Interest Rates

Interest Rates

Chapter 4

4.1

Goals of Chapter 4

4.2

Introduce three types of interest rates– Treasury rates (國庫券利率 ), LIBOR (倫敦銀行間拆款利率 ), and repo rates (附買回利率 )

Compounding frequency and continuous compounding (連續複利 )

Zero rates (零息利率 ) and bond prices Forward rates (遠期利率 ) and the term

structure of interest rates (利率期間結構 ) Forward rate agreement (遠期利率協定 )

– A derivative whose underlying variable is the forward rate

4.1 Three Types of Interest Rates

4.3

Three Types of Interest Rates

Why to study the interest rates?– Due to the classical discount cash flow (DCF)

pricing model, the interest rate is a factor in the valuation of virtually all assets (or derivatives)

– The theoretical futures and forward prices (will be introduced in Ch. 5) depend on the interest rate during the contract life

– The interest rates can be underlying variables of derivatives (Ch.6 introduces interest rate futures)

Treasury rates (國庫券利率 )– The rate of return an investor earns on Treasury

bills or Treasury bonds, which are government debts issued in its own currency 4.4


– Treasury rates are theoretically risk-free since the government is always able to pay the promised interest and principal payments in domestic currency

LIBOR and LIBID– The shorts for London Interbank Offered Rate and

London Interbank Bid Rate (倫敦銀行間拆款利率 )– A LIBOR (LIBID) quote is the interest rate at which an

AA-rated bank is prepared to make (accept) a wholesale lending (deposit) with other AA-rated banks LIBOR is higher than LIBID

– Large banks quote LIBOR and LIBID for maturities up to 12 months in all major currencies every day 4.5


Prices of Eurodollar futures (introduced in Ch.6) and swap rates (introduced in Ch. 7) can be used to imply the LIBOR rates beyond 12 months

– LIBOR and LIBID trade in the Eurocurrency market, which is outside the control of any government Eurocurrencies indicate the currencies that are traded

outside their home markets, e.g., trade US$ with a British bank in London market

Eurosterling, Euroyen, or Eurodollar– Credit risk issue:

The credit risk of a AA-rated financial institution is small for short-term loans

Thus, LIBOR rates are close to risk-free 4.6


Derivatives traders regard LIBOR rates as a better approximation of the “true” risk-free rate than Treasury rates– LIBOR rates reflect the opportunity cost of funds for AA-rated

bank traders– It is believed that Treasury rates are artificially low due to some

tax advantage and regulatory issues for financial institutions In the U.S., Treasury instruments are not taxed at the state level Treasury instruments must be purchased by financial institutions to

fulfill a variety of regulatory requirements Minimal capital requirements for Treasury instruments is lower than

those for other fixed-income securities ※ All the above reasons stimulate the demand of Treasury

instruments and thus bid up their prices The rates of return of investing in Treasury instruments are driven down

The overnight indexed swap rate is increasingly being used instead of LIBOR as the risk-free rate (introduced in Ch. 7) 4.7


Repo rates (附買回利率 )– Repurchase agreement (repo) (附買回合約 ): a

contract where a trader who owns securities agrees to sell them to a financial institutions now and buy them back at a slightly higher price Equivalent to borrow funds with securities as collaterals Thus, the repo loan involves very little credit risk

– Price margins reflect the interest earned by the financial institutions, which is known as the repo rate

– Overnight repos are the most common, but there are also longer-term arrangements, known as term repos

4.8

4.2 Compounding Frequency and Continuous Compounding

4.9

Compounding Frequency

There are different compounding frequencies used for an interest rate, for example, quarterly or annually compounding

The terminal value of the investment amount after years is

= number of compounding frequency per year = investment horizon in terms of years = annual interest rate※ Note that it is a market convention that the

interest rate is always quoted on an annual basis4.10

For , year, and , analyze the effect of different compounding frequencies

4.11


Compounding frequency Terminal value of $1 at the end of 1 year1.10000000

1.10250000

1.10381289

1.10471307

1.10506479

1.10515578

1.10517092

※ Due to the compounding effect, the terminal value increases with the compounding frequency, , although the interest rate for each period is

※ In the limit as we compound more and more frequently, we obtain continuously compounded interest rates, i.e.,

