{"id":1010,"date":"2020-08-17T16:39:43","date_gmt":"2020-08-17T20:39:43","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&#038;p=1010"},"modified":"2025-01-15T15:38:09","modified_gmt":"2025-01-15T20:38:09","slug":"perpetuities","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/perpetuities\/","title":{"raw":"5.8 Perpetuities","rendered":"5.8 Perpetuities"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Outcomes<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nCalculate the payment sizes or present values for regular and deferred perpetuities.\r\n\r\n<\/div>\r\n<\/div>\r\nA [pb_glossary id=\"3322\"]perpetuity[\/pb_glossary] is like a bond, but with no fixed term (no fixed [pb_glossary id=\"3289\"]maturity date[\/pb_glossary]).\u00a0 If a corporation issues a perpetuity to an investor, the perpetuity will continue making payments to this investor indefinitely[footnote]Information thanks to <a href=\"https:\/\/languages.oup.com\/google-dictionary-en\/\">Oxford Languages<\/a>; <a href=\"https:\/\/www.investopedia.com\/terms\/p\/perpetuity.asp\">Perpetuity: Financial Definition, Formula, and Examples (Investopedia)<\/a>; <a href=\"https:\/\/en.wikipedia.org\/wiki\/War_bond\">War Bond (Wikipedia)<\/a>; <a href=\"https:\/\/www.advisor.ca\/tax\/estate-planning\/design-a-dynasty-with-perpetual-trusts\/\">Design a dynasty with perpetual trusts (Advisor.ca)<\/a>; <a href=\"https:\/\/www.investopedia.com\/terms\/t\/trust-fund.asp\">Trust Funds (Investopedia)<\/a>; <a href=\"https:\/\/thephilanthropist.ca\/original-pdfs\/Philanthropist-21-3-370.pdf\">Charities and the Rule Against Perpetuities [PDF]<\/a>;[\/footnote]. Common examples of perpetuities are scholarship funds, perpetual trusts and war bonds, like the British Treasury Bonds issued for World War 1[footnote]Historically, governments issued bonds to raise money to meet the growing costs of war.\u00a0 Some of these bonds lasted in perpetuity.[\/footnote]. Finally, s<span style=\"text-align: initial;\">ome business investments can also be treated like perpetuities[footnote]Suppose a company invests a certain initial amount of money with the hope\/expectation that they will earn a return on their investment in the form of regular income from the business (possibly for an indefinite amount of time).\u00a0 If we assume the returns last for an indefinite amount of time, we treat this problem like a perpetuity as well.[\/footnote].<\/span>\r\n\r\nThere are several different types of perpetuities \u2014 see the sections below for the key formulas, tips and examples related to perpetuity calculations.\r\n<h1>payments for Ordinary Perpetuities<\/h1>\r\nIf the payments for a perpetuity are withdrawn at the end of each interval, we call this an [pb_glossary id=\"1012\"]ordinary perpetuity[\/pb_glossary].\u00a0 The payment (PMT) for an ordinary perpetuity can be given by <span style=\"text-align: initial;\">the following formula:<\/span>\r\n<p style=\"text-align: center;\">[latex]\\textrm{PMT}= \\textrm{PV} \\times i[\/latex]<\/p>\r\nwhere PV is the initial value of the perpetuity and <em>i<\/em> is the periodic rate (the rate per period).\r\n\r\nWe can also use the BAII Plus to calculate the PMT. There are a few 'tricks' to know when using the BAII Plus:\r\n<ul>\r\n \t<li>B\/E = \"END\" for ordinary perpetuities<\/li>\r\n \t<li>N = 1000 \u00d7 P\/Y. Use total # years = 1,000 for perpetuities[footnote]A perpetuity, technically, lasts forever. In the calculator, the closest to 'forever' we should enter for a number of years is 1,000. This number of years will work with all types of calculations in the TVM keys.[\/footnote].<\/li>\r\n \t<li>PV = the initial balance of the perpetuity.<\/li>\r\n \t<li>FV = 0. The funds are never withdrawn. The amount withdrawn equals 0 for this reason.<\/li>\r\n \t<li><strong>CPT<\/strong> PMT. Calculate the value of the payment (PMT).<\/li>\r\n<\/ul>\r\n<h2>Example 5.8.1<\/h2>\r\nSasha, a wealthy BCIT Alumnus, just donated $100,000 to create a scholarship at BCIT for first year business students. The scholarship will be awarded semi-annually and the fund will earn 5% compounded semi-annually. The first scholarship will be awarded in 6 months. What will be the size of the semi-annual scholarships awarded?\r\n\r\nUsing the formula:\r\n\r\n[latex]\r\n\\begin{align*}\r\n\\textrm{PMT}&amp;= \\textrm{PV} \u00d7 i = $100,000 \u00d7 \\frac{0.05}{2}\\\\\r\n\r\n&amp;= $100,000 \u00d7 0.025 = $2,500\r\n\\end{align*} [\/latex]\r\n\r\nUsing the BAII Plus:\r\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\" border=\"0\">\r\n<thead>\r\n<tr style=\"height: 17px;\">\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\r\n<th class=\"border\" style=\"width: 20%; height: 17px;\">N<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">I\/Y<\/th>\r\n<th class=\"border\" style=\"width: 17.662%; height: 17px;\">PV<\/th>\r\n<th class=\"border\" style=\"width: 20.338%; height: 17px;\">PMT<\/th>\r\n<th class=\"border\" style=\"width: 10%; height: 17px;\">FV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">END<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\r\n<td class=\"border\" style=\"width: 20%; height: 19px;\">1000\u00d72=2000<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">5<\/td>\r\n<td class=\"border\" style=\"width: 17.662%; height: 19px;\">+100,000<\/td>\r\n<td class=\"border\" style=\"width: 20.338%; height: 19px;\"><span style=\"color: #ff0000;\"><strong>CPT<\/strong> \u22122,500<\/span><\/td>\r\n<td class=\"border\" style=\"width: 10%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nConclusion: BCIT can pay out a scholarship of $2,500 every 6 months.\r\n\r\n[h5p id=\"71\"]\r\n<h1>the Present Value for Ordinary Perpetuities<\/h1>\r\nThere are also two ways to calculate the present value of an ordinary annuity. We can u<span style=\"text-align: initial;\">se the following formula:<\/span>\r\n<p style=\"text-align: center;\">[latex]\\textrm{PV}= \\frac{\\textrm{PMT}}{i}[\/latex]<\/p>\r\nor we can, again, use the BAII Plus and use the same tricks as when calculating the payment (PMT).\r\n\r\nLet us look again at Sasha's example but instead, let us figure out the amount Sasha needs to donate.\r\n<h2>Example 5.8.2<\/h2>\r\nHow much does Sasha (the wealthy BCIT alumnus) need to donate if he wants BCIT to give out $3,000 semi-annual scholarships with the first scholarship awarded in 6 months. Assume the fund still earns 5% compounded semi-annually.\r\n\r\nUsing the formula:\r\n<p style=\"text-align: center;\">[latex] \\textrm{PV}= \\frac{\\textrm{PMT}}{i} = \\frac{\\$3,000}{0.025}= \\$120,000 [\/latex]<\/p>\r\nUsing the BAII Plus:\r\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\" border=\"0\">\r\n<thead>\r\n<tr style=\"height: 17px;\">\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\r\n<th class=\"border\" style=\"width: 20%; height: 17px;\">N<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">I\/Y<\/th>\r\n<th class=\"border\" style=\"width: 20.4789%; height: 17px;\">PV<\/th>\r\n<th class=\"border\" style=\"width: 17.5211%; height: 17px;\">PMT<\/th>\r\n<th class=\"border\" style=\"width: 10%; height: 17px;\">FV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">END<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\r\n<td class=\"border\" style=\"width: 20%; height: 19px;\">1000\u00d72=2000<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">5<\/td>\r\n<td class=\"border\" style=\"width: 20.4789%; height: 19px;\"><strong>CPT<\/strong> +120,000<\/td>\r\n<td class=\"border\" style=\"width: 17.5211%; height: 19px;\"><span style=\"color: #ff0000;\">\u22123,000<\/span><\/td>\r\n<td class=\"border\" style=\"width: 10%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nConclusion: Sasha will need to donate $120,000 to the scholarship fund at BCIT.