Interest Rates
Chapter 4
4.1
Goals of Chapter 4

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
4.2
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
Three Types of Interest Rates
– 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
Three Types of Interest Rates

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
Three Types of Interest Rates

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
Three Types of Interest Rates

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
𝐴 after 𝑛 years is
𝑅 𝑚𝑛
𝐴 1+
𝑚
𝑚 = number of compounding frequency per year
𝑛 = investment horizon in terms of years
𝑅 = annual interest rate
※ Note that it is a market convention that the
interest rate is always quoted on an annual basis
4.10
Compounding Frequency

For 𝐴 = $1, 𝑛 = 1 year, and 𝑅 = 10%, analyze
the effect of different compounding frequencies
Compounding frequency Terminal value of $1 at the end of 1 year
𝑚=1
1.10000000
𝑚=2
1.10250000
𝑚=4
1.10381289
𝑚 = 12
1.10471307
𝑚 = 52
1.10506479
𝑚 = 365
1.10515578
𝑚=∞
1.10517092
※ Due to the compounding effect, the terminal value increases with the
compounding frequency, 𝑚, although the interest rate for each period is 𝑅/𝑚
※ In the limit as we compound more and more frequently, we obtain
continuously compounded interest rates, i.e., 𝑚 = ∞
4.11
Compounding Frequency

With continuous compounding, i.e., 𝑚 = ∞,
the terminal value for the amount 𝐴 approches
𝑅 𝑚𝑛
lim 𝐴 1 +
= 𝐴𝑒 𝑅𝑛
𝑚→∞
𝑚
where 𝑒 is a constant of 2.718281828
– The exponential function enjoys some advantages
of simplifying algebraic calculation, e.g., 1/𝑒 𝑥 = 𝑒 −𝑥
– Thus, it is convenient to employ the continuous
compounding to compute PVs and FVs


$100 grows to $100𝑒 𝑅𝑇 when invested at a continuously
compounded rate 𝑅 for time 𝑇
$100 received at time 𝑇 discounts to $100𝑒 −𝑅𝑇 at time 0
when the continuously compounded discount rate is 𝑅
4.12
Compounding Frequency

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
4.13
Compounding Frequency

Conversion formula to derive the equivalent
continuous compounding rate
𝑅𝑚 : interest rate compounded 𝑚 times per year
𝑅𝑐 : equivalent continuous compounding rate
𝑅𝑚 𝑚𝑛
=𝐴 1+
𝑚
𝑅𝑚 𝑚𝑛
𝑅𝑐 𝑛
⇒𝑒
= 1+
𝑚
𝑅
⇒ 𝑅𝑐 𝑛 = 𝑚𝑛 ln 1 + 𝑚
𝑚
𝑅𝑚
⇒ 𝑅𝑐 = 𝑚 ln 1 +
𝑚
(or 𝑅𝑚 = 𝑚 𝑒 𝑅𝑐/𝑚 − 1 )
𝐴𝑒 𝑅𝑐 𝑛
4.14
4.3 Zero Rates and Bond
Prices
4.15
Zero Rates (零息利率)


A zero rate (also known as a spot rate) for
maturity 𝑇 is the rate of interest earned on an
investment that provides a payoff only at time 𝑇
An example of zero rates with different times to
maturity
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 ∙ 𝑒 −5.0%∙0.5 = $0.9753
1.0
5.8%
$1 ∙ 𝑒 −5.8%∙1.0 = $0.9436
1.5
6.4%
$1 ∙ 𝑒 −6.4%∙1.5 = $0.9085
2.0
6.8%
$1 ∙ 𝑒 −6.8%∙2.0 = $0.8728
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 𝑡,
a zero rate with the time to maturity 𝑡 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
3𝑒 −0.05∙0.5 + 3𝑒 −0.058∙1.0 + 3𝑒 −0.064∙1.5
+ 103𝑒 −0.068∙2.0
= 98.39
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
3𝑒 −𝑦∙0.5 + 3𝑒 −𝑦∙1.0 + 3𝑒 −𝑦∙1.5 + 103𝑒 −𝑦∙2.0 = 98.39
– Solve the above equation by the bisection method
(二分逼近法) to obtain 𝑦 = 0.0676 or 6.76%
※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
𝑐 −0.05∙0.5
𝑒
2
𝑐
100 +
2
𝑐 −0.058∙1.0
𝑐 −0.064∙1.5
+ 𝑒
+ 𝑒
2
2
𝑒 −0.068∙2.0 = 100
+
to get 𝑐 = 6.87
4.19
Par Yield