4.12


With continuous compounding, i.e., , the terminal value for the amount approches

where is a constant of 2.718281828– The exponential function enjoys some advantages

of simplifying algebraic calculation, e.g., – Thus, it is convenient to employ the continuous

compounding to compute PVs and FVs $100 grows to when invested at a continuously

compounded rate for time $100 received at time discounts to at time 0 when the

continuously compounded discount rate is

Comparing to daily compounding frequency, the continuous compounding can provide accurate approximation (see Slide 4.11)– In financial markets, it is common to compound

interest rates daily, e.g., for deposit accounts or loans In derivatives markets, almost all formulae are

expressed with continuous compounding For a given interest rate that is compounded at

a lower frequency:– A conversion to find the equivalent continuous

compounding rate is needed before using formulae expressed with the continuous compounding 4.13


Conversion formula to derive the equivalent continuous compounding rate : interest rate compounded times per year: equivalent continuous compounding rate

(or )

4.14


4.3 Zero Rates and Bond Prices

4.15

Maturity (years) Zero rate (continuous compounding)

Current value of the corresponding zero coupon bond (零息債券 ) (face value = $1 paid at

maturity)0.5 5.0%

1.0 5.8%

1.5 6.4%

2.0 6.8%

Zero Rates (零息利率 )

A zero rate (also known as a spot rate) for maturity is the rate of interest earned on an investment that provides a payoff only at time

An example of zero rates with different times to maturity

4.16

Bond Pricing

The pricing of coupon-bearing bonds (附息債券 )– Each cash payment is discounted at the

appropriate zero rate More specifically, to discount a cash payment matured at ,

a zero rate with the time to maturity should be employed– Based on the table of zero rates on the previous

slide, the theoretical price of a two-year bond providing a 6% coupon paid semiannually is

4.17

Bond Yield (or Yield to Maturity)

The bond yield (or yield to maturity) is a constant discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond– Given the market price of the bond equals $98.39,

the bond yield satisfies the following equation

– Solve the above equation by the bisection method (二分逼近法 ) to obtain or

※Financial calculators cannot solve the bond yield correctly given continuous-compounding formulae

4.18

Par Yield

The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value– For the same example, we solve

to get

4.19

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

※ In the above example, , , and

4.20

Par Yield

4.21

The Bootstrap Method

Bootstrap method (拔靴法 ): to determine treasury zero rates sequentially from the shortest maturity to the longest maturity based on market prices of Treasury bills and bonds– The sequence must be followed because the

information of zero rates with shorter maturities is needed to solve the zero rate with a longer maturity (shown in the following numerical example)

– The name of “bootstrap”: In order to take off your shoes by unfastening the shoelace, make sure you first loosen the upper part of the shoelace and then loosen the lower part

4.22


Hypothetic data for Treasury bills (the first three quotes) and bonds (the last two quotes)

Find the zero rates corresponding to the time to maturities of 0.25, 0.5, 1, 1.5, and 2 years

Bond Principal ($)

Time to Maturity (years)

Annual Coupon ($)

Bond Price ($)

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


Step 1 (for ):– For this zero coupon bond, an amount of $2.5 can

be earned on the investment of $97.5 in 3 months– The 3-month rate is 4 times $2.5/$97.5 or

10.2564% with quarterly compounding– This is equivalent 10.1271% with continuous

compounding Method 1: Exploit the conversion formula on Slide 4.14

to solve with and to be 4 and 10.2564% Method 2: Solve from

Step 2 (for and ): – Similarly, the 6-month and 1-year continuous

compounding zero rates are 10.4693% and 10.5361% 4.23

Step 3 (for ):– Solve the following equation for

Step 4 (for ): – Solve the following equation for

4.24


Zero Curve Calculated From the Hypothetic Data

4.25

9

10

11

12

0 0.5 1 1.5 2 2.5

Zero Rate (%)

Maturity (yrs)

10.1271

10.469310.5316

10.6810 10.8082

※ The zero curve (零息利率曲線 ) is also known as the term structure (期間結構 ) of interest rates, i.e., the interest rate is a function of the time to maturity※ Bond prices are determined with the demand and supply Bond prices are

stochastic interest rates are stochastic

4.4 Forward Rates and Term Structure of Interest Rates

4.26

Forward Rates (遠期利率 )

The forward rate is the future zero rate implied by the term structure of interest rates today

Formula to calculate forward rates:– Suppose that the zero rates for time periods and

are and , respectively, with both rates continuously compounded

– Formula for the forward rate between and is,

which is the future zero rate at with the time to maturity implied from the current term structure

4.27

Calculation of Forward Rates

– The intuition for the formula is the equality of1. Cumulative return compounding at until 2. Cumulative return compounding at until and next

compounding at between and

– An example of calculation of forward rates

4.28

Years (T)