\r\n\r\n[h5p id=\"72\"]\r\n<h1>the Present Value for Perpetuitues Due<\/h1>\r\nIt is possible to have a perpetuity that gives out the first payment immediately.\u00a0 We call this a [pb_glossary id=\"1015\"]perpetuity due[\/pb_glossary].\u00a0 An example of this is a scholarship fund where the first scholarship is awarded as soon as the fund has been created. [footnote]The same principle applies as in ordinary perpetuities, the balance in the scholarship fund never increases nor decreases in value. This is because after a payment has been withdrawn from the account, interest is earned on the remaining balance in the scholarship fund.\u00a0 The payments are calculated such that the after the interest has been earned, the balance in the account increases back to its original value (PV).[\/footnote]\r\n\r\nIf payments are withdrawn from a perpetuity fund at the start of each payment interval (perpetuity due), there will need to be slightly more in the perpetuity fund to account for the initial balance in the account dropping in value right at the start of the perpetuity.\u00a0 That \u201cslightly more\u201d amount will be the value of the payment[footnote]We, again, need to be careful if the number of payments per year (P\/Y) is not equal to the number of compounding periods per year (C\/Y).\u00a0 If that is the case, we need to calculate the equivalent interest rate with the number of compounding periods equal to the number of payment intervals for the perpetuity.[\/footnote]:\r\n<p style=\"text-align: center;\">[latex]\\text{PV}_{due} = \\frac{PMT}{i} + PMT[\/latex]<\/p>\r\nLet us now revisit Sasha's scholarship example and determine the size of Sasha's donation if the scholarship fund is a perpetuity due.\r\n<h2>Example 5.8.3<\/h2>\r\nSuppose that Sasha would like to donate enough money such that the semi-annual scholarships are still $3,000 but the first scholarship will be awarded immediately.\u00a0 Assume the scholarship fund still earns 5% compounded semi-annually.\u00a0 How much more will Sasha need to donate?\r\n\r\nUsing the formula and <em>i<\/em> = 0.025 and PMT = 3,000 gives:\r\n<p style=\"text-align: center;\">[latex]PV_{due} = \\frac{$3,000}{0.025} + \\$3,000 = $123,000[\/latex]<\/p>\r\nUsing the BAII Calculator gives:\r\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\" border=\"0\">\r\n<thead>\r\n<tr style=\"height: 17px;\">\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\r\n<th class=\"border\" style=\"width: 20%; height: 17px;\">N<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">I\/Y<\/th>\r\n<th class=\"border\" style=\"width: 20.4789%; height: 17px;\">PV<\/th>\r\n<th class=\"border\" style=\"width: 17.5211%; height: 17px;\">PMT<\/th>\r\n<th class=\"border\" style=\"width: 10%; height: 17px;\">FV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">BGN<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\r\n<td class=\"border\" style=\"width: 20%; height: 19px;\">1000\u00d72=2000<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">5<\/td>\r\n<td class=\"border\" style=\"width: 20.4789%; height: 19px;\"><strong>CPT<\/strong> +123,000<\/td>\r\n<td class=\"border\" style=\"width: 17.5211%; height: 19px;\"><span style=\"color: #ff0000;\">\u22123,000<\/span><\/td>\r\n<td class=\"border\" style=\"width: 10%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSasha will need to donate $123,000 if the first scholarship is awarded immediately. Taking the difference between the required donation amount donated for Example 2 (an ordinary perpetuity) and Example 3 (a perpetuity due) gives:\r\n<p style=\"text-align: center;\">[latex] \\textrm{Difference} = \\$123,000 - \\$120,000 = \\$3,000 [\/latex]<\/p>\r\nConclusion: Sasha will need to donate $3,000 more if the first scholarship is awarded today instead of 6 months from now. It is no coincidence that the difference is equal to the payment size.\r\n<h1>the Present Value for Deferred Perpetuitues<\/h1>\r\nIt is possible to defer the payments received from a perpetuity.\u00a0 A first example is if a non-profit or charity receives a donation to cover future costs for the organization[footnote]They deposit the money once it\u2019s received but do not need to start withdrawing from the account until the non-profit opens.\u00a0 If we assume that the non-profit will remain open indefinitely, then we assume that they will need withdrawals to cover their costs for an undetermined amount of time as well.\u00a0 We treat this problem as a perpetuity.[\/footnote].\u00a0\u00a0Another example of a perpetuity is a business decision where there is an initial amount invested to start the business (PV<sub>1<\/sub>) and then it takes several years for the business to start earning income[footnote]In that case, the repayments of the initial investment are delayed (deferred) for a certain number of years.\u00a0\u00a0\u00a0 If we assume that there is no fixed end date to the business, we treat this problem as a perpetuity where the returns continue on indefinitely.[\/footnote].\r\n\r\nWhen calculating payment sizes for deferred perpetuities or the initial amount needed to be deposited at the start of the perpetuity, it is important to remember that there are actually two parts to the deferred perpetuity problem:\r\n<p style=\"text-align: center;\"><a href=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig0.jpg\"><img class=\"aligncenter wp-image-3345 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig0.jpg\" alt=\"Timeline Image for Deferred Perpetuities\" width=\"95%\" \/><\/a><\/p>\r\n<strong>Part 1<\/strong>: The initial balance gathers interest.\u00a0 There are no payments nor withdrawals in part 1 (the deferral period).\u00a0 The only money being added to the balance is the interest being earned (or charged).\u00a0 This problem is a compound interest problem (Chapter 4):\r\n\r\n<strong style=\"text-align: initial;\">Part 2<\/strong><span style=\"text-align: initial;\">: Payments are now being withdrawn. These payments exactly equal to the interest earned on the current balance in the perpetuity account. <\/span>The starting balance for the perpetuity (PV<sub>2<\/sub>) equals to the ending balance from the deferral period (FV<sub>1<\/sub>) .\r\n<h2>Example 5.8.4<\/h2>\r\nWhat if Sasha's scholarship fund (from Examples 1 to 3) doesn't give out its first scholarship for one year?\u00a0 How much would Sasha need to donate if the semi-annual scholarships are still $3,000 and the fund still earns 5% compounded semi-annually?