Fair price of a bond
Bond valuation
is the process by which an investor arrives at an estimate of the theoretical fair value, or intrinsic worth, of a
bond
. As with any security or capital investment, the theoretical fair value of a bond is the
present value
of the stream of cash flows it is expected to generate. Hence, the value of a bond is obtained by discounting the bond's expected cash flows to the present using an
appropriate discount rate
.
[1]
[2]
In practice, this discount rate is often determined by reference to similar instruments, provided that such instruments exist. Various related yield-measures are then calculated for the given price. Where the market price of bond is less than its
par value
, the bond is selling at a
discount
. Conversely, if the market price of bond is greater than its par value, the bond is selling at a
premium
. For this and other relationships between price and yield, see
below
.
If the bond includes
embedded options
, the valuation is more difficult and combines
option pricing
with discounting. Depending on the type of option, the
option price
as calculated is either added to or subtracted from the price of the "straight" portion.
[3]
See
further
under
Bond option
. This total is then the value of the bond.
Bond valuation
[
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The fair price of a "straight bond" (a bond with no
embedded options
; see
Bond (finance) § Features
) is usually determined by discounting its expected cash flows at the appropriate discount rate. Although this present value relationship reflects the theoretical approach to determining the value of a bond, in practice its price is (usually) determined with reference to other, more
liquid
instruments. The two main approaches here, Relative pricing and Arbitrage-free pricing, are discussed next. Finally, where it is important to recognise that future interest rates are uncertain and that the discount rate is not adequately represented by a single fixed number?for example
when an option is written on the bond in question
?stochastic calculus may be employed.
[4]
Present value approach
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The basic method for calculating a bond's theoretical fair value, or intrinsic worth, uses the
present value
(PV) formula shown below, using a single market interest rate to discount cash flows in all periods. A more complex approach would use different interest rates for cash flows in different periods.
[2]
: 294
The formula shown below assumes that a coupon payment has just been made (see
below
for adjustments on other dates).
- where:
- par value
- contractual interest rate
- coupon payment (periodic interest payment)
- number of payments
- market interest rate, or required yield, or observed / appropriate
yield to maturity
(see
below
)
- value at maturity, usually equals par value
- theoretical fair value
Relative price approach
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Under this approach?an extension, or application, of the above?the bond will be priced relative to a benchmark, usually a
government security
; see
Relative valuation
. Here, the yield to maturity on the bond is determined based on the bond's
Credit rating
relative to a government security with similar maturity or
duration
; see
Credit spread (bond)
. The better the quality of the bond, the smaller the spread between its required return and the YTM of the benchmark. This required return is then used to discount the bond cash flows, replacing
in the formula above, to obtain the price.
[5]
Arbitrage-free pricing approach
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As distinct from the two related approaches above, a bond may be thought of as a "package of cash flows"?coupon or face?with each cash flow viewed as a
zero-coupon
instrument maturing on the date it will be received. Thus, rather than using a single discount rate, one should use multiple discount rates, discounting each cash flow at its own rate.
[4]
Here, each cash flow is separately discounted at the same rate as a
zero-coupon bond
corresponding to the coupon date, and of equivalent credit worthiness (if possible, from the same issuer as the bond being valued, or if not, with the appropriate
credit spread
).
Under this approach, the bond price should reflect its "
arbitrage
-free" price, as any deviation from this price will be exploited and the bond will then quickly reprice to its correct level. Here, we apply the
rational pricing
logic relating to
"Assets with identical cash flows"
. In detail: (1) the bond's coupon dates and coupon amounts are known with certainty. Therefore, (2) some multiple (or fraction) of zero-coupon bonds, each corresponding to the bond's coupon dates, can be specified so as to produce identical cash flows to the bond. Thus (3) the bond price today must be equal to the sum of each of its cash flows discounted at the discount rate implied by the value of the corresponding ZCB.
Stochastic calculus approach
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When modelling a
bond option
, or other
interest rate derivative
(IRD), it is important to recognize that future interest rates are uncertain, and therefore, the discount rate(s) referred to above, under all three cases?i.e. whether for all coupons or for each individual coupon?is not adequately represented by a fixed (
deterministic
) number. In such cases,
stochastic calculus
is employed.
