UNSW Business School/ Banking & Finance
FINS2624 Lecture 1
Introduction to Bond Pricing and
the Term Structure of Interest Rates I
Lecture outline
❑
Bond characteristics
❑
Arbitrage pricing
❑
Bond pricing and yield to maturity
❑
Return measures
❑
Introduction to the term structure of interest rates
➢ Yield Curve
➢ Inferring future interest rates
2
Bond Characteristics
BKM 14.1
3
What is a Bond?
❑
❑
A bond is a certificate specifying a debt obligation between an Issuer (borrower) and Bondholders
(lenders)
➢
Issuers most often are government agencies and corporations
➢
Bondholders are most often institutional investors and hedge funds.
➢
The indenture is the contract between the issuer and the bondholders describing the terms,
conditions, and protections, if any
➢
The issuer makes payments on designated dates to repay the debt obligation until the bond
“matures” and makes its last payments.
➢
Coupon payments constitute a series of smaller cash flows (interest payments) given to bondholders
regularly up to and including maturity.
➢
Principal is repaid at maturity.
The risk that the issuer will not repay their debt obligations is called Default Risk
➢
Bondholders do not collect all of a bond’s expected cash flows if the issuer defaults
➢
While default risk is very important in practice, it is not a focus of this course
4
Bond cash flows
Problem: Determine the cash flows of a 5-year bond with a face value on $1,000 and 4% coupons
paid annually.
5
Bond cash flows
Problem: Determine the cash flows of a 5 year bond with a face value on $1,000 and 4% coupons
paid semi-annually.
6
Bond cash flows
Face Value (FV)
The principal or loan amount of the
bond, typically repaid in full as one
large cash flow at maturity. Also
called par value.
Coupon frequency
The number of times per annum
the coupon is paid.
A 5-year bond with a face value on $1,000 and 4% coupons paid semi-annually.
Term (T)
Coupon rate (C)
The period of time to maturity of
the bond (ie when the bond pays
its last cash flow) i.e. the “term to
maturity”
The total annual “interest”
payments of the bond
7
Bond characteristics
❑
A bond is a claim on (generally fixed) future cash flows (CF) with five key parameters in determining
the price (P):
➢
Four parameters determine the cash flows:
• Term (T): the period of time to maturity of the bond (i.e., when the bond pays its last cash flow).
This may be also called the “term to maturity.”
• Face Value (FV): the principal or loan amount of the bond, typically repaid in full as one large
cash flow at maturity. Also called par value. Bonds are typically denominated in $1,000 face
value lots.
• Coupon rate (C): The total of annual coupon payments expressed as a fraction or annual
percentage rate of the face value.
• Coupon frequency: the number of times per annum the coupon is paid. May be zero, one or
more coupons in a given year. Typically either annual or semi-annual payments.
➢
One parameter indicates how to discount the cash flows:
• Yield to Maturity (YTM): the discount rate on the bond’s principal and coupon cash flows that
gives the actual, market price
8
Arbitrage Pricing
BKM 14.2
9
Arbitrage: Motivation
Exemple 1:
Assume you can borrow money from bank A for one year at 3% interest and that you can invest
money in bank B for one year at 4% interest. What should you do?
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Arbitrage: Motivation
Exemple 2:
Assume you can borrow money from bank A for one year at 3% interest. You can also buy a bond
today for $96.15 that makes a single payment of $100 in one year. The bond has no default risk.
What should everyone do? What should happen as a result of this?
11
Arbitrage
An arbitrage is a set of trades that either:
➢ Has a positive and risk-free cash flow today with no net outlay (i.e., zero cash flow) at all
points in the future
➢ Or, requires no net capital outlay today and generates positive but risk-free cash flows in the
future
The no-arbitrage principle says that “true” risk-free arbitrage opportunities should not exist for
long:
➢ If prices of two identical assets differ, execute an arbitrage trade by buying the cheaper asset
and selling the more expensive one simultaneously
➢ Supply and demand suggest that the execution of arbitrage trades encourages the prices of
identical assets to converge
➢ For example, two identical bonds should sell for the same price
The Law of One Price states that two assets with identical cash flows should have the same price
in equilibrium
12
Arbitrage achieves price equilibrium
❑
True arbitrage opportunities are rare and short-lived as market participants eliminate them
through trading
❑
In our example:
➢
Traders borrow from the bank and use the money to buy the bond. This will:
• Increase demand for the bond, raising its price and lowering its yield
• Increase the bank borrowing rate
• Continue until no further arbitrage trades are possible. The two trades have the same
cash flows and prices are in equilibrium
❑
We can not say whether the bond price or the bank’s interest rate was wrong.
