1. NPV, ROI, and DCF

NPV (net present value) analysis is just another name for discounting. There is nothing fancy about this kind of valuation. In business practice it is usual to discount each year (or, sometimes, quarter) separately. For example, suppose the hurdle (discount) rate is 10%, i.e. r = 0.1. If some project initially costs $5000, returns $2000 per year for the next three years, and then will be scrapped at no cost, then its NPV is

$2000 / (1 + r)^1 + $2000 / (1 + r)^2 + $2000 / (1 + r)^3 - $5000

= $2000/1.1 + $2000/1.21 + $2000/1.331 - $5000

= - $26.

Since the NPV is negative, the project is rejected. Although the math does not even use the exponential function, some very large US companies use this method (possibly with modifications). Sometimes it might be assumed that the project will return a set amount per year after the end of the budgeted period. In the example above, it might be decided that after 3 years there will still be positive cash flows from the project of $10 per year for eternity (e.g., because the project will secure another loyal customer, and each loyal customer is worth, say, $10 per year). After three years, the value of this $10 perpetuity will be

$10 (1/1.1 + 1/1.1^2 + 1/1.1^3 + ... ) = $10x,

where (x – 1/1.1)(1.1) = x => x - 1/1.1 = x/1.1 => 1.1x – x = 1 => x = 10,

i.e., the value of the $10 perpetuity will be $100, which has net present value

$100/ (1 + r)^3 = $100/1.331 = $75.

Since the project now has a positive net present value, namely $75 - $26 = $49, it should be implemented. According to some business literature, if there are many potential projects then the one with the highest net present value should be undertaken. However, it is the combination of projects with the highest NPV that should be implemented. E.g., BioTel.com Corporation might have only 3 potential projects with positive NPV: two costing $100m with NPV of $2m, and one costing $200m with NPV of $3m. If BioTel.com has only $200m available, clearly going ahead with the first two projects (giving total NPV of $4m) is the most profitable choice. Some sources recommend that the projects with the highest NPV as a percentage of the cost (i.e., NPV return on investment or NPV ROI), but this is not strictly true either. E.g., suppose BioTel.com has a fourth potential project costing $110m with NPV of $3.3m. This project has an ROI of 3%, which is the highest, but if the company were to go ahead with this project, it would not have enough cash to fund any other projects with positive NPV. Hence, it would still be better to implement the two projects that cost $100m each. These techniques are forms of DCF (discounted cash flow) analysis.

2. Real option value

When there is uncertainty about future cash flows, NPV is practically useless because it assumes that cash flows are set in stone. In such cases of volatility, NPV is supplanted by real option value (ROV). The mathematics of financial derivatives applies to real options, although some specialized cases require other financial mathematics. In the following sections, the three basic types of real options are considered in turn.

3. The option to abandon

Suppose BioTel.com is considering investing in a Wobblyball Internet web site that will cost $100m straight up, $110m in exactly 12 months, and essentially nothing after that. Wobblyball, of which BioTel.com expects to corner the online market due to first mover advantage, is the latest fad among preschoolers. BioTel.com’s positive cash flows from latest fads generally has volatility sigma=2, and BioTel.com generally captures 10% of the market share of such fads (BioTel.com, whose CEO is 4 years old, always follows the latest fads among preschoolers). The Wobblyball market is currently $100m per annum. BioTel.com values its potential 10% share of the $140m Wobblyball market at $140m using geometric series. Because Wobblyball is expected to be so volatile, BioTel.com astutely realizes that ROV is the best measure of its value. After a year, BioTel.com can abandon its Internet web site, saving $110m ($100m in present value), if the Wobblyball market goes sour. This is a European call option with asset value S = $140m, exercise price E = $110m, risk-free rate r = 0.1, time to expiration t =1, and volatility sigma=2. Maple gives its value as $103m:

> restart; with (finance):

> evalf (blackscholes (140, 110, 0.1, 1, 2);


Since this option will cost only $100m, BioTel.com should make the initial investment in the web site.

4. The option to expand

Suppose BioTel.com realizes that for $110m it can change all instances of "Wobblyball" to "ICBM" in its source code and thus create an online Intercontinental Ballistic Missile auction house. Surprisingly, the international arms trade closely parallels the toy industry and also has volatility sigma=2. The option to build the ICBM auction site has time to expiration t = 1 because otherwise one of their programmers will steal the idea and set up their own company, thus obtaining first-mover advantage. Similarly to above, BioTel.com values its 0.1% commission on its expected 10% share of the $140b ICBM trade at $140m. Since it has the same parameters, this "option to expand" has the same value, namely $103m, as above.

5. The option to delay

Suppose BioTel.com has exclusive Internet rights to Wobblyball which it will forfeit if it does not begin building its web site within two years. BioTel.com sees that there is value in being able to wait and see how the Wobblyball market evolves. However, if Wobblyball delays its investment, it will have to contract out the web site development, which will cost $200m up front. This is like an American call option with asset value S = $140m, E = $200m, r = 0.1, t = 2, and volatility sigma=2, which has the same value as a European call option with the same parameters:

> evalf (blackscholes (140, 200, 0.1, 2, 2));


This option to delay is worth $116m to BioTel.com Corporation.

6. Decision trees

A business person not trained in financial mathematics would probably use a decision tree rather than real options. In the decision tree method, the analyst will estimate the probability of arbitrarily chosen outcomes. For example, in section 3 above BioTel.com might have decided that there is a 20% chance that project would be abandoned, and that if the project was not abandoned then the expected to be worth $160m per annum. The financial people at BioTel.com might use a normal-function-like graph of the expected market share to justify these figures. According to these figures, the present value of the option is 0.6*$160m/1.1 = $96m/1.1 = $87m. The figures used in decision tree analysis are usually quite conservative.

7. EVA: Economic Value Added

EVA, introduced by New York consulting firm Stern Stewart & Co. in 1989, is a fashionable alternative to NPV. EVA is given by

EVA = (operating profit) – (cost of capital) * (capital),

where operating profit is profit from operations after taxes but before interest and non-cash costs such as depreciation and amortization. EVA is probably a better way of quantifying the value of investments, but it is really just a modification of NPV.