The following is a
partial differential equation
(PDE) in stochastic calculus, which,
by arbitrage arguments
,
[6]
is satisfied by any zero-coupon bond
, over (instantaneous) time
, for corresponding changes in
, the
short rate
.
The solution to the PDE (i.e. the corresponding formula for bond value) ? given in Cox et al.
[7]
? is:
- where
is the
expectation
with respect to
risk-neutral probabilities
, and
is a random variable representing the discount rate; see also
Martingale pricing
.
To actually determine the bond price, the analyst must choose the specific
short-rate model
to be employed. The approaches commonly used are:
Note that depending on the model selected, a
closed-form
(
“Black like”
) solution may not be available, and a
lattice-
or
simulation-based
implementation of the model in question is then employed. See also
Bond option § Valuation
.
Clean and dirty price
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When the bond is not valued precisely on a coupon date, the calculated price, using the methods above, will incorporate
accrued interest
: i.e. any interest due to the owner of the bond over the "
stub period
" since the previous coupon date (see
day count convention
). The price of a bond which includes this accrued interest is known as the "
dirty price
" (or "full price" or "all in price" or "Cash price"). The "
clean price
" is the price excluding any interest that has accrued. Clean prices are generally more stable over time than dirty prices. This is because the dirty price will drop suddenly when the bond goes "ex interest" and the purchaser is no longer entitled to receive the next coupon payment.
In many markets, it is market practice to quote bonds on a clean-price basis. When a purchase is settled, the accrued interest is added to the quoted clean price to arrive at the actual amount to be paid.
Yield and price relationships
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Once the price or value has been calculated, various
yields
relating the price of the bond to its coupons can then be determined.
Yield to maturity
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The
yield to maturity
(YTM) is the discount rate which returns the
market price
of a bond without embedded optionality; it is identical to
(required return) in the
above equation
. YTM is thus the
internal rate of return
of an investment in the bond made at the observed price. Since YTM can be used to price a bond, bond prices are often quoted in terms of YTM.
To achieve a return equal to YTM, i.e. where it is the required return on the bond, the bond owner must:
- buy the bond at price
,
- hold the bond until maturity, and
- redeem the bond at par.
Coupon rate
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The
coupon rate
is the coupon payment
as a percentage of the face value
.
Coupon yield is also called
nominal yield
.
Current yield
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The
current yield
is the coupon payment
as a percentage of the (
current
) bond price
.
Relationship
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The concept of current yield is closely related to other bond concepts, including yield to maturity, and coupon yield. The relationship between yield to maturity and the coupon rate is as follows:
Relationship between yield to maturity and the coupon rate
Status
|
Connection
|
At a discount
|
YTM > current yield > coupon yield
|
At a premium
|
coupon yield > current yield > YTM
|
Sells at par
|
YTM = current yield = coupon yield
|
Price sensitivity
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The
sensitivity
of a bond's market price to interest rate (i.e. yield) movements is measured by its
duration
, and, additionally, by its
convexity
.
Duration is a
linear measure
of how the price of a bond changes in response to interest rate changes. It is approximately equal to the percentage change in price for a given change in yield, and may be thought of as the
elasticity
of the bond's price with respect to discount rates. For example, for small interest rate changes, the duration is the approximate percentage by which the value of the bond will fall for a 1% per annum increase in market interest rate. So the market price of a 17-year bond with a duration of 7 would fall about 7% if the market interest rate (or more precisely the corresponding
force of interest
) increased by 1% per annum.
Convexity is a measure of the "
curvature
" of price changes. It is needed because the price is not a linear function of the discount rate, but rather a
convex function
of the discount rate. Specifically, duration can be formulated as the
first derivative
of the price with respect to the interest rate, and convexity as the
second derivative
(see:
Bond duration closed-form formula
;
Bond convexity closed-form formula
;
Taylor series
). Continuing the above example, for a more accurate estimate of sensitivity, the convexity score would be multiplied by the square of the change in interest rate, and the result added to the value derived by the above linear formula.