❑
We can only say the prices are internally inconsistent and violate the no-arbitrage principle
and, consequently, the law of one price.
13
Arbitrage pricing
We use use arbitrage pricing for bonds and derivatives in this course:
➢
Build a replicating portfolio of assets (also called a synthetic asset) with known prices that
has the exact same future cash flows as those of the asset we want to price
➢
Replicating cash flows must have the same timing, amount, and risk as the asset to be
priced
➢
The market value of the asset we are trying to price will equal the market value of the
replicating portfolio under a no-arbitrage condition
Arbitrage pricing differs from fundamental pricing where:
➢
Prices are set in a supply-demand equilibrium
➢
The properties of an asset gives us the price
14
Bond pricing and yield to maturity
BKM 14.3
15
Bond pricing
❑
Arbitrage pricing requires constructing a replicating portfolio (or “synthetic asset”):
➢
❑
A collection of assets with known prices that exactly mimic the cash flows of the asset we
want to price
We can price the replicating portfolio and then use the no-arbitrage principle and law of one
price to price the original asset correctly
16
Bond pricing
Our motivating example for arbitrage pricing priced a zero-coupon bond, but arbitrage pricing
also holds for coupon-paying bonds.
A bond’s cash flow stream is:
A bond making payments for N periods is just a combination of N future cash flows (CFs)
➢
The bond price (P) is the sum of the present value (PV) of its future cash flows
➢
The PV of the CF stream must be the same as the PV of the replicating portfolio if arbitrage
is not possible
➢
The process of calculating the PV of future CFs is called discounting
17
Bond pricing
❑
Replicate each cash flow:
➢
Example: If C2 is a $5 coupon in year 2, deposit the correct amount in a bank account that
yields $5 in 2 years
❑
All deposits (N deposits for N cash flows) constitute the replicating portfolio
❑
The market determines the interest rate (or discount rate) for each future time horizon
➢
The interest rates may differ for each investment horizon — the 1-year interest rate can be
different from the 2-year interest rate
➢
We typically indicate the time horizon of an interest rate with a payment index subscript: yi
18
Bond pricing
❑
Replicating the coupon cash flows before maturity for any payment i:
Ci is a cash flow (coupon) payment i (i = 1, 2, …, N-1)
Today: Deposit Mi such that Mi(1 + yi)i = Ci
The correct deposit amount Mi is therefore: Mi = Ci / (1 + yi)i
❑
Replicating the cash flow at maturity, payment N:
CN is a cash flow consisting of a coupon payment and repayment of face value
Today: Deposit MN such that MN(1 + yN)N = FV + CN
The correct deposit amount MN is therefore: MN = (FV + CN) / (1 + yN)N
❑
To execute the complete strategy:
P = M1 + M2 + … + MN-1 + MN
Therefore:
19
Bond pricing
Example:
Price a three-year bond with a face value of $1,000 and annual coupons paid at a rate of 6% per
year. The one-year discount rate is 5%, the two-year rate is 7%, and the three-year rate is 8%.
20
Yield to maturity
❑
As we saw, the bond price is obtained by discounting its future cash flows:
❑
But how do we derive the appropriate discount rates yi ?
➢ These interest rates are determined by the market – essentially as a result
of the supply and demand for investments producing cash flows in year t
(with identical risks as those from this bond)
21
Yield to maturity
❑
If the discount rate is the same for each bond payment, the bond pricing
formula becomes:
❑
This constant interest rate y is called the Yield to Maturity (YTM)
22
Bond pricing
Example:
Price a three-year bond with a face value of $1,000 and annual coupons paid at a rate of 6% per
year. The yield to maturity is 7.8974%.
23
Yield to maturity
Bonds may be priced in two ways:
❑
Use an appropriate interest rate for each cash flow:
❑
Use the yield to maturity:
The bond can only have one price, both approaches MUST give the same price.
24
Yield to maturity
Yield to Maturity (YTM) is:
➢
The single rate which gives the bond price
➢
The bond’s internal rate of return, assuming that all coupon payments can be reinvested at
the YTM
➢
Expressed on an annual basis, and denoted y
25
Computing yield to maturity
Yield to maturity requires solving a non-linear equation.