\r\n\r\nLet us first draw out the timeline for this problem:\r\n\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig1-1.jpg\"><img class=\"aligncenter wp-image-3348 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig1-1.jpg\" alt=\"Timeline for deferred perpetuity in Example 4\" width=\"95%\" \/><\/a>\r\n\r\nBecause we know the payments for part 2, start there. We can either use the formula or the BAII Plus to do the calculations for this problem. Let us start by using the BAII Plus:\r\n\r\n<strong>Part 2: Perpetuity (using the BAII Plus)<\/strong>\r\n\r\nLet us 'start' part 2 exactly when the first scholarship is awarded. For this reason, B\/E will be set to BGN:\r\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\" border=\"0\">\r\n<thead>\r\n<tr style=\"height: 17px;\">\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\r\n<th class=\"border\" style=\"width: 20%; height: 17px;\">N<sub>2<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">I\/Y<\/th>\r\n<th class=\"border\" style=\"width: 20.4789%; height: 17px;\">PV<sub>2<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 17.5211%; height: 17px;\">PMT<sub>2<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 10%; height: 17px;\">FV<sub>2<\/sub><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">BGN<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\r\n<td class=\"border\" style=\"width: 20%; height: 19px;\">1000\u00d72=2000<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">5<\/td>\r\n<td class=\"border\" style=\"width: 20.4789%; height: 19px;\"><strong>CPT<\/strong> +123,000<\/td>\r\n<td class=\"border\" style=\"width: 17.5211%; height: 19px;\"><span style=\"color: #ff0000;\">\u22123,000<\/span><\/td>\r\n<td class=\"border\" style=\"width: 10%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that PV<sub>2<\/sub> equals to the amount Sasha would need to donate in Example 3. We will now enter that value into FV<sub>1<\/sub> and calculate the amount donated initially (PV<sub>1<\/sub>).\r\n\r\n<strong>Part 1: Deferral Period (using the BAII Plus)<\/strong>\r\n\r\nThere are no payments nor withdrawals for part 1 so enter '\u2013\u2013' for B\/E. Also, because the deferral period lasts for 1 year, N<sub>1 <\/sub>=1\u00d72 = 2.\r\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\" border=\"0\">\r\n<thead>\r\n<tr style=\"height: 17px;\">\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\r\n<th class=\"border\" style=\"width: 20%; height: 17px;\">N<sub>1<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">I\/Y<\/th>\r\n<th class=\"border\" style=\"width: 23.7183%; height: 17px;\">PV<sub>1<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 12.169%; height: 17px;\">PMT<sub>1<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 12.1127%; height: 17px;\">FV<sub>1<\/sub><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">\u2013\u2013<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\r\n<td class=\"border\" style=\"width: 20%; height: 19px;\">1\u00d72=2<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">5<\/td>\r\n<td class=\"border\" style=\"width: 23.7183%; height: 19px;\"><strong>CPT<\/strong> +117,073.17<\/td>\r\n<td class=\"border\" style=\"width: 12.169%; height: 19px;\">0<\/td>\r\n<td class=\"border\" style=\"width: 12.1127%; height: 19px;\"><span style=\"color: #ff0000;\">\u2212123,000<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nConclusion: Sasha will only need to donate $117,073.17 if the first scholarship is awarded in one year.\r\n\r\n<strong>Method 2 \u2014 Using Formulas:<\/strong>\r\n\r\nWe will again start with part 2. Let us rewrite the timeline, slightly differently, however:\r\n\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig2.jpg\"><img class=\"aligncenter size-full wp-image-3352\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig2.jpg\" alt=\"Timeline image for Deferred Perpetuity with start of perpetuity at 6 month mark\" width=\"95%\" \/><\/a>\r\n\r\nNotice that we will only run the deferral period for 6 months and start the perpetuity (part 2) 6 months before the first scholarship is withdrawn. This makes the calculation easier for part 2:\r\n\r\n<strong>Part 2: Perpetuity (using Formulas)<\/strong>\r\n\r\nWe now have 'started' part 2 one payment period earlier. This now makes the perpetuity in part 2 an ordinary annuity. We use the following formula to calculate its present value:\r\n<p style=\"text-align: center;\">[latex]\\text{PV}_2= \\frac{\\text{PMT}_2}{i} = \\frac{$3,000}{0.025}=$120,000[\/latex]<\/p>\r\nThere will need to be $120,000 in the scholarship fund in 6 months. We can find the present value of this amount using the compound interest formula from Chapter 4.\r\n\r\n<strong>Part 1: Deferral Period (using Formulas)<\/strong>\r\n<p style=\"text-align: center;\">[latex]\\textrm{PV}_{1}= \\frac{\\textrm{FV}_{1}}{(1+i)^{n}} = \\frac{$120,000}{(1+0.025)^{1}}=$117,073.17[\/latex]<\/p>\r\nConclusion: Again, we find that Sasha will need to donate $117,073.17 now if the first scholarship is awarded in 1 year.\r\n<h1>payments for Deferred Perpetuities<\/h1>\r\nIt is possible to use formulas to calculate the payment size for deferred perpetuities. We will however, just use the BAII Plus method for this section.\r\n\r\nLet us examine a similar donation example. In this example, Ezra, a wealthy philanthropist will make a large donation to help out a medical center that will not open for several years.\r\n<h2>Example 5.8.5<\/h2>\r\nEzra, a wealthy philanthropist, donates $1,000,000 to the Moshi Medical Diagnostics Center, located in Moshi, Tanzania. The Center is due to open in 3 years. The Center will use the donated funds to cover their annual equipment and testing costs. They will invest the funds at 4.25%, effective, into a perpetuity.\u00a0 They will withdraw the first perpetuity payment in exactly 3 years when the center opens.\u00a0 Calculate the size of the annual withdrawals.\r\n\r\nLet us first draw out a timeline for this problem:\r\n\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig3.jpg\"><img class=\"aligncenter wp-image-3356 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig3.jpg\" alt=\"Timeline for deferred perpetuity in Example 5\" width=\"95%\" \/><\/a>\r\n\r\nNext, determine which part to start with. In this case, because we know the initial deposit (PV<sub>1<\/sub>), start with part 1 (the deferral period).\r\n\r\n<strong>Part 1 (the deferral period):<\/strong>\r\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\" border=\"0\">\r\n<thead>\r\n<tr style=\"height: 17px;\">\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\r\n<th class=\"border\" style=\"width: 12.8168%; height: 17px;\">N<sub>1<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 11.8029%; height: 17px;\">I\/Y<\/th>\r\n<th class=\"border\" style=\"width: 16.2535%; height: 17px;\">PV<sub>1<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 10.9016%; height: 17px;\">PMT<sub>1<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 24.2252%; height: 17px;\">FV<sub>1<\/sub><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">\u2013\u2013<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">1<\/td>\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">1<\/td>\r\n<td class=\"border\" style=\"width: 12.8168%; height: 19px;\">3\u00d71=3<\/td>\r\n<td class=\"border\" style=\"width: 11.8029%; height: 19px;\">4.25<\/td>\r\n<td class=\"border\" style=\"width: 16.2535%; height: 19px;\">+1,000,000<\/td>\r\n<td class=\"border\" style=\"width: 10.9016%; height: 19px;\">0<\/td>\r\n<td class=\"border\" style=\"width: 24.2252%; height: 19px;\"><strong>CPT<\/strong><span style=\"color: #ff0000;\"> \u22121,132,995.52<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt does not matter what we enter in B\/E above because there are no payments. Let us now enter the ending value for the deferral period (FV<sub>1<\/sub>) into the starting value for the perpetuity (PV<sub>2<\/sub>).