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Bond valuation is the process of determining 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 determined by discounting the bond's expected cash flows to the present using the appropriate discount rate. Determining this rate in practice - i.e. "pricing" the bond - is done with reference to other instruments. Once the price or value has been calculated, the sensitivity of the price can then be estimated; the various yields, which relate the price of the bond to its coupons, can also be determined.
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Bond valuation
As above, the fair price of a straight bond (a bond with no embedded option; see Embedded Option) is determined by discounting its expected cash flows at the appropriate discount rate. The formula applied is as follows:
Present value relationship
- Cash flows:
- the periodic coupon payments C, each of which is made n times (n is usually 2) every year
- the par or face value F, which is payable at maturity of the bond after T years. (note: final year payments will include the par value plus the coupon payments for the year). In some of the bonds, their Maturity Redemption Price might be more than par value, in this case the F is actually the Redemption Price.
- Discount rate: the required (annually compounded) yield or rate of return r
- r is the market interest rate for bonds with similar terms and risk ratings
- m is the number of coupons to be paid over the remaining lifetime of the bond, ie n times T. (It is assumed that the previous coupon has just been paid.)
- u is (1 + r)^(1 / n) ie an interest accumulation factor over one coupon period
- Bond Price =
Because the price is the present value of the cash flows, there is an inverse relationship between price and discount rate: the higher the discount rate the lower the value of the bond
Clean and dirty price
When the bond is not valued precisely on a coupon date, the present value relationship as 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"; the "clean price" is the price excluding any interest that has accrued. The value returned by the above formula is thus the dirty price.
Clean prices are generally more stable over time than dirty prices. This is because clean prices change for economic reasons ( for instance a change in interest rates or in the bond issuer's credit quality), whereas dirty prices change day to day depending on where the current date is in relation to the coupon dates, in addition to any economic reasons.
It is market practice to quote bonds on a clean-price basis. When a bond settles the accrued interest is added to the value based on the clean price to reflect the full market value.
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 (YTM) is the discount rate which returns the market price of the bond - in other words, 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 / P0.
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
Bond Pricing
As above, the present value relationship reflects the theoretical approach to determining the value of a bond. In practice though, the bond's price is (usually) determined with reference to other, more liquid instruments. The two main approaches are as follows:
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. 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 as above to obtain the price. See Credit spread (bond).
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). 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 government securities 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.
Price sensitivity
The sensitivity of a bond's market price to interest rate (ie 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 interest 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 a 15-year bond with a duration of 7 would fall approximately 7% in value if the interest rate increased by 1% per annum.
Convexity is a measure of the "curvature" of price changes, and is thus a complement to duration. The necessity for this additional measure arises since, as mentioned, duration is a linear measure, whereas, in reality, as interest rates change, the price is a convex function of interest rates. (Specifically, duration can be formulated as the first derivative of the price function 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 added to the value of 7 for duration.
See also
- Bond duration
- Bond convexity
- Yield to maturity
- Clean price
- Dirty price
- Bond option
- Option-adjusted spread
- Immunization (finance)
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
- Bond Price Excel spreadsheet
- Bond Valuation Online Calculator (Perpetual, Zero Coupon and Nonzero Coupon Bonds)
- Fixed Coupon Bond Calculator
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