Mastery Problem: Time Value of Money Time value of money Due to both interest earnings and the fact that money put to good use should generate additional funds above and beyond the original investment, money tomorrow will be worth less than money today. Simple interest Ross Co., a company that you regularly do business with, gives you a $11,000 note. The note is due in three years and pays simple interest of 9% annually. How much will Ross pay you at the end of that term? Note: Enter the interest rate as a decimal. (i.e. 15% would be entered as .15) Principal + (Principal + ($ Feedback X Rate X Time Total X X years Check My Work Remember, the interest is only computed on the principal amount. Compound interest With compound interest, the interest is added to principal in the calculation of interest in future periods. This addition of interest to the principal is called compounding. This differs from simple interest, in which interest is computed based upon only the principal. The frequency with which interest is compounded per year will dictate how many interest computations are required (i.e. annually is once, semi-annually is twice, and quarterly is four times). Imagine that Ross Co., fearing that you wouldn't take its deal, decides instead to offer you compound interest on the same $11,000 note. How much will Ross pay you at the end of three years if interest is compounded annually at a rate of 9%? If required, round your answers to the nearest cent. Principal Amount at Beginning of Year Year Annual Amount of Interest (Principal at Beginning of Year x 9%) Accumulated Amount at End of Year (Principal at Beginning of Year + Annual Amount of Interest) 1 $11,000 $990 $11,990 2 $11,990 $ 3 If you were given the choice to receive more or less compounding periods, which would you choose in order to maximize your monetary situation? More Feedback Check My Work To solve compounding interest, interest is added to principal in the calculation of interest in future periods. compounding periods per year. APPLY THE CONCEPTS: Present value of a single amount in the future As it is important to know what a current investment will yield at a point in the future, it is equally important to understand what investment would be required today in order to yield a required future return. The following timeline displays what present investment is required in order to yield $8,000 three years from now, assuming annual compounding at 5%. Future Value: $8,000 Present Value: ? Year 1 Year 2 Year 3 The most straightforward method for calculating the present value of a future amount is to use the Present Value Table. By multiplying the future amount by the appropriate figure from the table, one may adequately determine the present value. Instructions for using present value tables + Present Value of a Future Amount Table1 - Present Value of $1 at Compound Interest Period 5% 6% 7% 8% 9% 10% 11% 12% 1 2 0.952 0.943 0.935 0.926 0.917 0.909 0.901 0.893 0.907 0.890 0.873 0.857 0.842 0.826 0.812 0.797 3 0.864 0.840 0.816 0.794 0.772 0.751 0.731 0.712 4 0.823 0.792 0.763 0.735 0.708 0.683 0.659 0.636 + Present Value of a Future Amount Table1 - Present Value of $1 at Compound Interest Period 5% 6% 7% 8% 9% 10% 11% 12% 1 0.952 0.943 0.935 0.926 0.917 0.909 0.901 0.893 2 0.907 0.890 0.873 0.857 0.842 0.826 0.812 0.797 3 0.864 0.840 0.816 0.794 0.772 0.751 0.731 0.712 4 0.823 0.792 0.763 0.735 0.708 0.683 0.659 0.636 5 0.784 0.747 0.713 0.681 0.650 0.621 0.593 0.567 6 0.746 0.705 0.666 0.630 0.596 0.564 0.535 0.507 7 0.711 0.665 0.623 0.583 0.547 0.513 0.482 0.452 8 0.677 0.627 0.582 0.540 0.502 0.467 0.434 0.404 9 0.645 0.592 0.544 0.500 0.460 0.424 0.391 0.361 10 0.614 0.558 0.508 0.463 0.422 0.386 0.352 0.322 11 0.585 0.527 0.475 0.429 0.388 0.350 0.317 0.287 12 0.557 0.497 0.444 0.397 0.356 0.319 0.286 0.257 13 0.530 0.469 0.415 0.368 0.326 0.290 0.258 0.229 14 0.505 0.442 0.388 0.340 0.299 0.263 0.232 0.205 15 0.481 0.417 0.362 0.315 0.275 0.239 0.209 0.183 16 0.458 0.394 0.339 0.292 0.252 0.218 0.188 0.163 17 0.436 0.371 0.317 0.270 0.231 0.198 0.170 0.146 18 0.416 0.350 0.296 0.250 0.212 0.180 0.153 0.130 19 0.396 0.331 0.277 0.232 0.194 0.164 0.138 0.116 20 0.377 0.312 0.258 0.215 0.178 0.149 0.124 0.104 Using the previous table, enter the correct factor for