Bond valuation is the determination of the
fair price
In accounting and in most schools of economic thought, fair value is a rational and unbiased estimate of the potential market price of a good, service, or asset. The derivation takes into account such objective factors as the costs associated ...
of a
bond
Bond or bonds may refer to:
Common meanings
* Bond (finance), a type of debt security
* Bail bond, a commercial third-party guarantor of surety bonds in the United States
* Chemical bond, the attraction of atoms, ions or molecules to form chemical ...
. As with any security or capital investment, the theoretical fair value of a bond is the
present value
In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
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.
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 face value (par value), the bond is selling at a discount. Conversely, if the market price of bond is greater than its face value, the bond is selling at a premium. For this and other relationships between price and yield, see
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
.
If the bond includes
embedded option
An embedded option is a component of a financial bond or other security, which provides the bondholder or the issuer the right to take some action against the other party. There are several types of options that can be embedded into a bond; common ...
s, 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. See
further under
Bond option. This total is then the value of the bond.
Bond valuation
As above, the fair price of a "straight bond" (a bond with no
embedded option
An embedded option is a component of a financial bond or other security, which provides the bondholder or the issuer the right to take some action against the other party. There are several types of options that can be embedded into a bond; common ...
s; see ) is usually determined by discounting its expected cash flows at the appropriate discount rate. The formula commonly applied is discussed initially. 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
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, ...
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.
[Fabozzi, 1998]
Present value approach
Below is the formula for calculating a bond's price, which uses the basic present value (PV) formula for a given discount rate.
This formula assumes that a coupon payment has just been made; see
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
for adjustments on other dates.
:
:where:
::F = face value
::i
F = contractual interest rate
::C = F * i
F = coupon payment (periodic interest payment)
::N = number of payments
::i = market interest rate, or required yield, or observed / appropriate
yield to maturity (see
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
)
::M = value at maturity, usually equals face value
::P = market price of bond.
Relative price approach
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.
Arbitrage-free pricing approach
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.
Here, each cash flow is separately discounted at the same rate as a
zero-coupon bond
A zero coupon bond (also discount bond or deep discount bond) is a bond in which the face value is repaid at the time of maturity. Unlike regular bonds, it does not make periodic interest payments or have so-called coupons, hence the term zero- ...
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
Credit (from Latin verb ''credit'', meaning "one believes") is the trust which allows one party to provide money or resources to another party wherein the second party does not reimburse the first party immediately (thereby generating a deb ...
).
Under this approach, the bond price should reflect its "
arbitrage
In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between t ...
-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. Were this not the case, (4) the arbitrageur could finance his purchase of whichever of the bond or the sum of the various ZCBs was cheaper, by
short selling the other, and meeting his cash flow commitments using the coupons or maturing zeroes as appropriate. Then (5) his "risk free", arbitrage profit would be the difference between the two values.
Stochastic calculus approach
When modelling a
bond option, or other
interest rate derivative
In finance, an interest rate derivative (IRD) is a derivative whose payments are determined through calculation techniques where the underlying benchmark product is an interest rate, or set of different interest rates. There are a multitude of diff ...
(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
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...
) number. In such cases,
stochastic calculus is employed.
The following is a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
(PDE) in stochastic calculus, which, by arbitrage arguments,
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.
[ John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross (1985)]
A Theory of the Term Structure of Interest Rates
, ''Econometrica
''Econometrica'' is a peer-reviewed academic journal of economics, publishing articles in many areas of economics, especially econometrics. It is published by Wiley-Blackwell on behalf of the Econometric Society. The current editor-in-chief is ...
'' 53:2 — is: