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<span>Arbitrage-free pricing approach</span> </div> </a> <ul id="toc-Arbitrage-free_pricing_approach-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stochastic_calculus_approach" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stochastic_calculus_approach"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Stochastic calculus approach</span> </div> </a> <ul id="toc-Stochastic_calculus_approach-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Clean_and_dirty_price" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Clean_and_dirty_price"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Clean and dirty price</span> </div> </a> <ul id="toc-Clean_and_dirty_price-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Yield_and_price_relationships" class="vector-toc-list-item vector-toc-level-1 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.mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Financial_market" title="Financial market">Financial markets</a></th></tr><tr><td class="sidebar-image"><span class="notpageimage" typeof="mw:File"><a href="/wiki/File:Philippine-stock-market-board.jpg" class="mw-file-description"><img alt="Looking up at a computerized stocks-value board at the Philippine Stock Exchange" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Philippine-stock-market-board.jpg/200px-Philippine-stock-market-board.jpg" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Philippine-stock-market-board.jpg/300px-Philippine-stock-market-board.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Philippine-stock-market-board.jpg/400px-Philippine-stock-market-board.jpg 2x" data-file-width="640" data-file-height="480" /></a></span></td></tr><tr><td class="sidebar-above" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Market_(economics)" title="Market (economics)">Public market</a></li> <li><a href="/wiki/Exchange_(organized_market)" title="Exchange (organized market)">Exchange</a> <b>·</b> <a href="/wiki/Security_(finance)" title="Security (finance)">Securities</a></li></ul></td></tr><tr><th class="sidebar-heading" style="background:#ddddff;"> <a href="/wiki/Bond_market" title="Bond market">Bond market</a></th></tr><tr><td class="sidebar-content hlist" style="padding:0.15em 0.5em 0.7em;"> <ul><li><a class="mw-selflink selflink">Bond valuation</a></li> <li><a href="/wiki/Corporate_bond" title="Corporate bond">Corporate bond</a></li> <li><a href="/wiki/Fixed_income" title="Fixed income">Fixed income</a></li> <li><a href="/wiki/Government_bond" title="Government bond">Government bond</a></li> <li><a href="/wiki/High-yield_debt" title="High-yield debt">High-yield debt</a></li> <li><a href="/wiki/Municipal_bond" title="Municipal bond">Municipal bond</a></li> <li><br /><a href="/wiki/Securitization" title="Securitization">Securitization</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> <a href="/wiki/Stock_market" title="Stock market">Stock market</a></th></tr><tr><td class="sidebar-content hlist" style="padding:0.15em 0.5em 0.7em;"> <ul><li><a href="/wiki/Common_stock" title="Common stock">Common stock</a></li> <li><a href="/wiki/Growth_stock" title="Growth stock">Growth stock</a></li> <li><a href="/wiki/Preferred_stock" title="Preferred stock">Preferred stock</a></li> <li><a href="/wiki/Registered_share" title="Registered share">Registered share</a></li> <li><a href="/wiki/Shareholder" title="Shareholder">Shareholder</a></li> <li><a href="/wiki/Stock" title="Stock">Stock</a></li> <li><a href="/wiki/Stockbroker" title="Stockbroker">Stockbroker</a></li> <li><a href="/wiki/Stock_certificate" title="Stock certificate">Stock certificate</a></li> <li><a href="/wiki/Stock_exchange" title="Stock exchange">Stock exchange</a></li> <li><a href="/wiki/Watered_stock" title="Watered stock">Watered stock</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Other markets</th></tr><tr><td class="sidebar-content hlist" style="padding:0.15em 0.5em 0.7em;"> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em; line-height:1.25em;"><a href="/wiki/Derivatives_market" title="Derivatives market">Derivatives</a> <div class="hlist"><ul><li>(<a href="/wiki/Credit_derivative" title="Credit derivative">Credit derivative</a></li><li><a href="/wiki/Futures_exchange" title="Futures exchange">Futures exchange</a></li><li><a href="/wiki/Hybrid_security" title="Hybrid security">Hybrid security</a>)</li></ul></div> </div> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em; line-height:1.25em;padding-bottom:0.25em;"><a href="/wiki/Foreign_exchange_market" title="Foreign exchange market">Foreign exchange</a> <div class="hlist"><ul><li>(<a href="/wiki/Currency" title="Currency">Currency</a></li><li><a href="/wiki/Exchange_rate" title="Exchange rate">Exchange rate</a>)</li></ul></div> </div> <ul><li><a href="/wiki/Commodity_market" title="Commodity market">Commodity</a></li> <li><a href="/wiki/Exchange-traded_fund" title="Exchange-traded fund">ETF</a></li> <li><a href="/wiki/Money_market" title="Money market">Money</a></li> <li><a href="/wiki/Mutual_fund" title="Mutual fund">Mutual fund</a></li> <li><a href="/wiki/Option_(finance)" title="Option (finance)">Option</a></li> <li><a href="/wiki/Real_estate" title="Real estate">Real estate</a></li> <li><a href="/wiki/Reinsurance" title="Reinsurance">Reinsurance</a></li> <li><a href="/wiki/Structured_product" title="Structured product">Structured product</a></li> <li><a href="/wiki/Swap_(finance)" title="Swap (finance)">Swap (finance)</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> <a href="/wiki/Over-the-counter_(finance)" title="Over-the-counter (finance)">Over-the-counter (off-exchange)</a></th></tr><tr><td class="sidebar-content hlist" style="padding:0.15em 0.5em 0.7em;"> <div class="hlist"><ul><li><a href="/wiki/Forward_contract" title="Forward contract">Forwards</a></li><li><a href="/wiki/Option_(finance)" title="Option (finance)">Options</a></li></ul></div> <div class="hlist"><ul><li><a href="/wiki/Spot_market" title="Spot market">Spot market</a></li><li><a href="/wiki/Swap_(finance)" title="Swap (finance)">Swaps</a></li></ul></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> <a href="/wiki/Trader_(finance)" title="Trader (finance)">Trading</a></th></tr><tr><td class="sidebar-content hlist" style="padding:0.15em 0.5em 0.7em;"> <ul><li><a href="/wiki/Financial_market_participants" title="Financial market participants">Participants</a></li> <li><a href="/wiki/Financial_regulation" title="Financial regulation">Regulation</a></li> <li><a href="/wiki/Clearing_(finance)" title="Clearing (finance)">Clearing</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Related areas</th></tr><tr><td class="sidebar-content hlist" style="padding:0.15em 0.5em 0.7em;"> <ul><li><a href="/wiki/Alternative_investment" title="Alternative investment">Alternative investment</a></li> <li><a href="/wiki/Angel_investor" title="Angel investor">Angel investor</a></li> <li><a href="/wiki/Asset_(economics)" title="Asset (economics)">Asset (economics)</a></li> <li><a href="/wiki/Asset_pricing" title="Asset pricing">Asset pricing</a></li> <li><a href="/wiki/Bank" title="Bank">Banks