# Bond valuation

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Bond valuation is the determination of the fair price 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. In finance, a bond is an instrument of indebtedness of the bond issuer to the holders. The most common types of bonds include municipal bonds and corporate bonds.

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 always less than or equal to the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be more than the future value. Time value can be described with the simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that its value is greater. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be compared to rent. Just as rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of the borrowed funds is less than the total amount of money paid to the lender.

## Contents

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.

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. See further under Bond option. This total is then the value of the bond.

An embedded option is a component of a financial bond or other security, and usually 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. Some common types of bonds with embedded options include callable bond, puttable bond, convertible bond, extendible bond, exchangeable bond, and capped floating rate note. A bond may have several options embedded if they are not mutually exclusive.

In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC.

## Bond valuation

As above, 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. 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 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. 

### 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 for adjustments on other dates.)

{\begin{aligned}P&={\begin{matrix}\left({\frac {C}{1+i}}+{\frac {C}{(1+i)^{2}}}+...+{\frac {C}{(1+i)^{N}}}\right)+{\frac {M}{(1+i)^{N}}}\end{matrix}}\\&={\begin{matrix}\left(\sum _{n=1}^{N}{\frac {C}{(1+i)^{n}}}\right)+{\frac {M}{(1+i)^{N}}}\end{matrix}}\\&={\begin{matrix}C\left({\frac {1-(1+i)^{-N}}{i}}\right)+M(1+i)^{-N}\end{matrix}}\end{aligned}} where:
F = face values
iF = contractual interest rate
C = F * iF = coupon payment (periodic interest payment)
N = number of payments
i = market interest rate, or required yield, or observed / appropriate yield to maturity (see 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 $i$ in the formula above, to obtain the price. A government bond or sovereign bond is a bond issued by a national government, generally with a promise to pay periodic interest payments called coupon payments and to repay the face value on the maturity date. The aim of a government bond is to support government spending. Government bonds are usually denominated in the country's own currency, in which case the government cannot be forced to default, although it may choose to do so. If a government is close to default on its debt the media often refer to this as a sovereign debt crisis.

Relative valuation also called valuation using multiples is the notion of comparing the price of an asset to the market value of similar assets. In the field of securities investment, the idea has led to important practical tools, which could presumably spot pricing anomalies. These tools have subsequently become instrumental in enabling analysts and investors to make vital decisions on asset allocation.

A credit rating is an evaluation of the credit risk of a prospective debtor, predicting their ability to pay back the debt, and an implicit forecast of the likelihood of the debtor defaulting. The credit rating represents an evaluation of a credit rating agency of the qualitative and quantitative information for the prospective debtor, including information provided by the prospective debtor and other non-public information obtained by the credit rating agency's analysts.

### 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 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. 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. See under Rational pricing #Fixed income securities.

### Stochastic calculus approach

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, is satisfied by any zero-coupon bond $P$ , over (instantaneous) time $t$ , for corresponding changes in $r$ , the short rate.

${\frac {1}{2}}\sigma (r)^{2}{\frac {\partial ^{2}P}{\partial r^{2}}}+[a(r)+\sigma (r)+\varphi (r,t)]{\frac {\partial P}{\partial r}}+{\frac {\partial P}{\partial t}}-rP=0$ The solution to the PDE (i.e. the corresponding formula for bond value) — given in Cox et al.  — is:

$P[t,T,r(t)]=E_{t}^{\ast }[e^{-R(t,T)}]$ where $E_{t}^{\ast }$ is the expectation with respect to risk-neutral probabilities, and $R(t,T)$ 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

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 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

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

The yield to maturity (YTM) is the discount rate which returns the market price of a bond without embedded optionality; it is identical to $i$ (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 $P_{0}$ ,
• hold the bond until maturity, and
• redeem the bond at par.

### Coupon rate

The coupon rate is simply the coupon payment $C$ as a percentage of the face value $F$ .

${\text{Coupon rate}}={\frac {C}{F}}$ Coupon yield is also called nominal yield.

### Current yield

The current yield is simply the coupon payment $C$ as a percentage of the (current) bond price $P$ .

${\text{Current yield}}={\frac {C}{P_{0}}}.$ ### Relationship

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:

• When a bond sells at a discount, YTM > current yield > coupon yield.
• When a bond sells at a premium, coupon yield > current yield > YTM.
• When a bond sells at par, YTM = current yield = coupon yield

## Price sensitivity

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.

