From Wikipedia, the free encyclopedia Bond valuation
Bond valuation
Bond valuation is the determination of the fair price of P = market price of bond
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 de- If the market price of bond is less than its face value
termined by reference to similar instruments, provided discount.
(par value), the bond is selling at a discount Conversely,
that such instruments exist. if the market price of bond is greater than its face value,
If the bond includes embedded options, the valuation the bond is selling at a premium [3]
premium.
is more difficult and combines option pricing with dis-
counting. Depending on the type of option, the option Relative price approach
price as calculated is either added to or subtracted from Under this approach, the bond will be priced relative to a
the price of the "straight" portion. This total is then the benchmark, usually a government security; see Relative
value of the bond; the various yields can then be calculat- valuation. Here, the yield to maturity on the bond is de-
ed for the total price. See further under Bond option. termined based on the bond’s Credit rating relative to a
government security with similar maturity or duration;
Bond valuation see Credit spread (bond). The better the quality of the
bond, the smaller the spread between its required return
[1] As above, the fair price of a straight bond (a bond and the YTM of the benchmark. This required return, i in
with no embedded option; see Embedded Option) is usu- the formula, is then used to discount the bond cash flows
ally determined by discounting its expected cash flows as above to obtain the price.
at the appropriate discount rate. The formula commonly
applied is discussed initially. Although this present value Arbitrage-free pricing approach
relationship reflects the theoretical approach to deter- Under this approach, the bond price will reflect its arbi-
mining the value of a bond, in practice its price is (usu- trage-free price. Here, each cash flow (coupon or face) is
ally) determined with reference to other, more liquid in- separately discounted at the same rate as a zero-coupon
struments. The two main approaches, Relative pricing bond corresponding to the coupon date, and of equiva-
and Arbitrage-free pricing, are discussed next. Finally, lent credit worthiness (if possible, from the same issuer
where it is important to recognise that future interest as the bond being valued, or if not, with the appropriate
rates are uncertain and that the discount rate is not ade- credit spread). Here, in general, we apply the rational
quately represented by a single fixed number - for exam- pricing logic relating to "Assets with identical cash
ple when an option is written on the bond in question - flows". In detail: (1) the bond’s coupon dates and coupon
stochastic calculus may be employed. amounts are known with certainty. Therefore (2) some
multiple (or fraction) of zero-coupon bonds, each corre-
Present value approach sponding to the bond’s coupon dates, can be specified so
Below is the formula for calculating a bond’s price, which as to produce identical cash flows to the bond. Thus (3)
uses the basic present value (PV) formula for a given dis- the bond price today must be equal to the sum of each
count rate:[2] (This formula assumes that a coupon pay- of its cash flows discounted at the discount rate implied
ment has just been made; see below for adjustments on by the value of the corresponding ZCB. Were this not
other dates.) the case, (4) the abitrageur could finance his purchase of
F = face value whichever of the bond or the sum of the various ZCBs was
iF = contractual interest rate cheaper, by short selling the other, and meeting his cash
C = F * iF = coupon payment (periodic interest pay- flow commitments using the coupons or maturing zeroes
ment) as appropriate. Then (5) his "risk free", arbitrage profit
N = number of payments would be the difference between the two values. See Ra-
i = market interest rate, or required yield, or observed tional pricing: Fixed income securities.
/ appropriate yield to maturity (see below)
M = value at maturity, usually equals face value
1
From Wikipedia, the free encyclopedia Bond valuation
Stochastic calculus approach Yield to Maturity
The following is a partial differential equation (PDE) in The yield to maturity is the discount rate which returns
stochastic calculus which is satisfied by any zero-coupon the market price of the bond; it is identical to r (required
bond. This methodology recognises that since future in- return) in the above equation. YTM is thus the internal
terest rates are uncertain, the discount rate referred to rate of return of an investment in the bond made at the
above is not adequately represented by a single fixed observed price. Since YTM can be used to price a bond,
number. 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:
The solution to the PDE is given in [4] • buy the bond at price P0,
• hold the bond until maturity, and
• redeem the bond at par.
where is the expectation with respect to risk-
neutral probabilities, and R(t,T) is a random variable rep- Coupon yield
resenting the discount rate; see also Martingale pricing. The coupon yield is simply the coupon payment (C) as a
Practically, to determine the bond price, specific percentage of the face value (F).
short rate models are employed here. However, when us- Coupon yield = C / F
ing these models, it is often the case that no closed form
solution exists, and a lattice- or simulation-based imple- Coupon yield is also called nominal yield.
mentation of the model in question is employed. The ap-
proaches commonly used are: Current yield
• the CIR model The current yield is simply the coupon payment (C) as a
• the Black-Derman-Toy model percentage of the (current) bond price (P).