three periods at 5%: Future value x Factor = Present value Using the previous table, enter the correct factor for three periods at 5%: Future value x Factor Present value $8,000 x 0.864 = $6,912 You may want to own a home one day. If you are 20 years old and plan on buying a $700,000 house when you turn 30, how much will you have to invest today, assuming your investment yields an 8% annual return? $ Feedback Check My Work Use the Present Value table to look up the appropriate value for the number of compounding periods and the interest rate per period. APPLY THE CONCEPTS: Present value of an ordinary annuity Many times future sums of money will not come in one payment but in a number of periodic payments. For example, imagine that you want to buy a house and know that you will have periodic mortgage payments and you need to know how much you would have to invest today in order to facilitate all of those payments into the future. This is called an ordinary annuity and it says that a certain value today at a stated interest rate is equal to a certain number of future payouts for a given amount per payment. The following timeline displays how an ordinary annuity pays out when distributed in three equal payments at an annually compounded interest rate of 5%. Present Value: ? Year 1 Payment: $6,000 Year 2 Payment: $6,000 Year 3 Payment: $6,000 The most simple and commonly used method of determining the present value of an ordinary annuity is to multiply the incremental payout by the appropriate rate found on the present value of an ordinary annuity table. + Present Value of an Ordinary Annuity Table 2 - Present Value of an Ordinary Annuity of $1 at Compound Interest Period 5% 6% 7% 8% 9% 10% 11% 12% 0.052 0.012 0.025 0.026 0.017 0.000 0.001 0.202 + Present Value of an Ordinary Annuity Table 2 - Present Value of an Ordinary Annuity of $1 at Compound Interest Period 5% 6% 7% 8% 9% 10% 11% 12% 1 0.952 0.943 0.935 0.926 0.917 0.909 0.901 0.893 2 1.859 1.833 1.808 1.783 1.759 1.736 1.713 1.690 3 2.723 2.673 2.624 2.577 2.531 2.487 2.444 2.402 4 3.546 3.465 3.387 3.312 3.240 3.170 3.102 3.037 5 4.329 4.212 4.100 3.993 3.890 6 5.076 4.917 4.767 4.623 3.791 3.696 3.605 4.486 4.355 4.231 4.111 7 5.786 5.582 5.389 5.206 5.033 4.868 4.712 4.564 8 6.463 6.210 5.971 5.747 5.535 5.335 5.146 4.968 9 7.108 6.802 6.515 6.247 5.995 5.759 5.537 5.328 10 7.722 7.360 7.024 6.710 6.418 6.145 5.889 5.650 11 8.306 7.887 7.499 7.139 6.805 6.495 6.207 5.938 12 8.863 8.384 7.943 7.536 7.161 6.814 6.492 6.194 13 9.394 8.853 8.358 7.904 7.487 7.103 6.750 6.424 14 9.899 9.295 8.745 8.244 7.786 7.367 6.982 6.628 15 10.380 9.712 9.108 8.559 8.061 7.606 7.191 6.811 16 10.838 10.106 9.447 8.851 17 11.274 10.477 9.763 9.122 8.544 8.313 7.824 7.379 6.974 8.022 7.549 7.120 18 11.690 10.828 10.059 9.372 8.756 8.201 7.702 7.250 19 12.085 11.158 10.336 9.604 8.950 8.365 7.839 7.366 20 12.462 11.470 10.594 9.818 9.129 8.514 7.963 7.469 469 Using the previous table, enter the correct factor for three periods at 5%: Periodic payment x Factor Present value 13 9.394 8.853 8.358 7.904 7.487 7.103 6.750 6.424 14 9.899 9.295 8.745 8.244 7.786 7.367 6.982 6.628 15 10.380 9.712 9.108 8.559 8.061 7.606 7.191 6.811 16 10.838 10.106 9.447 8.851 8.313 7.824 7.379 6.974 17 11.274 10.477 9.763 9.122 8.544 8.022 7.549 7.120 18 11.690 10.828 10.059 9.372 8.756 8.201 7.702 7.250 19 12.085 11.158 10.336 9.604 8.950 8.365 7.839 7.366 20 12.462 11.470 10.594 9.818 9.129 8.514 7.963 7.469 Using the previous table, enter the correct factor for three periods at 5%: Periodic payment x Factor $6,000 Present value x 2.723 $16,338 The controller at Ross has determined that the company could save $6,000 per year in engineering costs by purchasing a new machine. The new machine would last 12 years and provide the aforementioned annual monetary benefit throughout its entire life. Assuming the interest rate at which Ross purchases this type of machinery is 9%, what is the maximum amount the company should pay for the machine? $ 26,801 X (Hint: This is basically a present value of an ordinary annuity problem as highlighted above.) Assume that the actual cost of the machine is $50,000. Weighing the present value of the benefits against the cost of the machine, should Ross purchase this piece of machinery? No