\r\n\r\n<strong>Part 2 (the perpetuity):<\/strong>\r\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\" border=\"0\">\r\n<thead>\r\n<tr style=\"height: 17px;\">\r\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\r\n<th class=\"border\" style=\"width: 7.57746%; height: 17px;\">P\/Y<\/th>\r\n<th class=\"border\" style=\"width: 7.71831%; height: 17px;\">C\/Y<\/th>\r\n<th class=\"border\" style=\"width: 11.9717%; height: 17px;\">N<sub>2<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 10.8169%; height: 17px;\">I\/Y<\/th>\r\n<th class=\"border\" style=\"width: 18.7888%; height: 17px;\">PV<sub>2<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 24.7045%; height: 17px;\">PMT<sub>2<\/sub><\/th>\r\n<th class=\"border\" style=\"width: 10.4223%; height: 17px;\">FV<sub>1<\/sub><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<td class=\"border\" style=\"width: 8%; height: 19px;\">BGN<\/td>\r\n<td class=\"border\" style=\"width: 7.57746%; height: 19px;\">1<\/td>\r\n<td class=\"border\" style=\"width: 7.71831%; height: 19px;\">1<\/td>\r\n<td class=\"border\" style=\"width: 11.9717%; height: 19px;\">1000\u00d71=1000<\/td>\r\n<td class=\"border\" style=\"width: 10.8169%; height: 19px;\">4.25<\/td>\r\n<td class=\"border\" style=\"width: 18.7888%; height: 19px;\">+1,132,995.52<\/td>\r\n<td class=\"border\" style=\"width: 24.7045%; height: 19px;\"><strong>CPT <\/strong><span style=\"color: #ff0000;\">\u221246,189.27<\/span><\/td>\r\n<td class=\"border\" style=\"width: 10.4223%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that BGN is on in part 2. This is because we started part 2 (the perpetuity) right when the first payment was withdrawn.\r\n\r\nConclusion: The Moshi Medical Center will receive $46,189.27 annually to cover equipment and testing costs starting in 3 years when the center opens.\r\n<h1>Key Takeaways for Perpetuities<\/h1>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways for Perpetuities<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhen entering perpetuities into the BAII Plus:\r\n<ul>\r\n \t<li>FV is always equal to 0 for perpetuities.<\/li>\r\n \t<li>We always use 1,000 years for perpetuities.<\/li>\r\n \t<li>This gives N = 1000 \u00d7 P\/Y<\/li>\r\n \t<li>It does not matter what sign we use for PV or PMT (as we are computing the other one)[footnote]If ever both were being entered into the BAII plus - we would need to use opposite signs for PV and PMT[\/footnote]<\/li>\r\n<\/ul>\r\nWhen using formulas:\r\n<ul>\r\n \t<li>For ordinary perpetuities, [latex]\\textrm{PMT}= \\textrm{PV} \\times i[\/latex]<\/li>\r\n \t<li>For perpetuities due, [latex]PV_{due} = \\frac{PMT}{i} + PMT[\/latex]<\/li>\r\n<\/ul>\r\nFor deferred perpetuities:\r\n<ul>\r\n \t<li>Run the deferral period until the first payment occurs<\/li>\r\n \t<li>Turn BGN on for part 2<\/li>\r\n \t<li>Set PV<sub>2<\/sub> = FV<sub>1<\/sub><\/li>\r\n \t<li>Set FV<sub>2<\/sub> = 0.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h1>Your Own Notes<\/h1>\r\n<ul>\r\n \t<li>Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t<li>These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t<li>Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"1\"]\r\n<h1>The Footnotes<\/h1>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Outcomes<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Calculate the payment sizes or present values for regular and deferred perpetuities.<\/p>\n<\/div>\n<\/div>\n<p>A <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_1010_3322\">perpetuity<\/a> is like a bond, but with no fixed term (no fixed <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_1010_3289\">maturity date<\/a>).\u00a0 If a corporation issues a perpetuity to an investor, the perpetuity will continue making payments to this investor indefinitely<a class=\"footnote\" title=\"Information thanks to Oxford Languages; Perpetuity: Financial Definition, Formula, and Examples (Investopedia); War Bond (Wikipedia); Design a dynasty with perpetual trusts (Advisor.ca); Trust Funds (Investopedia); Charities and the Rule Against Perpetuities [PDF];\" id=\"return-footnote-1010-1\" href=\"#footnote-1010-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>. Common examples of perpetuities are scholarship funds, perpetual trusts and war bonds, like the British Treasury Bonds issued for World War 1<a class=\"footnote\" title=\"Historically, governments issued bonds to raise money to meet the growing costs of war.\u00a0 Some of these bonds lasted in perpetuity.\" id=\"return-footnote-1010-2\" href=\"#footnote-1010-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>. Finally, s<span style=\"text-align: initial;\">ome business investments can also be treated like perpetuities<a class=\"footnote\" title=\"Suppose a company invests a certain initial amount of money with the hope\/expectation that they will earn a return on their investment in the form of regular income from the business (possibly for an indefinite amount of time).\u00a0 If we assume the returns last for an indefinite amount of time, we treat this problem like a perpetuity as well.\" id=\"return-footnote-1010-3\" href=\"#footnote-1010-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a>.<\/span><\/p>\n<p>There are several different types of perpetuities \u2014 see the sections below for the key formulas, tips and examples related to perpetuity calculations.<\/p>\n<h1>payments for Ordinary Perpetuities<\/h1>\n<p>If the payments for a perpetuity are withdrawn at the end of each interval, we call this an <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_1010_1012\">ordinary perpetuity<\/a>.\u00a0 The payment (PMT) for an ordinary perpetuity can be given by <span style=\"text-align: initial;\">the following formula:<\/span><\/p>\n<p style=\"text-align: center;\">[latex]\\textrm{PMT}= \\textrm{PV} \\times i[\/latex]<\/p>\n<p>where PV is the initial value of the perpetuity and <em>i<\/em> is the periodic rate (the rate per period).<\/p>\n<p>We can also use the BAII Plus to calculate the PMT. There are a few &#8216;tricks&#8217; to know when using the BAII Plus:<\/p>\n<ul>\n<li>B\/E = &#8220;END&#8221; for ordinary perpetuities<\/li>\n<li>N = 1000 \u00d7 P\/Y. Use total # years = 1,000 for perpetuities<a class=\"footnote\" title=\"A perpetuity, technically, lasts forever. In the calculator, the closest to 'forever' we should enter for a number of years is 1,000. This number of years will work with all types of calculations in the TVM keys.\" id=\"return-footnote-1010-4\" href=\"#footnote-1010-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a>.<\/li>\n<li>PV = the initial balance of the perpetuity.<\/li>\n<li>FV = 0. The funds are never withdrawn. The amount withdrawn equals 0 for this reason.<\/li>\n<li><strong>CPT<\/strong> PMT. Calculate the value of the payment (PMT).<\/li>\n<\/ul>\n<h2>Example 5.8.1<\/h2>\n<p>Sasha, a wealthy BCIT Alumnus, just donated $100,000 to create a scholarship at BCIT for first year business students. The scholarship will be awarded semi-annually and the fund will earn 5% compounded semi-annually. The first scholarship will be awarded in 6 months. What will be the size of the semi-annual scholarships awarded?<\/p>\n<p>Using the formula:<\/p>\n<p>[latex]\\begin{align*}  \\textrm{PMT}&= \\textrm{PV} \u00d7 i = $100,000 \u00d7 \\frac{0.05}{2}\\\\    &= $100,000 \u00d7 0.025 = $2,500  \\end{align*}[\/latex]<\/p>\n<p>Using the BAII Plus:<\/p>\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\">\n<thead>\n<tr style=\"height: 17px;\">\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\n<th class=\"border\" style=\"width: 20%; height: 17px;\">N<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">I\/Y<\/th>\n<th class=\"border\" style=\"width: 17.662%; height: 17px;\">PV<\/th>\n<th class=\"border\" style=\"width: 20.338%; height: 17px;\">PMT<\/th>\n<th class=\"border\" style=\"width: 10%; height: 17px;\">FV<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 19px;\">\n<td class=\"border\" style=\"width: 8%; height: 19px;\">END<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\n<td class=\"border\" style=\"width: 20%; height: 19px;\">1000\u00d72=2000<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">5<\/td>\n<td class=\"border\" style=\"width: 17.662%; height: 19px;\">+100,000<\/td>\n<td class=\"border\" style=\"width: 20.338%; height: 19px;\"><span style=\"color: #ff0000;\"><strong>CPT<\/strong> \u22122,500<\/span><\/td>\n<td class=\"border\" style=\"width: 10%; height: 19px;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Conclusion: BCIT can pay out a scholarship of $2,500 every 6 months.