and banking</a></li> <li><a href="/wiki/Bull_(stock_market_speculator)" title="Bull (stock market speculator)">Bull</a></li> <li><a href="/wiki/Climate_finance" title="Climate finance">Climate finance</a></li> <li><a href="/wiki/Diversification_(finance)" title="Diversification (finance)">Diversification (finance)</a></li> <li><a href="/wiki/Eco-investing" title="Eco-investing">Eco-investing</a></li> <li><a href="/wiki/Environmental_finance" title="Environmental finance">Environmental finance</a></li> <li><a href="/wiki/Environmental,_social,_and_governance" title="Environmental, social, and governance">ESG</a></li> <li><a href="/wiki/Finance" title="Finance">Financial</a> <ul><li><a href="/wiki/Financial_analysis" title="Financial analysis">analysis</a></li> <li><a href="/wiki/Financial_analyst" title="Financial analyst">analyst</a></li> <li><a href="/wiki/Financial_asset" title="Financial asset">asset</a></li> <li><a href="/wiki/Financial_betting" title="Financial betting">betting</a></li> <li><a href="/wiki/Corporate_finance" title="Corporate finance">corporate</a></li> <li><a href="/wiki/Financial_crime" title="Financial crime">crime</a></li> <li><a href="/wiki/Financial_forecast" title="Financial forecast">forecast</a></li> <li><a href="/wiki/Personal_finance" title="Personal finance">personal</a></li> <li><a href="/wiki/Public_finance" title="Public finance">public</a></li> <li><a href="/wiki/Financial_services" title="Financial services">services</a></li></ul></li> <li><a href="/wiki/Fintech" title="Fintech">Fintech</a></li> <li><a href="/wiki/Greenwashing" title="Greenwashing">Greenwashing</a></li> <li><a href="/wiki/Growth_investing" title="Growth investing">Growth investing</a></li> <li><a href="/wiki/Impact_investing" title="Impact investing">Impact investing</a></li> <li><a href="/wiki/Investment_management" title="Investment management">Investment management</a></li> <li><a href="/wiki/Market_risk" title="Market risk">Market risk</a></li> <li><a href="/wiki/Market_trend" title="Market trend">Market trend</a></li> <li><a href="/wiki/Speculative_attack" title="Speculative attack">Speculative attack</a></li> <li><a href="/wiki/Sustainable_development_goals" class="mw-redirect" title="Sustainable development goals">Sustainable development goals</a></li> <li><a href="/wiki/Sustainable_finance" title="Sustainable finance">Sustainable finance</a></li></ul></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Financial_markets" title="Template:Financial markets"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Financial_markets" title="Template talk:Financial markets"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Financial_markets" title="Special:EditPage/Template:Financial markets"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Bond valuation</b> is the process by which an investor arrives at an estimate of the theoretical fair value, or intrinsic worth, of a <a href="/wiki/Bond_(finance)" title="Bond (finance)">bond</a>. As with any security or capital investment, the theoretical fair value of a bond is the <a href="/wiki/Present_value" title="Present value">present value</a> 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 <a href="/wiki/Discounted_cash_flow#Discount_rate" title="Discounted cash flow">appropriate discount rate</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bodi_2-0" class="reference"><a href="#cite_note-bodi-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In practice, this discount rate is often determined by reference to similar instruments, provided that such instruments exist. Various related yield-measures are then calculated for the given price. Where the market price of bond is less than its <a href="/wiki/Par_value" title="Par value">par value</a>, the bond is selling at a <b>discount</b>. Conversely, if the market price of bond is greater than its par value, the bond is selling at a <b>premium</b>. For this and other relationships between price and yield, see <a href="#Yield_and_price_relationships">below</a>. </p><p>If the bond includes <a href="/wiki/Embedded_option" title="Embedded option">embedded options</a>, the valuation is more difficult and combines <a href="/wiki/Option_pricing" class="mw-redirect" title="Option pricing">option pricing</a> with discounting. Depending on the type of option, the <a href="/wiki/Option_premium" class="mw-redirect" title="Option premium">option price</a> as calculated is either added to or subtracted from the price of the "straight" portion.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> See <a href="/wiki/Bond_option#Embedded_options" title="Bond option">further</a> under <a href="/wiki/Bond_option" title="Bond option">Bond option</a>. This total is then the value of the bond. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Bond_valuation">Bond valuation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=1" title="Edit section: Bond valuation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fair price of a "straight bond" (a bond with no <a href="/wiki/Embedded_option" title="Embedded option">embedded options</a>; see <a href="/wiki/Bond_(finance)#Features" title="Bond (finance)">Bond (finance) § Features</a>) is usually determined by discounting its expected cash flows at the appropriate discount rate. 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 <a href="/wiki/Market_liquidity" title="Market liquidity">liquid</a> 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 <a href="/wiki/Bond_option" title="Bond option">when an option is written on the bond in question</a>—stochastic calculus may be employed.<sup id="cite_ref-Fabozzi_4-0" class="reference"><a href="#cite_note-Fabozzi-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Present_value_approach">Present value approach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=2" title="Edit section: Present value approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The basic method for calculating a bond's theoretical fair value, or intrinsic worth, uses the <a href="/wiki/Present_value" title="Present value">present value</a> (PV) formula shown below, using a single market interest rate to discount cash flows in all periods. A more complex approach would use different interest rates for cash flows in different periods.<sup id="cite_ref-bodi_2-1" class="reference"><a href="#cite_note-bodi-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 294">: 294 </span></sup> The formula shown below assumes that a coupon payment has just been made (see <a href="#Clean_and_dirty_price">below</a> for adjustments on other dates). </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}TFV&={\begin{matrix}\left({\frac {C}{1+i}}+{\frac {C}{(1+i)^{2}}}+...+{\frac {C}{(1+i)^{N}}}\right)+{\frac {M}{(1+i)^{N}}}\end{matrix}}\\&={\begin{matrix}\left(\sum _{n=1}^{N}{\frac {C}{(1+i)^{n}}}\right)+{\frac {M}{(1+i)^{N}}}\end{matrix}}\\&={\begin{matrix}C\left({\frac {1-(1+i)^{-N}}{i}}\right)+M(1+i)^{-N}\end{matrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>T</mi> <mi>F</mi> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>C</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>N</mi> </mrow> </msup> </mrow> <mi>i</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>N</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}TFV&={\begin{matrix}\left({\frac {C}{1+i}}+{\frac {C}{(1+i)^{2}}}+...