{"id":998,"date":"2020-08-17T16:23:53","date_gmt":"2020-08-17T20:23:53","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&p=998"},"modified":"2021-07-09T12:10:20","modified_gmt":"2021-07-09T16:10:20","slug":"bonds","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/bonds\/","title":{"raw":"5.7 Bonds","rendered":"5.7 Bonds"},"content":{"raw":"
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Learning Outcomes<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nCalculate the coupon payment size, market value or gain\/loss for bonds.\r\n\r\n<\/div>\r\n<\/div>\r\nIf a corporation or government is looking to raise money (capital), they can issue bonds.\u00a0 The [pb_glossary id=\"1000\"]issuer[\/pb_glossary] (the company or government) must pay the [pb_glossary id=\"1001\"]bond holder[\/pb_glossary] (owner of the bond) a series of equal-sized regular interest payments for a fixed period of time.\u00a0 The size of these payments is determined by the agreed-upon interest rate ([pb_glossary id=\"1002\"]coupon rate[\/pb_glossary]). At the end of the fixed period of time, or [pb_glossary id=\"3289\"]maturity date,[\/pb_glossary] the issuer must repay the bond holder the [pb_glossary id=\"898\"]principal[\/pb_glossary] (the amount of money the bond issuer borrowed).\r\n\r\nWe consider the bond is a debt owed by the issuer to the bond holder. The amount owed never increases because the issuer pays the interest owed each period to the holder in the form of a [pb_glossary id=\"1003\"]coupon payment[\/pb_glossary].\u00a0 This is why the final amount owed by the issuer to the bond holder (the face value) will be the principal (amount borrowed).\r\n\r\nThere are several different types of calculations for bonds \u2014 see the sections below for the key formulas, tips and examples related to bond calculations.\r\n

Calculating Coupon Payments for Bonds<\/h1>\r\nThe coupon payment (PMT) is the interest earned in one period (6 months):\r\n

[latex]PMT = \\textrm{Principal}\\times i[\/latex]<\/p>\r\nwhere the principal is the initial purchase price of the bond (when the bond is purchased from the issuer) and i<\/em> is the [pb_glossary id=\"469\"]periodic rate[\/pb_glossary].\u00a0 The periodic rate is the interest rate for one period.\r\n

Example 5.7.1<\/h2>\r\nAmir purchased a $3,000 bond that has a 10-year term (will mature in 10 years).\u00a0 The bond has a coupon rate of 5%, compounded semi-annually. What is the size of the semi-annual coupon payment?\r\n
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Check your Knowledge for Example 1<\/p>\r\n\r\n<\/header>\r\n

[h5p id=\"61\"]\r\n[h5p id=\"62\"]<\/div>\r\n<\/div>\r\n \r\n

calculating the Fair Market Value Of a Bond<\/h1>\r\nOften, the bond holder can sell the bond (transfer the bond) to a [pb_glossary id=\"3297\"]secondary bond holder[\/pb_glossary] before the end of the [pb_glossary id=\"3300\"]term[\/pb_glossary].\u00a0 Because interest rates change throughout the term of the bond, the bond can be worth different amounts at different times throughout its term.\u00a0 The amount the purchaser (secondary bond holder) is willing to pay is called the [pb_glossary id=\"3298\"]fair market value of the bond[\/pb_glossary]. \u00a0The fair market value is determined by the current interest rate, the size of the coupon payment, the number of years remaining in the term (number of years before the maturity date) and the [pb_glossary id=\"1006\"]face value[\/pb_glossary] of the bond.\r\n\r\nWe will use the BAII Plus to calculate the fair market value of a bond:\r\n\r\n\r\n\r\n\r\n\r\n\r\n
PV<\/th>\r\nInterest<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Fair Market Value<\/strong><\/span><\/td>\r\n+ % Market Rate<\/strong><\/span><\/td>\r\n= Coupon Payments<\/strong><\/td>\r\n+ Face Value<\/strong><\/span><\/td>\r\n<\/tr>\r\n
\u2212<\/span><\/strong><\/td>\r\n\u2212<\/span><\/strong><\/td>\r\n+<\/span><\/strong><\/td>\r\n+<\/span><\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBonds are the ONLY type of investment in this text where we will have a negative value for PV and a positive value for FV.\u00a0 To make sense of this, imagine the bond from the perspective of the bond holder.\u00a0 The bond holder initially \u2018lends\u2019 money to the bond issuer when purchasing a bond from the issuer. We consider this amount lent (PV) to be negative because this is the amount the bond holder must outlay to purchase the bond.\r\n\r\nEach period, the issuer will owe the bond holder interest on that loan. \u00a0The issuer pays this interest (coupon payment) to the holder each period.\u00a0 The issuer also repays the face value of the bond (FV) at the end to the holder[footnote]Information thanks to https:\/\/www.investopedia.com\/terms\/b\/bond.asp<\/a> ; https:\/\/en.wikipedia.org\/wiki\/Bond_(finance)<\/a> and https:\/\/www.investopedia.com\/articles\/investing\/062813\/why-companies-issue-bonds.asp[\/footnote]. For this reason, we consider PMT and FV as positive because they are money that the bond holder receives.\r\n