<\/p>\n<div id=\"h5p-71\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-71\" class=\"h5p-iframe\" data-content-id=\"71\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"5.8.1 Calculating Payments for Ordinary Perpetuities Explanation for the BAII Plus\"><\/iframe><\/div>\n<\/div>\n<h1>the Present Value for Ordinary Perpetuities<\/h1>\n<p>There are also two ways to calculate the present value of an ordinary annuity. We can u<span style=\"text-align: initial;\">se the following formula:<\/span><\/p>\n<p style=\"text-align: center;\">[latex]\\textrm{PV}= \\frac{\\textrm{PMT}}{i}[\/latex]<\/p>\n<p>or we can, again, use the BAII Plus and use the same tricks as when calculating the payment (PMT).<\/p>\n<p>Let us look again at Sasha&#8217;s example but instead, let us figure out the amount Sasha needs to donate.<\/p>\n<h2>Example 5.8.2<\/h2>\n<p>How much does Sasha (the wealthy BCIT alumnus) need to donate if he wants BCIT to give out $3,000 semi-annual scholarships with the first scholarship awarded in 6 months. Assume the fund still earns 5% compounded semi-annually.<\/p>\n<p>Using the formula:<\/p>\n<p style=\"text-align: center;\">[latex]\\textrm{PV}= \\frac{\\textrm{PMT}}{i} = \\frac{\\$3,000}{0.025}= \\$120,000[\/latex]<\/p>\n<p>Using the BAII Plus:<\/p>\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\">\n<thead>\n<tr style=\"height: 17px;\">\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\n<th class=\"border\" style=\"width: 20%; height: 17px;\">N<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">I\/Y<\/th>\n<th class=\"border\" style=\"width: 20.4789%; height: 17px;\">PV<\/th>\n<th class=\"border\" style=\"width: 17.5211%; height: 17px;\">PMT<\/th>\n<th class=\"border\" style=\"width: 10%; height: 17px;\">FV<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 19px;\">\n<td class=\"border\" style=\"width: 8%; height: 19px;\">END<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\n<td class=\"border\" style=\"width: 20%; height: 19px;\">1000\u00d72=2000<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">5<\/td>\n<td class=\"border\" style=\"width: 20.4789%; height: 19px;\"><strong>CPT<\/strong> +120,000<\/td>\n<td class=\"border\" style=\"width: 17.5211%; height: 19px;\"><span style=\"color: #ff0000;\">\u22123,000<\/span><\/td>\n<td class=\"border\" style=\"width: 10%; height: 19px;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Conclusion: Sasha will need to donate $120,000 to the scholarship fund at BCIT.<\/p>\n<div id=\"h5p-72\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-72\" class=\"h5p-iframe\" data-content-id=\"72\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"5.8.2 Calculating PV for Ordinary Perpetuities Explanation for the BAII Plus\"><\/iframe><\/div>\n<\/div>\n<h1>the Present Value for Perpetuitues Due<\/h1>\n<p>It is possible to have a perpetuity that gives out the first payment immediately.\u00a0 We call this a <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_1010_1015\">perpetuity due<\/a>.\u00a0 An example of this is a scholarship fund where the first scholarship is awarded as soon as the fund has been created. <a class=\"footnote\" title=\"The same principle applies as in ordinary perpetuities, the balance in the scholarship fund never increases nor decreases in value. This is because after a payment has been withdrawn from the account, interest is earned on the remaining balance in the scholarship fund.\u00a0 The payments are calculated such that the after the interest has been earned, the balance in the account increases back to its original value (PV).\" id=\"return-footnote-1010-5\" href=\"#footnote-1010-5\" aria-label=\"Footnote 5\"><sup class=\"footnote\">[5]<\/sup><\/a><\/p>\n<p>If payments are withdrawn from a perpetuity fund at the start of each payment interval (perpetuity due), there will need to be slightly more in the perpetuity fund to account for the initial balance in the account dropping in value right at the start of the perpetuity.\u00a0 That \u201cslightly more\u201d amount will be the value of the payment<a class=\"footnote\" title=\"We, again, need to be careful if the number of payments per year (P\/Y) is not equal to the number of compounding periods per year (C\/Y).\u00a0 If that is the case, we need to calculate the equivalent interest rate with the number of compounding periods equal to the number of payment intervals for the perpetuity.\" id=\"return-footnote-1010-6\" href=\"#footnote-1010-6\" aria-label=\"Footnote 6\"><sup class=\"footnote\">[6]<\/sup><\/a>:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{PV}_{due} = \\frac{PMT}{i} + PMT[\/latex]<\/p>\n<p>Let us now revisit Sasha&#8217;s scholarship example and determine the size of Sasha&#8217;s donation if the scholarship fund is a perpetuity due.<\/p>\n<h2>Example 5.8.3<\/h2>\n<p>Suppose that Sasha would like to donate enough money such that the semi-annual scholarships are still $3,000 but the first scholarship will be awarded immediately.\u00a0 Assume the scholarship fund still earns 5% compounded semi-annually.\u00a0 How much more will Sasha need to donate?<\/p>\n<p>Using the formula and <em>i<\/em> = 0.025 and PMT = 3,000 gives:<\/p>\n<p style=\"text-align: center;\">[latex]PV_{due} = \\frac{$3,000}{0.025} + \\$3,000 = $123,000[\/latex]<\/p>\n<p>Using the BAII Calculator gives:<\/p>\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\">\n<thead>\n<tr style=\"height: 17px;\">\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\n<th class=\"border\" style=\"width: 20%; height: 17px;\">N<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">I\/Y<\/th>\n<th class=\"border\" style=\"width: 20.4789%; height: 17px;\">PV<\/th>\n<th class=\"border\" style=\"width: 17.5211%; height: 17px;\">PMT<\/th>\n<th class=\"border\" style=\"width: 10%; height: 17px;\">FV<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 19px;\">\n<td class=\"border\" style=\"width: 8%; height: 19px;\">BGN<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\n<td class=\"border\" style=\"width: 20%; height: 19px;\">1000\u00d72=2000<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">5<\/td>\n<td class=\"border\" style=\"width: 20.4789%; height: 19px;\"><strong>CPT<\/strong> +123,000<\/td>\n<td class=\"border\" style=\"width: 17.5211%; height: 19px;\"><span style=\"color: #ff0000;\">\u22123,000<\/span><\/td>\n<td class=\"border\" style=\"width: 10%; height: 19px;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Sasha will need to donate $123,000 if the first scholarship is awarded immediately. Taking the difference between the required donation amount donated for Example 2 (an ordinary perpetuity) and Example 3 (a perpetuity due) gives:<\/p>\n<p style=\"text-align: center;\">[latex]\\textrm{Difference} = \\$123,000 - \\$120,000 = \\$3,000[\/latex]<\/p>\n<p>Conclusion: Sasha will need to donate $3,000 more if the first scholarship is awarded today instead of 6 months from now. It is no coincidence that the difference is equal to the payment size.<\/p>\n<h1>the Present Value for Deferred Perpetuitues<\/h1>\n<p>It is possible to defer the payments received from a perpetuity.