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 $1 on each coupon
date
𝑐
100 − 100𝑑 𝑚
𝐴 + 100𝑑 = 100 ⇒ 𝑐 =
𝑚
𝐴
※In the above example, 𝑚 = 2, 𝑑 = 𝑒 −0.068∙2.0 , and
𝐴 = 𝑒 −0.05∙0.5 + 𝑒 −0.058∙1.0 + 𝑒 −0.064∙1.5 + 𝑒 −0.068∙2.0
4.20
The Bootstrap Method

Bootstrap method (拔靴法): to determine
treasury zero rates sequentially from the
shortest maturity to the longest maturity based
on market prices of Treasury bills and bonds
– The sequence must be followed because the
information of zero rates with shorter maturities is
needed to solve the zero rate with a longer maturity
(shown in the following numerical example)
– The name of “bootstrap”: In order to take off your
shoes by unfastening the shoelace, make sure you
first loosen the upper part of the shoelace and then
loosen the lower part
4.21
The Bootstrap Method

Hypothetic data for Treasury bills (the first
three quotes) and bonds (the last two quotes)
Bond Principal
Time to
($)
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
Find the zero rates corresponding to the time
to maturities of 0.25, 0.5, 1, 1.5, and 2 years
4.22
The Bootstrap Method

Step 1 (for 𝑅(0.25)):
– 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 𝑅𝑐 with 𝑚 and 𝑅𝑚 to be 4 and 10.2564%
Method 2: Solve 𝑅(0.25) from $97.5𝑒 𝑅(0.25)∙0.25 = $100
Step 2 (for 𝑅(0.5) and 𝑅(1)):
– Similarly, the 6-month and 1-year continuous
compounding zero rates are 10.4693% and 10.5361%
4.23
The Bootstrap Method

Step 3 (for 𝑅(1.5)):
– Solve the following equation for 𝑅(1.5)
4𝑒 −0.104693∙0.5 + 4𝑒 −0.105361∙1 + 104𝑒 −𝑅(1.5)∙1.5 = 96
⇒ 𝑅 1.5 = 10.6810%

Step 4 (for 𝑅(2)):
– Solve the following equation for 𝑅(2)
6𝑒 −0.104693∙0.5 + 6𝑒 −0.105361∙1 + 6𝑒 −0.106810∙1.5
+ 106𝑒 −𝑅(2)∙2 = 101.6
⇒ 𝑅 2 = 10.8082%
4.24
Zero Curve Calculated From
the Hypothetic Data
Zero Rate (%)
12
11
10.4693
10
10.5316
10.6810
10.8082
10.1271
Maturity (yrs)
9
0
0.5
1
1.5
2
2.5
※ 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
4.25
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 𝑇1 and
𝑇2 are 𝑅1 and 𝑅2 , respectively, with both rates
continuously compounded
– Formula for the forward rate between 𝑇1 and 𝑇2 is
𝑅𝐹 =
𝑅2 𝑇2 −𝑅1 𝑇1
,
𝑇2 −𝑇1
which is the future zero rate at 𝑇1 with the time to
maturity (𝑇2 −𝑇1 ) implied from the current term
structure
4.27
Calculation of Forward Rates
– The intuition for the formula is the equality of
1. Cumulative return compounding at 𝑅2 until 𝑇2
2. Cumulative return compounding at 𝑅1 until 𝑇1 and next
compounding at 𝑅𝐹 between 𝑇1 and 𝑇2
𝑒 𝑅2 𝑇2 = 𝑒 𝑅1 𝑇1 𝑒 𝑅𝐹 (𝑇2−𝑇1 )
– An example of calculation of forward rates
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%
4.28
Upward vs. Downward Sloping
Yield Curve

Rewrite the formula for the forward rate as
𝑇1
𝑅𝐹 = 𝑅2 + (𝑅2 − 𝑅1 )
𝑇2 − 𝑇1
– For an upward sloping zero curve, i.e., 𝑅2 > 𝑅1 :
forward rate 𝑅𝐹 (applicable for the interval [𝑇1 , 𝑇2 ])
> zero rate 𝑅2 (matured at 𝑇2 ) (see the table on
the previous slide)
– For a downward sloping zero curve, i.e., 𝑅2 < 𝑅1 :
forward rate 𝑅𝐹 (applicable for the interval
[𝑇1 , 𝑇2 ]) < zero rate 𝑅2 (matured at 𝑇2 )
4.29
Theories of the Term Structure