• the Hull-White model Current yield = C / P0.
• the HJM framework
• the Chen model. Relationship
The concept of current yield is closely related to other
Clean and dirty price bond concepts, including yield to maturity, and coupon
Main articles: Clean price and Dirty price yield. The relationship between yield to maturity and the
When the bond is not valued precisely on a coupon date, coupon rate is as follows:
the calculated price, using the methods above, will incor- • When a bond sells at a discount, YTM > current yield
porate accrued interest: i.e. any interest due to the own- > coupon yield.
er of the bond since the previous coupon date; see day • When a bond sells at a premium, coupon yield >
count convention. The price of a bond which includes current yield > YTM.
this accrued interest is known as the "dirty price" (or • When a bond sells at par, YTM = current yield =
"full price" or "all in price" or "Cash price"). The "clean coupon yield amt
price" is the price excluding any interest that has ac-
crued. Clean prices are generally more stable over time Price sensitivity
than dirty prices. This is because the dirty price will drop
Main articles: Bond duration and Bond convexity
suddenly when the bond goes "ex interest" and the pur-
The sensitivity of a bond’s market price to interest rate
chaser is no longer entitled to receive the next coupon
(i.e. yield) movements is measured by its duration, and,
payment. In many markets, it is market practice to quote
additionally, by its convexity.
bonds on a clean-price basis. When a purchase is settled,
Duration is a linear measure of how the price of a
the accrued interest is added to the quoted clean price to
bond changes in response to interest rate changes. It is
arrive at the actual amount to be paid.
approximately equal to the percentage change in price
for a given change in yield, and may be thought of as
Yield and price relationships the elasticity of the bond’s price with respect to discount
Once the price or value has been calculated, various rates. For example, for small interest rate changes, the
yields - which relate the price of the bond to its coupons duration is the approximate percentage by which the
- can then be determined. 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 mar-
2
From Wikipedia, the free encyclopedia Bond valuation
ket interest rate (or more precisely the corresponding • Bond convexity
force of interest) increased by 1% per annum. • Yield to maturity
Convexity is a measure of the "curvature" of price • Clean price
changes. It is needed because the price is not a linear • Dirty price
function of the discount rate, but rather a convex func- • Bond option
tion of the discount rate. Specifically, duration can be • Option-adjusted spread
formulated as the first derivative of the price with re-
spect to the interest rate, and convexity as the second de-
rivative (see: Bond duration closed-form formula; Bond
References
convexity closed-form formula). Continuing the above [1] Frank Fabozzi (1998). Valuation of fixed income
example, for a more accurate estimate of sensitivity, the securities and derivatives (3rd ed.). John Wiley.
convexity score would be multiplied by the square of the ISBN 978-1-883249-25-0.
change in interest rate, and the result added to the value [2] http://www.investopedia.com/university/
derived by the above linear formula. advancedbond/advancedbond2.asp
[3] http://www.investopedia.com/terms/a/
amortizable-bond-premium.asp
Accounting treatment [4] Cox et al. (1985b) - A Theory of the Term Structure
In accounting for liabilities, any bond discount or premi- of Interest Rates1985)
um 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
External links
amount in each period is calculated from the following References
formula: • Bond Valuation, Prof. Campbell R. Harvey, Duke
University
• A Primer on the Time Value of Money, Prof. Aswath
an + 1 = amortization amount in period number "n+1"
Damodaran, Stern School of Business
an + 1 = | iP − C | (1 + i)n
• Basic Bond Valuation Prof. Alan R. Palmiter, Wake
Bond Discount or Bond Premium = | F − P | = a1 + a2 + ... Forest University
+ aN • Bond Price Volatility Investment Analysts Society of
Bond Discount or Bond Premium = South Africa
• Duration and convexity Investment Analysts Society
of South Africa
Calculators
• Bond Price Excel spreadsheet
See also
• Bond duration
Retrieved from "http://en.wikipedia.org/w/index.php?title=Bond_valuation&oldid=441242814"
Categories:
• Bonds
• Fixed income analysis
This page was last modified on 24 July 2011 at 21:52. Text is available under the Creative Commons Attribution-
ShareAlike License; additional terms may apply. See Terms of use for details. Wikipedia® is a registered trademark of
the Wikimedia Foundation, Inc., a non-profit organization.Contact us
Privacy policy About Wikipedia Disclaimers Mobile view
3