Annuities Sinking Funds

Annuities Sinking Funds - CM 426 - Time Value of Money --...

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Unformatted text preview: CM 426 - Time Value of Money -- ANNUITIES The formula for determining the future value of an annuity (a stream of equal payments made at equal intervals compounding at a single interest rate) is represented by the following: ‘ FV of Annuity = Payment * [((1+Periodic Interest Rate)ANumber of Periods - 1) / Periodic Interest Rate] (“Pansy-t m: mm— FK/A : PMT Per‘fwf: Pffiiobic INTEREST RATE P6r__ir\+ N = Numeea at rations This formula assumes that the first payment into the annuity will be made at the END of the first period, and the final payment will be made at the END of the final compounding period. Note that in this case, the last payment made will not accrue any interest because it is made at the END of the final period. When your calculator is in the END mode, this is what happens. Problem: Jim will make regular payments of $100 every six months for four years into an account that pays 10%, compounded semi-annually. He will begin payments in six months. How much will he have in this account exactly four years from today? To make the problem more understandable and to help you as you enter the known variables into your calculator (HPlOB), it is my suggestion that you first diagram the information using a time line with arrows indicating the cash-flow directions, and then make some notes. For the problem stated above, a time line will look something like this: CF} of ‘15» EW Tao», E"D Ar E012. 15sz '-I W 4".) -100 -Ioo r —-Ioo —/00 .100 -/00 eloo Now, to program the information into your calculator, do the following: 1. In financial problems that involve payments, the first step is to indicate how many payments per year (P/Y R) there will be. For the problem above, enter 2, then SWITCH (YELLOW, ORANGE OR BLUE button on your calculator), and finally P/Y R. 2. The next step is be sure your calculator is in END mode (because the payments in this problem will be made at the end of the periods) -- or BEGIN mode (if the payments are to be made at the beginning of the periods). 3. Now key in the ANNUAL interest rate (I/Y R) of 10%. Because you have already programmed the number of payments per year (2), your calculator will automatically understand that the periodic interest rate is actually 5% (10% per year divided by 2 = 5%). 4. Next, enter the amount of the regular periodic payments (PMT). Key in 35—100 (remember that payments are going 9111, as indicated by the down arrows in your diagram); then enter PMT. Don’t forget that annuities have several characteristics that must be met: equal garments made at equal intervals compounded at an equal periodic interest rate. 5. Key in the number of periods (N). In this case, there are eight periods (four years of semi-annual payments). You can do this by simply entering 8 and then N; or you can take a shortcut and enter 4 (the number of years), SWITCH, and finally XP/Y R. My suggestion for most problems is to let the calculator do the “thinking” for you. Because you have already “told” your calculator that there are two payments per year the calculator automatically does the math: 4 years times 2 periods per year = 8 N. 6. A final step (not necessary, but a good idea) is to indicate the present value of your annuity; that is, how much money there is in this account today (PV). Of course, in this case, there is nothing. So enter 0, then PV. If you don’t enter anything for PV, the default by the calculator is that PV = 0. Afier entering this information (see below), you should be able to call for the future value (FV) calculation. The answer is $954.91. 7. Now, if you were making your first payment into this account at the beginning of the first period, instead of at the end of the first period, you would simply toggle your calculator to BEGIN mode instead of END mode. Do this by entering SWITCH, BEG/END. You should see the word begin in your register. If you enter SWITCH, BEG/END again, you will not see anything in the register and would be in END mode. After you have toggled to BEGIN mode, ask for the future value (FV), and the answer will be $1,002.66. 2 E] m ‘L/M : ,0 N ; Lf [ilk-$22; : END .. - .- PM‘ ' SINKING FUND The formula for determining the payment required for a sinking fund -- a stream of equal payments made at equal intervals compounding at a single interest rate that will yield a specified future value -- is the same as that used to determine the future value of an annuity; only in this case the future value is known, so we solve for the payment. Payment = FV / [ ((1+Periodic Interest Rate)"Number of Periods - 1) / Periodic Interest Rate] FVA M “‘1 PMT : (1+ Pen“) I /, per, ‘m‘l’ This formula assumes that the first payment into the sinking fund will be made at the END of the first period, and the final payment will be made at the END of the final compounding period. Note that in this case, the last payment made will not accrue any interest because it is made at the END of the final period. When your calculator is in the END mode, this is what happens. (See time line for annuity in END mode.) Problem: Jim predicts that he will need $1,000 three years from today and is willing to make equal deposits (payments) every three months into an account that pays 8%, compounded quarterly for the next three years. He will make his first deposit in three months. How much will his payments have to be in order to have $1,000 in this account exactly three years from today? To make the problem more understandable, and to help you as you enter the known variables into your calculator, it is my suggestion that you make some notes. For example: 1. First write down 4 periods per year (4 SWITCH P/YR) 2. Next, write down END to be sure your calculator is in the END mode (Payments are going to be made at the end of the periods) 3. Next, list the known variables. You can do this in any order. Future Value is $1,000; Interest per year is 8% (so periodic interest would be 2% (8% / 4 periods per year); Number of periods = 12 (Regular payments every three months for 3 years) Or better: N = 3 SWITCH XP/Y R 4. Payments = Unknown After entering this information, you should be able to call for the Payment calculation. The answer is $74.56. If the question called for the payments to be made at the beginning of each period (see time line below), you would key in BEGIN mode instead of END mode, and your calculator should compute an answer of $73.10. ...
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Annuities Sinking Funds - CM 426 - Time Value of Money --...

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