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.
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.
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] 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).
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
[5] 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.
[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).
Here, we apply the rational pricing logic relating to "Assets with identical cash flows".
whether for all coupons or for each individual coupon—is not adequately represented by a fixed (deterministic) number.
The following is a partial differential equation (PDE) in stochastic calculus, which, by arbitrage arguments,[6] is satisfied by any zero-coupon bond
The solution to the PDE (i.e. the corresponding formula for bond value) — given in Cox et al.[7] — is:
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.
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).
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.
When a purchase is settled, the accrued interest is added to the quoted clean price to arrive at the actual amount to be paid.
The yield to maturity (YTM) is the discount rate which returns the market price of a bond without embedded optionality; it is identical to
YTM is thus the internal rate of return of an investment in the bond made at the observed price.
To achieve a return equal to YTM, i.e. where it is the required return on the bond, the bond owner must: The coupon rate is the coupon payment
The relationship between yield to maturity and the coupon rate is as follows: 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.
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.
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.
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]