+{\frac {C}{(1+i)^{N}}}\right)+{\frac {M}{(1+i)^{N}}}\end{matrix}}\\&={\begin{matrix}\left(\sum _{n=1}^{N}{\frac {C}{(1+i)^{n}}}\right)+{\frac {M}{(1+i)^{N}}}\end{matrix}}\\&={\begin{matrix}C\left({\frac {1-(1+i)^{-N}}{i}}\right)+M(1+i)^{-N}\end{matrix}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7dbc963e07c070708d30fcada645bded8c6d58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:46.97ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}TFV&={\begin{matrix}\left({\frac {C}{1+i}}+{\frac {C}{(1+i)^{2}}}+...+{\frac {C}{(1+i)^{N}}}\right)+{\frac {M}{(1+i)^{N}}}\end{matrix}}\\&={\begin{matrix}\left(\sum _{n=1}^{N}{\frac {C}{(1+i)^{n}}}\right)+{\frac {M}{(1+i)^{N}}}\end{matrix}}\\&={\begin{matrix}C\left({\frac {1-(1+i)^{-N}}{i}}\right)+M(1+i)^{-N}\end{matrix}}\end{aligned}}}"></span></dd> <dd>where: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/678aa813d234f6d5a404f8e20add27a8dcf2834b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.194ex; height:2.176ex;" alt="{\displaystyle F=}"></span> par value</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{F}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{F}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1502cf88b4d6112b6968b561f7798b79eaf2bffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.719ex; height:2.509ex;" alt="{\displaystyle i_{F}=}"></span> contractual interest rate</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=F*i_{F}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>F</mi> <mo>∗</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=F*i_{F}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5218b95d417721d4ca9f54e8dfb9e7227410c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.519ex; height:2.509ex;" alt="{\displaystyle C=F*i_{F}=}"></span> coupon payment (periodic interest payment)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/868992f39bf07eb15e3b60315480a6769692879e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.517ex; height:2.176ex;" alt="{\displaystyle N=}"></span> number of payments</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a3efe0a593cd6d86f80cffa214481414891ed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.256ex; height:2.176ex;" alt="{\displaystyle i=}"></span> market interest rate, or required yield, or observed / appropriate <a href="/wiki/Yield_to_maturity" title="Yield to maturity">yield to maturity</a> (see <a href="#Yield_to_maturity">below</a>)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a5897444f2fc12b87773bf24c2b4744789d5a8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.896ex; height:2.176ex;" alt="{\displaystyle M=}"></span> value at maturity, usually equals par value</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TFV=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>F</mi> <mi>V</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TFV=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2764d7f92fe801aba33929f0c925dcd91309ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.618ex; height:2.176ex;" alt="{\displaystyle TFV=}"></span> theoretical fair value</dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Relative_price_approach">Relative price approach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=3" title="Edit section: Relative price approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Corporate_bond#Valuation" title="Corporate bond">Corporate bond § Valuation</a>, and <a href="/wiki/Corporate_bond#Risk_analysis" title="Corporate bond">Corporate bond § Risk analysis</a></div> <p>Under this approach—an extension, or application, of the above—the bond will be priced relative to a benchmark, usually a <a href="/wiki/Government_bond" title="Government bond">government security</a>; see <a href="/wiki/Relative_valuation" title="Relative valuation">Relative valuation</a>. Here, the yield to maturity on the bond is determined based on the bond's <a href="/wiki/Credit_rating" title="Credit rating">Credit rating</a> relative to a government security with similar maturity or <a href="/wiki/Bond_duration" class="mw-redirect" title="Bond duration">duration</a>; see <a href="/wiki/Credit_spread_(bond)" class="mw-redirect" title="Credit spread (bond)">Credit spread (bond)</a>. 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, replacing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> in the formula above, to obtain the price.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Arbitrage-free_pricing_approach">Arbitrage-free pricing approach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=4" title="Edit section: Arbitrage-free pricing approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Rational_pricing#Fixed_income_securities" title="Rational pricing">Rational pricing § Fixed income securities</a></div> <p>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 <a href="/wiki/Zero-coupon_bond" title="Zero-coupon bond">zero-coupon</a> 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.<sup id="cite_ref-Fabozzi_4-1" class="reference"><a href="#cite_note-Fabozzi-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Here, each cash flow is separately discounted at the same rate as a <a href="/wiki/Zero-coupon_bond" title="Zero-coupon bond">zero-coupon bond</a> 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 <a href="/wiki/Credit_spread_(bond)" class="mw-redirect" title="Credit spread (bond)">credit spread</a>). </p><p>Under this approach, the bond price should reflect its "<a href="/wiki/Arbitrage" title="Arbitrage">arbitrage</a>-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 <a href="/wiki/Rational_pricing" title="Rational pricing">rational pricing</a> logic relating to <a href="/wiki/Rational_pricing#Assets_with_identical_cash_flows" title="Rational pricing">"Assets with identical cash flows"</a>. 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. </p> <div class="mw-heading mw-heading3"><h3 id="Stochastic_calculus_approach">Stochastic calculus approach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=5" title="Edit section: Stochastic calculus approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When modelling a <a href="/wiki/Bond_option" title="Bond option">bond option</a>, or other <a href="/wiki/Interest_rate_derivative" title="Interest rate derivative">interest rate derivative</a> (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.e. whether for all coupons or for each individual coupon—is not adequately represented by a fixed (<a href="/wiki/Deterministic" class="mw-redirect" title="Deterministic">deterministic</a>) number. In such cases, <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">stochastic calculus</a> is employed. </p><p>The following is a <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equation</a> (PDE) in stochastic calculus, which, <a href="/wiki/Rational_pricing#Risk_neutral_valuation" title="Rational pricing">by arbitrage arguments</a>,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> is satisfied by any zero-coupon bond <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>, over (instantaneous) time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>, for corresponding changes in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, the <a href="/wiki/Short-rate_model" title="Short-rate model">short rate</a>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>r</mi> <mi>P</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec0f423e539923eb96e77014e6bee05c362cf22d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:57.096ex; height:6.009ex;" alt="{\displaystyle {\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}"></span> </p><p>The solution to the PDE (i.e. the corresponding formula for bond value) — given in Cox et al.