{"id":983,"date":"2020-08-17T15:18:16","date_gmt":"2020-08-17T19:18:16","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&p=983"},"modified":"2021-07-12T16:59:28","modified_gmt":"2021-07-12T20:59:28","slug":"saving-annuities","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/saving-annuities\/","title":{"raw":"5.1 Adding to a Savings Account","rendered":"5.1 Adding to a Savings Account"},"content":{"raw":"
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Learning Outcomes<\/p>\r\n\r\n<\/header>\r\n

Calculate the amount saved and interest earned when regular deposits are made.<\/div>\r\n<\/div>\r\n\"\"You make a series of equal-sized payments (deposits), at regular intervals over the course of a fixed time period. When you make these regular deposits, you are adding to the money in the account and building up a positive account balance. Therefore, both PV (your initial deposit) and PMT (your subsequent regular deposits) must have the same sign (positive). The future value (FV) must have the opposite sign (negative)[footnote]Other texts might treat PV and PMT as negative and FV as positive. What is most important is that we are aware that PV and PMT must have the same sign and FV must be opposite in sign.[\/footnote]:\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
Initial Deposit<\/strong><\/td>\r\n+ % Gain<\/strong><\/td>\r\n+ Regular Deposits<\/strong><\/td>\r\n= Final Withdrawal<\/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\nTo make sense of FV (your final withdrawal) being negative, you could imagine that we are closing the account at the end of the investment period. When we close it, we withdraw<\/em> all of the funds, making the final\/future value opposite in sign from the regular deposits (PMT) and the initial amount deposited (PV).\r\n\r\nSee the sections below for key formulas, tips and examples related to calculations when adding to a savings account.\r\n

Calculating An Amount Saved (FV)<\/h1>\r\nLet us start by calculating the ending account balance (FV) when regular deposits (PMT) are made into an account. For this scenario, it is possible that there is an initial balance (PV) or no money in the account (PV = 0). Let's see an example where you start saving for your retirement and we'll examine both scenarios \u2014 when you have no money in the account at the start and when you have an initial balance.\r\n

EXAMPLE 5.1.1\u2014 SAVING FOR RETIREMENT<\/h2>\r\nFor the next ten years, you deposit $400 each month into a retirement savings account. You deposit the first payment one month from now. The account will pay 4.5%, compounded monthly. You start with $0 in the account at the start of the ten years. How much money will you have in the account at the end of ten years?\r\n\r\n\r\n\r\n\r\n\r\n
B\/E<\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN<\/th>\r\nI\/Y<\/th>\r\nPV<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
END<\/td>\r\n12<\/td>\r\n12<\/td>\r\n10\u00d712=120<\/td>\r\n4.5<\/td>\r\n0<\/td>\r\n+400<\/td>\r\nCPT<\/strong> -60,479.23<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhy is B\/E set to END? This is because the first deposit is being made in one month, at the end of the first payment interval. When regular payments or deposits are made at the end of the payment interval, we call this an [pb_glossary id=\"892\"]ordinary annuity[\/pb_glossary]. We do not need to do anything in the calculator (it is set to END by default). We will talk more about changing this setting in the later sections.\r\n\r\nWhy are P\/Y and C\/Y equal to 12? P\/Y equals the number of payments or deposits per year. C\/Y equals the number of times the interest rate compounds per year and the interest. Since both the compounding and deposits are monthly, both P\/Y and C\/Y equal 12.\r\n\r\nWhy does N equal 120? N equals 120 because you will make monthly deposits for 10 years. This will give you a total of 10\u00d712 deposits.\r\nNote: N = The total number of payments or deposits = Number of years \u00d7 P\/Y\r\n\r\nWhy does PV equal 0? PV is set to 0 because there is no money in the account at the start. PV is the initial (starting) balance in the account.\r\n\r\nFinally, why does PMT equal +400? PMT equals to +400 because we make regular deposits of $400 into our savings account. We treat all deposits as positive because it is money going into a savings account[footnote]Some texts might treat deposits into a savings account as negative. If they do, the initial deposit, PV (if non-zero) will also be negative and FV (the ending balance) will be positive.[\/footnote].\r\n\r\nNow we can calculate the amount saved at the end of 10 years (FV). Notice that the BAII Plus will give us a negative value for the future value. This minus sign (negative sign) indicates that the future value (FV) is going in the opposite direction of the deposits (PMT and PV). Don't include the minus sign (negative sign) in your final answer.\r\n\r\nConclusion: You will have $60,479.23 in your savings account at the end of 10 years.\r\n
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Check Your Knowledge 5.1.1<\/p>\r\n\r\n<\/header>\r\n

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Redo the above example but instead, start with a balance of $1,000 in the account. What will be the ending balance in the account?\u00a0 Drag the values that you would enter into the BAII Plus Calculator Keys in the exercise below.<\/div>\r\n
<\/div>\r\n
[h5p id=\"11\"]<\/div>\r\n
<\/div>\r\n
[h5p id=\"3\"]<\/div>\r\n
<\/div>\r\n
Conclusion: You will have $62,046.22 in your savings account at the end of 10 years. Note that the initial deposit of $1,000 makes your ending balance $1,566.99 larger.<\/div>\r\n<\/div>\r\n<\/div>\r\n

Calculating the Interest Earned<\/h1>\r\nFor all investments, loans, etc., we can consider the following formula to be true when calculating the interest earned or charged on our investment or loan:\r\n

Interest = Money Out - Money In = $ OUT<\/span> - $ IN<\/p>\r\nIn the case of regular deposits into a savings account, in this textbook, we will consider the initial deposit, PV (if there is one) to be \"money in\" since we are depositing this money INTO the account.[footnote]Other texts or instructors might treat signs differently than we do here. What is most important is that we are aware that money flows in two different directions (in and out). It could be possible to consider the regular deposits as negative and the ending balance (FV) as positive. You will still get the correct answer because the deposits and final balance are opposite in sign.[\/footnote] We consider the regular deposits, PMT, to be \"money in\" since they are also deposits going INTO the savings account. Finally, we consider FV to be \"money out\" since it is being withdrawn from the account when we close it:\r\n\r\n\r\n\r\n\r\n\r\n\r\n
PV<\/th>\r\n\u00a0Interest<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Initial Deposit<\/strong><\/td>\r\n+ % Gain<\/strong><\/td>\r\n+\u00a0 Regular Deposits<\/strong><\/td>\r\n=\u00a0 Final Withdrawal<\/span><\/strong><\/td>\r\n<\/tr>\r\n
$ IN<\/span><\/td>\r\n$ IN<\/span><\/td>\r\n$ IN<\/span><\/td>\r\n$ OUT<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis gives us the following equation for the interest earned:\r\n

[latex]\\begin{align*}\r\n\\textrm{Interest Earned} &= \\textrm{\\$ Out}-\\textrm{\\$ IN}\\\\\r\n&=\\textrm{Final Withdrawal}-\\textrm{(Initial Deposit + Regular Deposits)}\\\\\r\n&=\\textrm{FV}-\\textrm{(PV + PMT \u00d7 N)}\r\n\\end{align*}[\/latex]<\/p>\r\n \r\n

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Check Your Knowledge 5.1.2<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n[h5p id=\"5\"]\r\n\r\n[h5p id=\"6\"]\r\n\r\n<\/div>\r\n<\/div>\r\n

Key Takeaways<\/h1>\r\n
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Key Takeaways: Regular Deposits into Savings Accounts<\/p>\r\n\r\n<\/header>\r\n

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Your Own Notes<\/h1>\r\n