Zero rate for an T-year investment Forward rate for the T-th year

1 3.0%

2 4.0% 5.0%

3 4.6% 5.8%

4 5.0% 6.2%

5 5.3% 6.5%

Upward vs. Downward SlopingYield Curve Rewrite the formula for the forward rate as

– For an upward sloping zero curve, i.e., :forward rate (applicable for the interval ) > zero rate (matured at ) (see the table on the previous slide)

– For a downward sloping zero curve, i.e., : forward rate (applicable for the interval ) < zero rate (matured at )

4.29

Theories of the Term Structure

Expectations Theory:

– () if and only if ()– Upward (downward) sloping zero curves indicate that

the market is expecting higher (lower) forward rates Liquidity Preference Theory:

– Explain upward sloping zero curves according to the liquidity preference of lenders and borrowers

– Lenders prefer to preserve their liquidity and invest funds for short periods of time Lenders demand lower (higher) rates for short- (long-) term loans

4.30

Theories of the Term Structure

– To avoid the re-borrowing interest rate risk, borrowers prefer to borrow at fixed rates for long periods of time Borrowers would like to pay lower (higher) rates for short- (long-) term loans

– The above two forces lead to a convergent result which is an upward sloping zero curve

The mixture of the above two theories can explain the occurrence of hump-shaped zero curves in markets– The hump-shaped zero curve is first rising and

then falling along the maturity dimension4.31

4.5 Forward Rate Agreement

4.32

Forward Rate Agreement

A forward rate agreement (FRA) (遠期利率協定 ) is an agreement made today that a fixed borrowing or lending rate will apply to a certain principal during a future time period– Illustration of a FRA from the viewpoint of the

lender

4.33

0t 1T 2T

lend L

L

fixed lending rate KR


Some details of FRAs– Traded in OTC markets– Commonly associated with LIBOR, e.g., at , to fix the

6-month lending or borrowing rate, which should be the prevailing 6-month LIBOR at without the FRA

– Market conventions for compounding frequency The compounding period for interest rates reflects the length

of the FRA period, i.e., the compounding period is for the reference interest rate and thus it is compounded once during the FRA period

More specifically, for any interest rate which is applied to , the corresponding interest payment at is if the principal is $1

4.34


– Payoff of the FRA at for the lender is

where is the actual LIBOR rate in Payoff for the lender if he lends to earn the actual

LIBOR rate for is

The net effect for the lender who enters into the FRA is to fixed the earned interest rate at

In practice, FRAs can be settled at and the settlement price equals the present value of the payoff at

4.35


– There is no cost to lock the forward rate for the period between and Zero-cost strategy to earn in : Borrow at for years and

invest this amount of at for years Cash outflow of at (considered as the initial investment) and cash inflow of at (considered as the final payoff) Earn the forward rate for the period between and (due to the definition of which can satisfy )

So, if is set as (with compounding period to be ), the value of a FRA should be zero, i.e., the value for the following payoff is zero

4.36


– If is set to be different from , the excess payoff (could be negative) of a FRA contributes to its value

– Thus, the present value of the excess payoff is the value of the FRA today

※ Note that , which is the continuous compounding LIBOR zero rate for , can be employed as the risk-free rate to discount the expected payoff (see Slide 4.7)

4.37

– Another way to derive the formula for the value of the FRA Note that can be viewed as the expectation of the most likely

value for based on today’s term structure, i.e., The general rule to price a FRA is the present value

(discounted at the continuous compounding risk-free rate) of its expected payoff

– It is common in practice to set as and thus the FRA is worth zero initially changes according to the demand and supply of funds in

Eurocurrency markets values of FRAs change randomly4.38


For the trading counterparty, i.e., the borrower of a FRA, – Payoff at

– Settlement price at

– Position value today

4.39


A pricing example of the FRA – A company has agreed that it will receive 4% () on

$100 million () for 3 months () starting after 3 years ()– The 3-year zero rate is 3% (), the 3.25-year zero rate

is 3.1% (), and the forward rate for the period between 3 and 3.25 years is 4.3% () (All of them are expressed with continuous compounding)

– According to Slide 4.14, the quarterly compounding is

4.40


– The current value of this FRA is

– Suppose 3-month LIBOR proves to be 4.5% () with quarterly compounding after 3 year At the 3.25-year point (i.e., at ), the payoff is

,which is equivalent to a payoff of

at the 3-year point (i.e., at 4.41


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Interest Rates