{"id":1036,"date":"2020-08-19T15:47:12","date_gmt":"2020-08-19T19:47:12","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&p=1036"},"modified":"2021-07-15T11:28:35","modified_gmt":"2021-07-15T15:28:35","slug":"back-to-back","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/back-to-back\/","title":{"raw":"5.6 Back to Back Annuities","rendered":"5.6 Back to Back Annuities"},"content":{"raw":"
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Learning Outcomes<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nCalculate the initial deposit value or payment sizes in back-to-back annuities.\r\n\r\n<\/div>\r\n<\/div>\r\nWhat are back-to-back annuities? They are a series of equal-sized, regular deposits (payments) over a fixed period of time (annuity 1) followed by a series of equal-size regular withdrawals for a fixed time period (annuity 2). In both cases, the balance in the account will be earning interest during the deposits and withdrawals.\u00a0 See the diagram below:\r\n\r\n\"Timeline<\/a>\r\n\r\nNotice that, unless there is a special deposit or withdrawal between the two annuities, the ending balance of the first annuity (FV1<\/sub>) becomes the starting value of annuity 2 (PV2<\/sub>) with one caution: we need to be careful of signs. FV1<\/sub> will be negative. PV2<\/sub> will be positive (they should be opposite in sign). We will talk more about this later during our first example.\r\n\r\nSee the sections below for key formulas, tips and examples related to back-to-back annuities calculations.\r\n

Examples of back-to-back annuities<\/h1>\r\nIt is common when saving for retirement, or for a child\u2019s education to save up by making regular deposits into an [pb_glossary id=\"3238\"]RRSP[\/pb_glossary] (registered retirement savings plan) or an [pb_glossary id=\"3239\"]RESP[\/pb_glossary] (registered education savings plan).\u00a0 Often, after making these regular deposits, the retiree (or student) starts making regular withdrawals from the account upon retirement (or upon starting school for the student).\r\n

The Signs of PV, PMT & FV for Back-to-Back Annuities<\/h1>\r\nWhen calculating deposit or withdrawal amounts for back-to-back annuities, it is important to be careful of the signs of each of the values (for PV, PMT and FV).\u00a0 Let\u2019s examine the signs below:\r\n\r\nAnnuity 1<\/strong>: The initial balance (PV1<\/sub>) is considered positive. This balance gathers interest. The subsequent payments (PMT1<\/sub>) add to the existing balance in the account and are therefore also positive. At the end of the annuity, we consider the future value (FV1<\/sub>) as the amount we would need to withdraw from the account to close the account. For this reason, the future value (FV1<\/sub>) is recorded as negative:\r\n\r\n\r\n\r\n\r\n\r\n\r\n
PV1<\/sub><\/th>\r\nInterest1<\/sub><\/th>\r\nPMT1<\/sub><\/th>\r\nFV1<\/sub><\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Initial Deposit<\/strong><\/td>\r\n+ % Gain<\/strong><\/td>\r\n+ Regular Deposits<\/strong><\/td>\r\n= Ending Balance<\/span><\/strong><\/td>\r\n<\/tr>\r\n
0 or +<\/span><\/strong><\/td>\r\n+<\/span><\/strong><\/td>\r\n+<\/span><\/strong><\/td>\r\n\u2212<\/span><\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAnnuity 2<\/strong>: The initial balance (PV2<\/sub>), in most cases, is the amount of money saved up in annuity 1 (FV1<\/sub>).\u00a0 There is one exception to this rule \u2013 when there is a lump-sum deposited or withdrawn between the end of Annuity1<\/sub> and the start of Annuity2<\/sub>. We will see examples of this later in this section.\r\n\r\n\r\n\r\n\r\n\r\n\r\n
PV2<\/sub><\/th>\r\nInterest2<\/sub><\/th>\r\nPMT2<\/sub><\/th>\r\nFV2<\/sub><\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Initial Deposit<\/strong><\/td>\r\n+ % Gain<\/strong><\/td>\r\n=\u00a0 \u00a0Regular Withdrawals<\/strong><\/span><\/td>\r\n+\u00a0 Final Withdrawal<\/strong><\/span><\/td>\r\n<\/tr>\r\n
+<\/span><\/strong><\/td>\r\n+<\/span><\/strong><\/td>\r\n\u2212<\/span><\/strong><\/td>\r\n0 or<\/span> \u2212<\/span><\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor annuity 2, both the regular withdrawals (PMT2<\/sub>) and the final ending balance (FV2<\/sub>) deduct from the balance in the account.\u00a0 They should both be negative:\r\n\r\n\u00a0<\/strong>Putting this all together gives:<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
PV1<\/sub><\/th>\r\nInterest1<\/sub><\/th>\r\nPMT1<\/sub><\/th>\r\nFV1<\/sub><\/th>\r\nPV2<\/sub><\/th>\r\nInterest2<\/sub><\/th>\r\nPMT2<\/sub><\/th>\r\nFV2<\/sub><\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Initial Deposit<\/strong><\/td>\r\n+ % Earned<\/strong><\/td>\r\n+ Regular Deposits<\/span><\/strong><\/td>\r\n=Ending Balance Part 1<\/strong><\/span><\/td>\r\n=Starting Balance Part 2<\/span><\/strong><\/td>\r\n+ % Earned<\/strong><\/td>\r\n=<\/strong> Regular Withdrawals<\/strong><\/span><\/td>\r\n+ Final<\/span> Wit<\/span>hdraw<\/span>al<\/span><\/strong><\/td>\r\n<\/tr>\r\n
+<\/strong><\/td>\r\n+<\/strong><\/td>\r\n+<\/strong><\/td>\r\n\u2212<\/span><\/strong><\/td>\r\n+<\/strong><\/td>\r\n+<\/strong><\/td>\r\n\u2212<\/span><\/strong><\/td>\r\n\u2212<\/span><\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that, we use the ending balance of annuity 1 (FV1<\/sub>) becomes the starting balance of annuity 2 (PV2<\/sub>) except we change the sign. FV1<\/sub> should be negative and PV2<\/sub> should be positive (they should be opposite in sign). We will talk more about this later during our first example.\r\n