Example 5.7.2<\/h2>\r\nFrancis purchases a $4,000 bond that has a 30-year term.\u00a0 The bond has a coupon rate of 5.25%, compounded semi-annually and a semi-annual coupon payment of $65. If Francis chooses to sell the bond after 10 years when the current interest rate is 3%, compounded semi-annually, what is the fair market value of the bond?\r\n
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Check your Knowledge for Example 2<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n[h5p id=\"63\"]\r\n\r\n[h5p id=\"64\"]\r\n\r\nConclusion: Francis can sell the bond for $4,149.58 in 10 years.\r\n\r\n<\/div>\r\n<\/div>\r\n

calculating the gain or loss when selling a bond<\/h1>\r\nIf the interest rate (coupon rate) on a bond is above the market rate then the bond increases in value [pb_glossary id=\"3308\"](sells at a premium)[\/pb_glossary]. If the interest rate (coupon rate) on a bond is below the market rate, then the bond goes down in value [pb_glossary id=\"3309\"](sells at a discount)[\/pb_glossary]. The gain (or loss) is calculated by:\r\n

[latex] \\textrm{Gain (or Loss)} = \\textrm{Selling Price} - \\textrm{Purchase Price} [\/latex]<\/p>\r\nLet us use this formula to determine Francis' gain or loss in Example 2.\r\n

Example 5.7.3: Selling at a Premium<\/h2>\r\nWhat is Francis' gain or loss on the sale of his bond in Example 2?\r\n
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Check your Knowledge for Example 2b<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n[h5p id=\"65\"]\r\n\r\n[h5p id=\"66\"]\r\n\r\nConclusion: Francis will make a $149.58 gain when he sells the bond in 10 years.\r\n\r\n<\/div>\r\n<\/div>\r\n

Example 5.7.4 \u2014 Selling at a Discount<\/h2>\r\nRedo Example 2 with an interest rate (market rate) of 6% when Francis chooses to sell the bond in 10 years. What is Francis' gain or loss on the sale of the bond?\r\n
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Check your Knowledge for Example 2c<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFirst, we need to calculate the fair market value of the bond in 10 years:\r\n\r\n[h5p id=\"67\"]\r\n\r\n[h5p id=\"68\"]\r\n\r\nThe fair market value of the bond will be $2,728.69 when Francis sells the bond in 10 years. We can use this value to now calculate Francis' gain or loss:\r\n\r\n[h5p id=\"70\"]\r\n\r\n[h5p id=\"69\"]\r\n\r\nConclusion: Francis will lose $1,271.31 on the sale of the bond.\r\n\r\n<\/div>\r\n<\/div>\r\n

Key Takeaways for Bonds<\/h1>\r\n
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Key Takeaways for Bonds<\/p>\r\n\r\n<\/header>\r\n

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