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 Stone Co., a company that you regularly do business with, gives you a $18,000 note. The note is due in three years and pays simple interest of 6% annually. How much will Stone 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 X Rate X Time Total $ + ($ X X years ) $ 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 Stone Co., fearing that you wouldn't take its deal, decides instead to offer you compound interest on the same $18,000 note. How much will Stone pay you at the end of three years if interest is compounded annually at a rate of 6%? If required, round your answers to the nearest cent. Principal Annual Amount of Accumulated Amount at Amount at Interest (Principal at End of Year (Principal at Beginning of Beginning of Year x Beginning of Year + Annual Year Year 6%) Amount of Interest) 1 $18,000 $1,080 $19,080 N $19,080 3APPLY 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%. Payment: 56,E|C|0 Payment: 56,000 Payment: 56,000 J, l l l Year 1 Year 2 Year 3 Present Value: 7 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% J 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 3.791 3.696 3.605 6 5.076 4.917 4.767 4.623 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 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 7.469Using the previous table, enter the correct factor for three periods at 5%: Periodic payment x Factor Present value $6,000 x 2.723 Incorrect The controller at Stone has determined that the company could save $8,000 per year in engineering costs by purchasing a new machine. The new machine would last 10 years and provide the aforementioned annual monetary benefit throughout its entire life. Assuming the interest rate at which Stone purchases this type of machinery is 8%, what is the maximum amount the company should pay for the machine? $ (Hint: This is basically a present value of an ordinary annuity problem as highlighted above.)