For embedded options, see
effective duration
and
effective convexity
.
Accounting treatment
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In
accounting
for
liabilities
, any bond discount or premium must be
amortized
over the life of the bond. A number of methods may be used for this depending on applicable accounting rules. One possibility is that amortization amount in each period is calculated from the following formula:
[
citation needed
]
= amortization amount in period number "n+1"
Bond Discount or Bond Premium =
=
Bond Discount or Bond Premium =
See also
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References
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- ^
Malkiel, Burton G. (1962).
"Expectations, Bond Prices, and the Term Structure of Interest Rates"
.
The Quarterly Journal of Economics
.
76
(2): 197?218.
doi
:
10.2307/1880816
.
ISSN
0033-5533
.
JSTOR
1880816
.
- ^
a
b
Bodi, Zvi; Kane, Alex.; Marcus, Alan J. (2010).
Essentials of Investments
(eighth ed.). New York: McGraw-Hill/Irwin.
ISBN
978-0-07-338240-1
.
{{
cite book
}}
: CS1 maint: multiple names: authors list (
link
)
- ^
Kalotay, Andrew J.; Williams, George O.; Fabozzi, Frank J. (1993).
"A Model for Valuing Bonds and Embedded Options"
.
Financial Analysts Journal
.
49
(3): 35?46.
doi
:
10.2469/faj.v49.n3.35
.
ISSN
0015-198X
– via Taylor & Francis.
- ^
a
b
Fabozzi, 1998
- ^
Jones, E. Philip; Mason, Scott P.; Rosenfeld, Eric (1984).
"Contingent Claims Analysis of Corporate Capital Structures: An Empirical Investigation"
.
The Journal of Finance
.
39
(3): 611?625.
doi
:
10.2307/2327919
.
ISSN
0022-1082
.
JSTOR
2327919
.
- ^
For a derivation,
analogous to Black-Scholes
, see: David Mandel (2015).
"Understanding Market Price of Risk"
,
Florida State University
- ^
John C. Cox
,
Jonathan E. Ingersoll
and
Stephen A. Ross
(1985).
A Theory of the Term Structure of Interest Rates
Archived
2011-10-03 at the
Wayback Machine
,
Econometrica
53:2
Selected bibliography
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- Guillermo L. Dumrauf (2012).
"Chapter 1: Pricing and Return"
.
Bonds, a Step by Step Analysis with Excel
. Kindle Edition.
- Frank Fabozzi
(1998).
Valuation of fixed income securities and derivatives
(3rd ed.).
John Wiley
.
ISBN
978-1-883249-25-0
.
- Frank J. Fabozzi (2005).
Fixed Income Mathematics: Analytical & Statistical Techniques
(4th ed.). John Wiley.
ISBN
978-0071460736
.
- R. Stafford Johnson (2010).
Bond Evaluation, Selection, and Management
(2nd ed.). John Wiley.
ISBN
978-0470478356
.
- Mayle, Jan (1993),
Standard Securities Calculation Methods: Fixed Income Securities Formulas for Price, Yield and Accrued Interest
, vol. 1 (3rd ed.),
Securities Industry and Financial Markets Association
,
ISBN
1-882936-01-9
- Donald J. Smith (2011).
Bond Math: The Theory Behind the Formulas
. John Wiley.
ISBN
978-1576603062
.
- Bruce Tuckman (2011).
Fixed Income Securities: Tools for Today's Markets
(3rd ed.). John Wiley.
ISBN
978-0470891698
.
- Pietro Veronesi (2010).
Fixed Income Securities: Valuation, Risk, and Risk Management
. John Wiley.
ISBN
978-0470109106
.
- Burton Malkiel
(1962).
"Expectations, Bond Prices, and the Term Structure of Interest Rates"
. The Quarterly Journal of Economics.
- Mark Mobius
(2012).
Bonds: An Introduction to the Core Concepts
. John Wiley.
ISBN
978-0470821473
.
External links
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