Introduction to Time Value of Money and Discounted Cash Flows

Suppose you discovered your grandmother left a trust fund for you in the amount of USD 200,000. The only catch is it will not be available to you for exactly 20 years. A financial firm offers to purchase the rights to your trust fund, so that you can have money today.

Absolutely! Since you'd rather have the money today, it must be the case that the same amount in some future time is worth less to you today.

Incorrect. Think about the fact that you would rather have USD 200,000 today, if you had your choice.

Incorrect. You would probably rather have USD 200,000 today. Think about what that means for the relative value of USD 200,000 which is "trapped" off in the future, and what that would be worth to you in comparison.

The USD 200,000 balance in the future is called the __future value__. This is often referred to as __FV__ for short, and sometimes subscripts are used to denote specific time periods. For example, > $$FV_{20} = 200{,}000$$ means the future value in year 20 is USD 200,000. This is exactly the example used so far. In contrast, the value of some future value in the present is called the __present value__, or __PV__ for short. In terms of years, the present value is the value in year zero. Year zero is today.

The financial firm offers you USD 62,360.95 today for the rights to your future trust fund balance, because at some discount rate r which it has determined to be fair, this amount would grow over time to reach USD 200,000 in 20 years. In this case, what has the financial firm calculated?

Using this same notation, which of the following would indicate a value of USD 500 in two years?

That's right! $$FV_2$$ is the notation for a future value in year two, or two years from now, and so this equation simply assigns a value of $500 to that year.

Incorrect. This actually tells you the future value in year 500 is USD 2.

Incorrect. This tells you the present value (today) is USD 500.

What is consistent throughout all time value of money calculations is that a present value is expected to grow to some future value at a positive discount rate r, or some future value can be discounted to the present value at a positive discount rate r. What can you then conclude about the relationship between present value and future value?

To summarize this discussion: [[summary]]

No, consider that some value in the present invested at a positive rate of return will grow to a different value over time, and how that would compare to the present value that you were given.

No, this is the case for investments that rise and fall in value, but think about some given, guaranteed rate of return r which is positive.

No, the firm has calculated the value of this future cash flow as of today. But this answer refers to the future value one year from today.

No, the firm has calculated the value of this future cash flow as of today. But this answer refers to the future value in 20 years. This is already given in the example as USD 200,000.

Obviously you would rather have USD 200,000 today than in a trust fund which is unavailable for 20 years. What do you think this USD 200,000 trust fund is worth today?

The financial firm offers to purchase your future trust fund balance for USD 62,360.95. You may be surprised at such a low offer, but 20 years is a long time. One way of saying this is the financial firm discounted the future cash flow of USD 200,000 to today. You can also think of the present value as some amount which will grow over time to reach a future value. The rate of growth used in time value of money calculations is some __discount rate__, r.

Absolutely!

Exactly! Only at a discount rate of zero could they be equal, and this is not an assumption that is often made. Outside of this odd case, a given present value will always equate to a larger future value, and a given future value will always equate to a smaller present value. This is only the beginning of a powerful tool set.

$$PV = 62{,}360.95$$

$$FV_1 = 62{,}360.95$$

$$FV_{20} = 62{,}360.95$$

$$PV = 500$$

$$FV_{500} = 2$$

$$FV_2 = 500$$

Present value is always smaller than future value

Present value is always larger than future value

Present value is sometimes smaller and sometimes larger than future value

Less than USD 200,000

Exactly USD 200,000

More than USD 200,000

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