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## 1.3.2 Single Cash Flow

We will first look at discounting a single cash flow or amount. The cash flow can be discounted back to a present value by using a discount rate that accounts for the factors mentioned above (present consumption preference, risk, and inflation). Conversely, cash flows in the present can be compounded to arrive at an expected future cash flow.

The present value of a single cash flow can be written as follows:

 PV = FVn / (1 + i)n PV the present value (or initial principal) FVn future value at the end of n periods i the interest rate paid each period n the number of periods

This means that if you know what a future payment will be, when it will be made and what interest rate that we would be paid to achieve comparable future payments -- you can compute that payment's present value! Armed with this basic formula, you can compute a present value quite easily if you know what the future payment will be (or is expected to be), when it will be made, and the discount rate applied.

To illustrate: What would you be willing to give up to have \$1,200 a year from now? Stated in our valuation lexicon we ask: What is the present value of a \$1,200 cash flow to be received one year from now? Assuming an appropriate discount rate of 20%, we can apply our present value formula:

 PV = FV / (1 + discount rate) = \$1,200 / (1 + .20).... = \$1,200 / 1.2 .............. = \$1,000....................... Notice that the higher the discount rate, the smaller the present value. This inverse relationship reflects the reality that an amount in the future is worth less today if present investment opportunities promise high returns (discount rate).

Simplifying present value calculations

You can simplify present value calculations by using a table that shows present value of \$1 discounted at i percent for n periods. Remember the attached tables:

 FV = (1 + i)n future value of \$1 compounded at i percent for n periods PV = 1/(1 + i)n present value of \$1 discounted at i percent for n periods FV = SUM [i=1 to n] (1 + i)n future value of \$1 deposited at the end of each of n periods compounded at i percent PV = SUM [i=1 to n] 1/(1 + i)n present value of \$1 deposited at the end of n periods discounted at i percent

Using the present value table, what is the present value of \$2,500 in 8 years, assuming a discount rate (the opportunity cost of other investments) of 12%? of 6%? Answers: \$2,500*.4039 = \$1,009.70 (12%); \$2,500*.6274 = \$1,568.50 (6%).

Notice a couple truths:
• The lower the discount rate, the more valuable are future amounts -- in a low-inflation economy, the promise of being a millionaire in ten years means something.
• The higher the discount rate, the less valuable are future amounts -- in a high-inflation economy, the promise of becoming a millionaire in ten years means little.

Example:

Suppose the government offered to pay you \$150,000 in five years. You determine that you can invest today in a five-year government note that yields 8.5%. What is the present value of this government offer?

We can solve our problem using a calculator, spreadsheet or table :

...PV = FVn / (1 + r)n

= \$150,000 / (1 + .085)5
= \$150,000 / 1.503657
= \$99,756.81

That is, having \$99,756.81 now (and investing it at 8.5% for five years) is the same as having \$150,000 in five years. Take your pick.

Notice that the problem assumed that the \$150,000 payment was a near certainty—just as certain as the government paying on a five-year note. Things change if the payment had been uncertain. We’ll begin to explore how uncertainty (risk) affects our choice of discount rate and ultimate valuation decisions in the next chapter. - see Chapter 2 (Risk and Return).

 1.3.1 Present Value - First Principles 1.3.3 Present Value of Cash Flow Streams