\u00a0 A first example is if a non-profit or charity receives a donation to cover future costs for the organization<a class=\"footnote\" title=\"They deposit the money once it\u2019s received but do not need to start withdrawing from the account until the non-profit opens.\u00a0 If we assume that the non-profit will remain open indefinitely, then we assume that they will need withdrawals to cover their costs for an undetermined amount of time as well.\u00a0 We treat this problem as a perpetuity.\" id=\"return-footnote-1010-7\" href=\"#footnote-1010-7\" aria-label=\"Footnote 7\"><sup class=\"footnote\">[7]<\/sup><\/a>.\u00a0\u00a0Another example of a perpetuity is a business decision where there is an initial amount invested to start the business (PV<sub>1<\/sub>) and then it takes several years for the business to start earning income<a class=\"footnote\" title=\"In that case, the repayments of the initial investment are delayed (deferred) for a certain number of years.\u00a0\u00a0\u00a0 If we assume that there is no fixed end date to the business, we treat this problem as a perpetuity where the returns continue on indefinitely.\" id=\"return-footnote-1010-8\" href=\"#footnote-1010-8\" aria-label=\"Footnote 8\"><sup class=\"footnote\">[8]<\/sup><\/a>.<\/p>\n<p>When calculating payment sizes for deferred perpetuities or the initial amount needed to be deposited at the start of the perpetuity, it is important to remember that there are actually two parts to the deferred perpetuity problem:<\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig0.jpg\"><img decoding=\"async\" class=\"aligncenter wp-image-3345 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig0.jpg\" alt=\"Timeline Image for Deferred Perpetuities\" width=\"95%\" srcset=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig0.jpg 916w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig0-300x40.jpg 300w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig0-768x102.jpg 768w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig0-65x9.jpg 65w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig0-225x30.jpg 225w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig0-350x47.jpg 350w\" sizes=\"(max-width: 916px) 100vw, 916px\" \/><\/a><\/p>\n<p><strong>Part 1<\/strong>: The initial balance gathers interest.\u00a0 There are no payments nor withdrawals in part 1 (the deferral period).\u00a0 The only money being added to the balance is the interest being earned (or charged).\u00a0 This problem is a compound interest problem (Chapter 4):<\/p>\n<p><strong style=\"text-align: initial;\">Part 2<\/strong><span style=\"text-align: initial;\">: Payments are now being withdrawn. These payments exactly equal to the interest earned on the current balance in the perpetuity account. <\/span>The starting balance for the perpetuity (PV<sub>2<\/sub>) equals to the ending balance from the deferral period (FV<sub>1<\/sub>) .<\/p>\n<h2>Example 5.8.4<\/h2>\n<p>What if Sasha&#8217;s scholarship fund (from Examples 1 to 3) doesn&#8217;t give out its first scholarship for one year?\u00a0 How much would Sasha need to donate if the semi-annual scholarships are still $3,000 and the fund still earns 5% compounded semi-annually?<\/p>\n<p>Let us first draw out the timeline for this problem:<\/p>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig1-1.jpg\"><img decoding=\"async\" class=\"aligncenter wp-image-3348 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig1-1.jpg\" alt=\"Timeline for deferred perpetuity in Example 4\" width=\"95%\" srcset=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig1-1.jpg 948w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig1-1-300x51.jpg 300w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig1-1-768x131.jpg 768w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig1-1-65x11.jpg 65w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig1-1-225x38.jpg 225w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig1-1-350x60.jpg 350w\" sizes=\"(max-width: 948px) 100vw, 948px\" \/><\/a><\/p>\n<p>Because we know the payments for part 2, start there. We can either use the formula or the BAII Plus to do the calculations for this problem. Let us start by using the BAII Plus:<\/p>\n<p><strong>Part 2: Perpetuity (using the BAII Plus)<\/strong><\/p>\n<p>Let us &#8216;start&#8217; part 2 exactly when the first scholarship is awarded. For this reason, B\/E will be set to BGN:<\/p>\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\">\n<thead>\n<tr style=\"height: 17px;\">\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\n<th class=\"border\" style=\"width: 20%; height: 17px;\">N<sub>2<\/sub><\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">I\/Y<\/th>\n<th class=\"border\" style=\"width: 20.4789%; height: 17px;\">PV<sub>2<\/sub><\/th>\n<th class=\"border\" style=\"width: 17.5211%; height: 17px;\">PMT<sub>2<\/sub><\/th>\n<th class=\"border\" style=\"width: 10%; height: 17px;\">FV<sub>2<\/sub><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 19px;\">\n<td class=\"border\" style=\"width: 8%; height: 19px;\">BGN<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\n<td class=\"border\" style=\"width: 20%; height: 19px;\">1000\u00d72=2000<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">5<\/td>\n<td class=\"border\" style=\"width: 20.4789%; height: 19px;\"><strong>CPT<\/strong> +123,000<\/td>\n<td class=\"border\" style=\"width: 17.5211%; height: 19px;\"><span style=\"color: #ff0000;\">\u22123,000<\/span><\/td>\n<td class=\"border\" style=\"width: 10%; height: 19px;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that PV<sub>2<\/sub> equals to the amount Sasha would need to donate in Example 3. We will now enter that value into FV<sub>1<\/sub> and calculate the amount donated initially (PV<sub>1<\/sub>).<\/p>\n<p><strong>Part 1: Deferral Period (using the BAII Plus)<\/strong><\/p>\n<p>There are no payments nor withdrawals for part 1 so enter &#8216;\u2013\u2013&#8217; for B\/E. Also, because the deferral period lasts for 1 year, N<sub>1 <\/sub>=1\u00d72 = 2.<\/p>\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\">\n<thead>\n<tr style=\"height: 17px;\">\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\n<th class=\"border\" style=\"width: 20%; height: 17px;\">N<sub>1<\/sub><\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">I\/Y<\/th>\n<th class=\"border\" style=\"width: 23.7183%; height: 17px;\">PV<sub>1<\/sub><\/th>\n<th class=\"border\" style=\"width: 12.169%; height: 17px;\">PMT<sub>1<\/sub><\/th>\n<th class=\"border\" style=\"width: 12.1127%; height: 17px;\">FV<sub>1<\/sub><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 19px;\">\n<td class=\"border\" style=\"width: 8%; height: 19px;\">\u2013\u2013<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">2<\/td>\n<td class=\"border\" style=\"width: 20%; height: 19px;\">1\u00d72=2<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">5<\/td>\n<td class=\"border\" style=\"width: 23.7183%; height: 19px;\"><strong>CPT<\/strong> +117,073.17<\/td>\n<td class=\"border\" style=\"width: 12.169%; height: 19px;\">0<\/td>\n<td class=\"border\" style=\"width: 12.1127%; height: 19px;\"><span style=\"color: #ff0000;\">\u2212123,000<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Conclusion: Sasha will only need to donate $117,073.17 if the first scholarship is awarded in one year.