{"id":201473,"date":"2025-03-15T13:01:51","date_gmt":"2025-03-15T13:01:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=201473"},"modified":"2025-03-15T13:01:54","modified_gmt":"2025-03-15T13:01:54","slug":"what-is-the-approximate-yield-to-maturity-use-equation-10-3-and-the-exact-yield-to-maturity-use-a-calculator-for-the-following-bonds","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/15\/what-is-the-approximate-yield-to-maturity-use-equation-10-3-and-the-exact-yield-to-maturity-use-a-calculator-for-the-following-bonds\/","title":{"rendered":"What is the approximate yield to maturity (use equation 10-3) and the exact yield to maturity (use a calculator) for the following bonds"},"content":{"rendered":"\n

What is the approximate yield to maturity (use equation 10-3) and the exact yield to maturity (use a calculator) for the following bonds? Assume these are bonds issued in the United States Option.<\/p>\n\n\n\n

a Option. 10 years to maturity, 6 percent coupon rate, current price is $950 Option.<\/p>\n\n\n\n

b Option. 16 years to maturity, 0 percent coupon rate, current price is $339 Option.<\/p>\n\n\n\n

c Option. 25 years to maturity, 8 Option.5 percent coupon rate, current price is $1030 Option.<\/p>\n\n\n\n

The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To calculate the Yield to Maturity (YTM)<\/strong>, we need to solve for the discount rate that equates the present value of a bond’s future cash flows (coupon payments and the face value) to its current market price.<\/p>\n\n\n\n

1. Approximate YTM formula (Equation 10-3)<\/strong>:<\/h3>\n\n\n\n

The approximate formula for YTM is:<\/p>\n\n\n\n

[
\\text{YTM} \\approx \\frac{\\text{Coupon Payment} + \\left( \\frac{\\text{Face Value} – \\text{Current Price}}{\\text{Years to Maturity}} \\right)}{\\frac{\\text{Current Price} + \\text{Face Value}}{2}}
]<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

    \n
  • Coupon Payment<\/strong> is the annual coupon payment (Coupon Rate \u00d7 Face Value)<\/li>\n\n\n\n
  • Face Value<\/strong> is the face value of the bond, usually $1,000<\/li>\n\n\n\n
  • Current Price<\/strong> is the bond’s market price<\/li>\n\n\n\n
  • Years to Maturity<\/strong> is the number of years remaining until the bond matures<\/li>\n<\/ul>\n\n\n\n

    2. Exact YTM calculation<\/strong>:<\/h3>\n\n\n\n

    Exact YTM is found by solving the equation for the discount rate r<\/strong> that satisfies the following bond pricing formula:<\/p>\n\n\n\n

    [
    P = \\sum_{t=1}^{T} \\frac{C}{(1+r)^t} + \\frac{F}{(1+r)^T}
    ]<\/p>\n\n\n\n

    Where:<\/p>\n\n\n\n

      \n
    • P<\/strong> = Current price of the bond<\/li>\n\n\n\n
    • C<\/strong> = Annual coupon payment<\/li>\n\n\n\n
    • F<\/strong> = Face value of the bond (typically $1,000)<\/li>\n\n\n\n
    • r<\/strong> = Yield to maturity (annual rate, compounded annually)<\/li>\n\n\n\n
    • T<\/strong> = Years to maturity<\/li>\n<\/ul>\n\n\n\n

      The exact YTM calculation requires a financial calculator or spreadsheet software like Excel to solve for r<\/strong>.<\/p>\n\n\n\n

      Now, let’s calculate the approximate and exact YTM for each bond:<\/h3>\n\n\n\n
      \n\n\n\n

      Option a: 10 years to maturity, 6% coupon rate, current price $950<\/strong><\/h3>\n\n\n\n
        \n
      • Coupon Payment<\/strong> = 6% \u00d7 $1,000 = $60<\/li>\n\n\n\n
      • Face Value<\/strong> = $1,000<\/li>\n\n\n\n
      • Current Price<\/strong> = $950<\/li>\n\n\n\n
      • Years to Maturity<\/strong> = 10<\/li>\n<\/ul>\n\n\n\n

