1.3.1
 Present value  first principles
To make decisions now  really the only kind we make
 it is useful to know the "present value"
of future money. Present value is
the current dollar value of a future amount  what
would have to be invested today (at a given interest
rate over a specified period) to equal the future amount.
What is a dollar in the future worth? It depends on
when it will be received and our current investment
opportunities. (More 1.3.1>>)
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. (More 1.3.2>>)
1.3.3  Present value of cash flow
streams
The future is sometimes bumpy and sometimes cyclical
and sometimes forever. Cash flows can come in a mixed
stream or a pattern of equal annual flows or even a
perpetual stream. (More 1.3.3>>)
1.3.4  Present value of an annuity
When equal payments are made over time (an annuity),
their present value can be determined using standard
valuation methodologies. This can also be done in reverse
 to calculate the equal payments over time equivalent
to a current lump sump payment (loan amortization).
Not surprisingly, questions of annuities often arise
in legal contexts. (More 1.3.4>>)
1.3.5  Present value of a perpetuity
Sometimes annuities last forever  or so we pretend.
What is the value of payments that are received indefinitely,
like proverbial AT&T dividends? On first reflection,
you might think that te value of perpetual payments
would be infinite. But remember that $1 paid in 10 years
may not be worth picking up from the sidewalk (assuming
normal inflation and the inherent time value of money)
and $1 paid in 100 years is worth even less. (More
1.3.5>>)
1.3.6  Constantly growing perpetuity
What if we expect that future returns will grow 
with inflation and as an investment progresses? If returns
grow at a constant rate (g), the DCF formula produces
one of the most oftenused formulas in stock valuation
 known as the "GordonShapiro dividend discount
model" or the "Gordon model." (More
1.3.6>>)
