵ for afixed ↵. The floating leg of the swap consists of payments↵LTi(Ti, Ti + ↵) at Ti + ↵. The fixed leg of the swap consists ofpayments ↵K at Ti + ↵.


Interest Rates


Interest rate swaps (IRS) II

The value of the fixed leg is given by V FXDK (t) = K

Pni=1 ↵Z(t, Ti).

The value of the floating leg isV FL(t) =

Pni=1 Lt(Ti�1, Ti)↵Z(t, Ti)

=Pn

i=1(Z(t, Ti�1)� Z(t, Ti)) = Z(t, T0)� Z(t, Tn).

The forward swap rate at time t for a swap form T0 to Tn is definedto be the value yt(T0, Tn) of the fixed rate K such that the value ofthe swap at t is zero, i.e.,

yt(T0, Tn) =

Pni=1 Lt(Ti�1, Ti)↵Z(t, Ti)Pn

i=1 ↵Z(t, Ti)=

Pni=1 Lt(Ti�1, Ti)↵Z(t, Ti)

Z(t, T0)� Z(t, Tn)

The value V SWK (t) at time t( T0) of a swap there we pay a fixed

rate K and receive LIBOR is given by

V SWK (t) = (yt(T0, Tn)�K)

nX

i=1

↵Z(t, Ti).


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Duration

The duration of a bond is a measure of how long on average theholder of the bond has to wait before receiving cash payments.

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

Consider a bond provides the holder with cash flows ci at time ti(1 i n). The bond price B and bond yield y are related by

B =nX

i=1

cie�yti .

The duration of the bond, D, is defined as

D =

Pni=1 ticie

�yti

B=

nX

i=1

tihcie�yti

B

i. (4)


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Duration

When a small change �y in the yield is considered, (4) implies that

�B = ��ynX

i=1

ticie�yti . (5)

From equations (4) and (5), we obtain an approximate relationshipbetween percentage changes in a bond price and changes in itsyields,

�B

B= �D�y. (6)

If y is expressed with a compounding frequency of m times a year,then

�B = �BD⇤�y,

where the variable D⇤, referred to as the bond’s modified duration

D⇤ =D

1 + y/m.


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Convexity

From Tyalor series expansions, we obtain an expression for �B as follows,

�B =@B

@y�y +

1

2

@2B

@y2��y

�2.

Figure 3: Two bond portoflios withthe same duration (Hull, 2014; Figure4.2).

Define the measure of convexityfor the bond,

C =1

B

@2B

@y2=

Pni=1 cit

2i e

�yti

B.

This leads to

�B

B= �D�y +

1

2C��y

�2.


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Theories of the term structure of interest rates

Question?

What determines the shape of the zero curve?

A number of di↵erent theories have been proposed.

Expectations theory: It conjectures that long-term interest ratesshould reflect expected future short-term interest rates.

Market segmentation theory: It conjectures that the short-,medium-, and long-term interest rates are determined by supply anddemand in the short-, medium-, and long-term bond markets,respectively. They are independent with each other.

Liquidity preference theory: It assumes that investors prefer topreserve their liquidity and invest funds for short periods of time.Borrowers, on the other hand, prefer to borrow at fixed rates forlong periods of time. This leads to a situation in which forward ratesare greater than expected future zero rates.


Interest Rates

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