## Accounting treatment

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:

$n\in \{0,1,...,N-1\}$ $a_{n+1}$ = amortization amount in period number "n+1"

$a_{n+1}=|iP-C|{(1+i)}^{n}$ Bond Discount or Bond Premium = $|F-P|$ = $a_{1}+a_{2}+...+a_{N}$ Bond Discount or Bond Premium = $F|i-i_{F}|({\frac {1-(1+i)^{-N}}{i}})$ ## Related Research Articles Discounting is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee. Essentially, the party that owes money in the present purchases the right to delay the payment until some future date. The discount, or charge, is the difference between the original amount owed in the present and the amount that has to be paid in the future to settle the debt.

In finance, the net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount rate. NPV accounts for the time value of money. It provides a method for evaluating and comparing capital projects or financial products with cash flows spread over time, as in loans, investments, payouts from insurance contracts plus many other applications.

The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return. The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. It is widely used, although often with some adjustments, by options market participants. The time value of money is the greater benefit of receiving money now rather than an identical sum later. It is founded on time preference.

The Black model is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing options on future contracts, bond options, Interest rate cap and floors, and swaptions. It was first presented in a paper written by Fischer Black in 1976. The yield to maturity (YTM), book yield or redemption yield of a bond or other fixed-interest security, such as gilts, is the (theoretical) internal rate of return earned by an investor who buys the bond today at the market price, assuming that the bond is held until maturity, and that all coupon and principal payments are made on schedule. Yield to maturity is the discount rate at which the sum of all future cash flows from the bond is equal to the current price of the bond. The YTM is often given in terms of Annual Percentage Rate (A.P.R.), but more often market convention is followed. In a number of major markets the convention is to quote annualized yields with semi-annual compounding ; thus, for example, an annual effective yield of 10.25% would be quoted as 10.00%, because 1.05 × 1.05 = 1.1025 and 2 × 5 = 10.

Rational pricing is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments. In finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields.

In finance, bond convexity is a measure of the non-linear relationship of bond prices to changes in interest rates, the second derivative of the price of the bond with respect to interest rates. In general, the higher the duration, the more sensitive the bond price is to the change in interest rates. Bond convexity is one of the most basic and widely used forms of convexity in finance.

The current yield, interest yield, income yield, flat yield, market yield, mark to market yield or running yield is a financial term used in reference to bonds and other fixed-interest securities such as gilts. It is the ratio of the annual interest payment and the bond's current clean price:

Fixed income analysis is the valuation of fixed income or debt securities, and the analysis of their interest rate risk, credit risk, and likely price behavior in hedging portfolios. The analyst might conclude to buy, sell, hold, hedge or stay out of the particular security. In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on the option's maturity date. For interest rate derivatives lattices are additionally useful in that they address many of the issues encountered with continuous models, such as pull to par. The method is also used for valuing certain exotic options, where because of path dependence in the payoff, Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise., though methods now exist for solving this problem.

The Z-spread, ZSPRD, zero-volatility spread or yield curve spread of a mortgage-backed security (MBS) is the parallel shift or spread over the zero-coupon Treasury yield curve required for discounting a pre-determined cash flow schedule to arrive at its present market price. The Z-spread is also widely used in the credit default swap (CDS) market as a measure of credit spread that is relatively insensitive to the particulars of specific corporate or government bonds.

The following outline is provided as an overview of and topical guide to finance:

In finance, bootstrapping is a method for constructing a (zero-coupon) fixed-income yield curve from the prices of a set of coupon-bearing products, e.g. bonds and swaps.

In finance, mortgage yield is a measure of yield of mortgage-backed bonds. It is also known as cash flow yield. The mortgage yield, or cash flow yield, of a mortgage-backed bond is the monthly compounded discount rate at which net present value of all future cash flows from the bond will be equal to the present price of the bond.

In finance, the weighted-average life (WAL) of an amortizing loan or amortizing bond, also called average life, is the weighted average of the times of the principal repayments: it's the average time until a dollar of principal is repaid.

In finance, a zero coupon swap (ZCS) is an interest rate derivative (IRD). In particular it is a linear IRD, that in its specification is very similar to the much more widely traded interest rate swap (IRS).

1. Staff, Investopedia (8 May 2008). "Amortizable Bond Premium".
2. Fabozzi, 1998
3. "Advanced Bond Concepts: Bond Pricing". 6 September 2016.
4. 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