Bond valuation
Encyclopedia
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 practice, this discount rate is often determined by reference to similar instruments, provided that such instruments exist.
If the bond includes embedded option
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. This total is then the value of the bond; the various yields can then be calculated for the total price. See further under Bond option
.
; see Embedded Option) 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, 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.
(This formula assumes that a coupon payment has just been made; see below for adjustments on other dates.)
If 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.
; 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,
in the formula, is then used to discount the bond cash flows as above to obtain the price.
-free price. Here, each cash flow (coupon or face) 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, in general, 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 abitrageur 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 Rational pricing: Fixed income securities.
(PDE) in stochastic calculus
which is satisfied by any zero-coupon bond. This methodology recognises that since future interest rates are uncertain, the discount rate referred to above is not adequately represented by a single fixed number.
The solution to the PDE is given in
where
is the expectation with respect to risk-neutral probabilities
, and
is a random variable representing the discount rate; see also Martingale pricing
.
Practically, to determine the bond price, specific short rate model
s are employed here. However, when using these models, it is often the case that no closed form
solution exists, and a lattice-
or simulation-based implementation of the model in question is employed. The approaches commonly used are:
: 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.
- which relate the price of the bond to its coupons - can then be determined.
is the discount rate which returns the market price
of the bond; it is identical to r (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:
is simply the coupon payment (C) as a percentage of the face value (F).
Coupon yield is also called nominal yield
.
is simply the coupon payment (C) as a percentage of the (current) bond price (P).
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). 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.
, any bond discount or premium must be amortized
over the life of 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:
= amortization amount in period number "n+1"
Bond Discount or Bond Premium =
= 
Bond Discount or Bond Premium =
Calculators
Bond (finance)
In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...
. As with any security or capital investment, the theoretical fair value of a bond is the present value
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
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
Discount rate
The discount rate can mean*an interest rate a central bank charges depository institutions that borrow reserves from it, for example for the use of the Federal Reserve's discount window....
. In practice, this discount rate is often determined by reference to similar instruments, provided that such instruments exist.
If the bond includes embedded option
Embedded option
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...
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. This total is then the value of the bond; the various yields can then be calculated for the total price. See further under Bond option
Bond option
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 optionOption (finance)
In finance, an option is a derivative financial instrument that specifies a contract between two parties for a future transaction on an asset at a reference price. The buyer of the option gains the right, but not the obligation, to engage in that transaction, while the seller incurs the...
; see Embedded Option) is usually determined by discounting its expected cash flows at the appropriate discount rate
Discount rate
The discount rate can mean*an interest rate a central bank charges depository institutions that borrow reserves from it, for example for the use of the Federal Reserve's discount window....
. 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, 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
Bond option
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....
- 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.)
- where:
- F = face value
- 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 maturityYield to maturityThe Yield to maturity or redemption yield of a bond or other fixed-interest security, such as gilts, is the internal rate of return earned by an investor who buys the bond today at the market price, assuming that the bond will be held until maturity, and that all coupon and principal payments...
(see below) - M = value at maturity, usually equals face value
- P = market price of bond.
If 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.
Relative price approach
Under this approach, the bond will be priced relative to a benchmark, usually a government securityGovernment bond
A government bond is a bond issued by a national government denominated in the country's own currency. Bonds are debt investments whereby an investor loans a certain amount of money, for a certain amount of time, with a certain interest rate, to a company or country...
; see Relative valuation
Relative valuation
Relative valuation is a generic term that refers to 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...
. Here, the yield to maturity on the bond is determined based on the bond's Credit rating
Credit rating
A credit rating evaluates the credit worthiness of an issuer of specific types of debt, specifically, debt issued by a business enterprise such as a corporation or a government. It is an evaluation made by a credit rating agency of the debt issuers likelihood of default. Credit ratings are...
relative to a government security with similar maturity or duration
Bond duration
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....
; see Credit spread (bond)
Credit spread (bond)
The financial term, credit spread is the yield spread, or difference in yield between different securities, due to different credit quality. The credit spread reflects the additional net yield an investor can earn from a security with more credit risk relative to one with less credit risk...
. The better the quality of the bond, the smaller the spread between its required return and the YTM of the benchmark. This required return,
in the formula, is then used to discount the bond cash flows as above to obtain the price.Arbitrage-free pricing approach
Under this approach, the bond price will reflect its arbitrageArbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
-free price. Here, each cash flow (coupon or face) 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
Credit spread (bond)
The financial term, credit spread is the yield spread, or difference in yield between different securities, due to different credit quality. The credit spread reflects the additional net yield an investor can earn from a security with more credit risk relative to one with less credit risk...
). Here, in general, we apply the rational pricing
Rational pricing
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"...