Expectations Theory:
𝑇2
𝑅𝐹 = 𝑅1 + (𝑅2 − 𝑅1 )
𝑇2 − 𝑇1
– 𝑅2 > 𝑅1 (𝑅2 < 𝑅1 ) if and only if 𝑅𝐹 > 𝑅1 (𝑅𝐹 < 𝑅1 )
– 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
4.30
lower (higher) rates for short- (long-) term loans
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 dimension
4.31
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 𝑅𝐾 will apply to a
certain principal during a future time period
– Illustration of a FRA from the viewpoint of the
lender
L
fixed lending rate RK
t 0
T1
lend L
T2
4.33
Forward Rate Agreement

Some details of FRAs
– Traded in OTC markets
– Commonly associated with LIBOR, e.g., at 𝑇1 , to fix the
6-month lending or borrowing rate, which should be
the prevailing 6-month LIBOR at 𝑇1 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 𝑇2 − 𝑇1
for the reference interest rate and thus it is compounded once
during the FRA period
More specifically, for any interest rate 𝑅 which is applied to
(𝑇1 , 𝑇2 ], the corresponding interest payment at 𝑇2 is 𝑅 𝑇2 − 𝑇1
if the principal is $1
4.34
Forward Rate Agreement
– Payoff of the FRA at 𝑇2 for the lender is
𝑃𝐹𝑅𝐴 = 𝐿(𝑅𝐾 − 𝑅𝑀 ) 𝑇2 − 𝑇1
where 𝑅𝑀 is the actual LIBOR rate in (𝑇1 , 𝑇2 ]

Payoff for the lender if he lends 𝐿 to earn the actual
LIBOR rate for (𝑇1 , 𝑇2 ] is
𝐿𝑅𝑀 𝑇2 − 𝑇1

The net effect for the lender who enters into the FRA is
to fixed the earned interest rate at 𝑅𝐾
𝐿𝑅𝑀 𝑇2 − 𝑇1 + 𝐿 𝑅𝐾 − 𝑅𝑀 𝑇2 − 𝑇1 = 𝐿𝑅𝐾 (𝑇2 − 𝑇1 )

In practice, FRAs can be settled at 𝑇1 and the settlement
price equals the present value of the payoff at 𝑇2
𝐿(𝑅𝐾 −𝑅𝑀 ) 𝑇2 −𝑇1
1+𝑅𝑀 𝑇2 −𝑇1
4.35
Forward Rate Agreement
– There is no cost to lock the forward rate for the
period between 𝑇1 and 𝑇2


Zero-cost strategy to earn 𝑅𝐹 in [𝑇1 , 𝑇2 ]: Borrow 𝐿 at 𝑅1
for 𝑇1 years and invest this amount of 𝐿 at 𝑅2 for 𝑇2
years  Cash outflow of 𝐿𝑒 𝑅1 𝑇1 at 𝑇1 (considered as the
initial investment) and cash inflow of 𝐿𝑒 𝑅2 𝑇2 at 𝑇2
(considered as the final payoff)  Earn the forward rate
𝑅𝐹 for the period between 𝑇1 and 𝑇2 (due to the definition
of 𝑅𝐹 which can satisfy 𝐿𝑒 𝑅2 𝑇2 = 𝐿𝑒 𝑅1 𝑇1 𝑒 𝑅𝐹(𝑇2 −𝑇1 ) )
So, if 𝑅𝐾 is set as 𝑅𝐹 (with compounding period to be
𝑇2 − 𝑇1 ), the value of a FRA should be zero, i.e., the
value for the following payoff is zero
𝐿(𝑅𝐹 − 𝑅𝑀 ) 𝑇2 − 𝑇1
4.36
Forward Rate Agreement
– If 𝑅𝐾 is set to be different from 𝑅𝐹 , the excess
payoff (could be negative) of a FRA contributes to
its value
𝐿 𝑅𝐾 − 𝑅𝑀 𝑇2 − 𝑇1 − 𝐿(𝑅𝐹 − 𝑅𝑀 ) 𝑇2 − 𝑇1
= 𝐿 𝑅𝐾 − 𝑅𝐹 𝑇2 − 𝑇1
– Thus, the present value of the excess payoff is the
value of the FRA today
𝑉𝐹𝑅𝐴 = 𝑒 −𝑅2 𝑇2 𝐿(𝑅𝐾 − 𝑅𝐹 ) 𝑇2 − 𝑇1
※ Note that 𝑅2 , which is the continuous compounding LIBOR
zero rate for 𝑇2 , can be employed as the risk-free rate to
discount the expected payoff (see Slide 4.7)
4.37
Forward Rate Agreement
– Another way to derive the formula for the value of the
FRA


