• Table of Contents • Introduction • 1-Time Value • 2-Risk/Return • 3-Accounting • 4-Securities • 5-Business • 6-Regulatory • Case Studies • Student Papers

## 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.

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 decision-makers 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 back-end 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 net-after-tax 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%?

Although it would seem appropriate to value the business by determining the present value of a 20-year stream of earnings discounted at rate of 25%, courts have generally decided that such a valuation in an equitable distribution would inappropriately constitute post-divorce 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, comparable-company coal prices, likely operating costs over a 24-year 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. Rent-gathering 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:

 P0 =D0×(1+g)1 / (1+i)1 +D0×(1+g)2 / (1+i)2 + ... + D0×(1+g)inf / (1+i)inf P0 = D1 / (i - g) or P0 = D0 (1 +g) / (i - g) P0 present value of common stock (with constant growth returns) D0 most recent per-share dividend D1 per-share 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 large-scale real estate investments. The law that created the REIT structure allowed the REIT entities to pass the real estate investment income tax-free 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>>)

 1.3.2 Single Cash Flow 1.3.4 Present Value of an Annuity