To find the YTM:
➢
Determine the bond’s cash flows,
➢
In the equation below where y is constant, take the bond price P as given and solve for y
(typically with a financial calculator or spreadsheet):
26
Bond pricing
A coupon paying bond can be viewed as the sum of two parts:
❑
An annuity consisting of all the bond’s coupon payments
❑
A single payment of face value at maturity
Under no-arbitrage, the bond price must be the present value of the annuity and the repayment of
principal:
PV of principal
repayment
PV of coupon
payments
Coupon amounts and yield to maturity are per period, not years.
N is the number of payment periods, not years.
27
Bond pricing
Example:
Price a 20-year bond with a face value of $1,000 and semi-annual coupons paid at a rate of 8%
per year. The yield to maturity is 10%.
28
YTM and bond prices
❑
Bond prices have an inverse relationship with yield to maturity
(i.e., as YTM é P ê and vice versa)
❑
The graph shows the price of a 30-year bond with a FV of $100 and a
coupon rate of 10 % for different YTMs.
Premium Bond:
P > FV (C > YTM)
400.
Price
300.
Par Bond:
P = FV (C = YTM)
200.
100.
0.
0%
5%
10%
15%
20%
YTM
29
25%
Discount Bond:
P < FV (C < YTM)
Return Measures
BKM 14.3
30
Return measures
How do we measure the return on a bond investment?
Consider the following assumptions:
❑
Coupon bonds make regular payments until maturity
❑
Bondholders reinvest coupon cash flows at the prevailing interest rates and hold the
reinvested cash flows until the maturity date of the original bond
❑
The outcome from the investment is the value of the final bond payment (coupon and face
value) plus the total value of reinvested coupon payments
31
Return measure example
Example:
Assume you can buy a bond with $1,000 face value, 10% annual coupons, and two years until
maturity at a 12% yield to maturity. What is your investment worth at maturity if coupon payments
can be reinvested at 8%?
32
Realised compound yield
❑
The steps to derive the realised compound yield are:
① Reinvest all interim cash flows at the market interest rates available at different horizons
until the end of the holding period
② Calculate the total bond proceeds as the value of all cash flow at the end of the holding
period
③ Calculate the holding period return (HPR) by dividing Total Bond Proceeds by the
original bond price P0:
▪
HPR = Total Bond Proceeds / P0
④ Annualise the return to get the realised compound yield
❑
Note that:
❑
The holding period does not need to be the bond maturity. We can assume the bond is
sold prior to maturity at the prevailing market rate.
33
Realised compound yield
0
1
2
Assume the following bond parameters
Bond
Ø FV:
$100 face value
96.62
10
10
Ø T:
2 years of time to maturity
100
Ø C:
$10 annual coupons
Ø YTM:
12% yield to maturity (calculate and derive price of $96.62)
q Suppose we can reinvest the year 1 coupon at 10%
q
Calculation
Ø The Total Bond Proceeds (aggregate cash flow) at T (note: T=2) is:
CF2 = 100 + 10 + 10(1+0.1) = 121
Ø The 2-year HPR = 121/96.62 - 1 = 0.25 (25%)
Ø The realized compound yield = (1.25)1/2 – 1 ≈ 11.9%
34
Realised compound yield
Example:
Assume you can buy a bond with $1,000 face value, 6% annual coupons, and 10 years until
maturity at a 6% yield to maturity.
What is your realized compound yield if the first coupon is reinvested at 7% per annum, the
second coupon is reinvested at 8% per annum, and you sell the bond after three years at a yield to
maturity of 9%?
35
YTM and realised returns
❑
Special case: If actual interest rates are constant and equal to the YTM, then the annualized
actual return over any holding period will be the YTM
❑
However, YTM does not generally equal the realized return over a specific holding period
because:
❑
➢
Actual market interest rates do not equal the YTM
➢
Actual market interest rates do not remain constant and change over time
The realised compound yield has an explicit relationship with yield to maturity.