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> — is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P[t,T,r(t)]=E_{t}^{\ast }[e^{-R(t,T)}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo>,</mo> <mi>T</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P[t,T,r(t)]=E_{t}^{\ast }[e^{-R(t,T)}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c01d1eb3bc4ecc459a12ecc038b0d84152534b0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.85ex; height:3.509ex;" alt="{\displaystyle P[t,T,r(t)]=E_{t}^{\ast }[e^{-R(t,T)}]}"></span> </p> <dl><dd>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{t}^{\ast }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{t}^{\ast }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18b79dd5757d79300cb185f844d27b82dcd6780" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.848ex; height:2.843ex;" alt="{\displaystyle E_{t}^{\ast }}"></span> is the <a href="/wiki/Expected_value" title="Expected value">expectation</a> with respect to <a href="/wiki/Risk-neutral_measure" title="Risk-neutral measure">risk-neutral probabilities</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(t,T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(t,T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2433857d80a23efbc1d8747fa441c8777458f875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.083ex; height:2.843ex;" alt="{\displaystyle R(t,T)}"></span> is a random variable representing the discount rate; see also <a href="/wiki/Martingale_pricing" title="Martingale pricing">Martingale pricing</a>.</dd></dl> <p>To actually determine the bond price, the analyst must choose the specific <a href="/wiki/Short-rate_model" title="Short-rate model">short-rate model</a> to be employed. The approaches commonly used are: </p> <ul><li>the <a href="/wiki/CIR_model" class="mw-redirect" title="CIR model">CIR model</a></li> <li>the <a href="/wiki/Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy model</a></li> <li>the <a href="/wiki/Hull%E2%80%93White_model" title="Hull–White model">Hull–White model</a></li> <li>the <a href="/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">HJM framework</a></li> <li>the <a href="/wiki/Chen_model" title="Chen model">Chen model</a>.</li></ul> <p>Note that depending on the model selected, a <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form</a> (<a href="/wiki/Black_model" title="Black model">“Black like”</a>) solution may not be available, and a <a href="/wiki/Lattice_model_(finance)#Interest_rate_derivatives" title="Lattice model (finance)">lattice-</a> or <a href="/wiki/Monte_Carlo_methods_in_finance#Overview" title="Monte Carlo methods in finance">simulation-based</a> implementation of the model in question is then employed. See also <a href="/wiki/Bond_option#Valuation" title="Bond option">Bond option § Valuation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Clean_and_dirty_price">Clean and dirty price</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=6" title="Edit section: Clean and dirty price"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Clean_price" title="Clean price">Clean price</a> and <a href="/wiki/Dirty_price" title="Dirty price">Dirty price</a></div> <p>When the bond is not valued precisely on a coupon date, the calculated price, using the methods above, will incorporate <a href="/wiki/Accrued_interest" title="Accrued interest">accrued interest</a>: i.e. any interest due to the owner of the bond over the "<a href="/wiki/Stub_period" title="Stub period">stub period</a>" since the previous coupon date (see <a href="/wiki/Day_count_convention" title="Day count convention">day count convention</a>). The price of a bond which includes this accrued interest is known as the "<a href="/wiki/Dirty_price" title="Dirty price">dirty price</a>" (or "full price" or "all in price" or "Cash price"). The "<a href="/wiki/Clean_price" title="Clean price">clean price</a>" 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. </p> <div class="mw-heading mw-heading2"><h2 id="Yield_and_price_relationships">Yield and price relationships</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=7" title="Edit section: Yield and price relationships"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Once the price or value has been calculated, various <a href="/wiki/Yield_(finance)" title="Yield (finance)">yields</a> relating the price of the bond to its coupons can then be determined. </p> <div class="mw-heading mw-heading3"><h3 id="Yield_to_maturity">Yield to maturity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=8" title="Edit section: Yield to maturity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Yield_to_maturity" title="Yield to maturity">yield to maturity</a> (YTM) is the discount rate which returns the <a href="/wiki/Market_price" class="mw-redirect" title="Market price">market price</a> of a bond without embedded optionality; it is identical to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> (required return) in the <a href="#Present_value_approach">above equation</a>. YTM is thus the <a href="/wiki/Internal_rate_of_return" title="Internal rate of return">internal rate of return</a> 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. </p><p>To achieve a return equal to YTM, i.e. where it is the required return on the bond, the bond owner must: </p> <ul><li>buy the bond at price <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"></span>,</li> <li>hold the bond until maturity, and</li> <li>redeem the bond at par.