Bond valuation

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 the appropriate discount rate. In practice this discount rate is often determined by reference to other similar instruments, provided that such instruments exist.

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. 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 valuation *

As above, the fair price of a straight bond (a bond with no embedded 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.

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:^[1] (This formula assumes that a coupon payment has just been made; see below for adjustments on other dates.)

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 yield to maturity (see below)

M = value at maturity, usually equals face value

P = market price of bond

$P=\left({\frac {C}{1+i}}+{\frac {C}{(1+i)^{2}}}+...+{\frac {C}{(1+i)^{N}}}\right)+{\frac {M}{(1+i)^{N}}}=\left(\sum _{n=1}^{N}{\frac {C}{(1+i)^{n}}}\right)+{\frac {M}{(1+i)^{N}}}$

$P=C\left({\frac {1-(1+i)^{-N}}{i}}\right)+M(1+i)^{-N}$

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.^[2]

Relative price approach

Under this approach, 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, $i$ 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 arbitrage-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.

Stochastic calculus approach

The following is a partial differential equation (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.

${\frac {1}{2}}\sigma (r)^{2}{\frac {\partial ^{2}P}{\partial r^{2}}}+[a(r)+\sigma (r)+\varphi (r,t)]{\frac {\partial P}{\partial r}}+{\frac {\partial P}{\partial t}}-rP=0$

The solution to the PDE is given in ^[3]

$P[t,T,r(t)]=E_{t}^{\ast }[e^{-R(t,T)}]$

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.

Practically, to determine the bond price, specific short rate models 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:

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 da club for beufss. 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 - which relate the price of the bond to its coupons - can then be determined.

Yield to Maturity

The yield to maturity 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:

buy the bond at price P₀,
hold the bond until maturity, and
redeem the bond at par.

Coupon yield

The coupon yield 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.

Current yield

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

Current yield =

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 amt

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). 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 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 = $F|i-i_{F}|({\frac {1-(1+i)^{-N}}{i}})$

References

External links

References

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

Calculators

[1] ttp://www.investopedia.com/university/advancedbond/advancedbond2.asp

[2] ttp://www.investopedia.com/terms/a/amortizable-bond-premium.asp

[3] Cox et al. (1985b) - A Theory of the Term Structure of Interest Rates1985)

[1]

[2]

[3]

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)