To calculate the value of a zero-coupon bond, we only need to find the present value of the face value. Fixed Income Essentials.

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### What is Bond Pricing Formula?

Login Newsletters. Bonds Fixed Income Essentials. What Is Bond Valuation? Key Takeaways Bond valuation is a way to determine the theoretical fair value or par value of a particular bond. It involves calculating the present value of a bond's expected future coupon payments, or cash flow, and the bond's value upon maturity, or face value. As a bond's par value and interest payments are set, bond valuation helps investors figure out what rate of return would make a bond investment worth the cost.

Coupon rate: Some bonds have an interest rate, also known as the coupon rate, which is paid to bondholders semi-annually. The coupon rate is the fixed return that an investor earns periodically until it matures. Maturity date: All bonds have maturity dates , some short-term, others long-term. When a bond matures, the bond issuer repays the investor the full face value of the bond.

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 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. Conversely, if the market price of bond is greater than its face value, the bond is selling at a premium.

Below is the formula for calculating a bond's price, which uses the basic present value PV formula for a given discount rate: [3] This formula assumes that a coupon payment has just been made; see below for adjustments on other dates. 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.

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- Bond valuation and bond yields!

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. 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.

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. Were this not the case, 4 the arbitrageur 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.

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## Bond Valuation

Then 5 his "risk free", arbitrage profit would be the difference between the two values. See under Rational pricing Fixed income securities. 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. In such cases, stochastic calculus is employed.

### Definition

The solution to the PDE—given in Cox et al. To actually determine the bond price, the analyst must choose the specific short rate model to be employed. The approaches commonly used are:.

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- Bond Price | Definition, Formula and Example!
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When the bond is not valued precisely on a coupon date, the calculated price, using the methods above, will incorporate accrued interest : i. 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". When considering this site as a source for academic reasons, please remember that this site is not subject to the same rigor as academic journals, course materials, and similar publications.

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