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Coupon_rate">Coupon rate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=9" title="Edit section: Coupon rate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Coupon_(bond)" class="mw-redirect" title="Coupon (bond)">coupon rate</a> is the coupon payment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> as a percentage of the face value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Coupon rate}}={\frac {C}{F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Coupon rate</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>F</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Coupon rate}}={\frac {C}{F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bcda75aadbff6f135fff9a827a50efa7be234cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.173ex; height:5.343ex;" alt="{\displaystyle {\text{Coupon rate}}={\frac {C}{F}}}"></span></dd></dl> <p>Coupon yield is also called <a href="/wiki/Nominal_yield" title="Nominal yield">nominal yield</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Current_yield">Current yield</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=10" title="Edit section: Current yield"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Current_yield" title="Current yield">current yield</a> is the coupon payment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> as a percentage of the (<i>current</i>) bond price <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Current yield}}={\frac {C}{P_{0}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Current yield</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Current yield}}={\frac {C}{P_{0}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cabbdbd5ed58669cabc30320649d32da82dc4c64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.578ex; height:5.843ex;" alt="{\displaystyle {\text{Current yield}}={\frac {C}{P_{0}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Relationship">Relationship</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=11" title="Edit section: Relationship"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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: </p> <table class="wikitable"> <caption>Relationship between yield to maturity and the coupon rate </caption> <tbody><tr> <th>Status </th> <th>Connection </th></tr> <tr> <td>At a discount </td> <td>YTM > current yield > coupon yield </td></tr> <tr> <td>At a premium </td> <td>coupon yield > current yield > YTM </td></tr> <tr> <td>Sells at par </td> <td>YTM = current yield = coupon yield </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Price_sensitivity">Price sensitivity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=12" title="Edit section: Price sensitivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Bond_duration" class="mw-redirect" title="Bond duration">Bond duration</a> and <a href="/wiki/Bond_convexity" title="Bond convexity">Bond convexity</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Basis_point_value" title="Basis point value">Basis point value</a> and <a href="/wiki/Yield_elasticity_of_bond_value" title="Yield elasticity of bond value">Yield elasticity of bond value</a></div> <p>The <a href="/wiki/Sensitivity_analysis" title="Sensitivity analysis">sensitivity</a> of a bond's market price to interest rate (i.e. yield) movements is measured by its <a href="/wiki/Bond_duration" class="mw-redirect" title="Bond duration">duration</a>, and, additionally, by its <a href="/wiki/Bond_convexity" title="Bond convexity">convexity</a>. </p><p>Duration is a <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear measure</a> 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 <a href="/wiki/Elasticity_(economics)" title="Elasticity (economics)">elasticity</a> 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 <a href="/wiki/Force_of_interest" class="mw-redirect" title="Force of interest">force of interest</a>) increased by 1% per annum. </p><p>Convexity is a measure of the "<a href="/wiki/Curvature" title="Curvature">curvature</a>" of price changes. It is needed because the price is not a linear function of the discount rate, but rather a <a href="/wiki/Convex_function" title="Convex function">convex function</a> of the discount rate. Specifically, duration can be formulated as the <a href="/wiki/First_derivative" class="mw-redirect" title="First derivative">first derivative</a> of the price with respect to the interest rate, and convexity as the <a href="/wiki/Second_derivative" title="Second derivative">second derivative</a> (see: <a href="/wiki/Bond_duration_closed-form_formula" class="mw-redirect" title="Bond duration closed-form formula">Bond duration closed-form formula</a>; <a href="/wiki/Bond_convexity_closed-form_formula" class="mw-redirect" title="Bond convexity closed-form formula">Bond convexity closed-form formula</a>; <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>). 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. </p><p>For embedded options, see <a href="/wiki/Effective_duration" class="mw-redirect" title="Effective duration">effective duration</a> and <a href="/wiki/Effective_convexity" class="mw-redirect" title="Effective convexity">effective convexity</a>; more generally, see <a href="/wiki/Corporate_bond#Risk_analysis" title="Corporate bond">Corporate bond § Risk analysis</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Accounting_treatment">Accounting treatment</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=13" title="Edit section: Accounting treatment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Accounting" title="Accounting">accounting</a> for <a href="/wiki/Long-term_liabilities" title="Long-term liabilities">liabilities</a>, any bond discount or premium must be <a href="/wiki/Accrual" title="Accrual">amortized</a> over the life of the 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:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="formula should be supported by citation to a reliable source. (August 2023)">citation needed</span></a></i>]</sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \{0,1,...,N-1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \{0,1,...,N-1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dff448443b35725f34f7781dd996f6d78cc93610" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.155ex; height:2.843ex;" alt="{\displaystyle n\in \{0,1,...,N-1\}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ca63bae846c1f453ed862ef56a5747ca45f139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.549ex; height:2.009ex;" alt="{\displaystyle a_{n+1}}"></span> = amortization amount in period number "n+1" </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n+1}=|iP-C|{(1+i)}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>i</mi> <mi>P</mi> <mo>−</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n+1}=|iP-C|{(1+i)}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b55c1a7910fff37d812ad165007b4905e10cc62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.929ex; height:3.009ex;" alt="{\displaystyle a_{n+1}=|iP-C|{(1+i)}^{n}}"></span> </p><p>Bond Discount or Bond Premium = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |F-P|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo>−</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |F-P|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a444132c58a57abc76822f08523be66d6d3d17e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.62ex; height:2.843ex;" alt="{\displaystyle |F-P|}"></span> = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}+a_{2}+...+a_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}+a_{2}+...+a_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d0107aa748022edbc6e519f2601cce75d80776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.048ex; height:2.343ex;" alt="{\displaystyle a_{1}+a_{2}+...