<\/p>\n<p><strong>Method 2 \u2014 Using Formulas:<\/strong><\/p>\n<p>We will again start with part 2. Let us rewrite the timeline, slightly differently, however:<\/p>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig2.jpg\"><img decoding=\"async\" class=\"aligncenter size-full wp-image-3352\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig2.jpg\" alt=\"Timeline image for Deferred Perpetuity with start of perpetuity at 6 month mark\" width=\"95%\" srcset=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig2.jpg 829w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig2-300x58.jpg 300w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig2-768x148.jpg 768w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig2-65x13.jpg 65w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig2-225x43.jpg 225w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig2-350x68.jpg 350w\" sizes=\"(max-width: 829px) 100vw, 829px\" \/><\/a><\/p>\n<p>Notice that we will only run the deferral period for 6 months and start the perpetuity (part 2) 6 months before the first scholarship is withdrawn. This makes the calculation easier for part 2:<\/p>\n<p><strong>Part 2: Perpetuity (using Formulas)<\/strong><\/p>\n<p>We now have &#8216;started&#8217; part 2 one payment period earlier. This now makes the perpetuity in part 2 an ordinary annuity. We use the following formula to calculate its present value:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{PV}_2= \\frac{\\text{PMT}_2}{i} = \\frac{$3,000}{0.025}=$120,000[\/latex]<\/p>\n<p>There will need to be $120,000 in the scholarship fund in 6 months. We can find the present value of this amount using the compound interest formula from Chapter 4.<\/p>\n<p><strong>Part 1: Deferral Period (using Formulas)<\/strong><\/p>\n<p style=\"text-align: center;\">[latex]\\textrm{PV}_{1}= \\frac{\\textrm{FV}_{1}}{(1+i)^{n}} = \\frac{$120,000}{(1+0.025)^{1}}=$117,073.17[\/latex]<\/p>\n<p>Conclusion: Again, we find that Sasha will need to donate $117,073.17 now if the first scholarship is awarded in 1 year.<\/p>\n<h1>payments for Deferred Perpetuities<\/h1>\n<p>It is possible to use formulas to calculate the payment size for deferred perpetuities. We will however, just use the BAII Plus method for this section.<\/p>\n<p>Let us examine a similar donation example. In this example, Ezra, a wealthy philanthropist will make a large donation to help out a medical center that will not open for several years.<\/p>\n<h2>Example 5.8.5<\/h2>\n<p>Ezra, a wealthy philanthropist, donates $1,000,000 to the Moshi Medical Diagnostics Center, located in Moshi, Tanzania. The Center is due to open in 3 years. The Center will use the donated funds to cover their annual equipment and testing costs. They will invest the funds at 4.25%, effective, into a perpetuity.\u00a0 They will withdraw the first perpetuity payment in exactly 3 years when the center opens.\u00a0 Calculate the size of the annual withdrawals.<\/p>\n<p>Let us first draw out a timeline for this problem:<\/p>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig3.jpg\"><img decoding=\"async\" class=\"aligncenter wp-image-3356 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig3.jpg\" alt=\"Timeline for deferred perpetuity in Example 5\" width=\"95%\" srcset=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig3.jpg 958w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig3-300x49.jpg 300w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig3-768x126.jpg 768w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig3-65x11.jpg 65w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig3-225x37.jpg 225w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/08\/5-8-Fig3-350x57.jpg 350w\" sizes=\"(max-width: 958px) 100vw, 958px\" \/><\/a><\/p>\n<p>Next, determine which part to start with. In this case, because we know the initial deposit (PV<sub>1<\/sub>), start with part 1 (the deferral period).<\/p>\n<p><strong>Part 1 (the deferral period):<\/strong><\/p>\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\">\n<thead>\n<tr style=\"height: 17px;\">\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">P\/Y<\/th>\n<th class=\"border\" style=\"width: 8%; height: 17px;\">C\/Y<\/th>\n<th class=\"border\" style=\"width: 12.8168%; height: 17px;\">N<sub>1<\/sub><\/th>\n<th class=\"border\" style=\"width: 11.8029%; height: 17px;\">I\/Y<\/th>\n<th class=\"border\" style=\"width: 16.2535%; height: 17px;\">PV<sub>1<\/sub><\/th>\n<th class=\"border\" style=\"width: 10.9016%; height: 17px;\">PMT<sub>1<\/sub><\/th>\n<th class=\"border\" style=\"width: 24.2252%; height: 17px;\">FV<sub>1<\/sub><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 19px;\">\n<td class=\"border\" style=\"width: 8%; height: 19px;\">\u2013\u2013<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">1<\/td>\n<td class=\"border\" style=\"width: 8%; height: 19px;\">1<\/td>\n<td class=\"border\" style=\"width: 12.8168%; height: 19px;\">3\u00d71=3<\/td>\n<td class=\"border\" style=\"width: 11.8029%; height: 19px;\">4.25<\/td>\n<td class=\"border\" style=\"width: 16.2535%; height: 19px;\">+1,000,000<\/td>\n<td class=\"border\" style=\"width: 10.9016%; height: 19px;\">0<\/td>\n<td class=\"border\" style=\"width: 24.2252%; height: 19px;\"><strong>CPT<\/strong><span style=\"color: #ff0000;\"> \u22121,132,995.52<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It does not matter what we enter in B\/E above because there are no payments. Let us now enter the ending value for the deferral period (FV<sub>1<\/sub>) into the starting value for the perpetuity (PV<sub>2<\/sub>).<\/p>\n<p><strong>Part 2 (the perpetuity):<\/strong><\/p>\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse; width: 90%; height: 36px;\">\n<thead>\n<tr style=\"height: 17px;\">\n<th class=\"border\" style=\"width: 8%; height: 17px;\">B\/E<\/th>\n<th class=\"border\" style=\"width: 7.57746%; height: 17px;\">P\/Y<\/th>\n<th class=\"border\" style=\"width: 7.71831%; height: 17px;\">C\/Y<\/th>\n<th class=\"border\" style=\"width: 11.9717%; height: 17px;\">N<sub>2<\/sub><\/th>\n<th class=\"border\" style=\"width: 10.8169%; height: 17px;\">I\/Y<\/th>\n<th class=\"border\" style=\"width: 18.7888%; height: 17px;\">PV<sub>2<\/sub><\/th>\n<th class=\"border\" style=\"width: 24.7045%; height: 17px;\">PMT<sub>2<\/sub><\/th>\n<th class=\"border\" style=\"width: 10.4223%; height: 17px;\">FV<sub>1<\/sub><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 19px;\">\n<td class=\"border\" style=\"width: 8%; height: 19px;\">BGN<\/td>\n<td class=\"border\" style=\"width: 7.57746%; height: 19px;\">1<\/td>\n<td class=\"border\" style=\"width: 7.71831%; height: 19px;\">1<\/td>\n<td class=\"border\" style=\"width: 11.9717%; height: 19px;\">1000\u00d71=1000<\/td>\n<td class=\"border\" style=\"width: 10.8169%; height: 19px;\">4.25<\/td>\n<td class=\"border\" style=\"width: 18.7888%; height: 19px;\">+1,132,995.52<\/td>\n<td class=\"border\" style=\"width: 24.7045%; height: 19px;\"><strong>CPT <\/strong><span style=\"color: #ff0000;\">\u221246,189.27<\/span><\/td>\n<td class=\"border\" style=\"width: 10.4223%; height: 19px;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that BGN is on in part 2. This is because we started part 2 (the perpetuity) right when the first payment was withdrawn.<\/p>\n<p>Conclusion: The Moshi Medical Center will receive $46,189.27 annually to cover equipment and testing costs starting in 3 years when the center opens.<\/p>\n<h1>Key Takeaways for Perpetuities<\/h1>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways for Perpetuities<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>When entering perpetuities into the BAII Plus:<\/p>\n<ul>\n<li>FV is always equal to 0 for perpetuities.<\/li>\n<li>We always use 1,000 years for perpetuities.