# Complete Guide To Corporate Finance

## Time Value Of Money - Present Value And Discounting

Present value, also called "discounted value," is the current worth of a future sum of money or stream of cash flow given a specified rate of return. Future cash flows are discounted at the discount rate; the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to properly valuing future cash flows, whether they are earnings or obligations.
If you received $10,000 today, the present value would be $10,000 because present value is what your investment gives you if you were to spend it today. If you received $10,000 in a year, the present value of the amount would not be $10,000 because you do not have it in your hand now, in the present. To find the present value of the $10,000 you will receive in the future, you need to pretend that the $10,000 is the total future value of an amount that you invested today. In other words, to find the present value of the future $10,000, we need to find out how much we would have to invest today in order to receive that $10,000 in the future.

To calculate present value, or the amount that we would have to invest today, you must subtract the (hypothetical) accumulated interest from the $10,000. To achieve this, we can discount the future payment amount ($10,000) by the interest rate for the period. In essence, all you are doing is rearranging the future value equation above so that you may solve for P. The above future value equation can be rewritten by replacing the P variable with present value (PV) and manipulating the equation as follows:

Let's walk backwards from the $10,000 offered in Option B. Remember, the $10,000 to be received in three years is really the same as the future value of an investment. If today we were at the two-year mark, we would discount the payment back one year. At the two-year mark, the present value of the $10,000 to be received in one year is represented as the following:

Note that if we were at the one-year mark today, the above $9,569.38 would be considered the future value of our investment one year from now.

At the end of the first year we would be expecting to receive the payment of $10,000 in two years. At an interest rate of 4.5%, the calculation for the present value of a $10,000 payment expected in two years would be the following:

Of course, because of the rule of exponents, we don't have to calculate the future value of the investment every year counting back from the $10,000 investment at the third year. We could put the equation more concisely and use the $10,000 as the future value. So, here is how you can calculate today's present value of the $10,000 expected from a three-year investment earning 4.5%:

The present value of a future payment of $10,000 is worth $8,762.97 today if interest rates are 4.5% per year. In other words, choosing Option B is like taking $8,762.97 now and then investing it for three years. The equations above illustrate that Option A is better not only because it offers you money right now but because it offers you $1,237.03 ($10,000 - $8,762.97) more in cash! Furthermore, if you invest the $10,000 that you receive from Option A, your choice gives you a future value that is $1,411.66 ($11,411.66 - $10,000) greater than the future value of Option B.

Let's add a little spice to our investment knowledge. What if the payment in three years is more than the amount you'd receive today? Say you could receive either $15,000 today or $18,000 in four years. Which would you choose? The decision is now more difficult. If you choose to receive $15,000 today and invest the entire amount, you may actually end up with an amount of cash in four years that is less than $18,000. You could find the future value of $15,000, but since we are always living in the present, let's find the present value of $18,000 if interest rates are currently 4%. Remember that the equation for present value is the following:

In the equation above, all we are doing is discounting the future value of an investment. Using the numbers above, the present value of an $18,000 payment in four years would be calculated as the following:

From the above calculation we now know our choice is between receiving $15,000 or $15,386.48 today. Of course we should choose to postpone payment for four years! (For related reading, see

These calculations demonstrate that time literally is money - the value of the money you have now is not the same as it will be in the future and vice versa. It is important to know how to calculate the time value of money so that you can distinguish between the worth of investments that offer you returns at different times.

To calculate present value, or the amount that we would have to invest today, you must subtract the (hypothetical) accumulated interest from the $10,000. To achieve this, we can discount the future payment amount ($10,000) by the interest rate for the period. In essence, all you are doing is rearranging the future value equation above so that you may solve for P. The above future value equation can be rewritten by replacing the P variable with present value (PV) and manipulating the equation as follows:

Let's walk backwards from the $10,000 offered in Option B. Remember, the $10,000 to be received in three years is really the same as the future value of an investment. If today we were at the two-year mark, we would discount the payment back one year. At the two-year mark, the present value of the $10,000 to be received in one year is represented as the following:

Present value of future payment of $10,000 at end of year two: |

Note that if we were at the one-year mark today, the above $9,569.38 would be considered the future value of our investment one year from now.

At the end of the first year we would be expecting to receive the payment of $10,000 in two years. At an interest rate of 4.5%, the calculation for the present value of a $10,000 payment expected in two years would be the following:

Present value of $10,000 in one year: |

Of course, because of the rule of exponents, we don't have to calculate the future value of the investment every year counting back from the $10,000 investment at the third year. We could put the equation more concisely and use the $10,000 as the future value. So, here is how you can calculate today's present value of the $10,000 expected from a three-year investment earning 4.5%:

The present value of a future payment of $10,000 is worth $8,762.97 today if interest rates are 4.5% per year. In other words, choosing Option B is like taking $8,762.97 now and then investing it for three years. The equations above illustrate that Option A is better not only because it offers you money right now but because it offers you $1,237.03 ($10,000 - $8,762.97) more in cash! Furthermore, if you invest the $10,000 that you receive from Option A, your choice gives you a future value that is $1,411.66 ($11,411.66 - $10,000) greater than the future value of Option B.

**Present Value of a Future Payment**Let's add a little spice to our investment knowledge. What if the payment in three years is more than the amount you'd receive today? Say you could receive either $15,000 today or $18,000 in four years. Which would you choose? The decision is now more difficult. If you choose to receive $15,000 today and invest the entire amount, you may actually end up with an amount of cash in four years that is less than $18,000. You could find the future value of $15,000, but since we are always living in the present, let's find the present value of $18,000 if interest rates are currently 4%. Remember that the equation for present value is the following:

In the equation above, all we are doing is discounting the future value of an investment. Using the numbers above, the present value of an $18,000 payment in four years would be calculated as the following:

Present Value |

From the above calculation we now know our choice is between receiving $15,000 or $15,386.48 today. Of course we should choose to postpone payment for four years! (For related reading, see

*Anything But Ordinary: Calculating The Present And Future Value Of Annuities*.)These calculations demonstrate that time literally is money - the value of the money you have now is not the same as it will be in the future and vice versa. It is important to know how to calculate the time value of money so that you can distinguish between the worth of investments that offer you returns at different times.

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