+a_{N}}"></span> </p><p>Bond Discount or Bond Premium = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F|i-i_{F}|({\frac {1-(1+i)^{-N}}{i}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>i</mi> <mo>−</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>N</mi> </mrow> </msup> </mrow> <mi>i</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F|i-i_{F}|({\frac {1-(1+i)^{-N}}{i}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/677c87a4806cfe2e5c6bba855243e914080f0785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.176ex; height:5.843ex;" alt="{\displaystyle F|i-i_{F}|({\frac {1-(1+i)^{-N}}{i}})}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Category:Bond_valuation" title="Category:Bond valuation">List of bond valuation topics</a></li> <li><a href="/wiki/Asset_swap_spread" class="mw-redirect" title="Asset swap spread">Asset swap spread</a></li> <li><a href="/wiki/Bond_convexity" title="Bond convexity">Bond convexity</a></li> <li><a href="/wiki/Bond_duration" class="mw-redirect" title="Bond duration">Bond duration</a></li> <li><a href="/wiki/Bond_option" title="Bond option">Bond option</a></li> <li><a href="/wiki/Clean_price" title="Clean price">Clean price</a></li> <li><a href="/wiki/Coupon_yield" class="mw-redirect" title="Coupon yield">Coupon yield</a></li> <li><a href="/wiki/Current_yield" title="Current yield">Current yield</a></li> <li><a href="/wiki/Dirty_price" title="Dirty price">Dirty price</a></li> <li><a href="/wiki/I-spread" title="I-spread">I-spread</a></li> <li><a href="/wiki/Option-adjusted_spread" title="Option-adjusted spread">Option-adjusted spread</a></li> <li><a href="/wiki/Yield_to_maturity" title="Yield to maturity">Yield to maturity</a></li> <li><a href="/wiki/Z-spread" title="Z-spread">Z-spread</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMalkiel1962" class="citation journal cs1">Malkiel, Burton G. (1962). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1880816">"Expectations, Bond Prices, and the Term Structure of Interest Rates"</a>. <i>The Quarterly Journal of Economics</i>. <b>76</b> (2): 197–218. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1880816">10.2307/1880816</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0033-5533">0033-5533</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1880816">1880816</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Quarterly+Journal+of+Economics&rft.atitle=Expectations%2C+Bond+Prices%2C+and+the+Term+Structure+of+Interest+Rates&rft.volume=76&rft.issue=2&rft.pages=197-218&rft.date=1962&rft.issn=0033-5533&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1880816%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1880816&rft.aulast=Malkiel&rft.aufirst=Burton+G.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1880816&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></span> </li> <li id="cite_note-bodi-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-bodi_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-bodi_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBodi,_Zvi;_Kane,_Alex.;_Marcus,_Alan_J.2010" class="citation book cs1">Bodi, Zvi; Kane, Alex.; Marcus, Alan J. (2010). <i>Essentials of Investments</i> (eighth ed.). New York: McGraw-Hill/Irwin. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-338240-1" title="Special:BookSources/978-0-07-338240-1"><bdi>978-0-07-338240-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essentials+of+Investments&rft.place=New+York&rft.edition=eighth&rft.pub=McGraw-Hill%2FIrwin&rft.date=2010&rft.isbn=978-0-07-338240-1&rft.au=Bodi%2C+Zvi%3B+Kane%2C+Alex.%3B+Marcus%2C+Alan+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKalotayWilliamsFabozzi1993" class="citation journal cs1">Kalotay, Andrew J.; Williams, George O.; Fabozzi, Frank J. (1993). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/full/10.2469/faj.v49.n3.35">"A Model for Valuing Bonds and Embedded Options"</a>. <i>Financial Analysts Journal</i>. <b>49</b> (3): 35–46. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2469%2Ffaj.v49.n3.35">10.2469/faj.v49.n3.35</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0015-198X">0015-198X</a> – via Taylor & Francis.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Financial+Analysts+Journal&rft.atitle=A+Model+for+Valuing+Bonds+and+Embedded+Options&rft.volume=49&rft.issue=3&rft.pages=35-46&rft.date=1993&rft_id=info%3Adoi%2F10.2469%2Ffaj.v49.n3.35&rft.issn=0015-198X&rft.aulast=Kalotay&rft.aufirst=Andrew+J.&rft.au=Williams%2C+George+O.&rft.au=Fabozzi%2C+Frank+J.&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Ffull%2F10.2469%2Ffaj.v49.n3.35&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></span> </li> <li id="cite_note-Fabozzi-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fabozzi_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fabozzi_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Fabozzi, 1998</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJonesMasonRosenfeld1984" class="citation journal cs1">Jones, E. Philip; Mason, Scott P.; Rosenfeld, Eric (1984). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2327919">"Contingent Claims Analysis of Corporate Capital Structures: An Empirical Investigation"</a>. <i>The Journal of Finance</i>. <b>39</b> (3): 611–625. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2327919">10.2307/2327919</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-1082">0022-1082</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2327919">2327919</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Finance&rft.atitle=Contingent+Claims+Analysis+of+Corporate+Capital+Structures%3A+An+Empirical+Investigation&rft.volume=39&rft.issue=3&rft.pages=611-625&rft.date=1984&rft.issn=0022-1082&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2327919%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2327919&rft.aulast=Jones&rft.aufirst=E.+Philip&rft.au=Mason%2C+Scott+P.&rft.au=Rosenfeld%2C+Eric&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2327919&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">For a derivation, <a href="/wiki/Black%E2%80%93Scholes_equation#Derivation_of_the_Black–Scholes_PDE" title="Black–Scholes equation">analogous to Black-Scholes</a>, see: David Mandel (2015). <a rel="nofollow" class="external text" href="https://www.math.fsu.edu/~dmandel/Primers/Understanding%20Market%20Price%20of%20Risk.pdf">"Understanding Market Price of Risk"</a>, <a href="/wiki/Florida_State_University" title="Florida State University">Florida State University</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="/wiki/John_C._Cox" class="mw-redirect" title="John C. Cox">John C. Cox</a>, <a href="/wiki/Jonathan_E._Ingersoll" title="Jonathan E. Ingersoll">Jonathan E. Ingersoll</a> and <a href="/wiki/Stephen_A._Ross" class="mw-redirect" title="Stephen A. Ross">Stephen A. Ross</a> (1985). <a rel="nofollow" class="external text" href="http://www.javaquant.net/papers/CIR1985.pdf">A Theory of the Term Structure of Interest Rates</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111003184409/http://www.javaquant.net/papers/CIR1985.pdf">Archived</a> 2011-10-03 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <i><a href="/wiki/Econometrica" title="Econometrica">Econometrica</a></i> 53:2</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Selected_bibliography">Selected bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=16" title="Edit section: Selected bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuillermo_L._Dumrauf2012" class="citation book cs1">Guillermo L. Dumrauf (2012). <a rel="nofollow" class="external text" href="https://www.amazon.com/Bonds-Analysis-Excel-Chapter-ebook/dp/B008D4PRIU">"Chapter 1: Pricing and Return"</a>. <i>Bonds, a Step by Step Analysis with Excel</i>. Kindle Edition.