Bond valuation

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

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).

{\begin{aligned}TFV&={\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=

par 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)

M=

value at maturity, usually equals par value

TFV=

theoretical fair value

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.^[5]

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.^[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

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 $P$ , 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.^[7] — 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 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

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 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 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:

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

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

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 ]}

$n\in \{0,1,...,N-1\}$

$a_{n+1}$ = 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

In finance, discounting is a 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. This transaction is based on the fact that most people prefer current interest to delayed interest because of mortality effects, impatience effects, and salience effects. 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 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 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 equal or 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 than tomorrow. 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 borrowed funds is less than the total amount of money paid to the lender.

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 parabolic 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 given the risk of the security and its expected return. The equation and model are named after economists Fischer Black and Myron Scholes. Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited.

The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference.

In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of a derivative instrument such as an option to changes in one or more underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are denoted by Greek letters. Collectively these have also been called the risk sensitivities, risk measures or hedge parameters.

In finance, a perpetuity is an annuity that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence. For example, the United Kingdom (UK) government issued them in the past; these were known as consols and were all finally redeemed in 2015.

The yield to maturity (YTM), book yield or redemption yield of a fixed-interest security is an estimate of the total rate of return anticipated to be earned by an investor who buys it at a given market price, holds it to maturity, and receives all interest payments and the capital redemption on schedule.

Rational pricing is the assumption in financial economics that asset prices – and hence asset pricing models – 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, such as 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, and is defined as 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. Convexity was based on the work of Hon-Fei Lai and popularized by Stanley Diller.

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 (coupon) payment and the bond's price:

In finance, interest rate immunization is a portfolio management strategy designed to take advantage of the offsetting effects of interest rate risk and reinvestment risk.

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 bond 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.

Fixed-income attribution is the process of measuring returns generated by various sources of risk in a fixed income portfolio, particularly when multiple sources of return are active at the same time.

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, par yield is the yield on a fixed income security assuming that its market price is equal to par value. Par yield is used to derive the U.S. Treasury’s daily official “Treasury Par Yield Curve Rates”, which are used by investors to price debt securities traded in public markets, and by lenders to set interest rates on many other types of debt, including bank loans and mortgages.

Option-adjusted spread (OAS) is the yield spread which has to be added to a benchmark yield curve to discount a security's payments to match its market price, using a dynamic pricing model that accounts for embedded options. OAS is hence model-dependent. This concept can be applied to a mortgage-backed security (MBS), or another bond with embedded options, or any other interest rate derivative or option. More loosely, the OAS of a security can be interpreted as its "expected outperformance" versus the benchmarks, if the cash flows and the yield curve behave consistently with the valuation model.

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).

References

↑ 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.
1 2 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.
1 2 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

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

Bond Valuation, Prof. Campbell R. Harvey, Duke University
A Primer on the Time Value of Money, Prof. Aswath Damodaran, Stern School of Business
Bond Price Volatility Investment Analysts Society of South Africa
Duration and convexity Investment Analysts Society of South Africa

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[1] 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.

[bodi-2] 1 2 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)

[3] 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.

[Fabozzi-4] 1 2 Fabozzi, 1998

[5] 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.

[6] For a derivation, analogous to Black-Scholes, see: David Mandel (2015). "Understanding Market Price of Risk", Florida State University

[7] 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

[1]

[2]

[3]

[4]

[5]

[6]

[7]

v t e Bond market
Bond Debenture Fixed income
Types of bonds by issuer	Agency bond Corporate bond Senior debt Subordinated debt Distressed debt Government bond Infrastructure bond Municipal bond Global bond
Types of bonds by payout	Accrual bond Auction rate security Callable bond Commercial paper Consol Contingent convertible bond Convertible bond Exchangeable bond Extendible bond Fixed rate bond Floating rate note High-yield debt Inflation-indexed bond Inverse floating rate note Lottery bond Perpetual bond Puttable bond Reverse convertible securities Zero-coupon bond
Bond valuation	Clean price Convexity Coupon Credit spread Current yield Dirty price Duration I-spread Mortgage yield Nominal yield Option-adjusted spread Risk-free bond Weighted-average life Yield curve Yield spread Yield to maturity Z-spread
Securitized products	Asset-backed security Collateralized debt obligation Collateralized mortgage obligation Commercial mortgage-backed security Mortgage-backed security
Bond options	Callable bond Convertible bond Embedded option Exchangeable bond Extendible bond Puttable bond
Institutions	Commercial Mortgage Securities Association (CMSA) International Capital Market Association (ICMA) Securities Industry and Financial Markets Association (SIFMA)

Bond valuation

Contents

Bond valuation

Present value approach

Relative price approach

Arbitrage-free pricing approach

Stochastic calculus approach

Clean and dirty price

Yield and price relationships

Yield to maturity

Coupon rate

Current yield

Relationship

Price sensitivity

Accounting treatment

See also

Related Research Articles

References

Selected bibliography

External links