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
Discount rate
The discount rate can mean*an interest rate a central bank charges depository institutions that borrow reserves from it, for example for the use of the Federal Reserve's discount window....
implied by the value of the corresponding ZCB. Were this not the case, (4) the abitrageur could finance his purchase of whichever of the bond or the sum of the various ZCBs was cheaper, by short selling
Short selling
In finance, short selling is the practice of selling assets, usually securities, that have been borrowed from a third party with the intention of buying identical assets back at a later date to return to that third party...
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 Rational pricing: Fixed income securities.
Stochastic calculus approach
The following is a partial differential equationPartial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
(PDE) in stochastic calculus
Stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes...
which is satisfied by any zero-coupon bond. This methodology recognises that since future interest rates are uncertain, the discount rate referred to above is not adequately represented by a single fixed number.
The solution to the PDE is given in
where
is the expectation with respect to risk-neutral probabilitiesRisk-neutral measure
In mathematical finance, a risk-neutral measure, is a prototypical case of an equivalent martingale measure. It is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted...
, and
is a random variable representing the discount rate; see also Martingale pricingMartingale pricing
Martingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g...
.
Practically, to determine the bond price, specific short rate model
Short rate model
In the context of interest rate derivatives, a short-rate model is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,.-The short rate:...
s are employed here. However, when using these models, it is often the case that no closed form
Closed form
-Maths:* Closed-form expression, a finitary expression* Closed differential form, a differential form \alpha with the property that d\alpha = 0-Poetry:* In poetry analysis, a type of poetry that exhibits regular structure, such as meter or a rhyming pattern;...
solution exists, and a lattice-
Lattice model (finance)
In finance, a lattice model can be used to find the fair value of a stock option; variants also exist for interest rate derivatives.The model divides time between now and the option's expiration into N discrete periods...
or simulation-based implementation of the model in question is employed. The approaches commonly used are:
- the CIR model
- the Black-Derman-Toy model
- the Hull-White modelHull-White modelIn financial mathematics, the Hull–White model is a model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates...
- the HJM framework
- the Chen modelChen modelIn finance, the Chen model is a mathematical model describing the evolution of interest rates. It is a type of "three-factor model" as it describes interest rate movements as driven by three sources of market risk...
.
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 interestAccrued interest
In finance, accrued Interest is the interest that has accumulated since the principal investment, or since the previous interest payment if there has been one already. For a financial instrument such as a bond, interest is calculated and paid in set intervals...
: i.e. any interest due to the owner of the bond since the previous coupon date; see day count convention
Day count convention
In finance, a day count convention determines how interest accrues over time for a variety of investments, including bonds, notes, loans, mortgages, medium-term notes, swaps, and forward rate agreements . This determines the amount transferred on interest payment dates, and also the calculation of...
. The price of a bond which includes this accrued interest is known as the "dirty price
Dirty price
The dirty price of a bond represents the value of a bond, exclusive of any commissions or fees. The dirty price is also called the "full price" or the "all in price."-Bond Pricing:...
" (or "full price" or "all in price" or "Cash price"). The "clean price
Clean price
In finance, the clean price is the price of a bond excluding any interest that has accrued since issue or the most recent coupon payment. This is to be compared with the dirty price, which is the price of a bond including the accrued interest....
" 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 yieldsYield (finance)
In finance, the term yield describes the amount in cash that returns to the owners of a security. Normally it does not include the price variations, at the difference of the total return...
- which relate the price of the bond to its coupons - can then be determined.
Yield to Maturity
The yield to maturityYield to maturity
The Yield to maturity or redemption yield of a bond or other fixed-interest security, such as gilts, is the internal rate of return earned by an investor who buys the bond today at the market price, assuming that the bond will be held until maturity, and that all coupon and principal payments...
is the discount rate which returns the market price
Market price
In economics, market price is the economic price for which a good or service is offered in the marketplace. It is of interest mainly in the study of microeconomics...
of the bond; it is identical to r (required return) in the above equation. YTM is thus the internal rate of return
Internal rate of return
The internal rate of return is a rate of return used in capital budgeting to measure and compare the profitability of investments. It is also called the discounted cash flow rate of return or the rate of return . In the context of savings and loans the IRR is also called the effective interest rate...
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 P0,
- hold the bond until maturity, and
- redeem the bond at par.
Coupon yield
The coupon yieldCoupon (bond)
A coupon payment on a bond is a periodic interest payment that the bondholder receives during the time between when the bond is issued and when it matures. Coupons are normally described in terms of the coupon rate, which is calculated by adding the total amount of coupons paid per year and...
is simply the coupon payment (C) as a percentage of the face value (F).
- Coupon yield = C / F
Coupon yield is also called nominal yield
Nominal yield
Nominal yield or coupon yield is the coupon rate of a fixed income security, which is a fixed percentage of the par value. Unlike current yield, it does not vary with the market price of the security....