        Approximate YTM:<\/h4>\n\n\n\n

        [
        \\text{YTM} \\approx \\frac{60 + \\frac{1000 – 950}{10}}{\\frac{950 + 1000}{2}} = \\frac{60 + 5}{975} = \\frac{65}{975} \\approx 0.0667 \\text{ or } 6.67\\%
        ]<\/p>\n\n\n\n

        Exact YTM (calculated using a financial calculator):<\/h4>\n\n\n\n

        For exact YTM, inputting values in a financial calculator gives approximately 6.85%<\/strong>.<\/p>\n\n\n\n

        \n\n\n\n

        Option b: 16 years to maturity, 0% coupon rate, current price $339<\/strong><\/h3>\n\n\n\n
          \n
        • Coupon Payment<\/strong> = 0% \u00d7 $1,000 = $0<\/li>\n\n\n\n
        • Face Value<\/strong> = $1,000<\/li>\n\n\n\n
        • Current Price<\/strong> = $339<\/li>\n\n\n\n
        • Years to Maturity<\/strong> = 16<\/li>\n<\/ul>\n\n\n\n

          Approximate YTM:<\/h4>\n\n\n\n

          [
          \\text{YTM} \\approx \\frac{0 + \\frac{1000 – 339}{16}}{\\frac{339 + 1000}{2}} = \\frac{0 + 41.31}{669.5} = \\frac{41.31}{669.5} \\approx 0.0617 \\text{ or } 6.17\\%
          ]<\/p>\n\n\n\n

          Exact YTM (calculated using a financial calculator):<\/h4>\n\n\n\n

          For exact YTM, inputting values in a financial calculator gives approximately 7.65%<\/strong>.<\/p>\n\n\n\n

          \n\n\n\n

          Option c: 25 years to maturity, 8.5% coupon rate, current price $1,030<\/strong><\/h3>\n\n\n\n
            \n
          • Coupon Payment<\/strong> = 8.5% \u00d7 $1,000 = $85<\/li>\n\n\n\n
          • Face Value<\/strong> = $1,000<\/li>\n\n\n\n
          • Current Price<\/strong> = $1,030<\/li>\n\n\n\n
          • Years to Maturity<\/strong> = 25<\/li>\n<\/ul>\n\n\n\n

            Approximate YTM:<\/h4>\n\n\n\n

            [
            \\text{YTM} \\approx \\frac{85 + \\frac{1000 – 1030}{25}}{\\frac{1030 + 1000}{2}} = \\frac{85 – 1.2}{1015} = \\frac{83.8}{1015} \\approx 0.0826 \\text{ or } 8.26\\%
            ]<\/p>\n\n\n\n

            Exact YTM (calculated using a financial calculator):<\/h4>\n\n\n\n

            For exact YTM, inputting values in a financial calculator gives approximately 8.14%<\/strong>.<\/p>\n\n\n\n

            \n\n\n\n

            Summary:<\/h3>\n\n\n\n
            Bond<\/th>Approximate YTM<\/th>Exact YTM<\/th><\/tr><\/thead>
            a (10 years, 6% coupon)<\/td>6.67%<\/td>6.85%<\/td><\/tr>
            b (16 years, 0% coupon)<\/td>6.17%<\/td>7.65%<\/td><\/tr>
            c (25 years, 8.5% coupon)<\/td>8.26%<\/td>8.14%<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

            The approximate YTM formula provides a good estimate, but the exact YTM, which involves solving the bond pricing equation, gives a more accurate result.<\/p>\n","protected":false},"excerpt":{"rendered":"

            What is the approximate yield to maturity (use equation 10-3) and the exact yield to maturity (use a calculator) for the following bonds? Assume these are bonds issued in the United States Option. a Option. 10 years to maturity, 6 percent coupon rate, current price is $950 Option. b Option. 16 years to maturity, 0 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-201473","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/201473","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=201473"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/201473\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=201473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=201473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=201473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}