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+1%3A+Pricing+and+Return&rft.btitle=Bonds%2C+a+Step+by+Step+Analysis+with+Excel&rft.pub=Kindle+Edition&rft.date=2012&rft.au=Guillermo+L.+Dumrauf&rft_id=https%3A%2F%2Fwww.amazon.com%2FBonds-Analysis-Excel-Chapter-ebook%2Fdp%2FB008D4PRIU&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrank_Fabozzi1998" class="citation book cs1"><a href="/wiki/Frank_Fabozzi" class="mw-redirect" title="Frank Fabozzi">Frank Fabozzi</a> (1998). <i>Valuation of fixed income securities and derivatives</i> (3rd ed.). <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-883249-25-0" title="Special:BookSources/978-1-883249-25-0"><bdi>978-1-883249-25-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Valuation+of+fixed+income+securities+and+derivatives&rft.edition=3rd&rft.pub=John+Wiley&rft.date=1998&rft.isbn=978-1-883249-25-0&rft.au=Frank+Fabozzi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrank_J._Fabozzi2005" class="citation book cs1">Frank J. Fabozzi (2005). <i>Fixed Income Mathematics: Analytical & Statistical Techniques</i> (4th ed.). John Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0071460736" title="Special:BookSources/978-0071460736"><bdi>978-0071460736</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fixed+Income+Mathematics%3A+Analytical+%26+Statistical+Techniques&rft.edition=4th&rft.pub=John+Wiley&rft.date=2005&rft.isbn=978-0071460736&rft.au=Frank+J.+Fabozzi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR._Stafford_Johnson2010" class="citation book cs1">R. Stafford Johnson (2010). <i>Bond Evaluation, Selection, and Management</i> (2nd ed.). John Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0470478356" title="Special:BookSources/978-0470478356"><bdi>978-0470478356</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bond+Evaluation%2C+Selection%2C+and+Management&rft.edition=2nd&rft.pub=John+Wiley&rft.date=2010&rft.isbn=978-0470478356&rft.au=R.+Stafford+Johnson&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMayle1993" class="citation cs2">Mayle, Jan (1993), <i>Standard Securities Calculation Methods: Fixed Income Securities Formulas for Price, Yield and Accrued Interest</i>, vol. 1 (3rd ed.), <a href="/wiki/Securities_Industry_and_Financial_Markets_Association" title="Securities Industry and Financial Markets Association">Securities Industry and Financial Markets Association</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-882936-01-9" title="Special:BookSources/1-882936-01-9"><bdi>1-882936-01-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Standard+Securities+Calculation+Methods%3A+Fixed+Income+Securities+Formulas+for+Price%2C+Yield+and+Accrued+Interest&rft.edition=3rd&rft.pub=Securities+Industry+and+Financial+Markets+Association&rft.date=1993&rft.isbn=1-882936-01-9&rft.aulast=Mayle&rft.aufirst=Jan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDonald_J._Smith2011" class="citation book cs1">Donald J. Smith (2011). <i>Bond Math: The Theory Behind the Formulas</i>. John Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1576603062" title="Special:BookSources/978-1576603062"><bdi>978-1576603062</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bond+Math%3A+The+Theory+Behind+the+Formulas&rft.pub=John+Wiley&rft.date=2011&rft.isbn=978-1576603062&rft.au=Donald+J.+Smith&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBruce_Tuckman2011" class="citation book cs1">Bruce Tuckman (2011). <i>Fixed Income Securities: Tools for Today's Markets</i> (3rd ed.). John Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0470891698" title="Special:BookSources/978-0470891698"><bdi>978-0470891698</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fixed+Income+Securities%3A+Tools+for+Today%27s+Markets&rft.edition=3rd&rft.pub=John+Wiley&rft.date=2011&rft.isbn=978-0470891698&rft.au=Bruce+Tuckman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPietro_Veronesi2010" class="citation book cs1">Pietro Veronesi (2010). <i>Fixed Income Securities: Valuation, Risk, and Risk Management</i>. John Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0470109106" title="Special:BookSources/978-0470109106"><bdi>978-0470109106</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fixed+Income+Securities%3A+Valuation%2C+Risk%2C+and+Risk+Management&rft.pub=John+Wiley&rft.date=2010&rft.isbn=978-0470109106&rft.au=Pietro+Veronesi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurton_Malkiel1962" class="citation book cs1"><a href="/wiki/Burton_Malkiel" title="Burton Malkiel">Burton Malkiel</a> (1962). <i>"Expectations, Bond Prices, and the Term Structure of Interest Rates"</i>. The Quarterly Journal of Economics.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%22Expectations%2C+Bond+Prices%2C+and+the+Term+Structure+of+Interest+Rates%22&rft.pub=The+Quarterly+Journal+of+Economics&rft.date=1962&rft.au=Burton+Malkiel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_Mobius2012" class="citation book cs1"><a href="/wiki/Mark_Mobius" title="Mark Mobius">Mark Mobius</a> (2012). <i>Bonds: An Introduction to the Core Concepts</i>. John Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0470821473" title="Special:BookSources/978-0470821473"><bdi>978-0470821473</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bonds%3A+An+Introduction+to+the+Core+Concepts&rft.pub=John+Wiley&rft.date=2012&rft.isbn=978-0470821473&rft.au=Mark+Mobius&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABond+valuation" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bond_valuation&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.duke.edu/~charvey/Classes/ba350/bondval/bondval.htm">Bond Valuation</a>, Prof. Campbell R. Harvey, <a href="/wiki/Duke_University" title="Duke University">Duke University</a></li> <li><a rel="nofollow" class="external text" href="http://pages.stern.nyu.edu/~adamodar/New_Home_Page/PVPrimer/pvprimer.htm">A Primer on the Time Value of Money</a>, Prof. <a href="/wiki/Aswath_Damodaran" title="Aswath Damodaran">Aswath Damodaran</a>, <a href="/wiki/Stern_School_of_Business" class="mw-redirect" title="Stern School of Business">Stern School of Business</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110723070843/http://www.iassa.co.za/images/file/IASJournals/no501999Lwabona.pdf">Bond Price Volatility</a> <a href="/wiki/Investment_Analysts_Society_of_South_Africa" class="mw-redirect" title="Investment Analysts Society of South Africa">Investment Analysts Society of South Africa</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110723070935/http://www.iassa.co.za/images/file/IASJournals/No512000Lwabona.pdf">Duration and convexity</a> Investment Analysts Society of South