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.
Mixed flows
To compute the present value of a mixed stream,
a spreadsheet is invaluable. Consider this problem,
where cash flows vary and discount rates change
according to the length of maturity:
Year 
Cash flow 
Discount rate 
Present value 
1 
$500 
7.5% 
$ 465.12 
3 
$1,000 
9.0% 
$ 772.18 
5 
$1,500 
10.0% 
$ 931.38 
10 
$2,500 
11.0% 
$ 880.46 
20 
$2,000 
12.5% 
$ 189.66 
Total 
$3,238.80 
Although it might seem that a prediction of mixed
flows (and discount rates) would best model the
real world, the general practice by financial
and legal decisionmakers has been to assume constant
discount rates throughout the life of an asset
and constant cash flows, particularly after a
few years out. This assumption can result in differences
in results  sometimes significant. Consider
our earlier example, this time where cash flows
and discount rates are averaged and evened out:
Year 
Cash flow 
Discount rate 
Present value 
0 
$1500 
10.0% 
$1,500.00 
5 
$1500 
10.0% 
$ 931.38 
10 
$1500 
10.0% 
$ 578.32 
15 
$1500 
10.0% 
$ 359.09 
20 
$1500 
10.0% 
$ 321.82 
Total 
$3,690.61 
Notice that more even flows and a steady discount
rate, compared to higher backend flows and increasing
discount rates, produce a difference of more than
10%  the devil is in the details.

Example: Equitable Distribution
Husband and wife divorce. Husband
is a lawyer with a solo practice, with netaftertax
earnings for the six most recent years:
Year 
Earnings 
1974 
$37,400 
1975 
$47,500 
1976 
$52,600 
1977 
$75,900 
1978 
$38,800 
1979 
$37,800 
Average 
$48,333 
In addition, the office's tangible
assets (based on current market value, in excess
of its liabilities) was $3,000 on the date of
separation. What is the value of the husband's
practice in an equitable distribution, assuming
that husband's earnings will remain constant
for another 20 years and that similar solo law
practices have been valued with a discount rate
of 25%?
Answer: $3,000!
(Forget about his earnings potential.)
Although it would seem appropriate
to value the business by determining the present
value of a 20year stream of earnings discounted
at rate of 25%, courts have generally decided
that such a valuation in an equitable distribution
would inappropriately constitute postdivorce
division of income. As the Delaware Supreme
Court stated in EEC
v. EJC, 457 A.2d 688 (Del. 1983):
We see little substantive difference between
the technique of discounting a future flow
of income to determine present value and capitalizing
an annualized average income. Both valuation
techniques seek to arrive at present value
by reference to future income. We agree with
Stern that earning capacity is irrelevant
in determining the present value of marital
assets for purpose of division of marital
property under 13 Del. C. §
1513. It follows that wife's technique of
valuing husband's sole proprietorship professional
practice based on a capitalization of husband's
earnings must be rejected.
The court rejected the wife's
expert's valuation which capitalized the earnings
using a capitalization multiplier of 2, not because
this misrepresents the value of the business assuming
the husband continues in practice, but rather
because the wife is not entitled to the husband's
labors after divorce  in an equitable distribution

Example:
Congress passes the Surface Mining Control and
Reclamation Act, which prohibits StripCo from
continuing mining operations on its property.
StripCo claims this governmental regulation is
a "taking" under the US Constitution,
which requires "fair compensation."
What has been taken and how much is it worth?
The company provides evidence of likely coal production,
comparablecompany coal prices, likely operating
costs over a 24year period, and the prevailing
average industry discount rate for proved coal
reserves of 10%  to come to a "discounted
cash flow" estimate of the value of the company's
coal reserves. The government counters with evidence
of how much similar coal properties had sold for.
(More>>)

Equal flows
The most straightforward application of the DCF
method is when expected cash flows will be received
in equal payments during the time period in question.
Rentgathering property and bond issues are common
examples of situations where we can expect equal
flows during the valuation period. 
Example: The State Tobacco Settlements
On November 28, 1998, 46 states, five territories,
and the District of Columbia entered into a Master
Settlement Agreement (“MSA”) with
the major tobacco companies, releasing the tobacco
industry from its liability for each state’s
past and future costs for treating tobacco related
diseases. The total amount to be paid: $206 billion.
At first glance, the settlement would appear
to be an unqualified boon to the states—filling
their coffers with money for disease research,
treatment and prevention. Actually obtaining the
settlements funds, however, has proven to be a
little tricky. The states are to receive the settlement
funds in equal payments over the course of 25
years. (More>>)

Constant growth
What happens if future values are growing at
a constant, consistent rate? Our methods for finding
the present value of mixed and equal cash flows
will not work in these circumstances.
When appraisers confront this issue, it is almost
invariably in the context of dividend growth.
Remember that a dividend is simply a special type
of cash flow paid periodically to the holders
of a company’s common stock. Dividend payments
can be an extremely important window into the
value of a company. Unless you derive salary or
fringe benefits from ownership in a company, the
only cash flow you receive from a firm when you
buy its stock is the dividend. The stock’s
value is therefore determined solely by the amount
of the expected stream of dividend payments.
Ideally, as a company grows and becomes more
successful, its dividend payments will also steadily
grow. But how do we value the prospect of constantly
growing dividends (or any other constantly growing
cash flow)? The DCF methods used for valuing equal
and mixed cash flows will not fit.

Drawing on the principles and intuition that
drives discounted cash flow valuation, Professor
Myron J. Gordon developed, in the early 1960s,
a DCF model that captures the value in a stream
of continuously growing cash flows. This simple
and powerful tool can be expressed as follows:
P_{0} =D_{0}×(1+g)^{1}
/ (1+i)^{1} +D_{0}×(1+g)^{2}
/ (1+i)^{2} + ... + D_{0}×(1+g)inf
/ (1+i)inf
P_{0} = D_{1}
/ (i  g)
or
P_{0} = D_{0}
(1 +g) / (i  g)

P_{0} 
present value of common stock (with constant
growth returns) 
D_{0} 
most recent pershare dividend 
D_{1} 
pershare dividend after one period of growth
[D1 = D0 (1 + g)] 
i 
required return (discount rate) for each
year t 
g 
rate of growth 
inf 
infinite time period 
The Gordon model is widely used in legal valuations
where a company's growth (driven by inflation
or business success) is accounted for. The model
then assumes that the discount rate will reflect
an inflation component. For a basic application
of the Gordon growth model, review this spreadsheet.

Example: Valuing a Real Estate Investment
Trust
Real estate investment trusts (REITs) were created
in the 1960s to allow small investors the opportunity
to participate in largescale real estate investments.
The law that created the REIT structure allowed
the REIT entities to pass the real estate investment
income taxfree to their investors.In exchange
for these tax benefits, REITS must distribute
virtually all of their earnings (at least 95%)
to their investors. Because the REIT entity’s
accumulation of earnings is so severely constrained,
it must pay dividends constantly, and any dividend
growth will, at best, be stable and modest. Because
of the dividend growth pattern inherent in the
REIT structure, it is a good candidate for application
of the Gordon model. (More>>)

