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Almost all the material is related to time value of money issues.
Another way to look at time value of money is that money earns money and the sooner it is received or invested the better. The sooner the money is lent or invested the more it will accumulate; conversely, money invested or lent much later will not earn as much, all other things being equal.
Time value of money can have numerous categories, but here the discussion is restricted to four basic categories for employing time value of money calculations and a few typical illustrations using the four basic categories.
The four basic categories for doing the calculations are given below.
In these four cases, there is an additional issue of whether the activity takes place at the beginning or end of the period. For purposes of simplicity, all activity is assumed to occur at the beginning of the period.
In all these categories, the amount used is $1 since knowing values related to $1 can translate to any amount.
Although there are formula for all these calculations, time value of money tables are often used. In this section, the [time value of money table are used and are made available].<!button to the tables, which can be scanned since they are so detailed>
In order to bring amounts back to the present, a discount factor is needed. The discount is the interest rate one could reasonably earn or expect to earn if the money were invested at the present for a given risk.
In both cases ($100,000, 15 years from now and $150,000, 20 years from now), the amounts will be worth considerably less since the present amount will have to be invested to reach the amounts promised 15 or 20 years from now. In more general terms, the earlier money is received the better.
amounts 100,000 150,000 discount factor 0.06 0.06 years from now 15 20 pv of $1 0.41726 0.3118 present value $41,726 $46,770
The present value of $1 (pv of $1) can easily be found in the tables first by finding the [pv of $1 table]<!button to pv of $1 table> and then finding the % column (here called the discount) and period row (years in this case).
When both amounts are discounted (.41726 for 15 years and 6% and .3118 for 20 years and 6%) the present value can be calculated simply by multiply the pv of $1 times the total amount (eg, .41427 * 100,000 = 41,426). In this illustration it would be better to wait the 20 years becasue enough is added to compensate for the extra 5 years.
Practical applications of using the present value of $1 for decision making include helping to decide on capital projects or pension plans. If the above were alternative pension plans and money were the only consideration, then the 20 deal is better.
annuity amount 5000 7000 discount 0.06 0.06 years 7 5 pv of an annuity 5.5824 4.2123 present value $27,912 $29,486
In this case, it is preferable to take the 7,000 for only 5 years rather than the 5,000 for as long as 7 years since in actuality the 5 year deal produces a higher present value amount. The [present value of an annuity can be found in the tables].< !button to present value of annuity table >
amount 41726 interest rate 0.06 years 15 fv of $1 2.3966 future value 100000.53
Here, $41,726 is invested at 6% for 15 years, yielding a total of essentially $100,000.
Note how present value and future value are related. In the present value of $1 illustration, the $100,000 comes back to today at a value of $41,726 given a period of 15 years and discount of 6%. The future value of $1 does just the opposite, it takes the $41,726 investment, invests it for 15 years at 6% and yields, essentially, $100,000.
Again, future values are essentially investment decisions.
annuity amount 5000 interest rate 0.06 years 7 fv of an annuity 8.394 future amount $41,970
Here is a situation one could use for personal investing. If I invested $5,000 at a fixed interest rate of 6% for 7 years how much would I have at the end of that period?
One project requires an initial investment of $1,000,000 and will generate $130,000 in extra taxes for 9 years. The other project requires an investment of $750,000 but will generate $77,000 in extra taxes for 15 years. Which is the better project, strictly on economic grounds?
project 1 project 2
investment 1,000,000 750,000
interest rate 0.06 0.06
return 135000 77000
years 9 15
pv of an annuity 6.8017 9.7122
present amounts $918,230 $747,839
Here is an interesting case where the first project is more productive but one must ask if it is worthwhile to take on either of these projects given that neither will return an amount equal to the original investment?
In terms of the calculations, the math is largely that of the present value of an annuity. There is a periodic stream of cash and it needs to be broght back to the present for comparison. Find the present value of an annuity for 6% and 9 years (periods) and multiple that by $135,000. Do the same for the $77,000 stream of cash.
debt 500000 current amount 25000 fv for current amount for 10 years at 5% 1.6289 future amount 40722.5 amount still needed 459277.5 fv of an annuity for 10 years at 5% 12.578 investment needed 36514.35
Parts of this problem are fairly complicated; others, not. Figuring the amount that will result from the $25,000 is a straight forward future value of $1 problem. Just multiply the $25,000 times the 1.6289. Then subtract the result, $40,722.50 from $500,000 to get what is still needed -- $459,277.50. Here is where some of the complexity enters.
However, a ratio could be set up to do the calculations for any amount. The algebra is given below.
x 1
-------- = ------
459277.5 12.578
Although we do not know how much to invest each year to secure $459,277.50, one dollar ($1) invested for 10 years at 5% will produce 12.578 so a ratio as above can be set up to do the calculation for any other amount.
59277.5x -------- = 459277.5(12.578) 59277.5
459277.5
x = --------
12.578
x, which is the amount to be invested annually = 36514.35
The process can also be thought of as a partitioning; the amount $459,277.50 needs to be partitioned so just think about dividing and partitioning as the same, and divide the $459,277.5 by the future value of the annuity.
This example just discussed can be used for any kind of debt that needs to paid -- pensions, accrued sick leave, legal bills, or bonds. The step are:
Or, algebraically:
debt amount / fv of an annuity for appropriate number of years and rate