Determining the size of the Regular deposits (PMT1<\/sub>)<\/h1>\r\nSome people plan for their retirement by deciding on the size of the withdrawals they would like to receive upon retirement (PMT2<\/sub>). They then back-calculate the size of the deposits (PMT1<\/sub>) they will need to make to achieve their retirement goals.\r\n\r\nLet's have a look at Raj's retirement plan in the next example. Raj is very wise and starts saving when he turns 25!\r\n

Example 5.6.1<\/h2>\r\nToday is Raj\u2019s 25th<\/sup> birthday, and he has opened an account to start his retirement savings with an initial deposit of $1,000. He plans to make regular deposits into the account on a monthly basis, with the first deposit today. He estimated that, the retirement account will earn an average interest rate of 6% compounded annually. At age 65 he will turn his retirement saving into an annuity paying 4% compounded annually and he will be able to withdraw $4,000 per month for 30 years with the first withdrawal occurring on his 65th<\/sup> birthday.\u00a0 How much does Raj\u2019s monthly deposit need to be in order to meet his retirement goals?\r\n\r\nLet us first organize this information into a time diagram:\r\n\r\n\"Timeline<\/a>\r\n\r\nNext, we need to determine where to start. For back-to-back annuities, always start where the known payment is. In this example, we know the size of Raj's withdrawals during his retirement (PMT2<\/sub>), therefore, we will \"start\" with part 2.\r\n\r\nLet us now fill in the BAII Plus table for Part 2:\r\n\r\n\r\n\r\n\r\n\r\n
B\/E<\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN2<\/sub><\/th>\r\nI\/Y<\/th>\r\nPV2<\/sub><\/th>\r\nPMT2<\/sub><\/th>\r\nFV2<\/sub><\/th>\r\n<\/tr>\r\n<\/thead>\r\n
BGN<\/td>\r\n12<\/td>\r\n1<\/td>\r\n30\u00d712=360<\/td>\r\n4<\/td>\r\nCPT<\/strong> +847,893.56<\/td>\r\n\u22124,000<\/span><\/td>\r\n0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n