<\/li>\n<li>This gives N = 1000 \u00d7 P\/Y<\/li>\n<li>It does not matter what sign we use for PV or PMT (as we are computing the other one)<a class=\"footnote\" title=\"If ever both were being entered into the BAII plus - we would need to use opposite signs for PV and PMT\" id=\"return-footnote-1010-9\" href=\"#footnote-1010-9\" aria-label=\"Footnote 9\"><sup class=\"footnote\">[9]<\/sup><\/a><\/li>\n<\/ul>\n<p>When using formulas:<\/p>\n<ul>\n<li>For ordinary perpetuities, [latex]\\textrm{PMT}= \\textrm{PV} \\times i[\/latex]<\/li>\n<li>For perpetuities due, [latex]PV_{due} = \\frac{PMT}{i} + PMT[\/latex]<\/li>\n<\/ul>\n<p>For deferred perpetuities:<\/p>\n<ul>\n<li>Run the deferral period until the first payment occurs<\/li>\n<li>Turn BGN on for part 2<\/li>\n<li>Set PV<sub>2<\/sub> = FV<sub>1<\/sub><\/li>\n<li>Set FV<sub>2<\/sub> = 0.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Your Own Notes<\/h1>\n<ul>\n<li>Are there any notes you want to take from this section? Is there anything you&#8217;d like to copy and paste below?<\/li>\n<li>These notes are for you only (they will not be stored anywhere)<\/li>\n<li>Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n<div id=\"h5p-1\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-1\" class=\"h5p-iframe\" data-content-id=\"1\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key takeaways, notes and comments from this section document tool.\"><\/iframe><\/div>\n<\/div>\n<h1>The Footnotes<\/h1>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1010-1\">Information thanks to <a href=\"https:\/\/languages.oup.com\/google-dictionary-en\/\">Oxford Languages<\/a>; <a href=\"https:\/\/www.investopedia.com\/terms\/p\/perpetuity.asp\">Perpetuity: Financial Definition, Formula, and Examples (Investopedia)<\/a>; <a href=\"https:\/\/en.wikipedia.org\/wiki\/War_bond\">War Bond (Wikipedia)<\/a>; <a href=\"https:\/\/www.advisor.ca\/tax\/estate-planning\/design-a-dynasty-with-perpetual-trusts\/\">Design a dynasty with perpetual trusts (Advisor.ca)<\/a>; <a href=\"https:\/\/www.investopedia.com\/terms\/t\/trust-fund.asp\">Trust Funds (Investopedia)<\/a>; <a href=\"https:\/\/thephilanthropist.ca\/original-pdfs\/Philanthropist-21-3-370.pdf\">Charities and the Rule Against Perpetuities [PDF]<\/a>; <a href=\"#return-footnote-1010-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-1010-2\">Historically, governments issued bonds to raise money to meet the growing costs of war.\u00a0 Some of these bonds lasted in perpetuity. <a href=\"#return-footnote-1010-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-1010-3\">Suppose a company invests a certain initial amount of money with the hope\/expectation that they will earn a return on their investment in the form of regular income from the business (possibly for an indefinite amount of time).\u00a0 If we assume the returns last for an indefinite amount of time, we treat this problem like a perpetuity as well. <a href=\"#return-footnote-1010-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-1010-4\">A perpetuity, technically, lasts forever. In the calculator, the closest to 'forever' we should enter for a number of years is 1,000. This number of years will work with all types of calculations in the TVM keys. <a href=\"#return-footnote-1010-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><li id=\"footnote-1010-5\">The same principle applies as in ordinary perpetuities, the balance in the scholarship fund never increases nor decreases in value. This is because after a payment has been withdrawn from the account, interest is earned on the remaining balance in the scholarship fund.\u00a0 The payments are calculated such that the after the interest has been earned, the balance in the account increases back to its original value (PV). <a href=\"#return-footnote-1010-5\" class=\"return-footnote\" aria-label=\"Return to footnote 5\">&crarr;<\/a><\/li><li id=\"footnote-1010-6\">We, again, need to be careful if the number of payments per year (P\/Y) is not equal to the number of compounding periods per year (C\/Y).\u00a0 If that is the case, we need to calculate the equivalent interest rate with the number of compounding periods equal to the number of payment intervals for the perpetuity. <a href=\"#return-footnote-1010-6\" class=\"return-footnote\" aria-label=\"Return to footnote 6\">&crarr;<\/a><\/li><li id=\"footnote-1010-7\">They deposit the money once it\u2019s received but do not need to start withdrawing from the account until the non-profit opens.\u00a0 If we assume that the non-profit will remain open indefinitely, then we assume that they will need withdrawals to cover their costs for an undetermined amount of time as well.\u00a0 We treat this problem as a perpetuity. <a href=\"#return-footnote-1010-7\" class=\"return-footnote\" aria-label=\"Return to footnote 7\">&crarr;<\/a><\/li><li id=\"footnote-1010-8\">In that case, the repayments of the initial investment are delayed (deferred) for a certain number of years.\u00a0\u00a0\u00a0 If we assume that there is no fixed end date to the business, we treat this problem as a perpetuity where the returns continue on indefinitely. <a href=\"#return-footnote-1010-8\" class=\"return-footnote\" aria-label=\"Return to footnote 8\">&crarr;<\/a><\/li><li id=\"footnote-1010-9\">If ever both were being entered into the BAII plus - we would need to use opposite signs for PV and PMT <a href=\"#return-footnote-1010-9\" class=\"return-footnote\" aria-label=\"Return to footnote 9\">&crarr;<\/a><\/li><\/ol><\/div><div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_1010_3322\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_1010_3322\"><div tabindex=\"-1\"><p>An annuity that has no end or an annuity with regular cash flows that continue forever.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_1010_3289\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_1010_3289\"><div tabindex=\"-1\"><p>The termination or ending date for which a loan, bond, or any amount borrowed must be paid back in full.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_1010_1012\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_1010_1012\"><div tabindex=\"-1\"><p>A perpetuity where the first payment comes at the <em>end<\/em> of the first period<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_1010_1015\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_1010_1015\"><div tabindex=\"-1\"><p>A <a href=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/perpetuities\/\">perpetuity<\/a> where the first payment is at the <em>beginning<\/em> of the first period.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":883,"menu_order":8,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1010","chapter","type-chapter","status-publish","hentry"],"part":46,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/1010","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/users\/883"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/1010\/revisions"}],"predecessor-version":[{"id":3988,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/1010\/revisions\/3988"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/parts\/46"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/1010\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/media?parent=1010"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapter-type?post=1010"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/contributor?post=1010"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/license?post=1010"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}