➢
If interest rates fall then the realised compound yield is less than the YTM
• Why? Coupons will be reinvested at lower rates
➢
If interest rates are constant then the realised compound yield is equal to the YTM
➢
If interest rates rise then the realised compound yield is greater than the YTM
36
Return measure comparison
Yield to maturity
❑
The annualized average return if the bond is held to maturity, assuming a constant future reinvestment rate
❑
Depends on the face value, maturity, coupon payments, and bond price
❑
All of these variables are readily observable
Realized compound yield
❑
The annualised rate of return over a particular investment period, allowing for changing future reinvestment rate
❑
Depends on the face value, maturity, coupon payments, and original bond price parameters plus
future interest rates and bond price at the end of the holding period (if sold)
❑
Can only be forecast, not readily observable
37
Term Structure of Interest Rates
BKM 15.1
38
Example of bond price quotes
Source: FactSet
39
The yield curve
❑
The term structure of interest rates describes how interest rates are
expected to vary over different investment horizons
❑
The term structure of interest rates is often called the Yield Curve
➢ The yield curve displays the relationship between yield and maturity
➢ The yield curve reflect market expectations on future interest rates
40
❑
Interest rates can be thought
of as the prices of future cash
flows. When these prices
change, the yield curve shifts
❑
As seen in the graph, the yield
curve is typically upwardsloping but can also be flat or
downward-sloping (“inverted”)
Zero coupon bonds
❑
A zero coupon bond (or « zero ») is a bond that makes no coupon payments,
making a single payment of principal at maturity
❑
As the bond’s coupon rate is zero, the entire return comes from price appreciation
➢ Zero coupon bonds avoid reinvestment risk as there are no coupons to reinvest
➢ They trade at a deep discount to par value
❑
A coupon bond can be viewed as a portfolio of zero coupon bonds
➢ Each coupon can be considered a maturing zero-coupon bond of the same term
➢ Bond stripping is the process of repacking each coupon and principal payment
from a coupon bond as a new, separate zero coupon bond
41
Yield curve types
Pure yield curve
❑
Derived from zero-coupon or stripped bond prices
❑
May be difficult to derive, as relevant zero coupon bonds may not be available in the
market
On-the-run Yield Curve
❑
Derived from recently issued (on-the-run) coupon bonds selling at or near par
❑
On-the-run bonds have the greatest liquidity in the market
❑
The financial press typically publishes on-the-run yield curves
Note: The pure yield curve may differ significantly from the on-the-run yield curve
42
Spot rates
❑
Yields for different maturity zero-coupon bonds shown on a pure yield curve are
known as spot rates
❑
The spot rate yt. is the interest rate today (at time 0) for a t-period zero coupon
bond. That is a bond that is purchased today and maturing at time t.
❑
The spot rate yt specifically refers to zero-coupon bonds and, as a result, may
differ from the YTM (y) of a t-period coupon bond.
❑
The spot rate is sometimes informally called the “pure yield”
43
Inferring the term structure
If we have zero coupon bond prices for all maturities, the term structure is found by solving for the
yield at each maturity individually.
❑
Assume a 1-year zero coupon bond trades for $90.91 and a 2-year zero has a price of $79.72.
❑
What are the 1-y1 and 2-year yields?
y1
y2
y2
❑
The zero-coupon bond pricing formula is:
❑
The yields are:
44
Inferring the term structure
q
Now consider a 1-year zero coupon bond and a 2-year coupon bond:
y1
Bond A
-PA
FVA
y2
-PB
c
Bond B
q
C+FV B
How do we derive the 1-year and 2-year spot rates – y1 and y2?
Ø y1 can be inferred from bond A’s price as shown on the previous slide
Ø Bond B’s pricing equation is given by:
P =
"!
#$%!
!
+
('($ "")
#$%" "
Ø So we can easily calculate 𝑦2 based on Bond B’s price given y1
45
Inferring the term structure
qConsider the following 1-year zero coupon bond and 2-year coupon bond:
y1
Bond A
-90.91
100
y2
Bond B
10
-95.78
110
qWe derive y1 and y2 (and thus the term structure) as follows:
Ø y1 = 10% as shown in the previous example
Ø y2 is derived as:
"#
""#
𝑃! = 95.78 =
+
("%"#%)
("%(! )!
46
→ 𝑦) ≅ 12.65%
Inferring the term structure
A generalised method:
❑
Start with the 1-period zero and compute y1 from the price equation of the 1-period
zero coupon bond
❑
Substitute y1 into the price equation for the 2-period coupon bond and solve for y2
❑
Repeat to bootstrap this process to find the spot rate yt by substituting y1, y2, …, yt-1
into the price equation of t-period coupon bond
47
Next week
48
Next lecture
Future Interest Rates and Duration
❑
❑
Key Concepts
➢
Bond Arbitrage
➢
Future interest rates – short rates and forward rates
➢
Term structure hypotheses
➢
Interest rate risk
➢
Duration and convexity
Readings
➢
BKM 15 Term Structure of Interest Rates: 15.2 – 15.4
➢
BKM 16 Managing Bond Portfolios: 16.1 – 16.3
49