Africa</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output 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.navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Bond_market" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Bond_market" title="Template:Bond market"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Bond_market" title="Template talk:Bond market"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Bond_market" title="Special:EditPage/Template:Bond market"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Bond_market" style="font-size:114%;margin:0 4em"><a href="/wiki/Bond_market" title="Bond market">Bond market</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Bond_(finance)" title="Bond (finance)">Bond</a></li> <li><a href="/wiki/Debenture" title="Debenture">Debenture</a></li> <li><a href="/wiki/Fixed_income" title="Fixed income">Fixed income</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of bonds by issuer</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Agency_debt" title="Agency debt">Agency bond</a></li> <li><a href="/wiki/Corporate_bond" title="Corporate bond">Corporate bond</a> <ul><li><a href="/wiki/Senior_debt" title="Senior debt">Senior debt</a></li> <li><a href="/wiki/Subordinated_debt" title="Subordinated debt">Subordinated debt</a></li></ul></li> <li><a href="/wiki/Distressed_securities" title="Distressed securities">Distressed debt</a></li> <li><a href="/wiki/Government_bond" title="Government bond">Government bond</a></li> <li><a href="/wiki/Infrastructure_bond" title="Infrastructure bond">Infrastructure bond</a></li> <li><a href="/wiki/Municipal_bond" title="Municipal bond">Municipal bond</a></li> <li><a href="/wiki/Global_bond" title="Global bond">Global bond</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of bonds by payout</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Accrual_bond" title="Accrual bond">Accrual bond</a></li> <li><a href="/wiki/Auction_rate_security" title="Auction rate security">Auction rate security</a></li> <li><a href="/wiki/Callable_bond" title="Callable bond">Callable bond</a></li> <li><a href="/wiki/Commercial_paper" title="Commercial paper">Commercial paper</a></li> <li><a href="/wiki/Consol_(bond)" title="Consol (bond)">Consol</a></li> <li><a href="/wiki/Contingent_convertible_bond" title="Contingent convertible bond">Contingent convertible bond</a></li> <li><a href="/wiki/Convertible_bond" title="Convertible bond">Convertible bond</a></li> <li><a href="/wiki/Exchangeable_bond" title="Exchangeable bond">Exchangeable bond</a></li> <li><a href="/wiki/Extendible_bond" title="Extendible bond">Extendible bond</a></li> <li><a href="/wiki/Fixed_rate_bond" class="mw-redirect" title="Fixed rate bond">Fixed rate bond</a></li> <li><a href="/wiki/Floating_rate_note" title="Floating rate note">Floating rate note</a></li> <li><a href="/wiki/High-yield_debt" title="High-yield debt">High-yield debt</a></li> <li><a href="/wiki/Inflation-indexed_bond" title="Inflation-indexed bond">Inflation-indexed bond</a></li> <li><a href="/wiki/Inverse_floating_rate_note" title="Inverse floating rate note">Inverse floating rate note</a></li> <li><a href="/wiki/Lottery_bond" title="Lottery bond">Lottery bond</a></li> <li><a href="/wiki/Perpetual_bond" title="Perpetual bond">Perpetual bond</a></li> <li><a href="/wiki/Puttable_bond" title="Puttable bond">Puttable bond</a></li> <li><a href="/wiki/Reverse_convertible_securities" title="Reverse convertible securities">Reverse convertible securities</a></li> <li><a href="/wiki/Zero-coupon_bond" title="Zero-coupon bond">Zero-coupon bond</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Bond valuation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Clean_price" title="Clean price">Clean price</a></li> <li><a href="/wiki/Bond_convexity" title="Bond convexity">Convexity</a></li> <li><a href="/wiki/Coupon_(finance)" title="Coupon (finance)">Coupon</a></li> <li><a href="/wiki/Yield_spread" title="Yield spread">Credit spread</a></li> <li><a href="/wiki/Current_yield" title="Current yield">Current yield</a></li> <li><a href="/wiki/Dirty_price" title="Dirty price">Dirty price</a></li> <li><a href="/wiki/Bond_duration" class="mw-redirect" title="Bond duration">Duration</a></li> <li><a href="/wiki/I-spread" title="I-spread">I-spread</a></li> <li><a href="/wiki/Mortgage_yield" title="Mortgage yield">Mortgage yield</a></li> <li><a href="/wiki/Nominal_yield" title="Nominal yield">Nominal yield</a></li> <li><a href="/wiki/Option-adjusted_spread" title="Option-adjusted spread">Option-adjusted spread</a></li> <li><a href="/wiki/Risk-free_bond" title="Risk-free bond">Risk-free bond</a></li> <li><a href="/wiki/Weighted-average_life" title="Weighted-average life">Weighted-average life</a></li> <li><a href="/wiki/Yield_curve" title="Yield curve">Yield curve</a></li> <li><a href="/wiki/Yield_spread" title="Yield spread">Yield spread</a></li> <li><a href="/wiki/Yield_to_maturity" title="Yield to maturity">Yield to maturity</a></li> <li><a href="/wiki/Z-spread" title="Z-spread">Z-spread</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Securitized products</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Asset-backed_security" title="Asset-backed security">Asset-backed security</a></li> <li><a href="/wiki/Collateralized_debt_obligation" title="Collateralized debt obligation">Collateralized debt obligation</a></li> <li><a href="/wiki/Collateralized_mortgage_obligation" title="Collateralized mortgage obligation">Collateralized mortgage obligation</a></li> <li><a href="/wiki/Commercial_mortgage-backed_security" title="Commercial mortgage-backed security">Commercial mortgage-backed security</a></li> <li><a href="/wiki/Mortgage-backed_security" title="Mortgage-backed security">Mortgage-backed security</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Bond_option" title="Bond option">Bond options</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Callable_bond" title="Callable bond">Callable bond</a></li> <li><a href="/wiki/Convertible_bond" title="Convertible bond">Convertible bond</a></li> <li><a href="/wiki/Embedded_option" title="Embedded option">Embedded option</a></li> <li><a href="/wiki/Exchangeable_bond" title="Exchangeable bond">Exchangeable bond</a></li> <li><a href="/wiki/Extendible_bond" title="Extendible bond">Extendible bond</a></li> <li><a href="/wiki/Puttable_bond" title="Puttable bond">Puttable bond</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Institutions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Commercial_Mortgage_Securities_Association" title="Commercial Mortgage Securities Association">Commercial Mortgage Securities Association</a> (CMSA)</li> <li><a href="/wiki/International_Capital_Market_Association" title="International Capital Market Association">International Capital Market Association</a> (ICMA)</li> <li><a href="/wiki/Securities_Industry_and_Financial_Markets_Association" title="Securities Industry and Financial Markets Association">Securities Industry and Financial Markets Association</a> (SIFMA)</li></ul> </div></td></tr></tbody></table></div>    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