.
Current yield
The current yieldCurrent yield
The current yield, interest yield, income yield, flat yield or running yield is a financial term used in reference to bonds and other fixed-interest securities such as gilts...
is simply the coupon payment (C) as a percentage of the (current) bond price (P).
- Current yield =
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 amt
Price sensitivity
The sensitivitySensitivity analysis
Sensitivity analysis is the study of how the variation in the output of a statistical model can be attributed to different variations in the inputs of the model. Put another way, it is a technique for systematically changing variables in a model to determine the effects of such changes.In any...
of a bond's market price to interest rate (i.e. yield) movements is measured by its duration
Bond duration
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....
, and, additionally, by its convexity
Bond convexity
In finance, convexity is a measure of the sensitivity of the duration of a bond to changes in interest rates, the second derivative of the price of the bond with respect to interest rates . In general, the higher the convexity, the more sensitive the bond price is to decreasing interest rates and...
.
Duration is a linear measure
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
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
Elasticity (economics)
In economics, elasticity is the measurement of how changing one economic variable affects others. For example:* "If I lower the price of my product, how much more will I sell?"* "If I raise the price, how much less will I sell?"...
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
Convex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
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
Second derivative
In calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is...
(see: Bond duration closed-form formula; Bond convexity closed-form formula). 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 liabilitiesLong-term liabilities
Long-term liabilities are liabilities with a future benefit over one year, such as notes payable that mature longer than one year.In accounting, the long-term liabilities are shown on the right wing of the balance-sheet representing the sources of funds, which are generally bounded in form of...
, any bond discount or premium must be amortized
Accrual
Accrual of something is, in finance, the adding together of interest or different investments over a period of time. It holds specific meanings in accounting, where it can refer to accounts on a balance sheet that represent liabilities and non-cash-based assets used in accrual-based accounting...
over the life of 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:
= amortization amount in period number "n+1"Bond Discount or Bond Premium =
= Bond Discount or Bond Premium =
See also
- Bond durationBond durationIn 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....
- Bond convexityBond convexityIn finance, convexity is a measure of the sensitivity of the duration of a bond to changes in interest rates, the second derivative of the price of the bond with respect to interest rates . In general, the higher the convexity, the more sensitive the bond price is to decreasing interest rates and...
- Yield to maturityYield to maturityThe Yield to maturity or redemption yield of a bond or other fixed-interest security, such as gilts, is the internal rate of return earned by an investor who buys the bond today at the market price, assuming that the bond will be held until maturity, and that all coupon and principal payments...
- Clean priceClean priceIn finance, the clean price is the price of a bond excluding any interest that has accrued since issue or the most recent coupon payment. This is to be compared with the dirty price, which is the price of a bond including the accrued interest....
- Dirty priceDirty priceThe dirty price of a bond represents the value of a bond, exclusive of any commissions or fees. The dirty price is also called the "full price" or the "all in price."-Bond Pricing:...
- Bond optionBond optionIn 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....
- Option-adjusted spreadOption-adjusted spreadOption adjusted spread is the flat spread which has to be added to the treasury yield curve in a pricing model to discount a security payment to match its market price. OAS is hence model dependent. This concept can be applied to a mortgage-backed security , option, bond and any other interest...
External links
References- Bond Valuation, Prof. Campbell R. Harvey, Duke UniversityDuke UniversityDuke University is a private research university located in Durham, North Carolina, United States. Founded by Methodists and Quakers in the present day town of Trinity in 1838, the school moved to Durham in 1892. In 1924, tobacco industrialist James B...
- A Primer on the Time Value of Money, Prof. Aswath DamodaranAswath DamodaranAswath Damodaran is a Professor of Finance at the Stern School of Business at New York University , where he teaches corporate finance and equity valuation...
, Stern School of Business - Basic Bond Valuation Prof. Alan R. Palmiter, Wake Forest UniversityWake Forest UniversityWake Forest University is a private, coeducational university in the U.S. state of North Carolina, founded in 1834. The university received its name from its original location in Wake Forest, north of Raleigh, North Carolina, the state capital. The Reynolda Campus, the university's main campus, is...
- Bond Price Volatility Investment Analysts Society of South Africa
- Duration and convexity Investment Analysts Society of South Africa
Calculators
- Bond Price ExcelMicrosoft ExcelMicrosoft Excel is a proprietary commercial spreadsheet application written and distributed by Microsoft for Microsoft Windows and Mac OS X. It features calculation, graphing tools, pivot tables, and a macro programming language called Visual Basic for Applications...
spreadsheetSpreadsheetA spreadsheet is a computer application that simulates a paper accounting worksheet. It displays multiple cells usually in a two-dimensional matrix or grid consisting of rows and columns. Each cell contains alphanumeric text, numeric values or formulas...