Note that 𝑅𝐹 can be viewed as the expectation of the most
likely value for 𝑅𝑀 based on today’s term structure, i.e.,
𝐸 𝑅𝑀 = 𝑅𝐹
The general rule to price a FRA is the present value
(discounted at the continuous compounding risk-free rate) of its
expected payoff
𝑉𝐹𝑅𝐴 = 𝑒 −𝑅2 𝑇2 𝐸 𝐿 𝑅𝐾 − 𝑅𝑀 𝑇2 − 𝑇1
= 𝑒 −𝑅2𝑇2 𝐿(𝑅𝐾 − 𝑅𝐹 ) 𝑇2 − 𝑇1
– It is common in practice to set 𝑅𝐾 as 𝑅𝐹 and thus the
FRA is worth zero initially

𝑅𝐹 changes according to the demand and supply of funds in
Eurocurrency markets  values of FRAs change randomly
4.38
Forward Rate Agreement

For the trading counterparty, i.e., the borrower
of a FRA,
– Payoff at 𝑇2
−𝐿 𝑅𝐾 − 𝑅𝑀 𝑇2 − 𝑇1 = 𝐿(𝑅𝑀 − 𝑅𝐾 ) 𝑇2 − 𝑇1
– Settlement price at 𝑇1
𝐿 𝑅𝐾 − 𝑅𝑀 𝑇2 − 𝑇1
𝐿(𝑅𝑀 − 𝑅𝐾 ) 𝑇2 − 𝑇1
−
=
1 + 𝑅𝑀 𝑇2 − 𝑇1
1 + 𝑅𝑀 𝑇2 − 𝑇1
– Position value today
−𝑒 −𝑅2𝑇2 𝐿 𝑅𝐾 − 𝑅𝐹 𝑇2 − 𝑇1
= 𝑒 −𝑅2 𝑇2 𝐿(𝑅𝐹 − 𝑅𝐾 ) 𝑇2 − 𝑇1
4.39
Forward Rate Agreement

A pricing example of the FRA
– A company has agreed that it will receive 4% (= 𝑅𝐾 )
on $100 million (= 𝐿) for 3 months (= 𝑇2 − 𝑇1 )
starting after 3 years (= 𝑇1 )
– The 3-year zero rate is 3% (= 𝑅1 ), the 3.25-year
zero rate is 3.1% (= 𝑅2 ), and the forward rate for the
period between 3 and 3.25 years is 4.3% (= 𝑅𝐹 ) (All
of them are expressed with continuous compounding)
– According to Slide 4.14, the quarterly compounding
𝑅𝐹 is
𝑅𝐹 = 4 𝑒 4.3%/4 − 1 = 4.3232%
4.40
Forward Rate Agreement
– The current value of this FRA is
𝑉𝐹𝑅𝐴 = 𝑒 −𝑅2 𝑇2 𝐿 𝑅𝐾 − 𝑅𝐹 𝑇2 − 𝑇1
= 𝑒 −3.1%∙3.25 $100,000,000 4% − 4.3232% 3.25 − 3
= −$73,056.05
– Suppose 3-month LIBOR proves to be 4.5% (= 𝑅𝑀 )
with quarterly compounding after 3 year

At the 3.25-year point (i.e., at 𝑇2 ), the payoff is
𝑃𝐹𝑅𝐴 = $100,000,000 4% − 4.5% 3.25 − 3 = −$125,000,
which is equivalent to a payoff of
𝑃𝐹𝑅𝐴
= −$123,609
1 + 4.5% 3.25 − 3
at the 3-year point (i.e., at 𝑇1 )
4.41