The discounted cash flow (DCF) analysis, in financial analysis, is a method used to value a security, project, company, or asset, that incorporates the time value of money. Discounted cash flow analysis is widely used in investment finance, real estate development, corporate financial management, and patent valuation. Used in industry as early as the 1700s or 1800s, it was widely discussed in financial economics in the 1960s, and U.S. courts began employing the concept in the 1980s and 1990s.

Application

In discount cash flow analysis, all future cash flows are estimated and discounted by using cost of capital to give their present values (PVs). The sum of all future cash flows, both incoming and outgoing, is the net present value (NPV), which is taken as the value of the cash flows in question; see aside. For further context see ; and for the mechanics see valuation using discounted cash flows, which includes modifications typical for startups, private equity and venture capital, corporate finance "projects", and mergers and acquisitions. Using DCF analysis to compute the NPV takes as input cash flows and a discount rate and gives as output a present value. The opposite process takes cash flows and a price (present value) as inputs, and provides as output the discount rate; this is used in bond markets to obtain the yield.

History

Discounted cash flow calculations have been used in some form since money was first lent at interest in ancient times. Studies of ancient Egyptian and Babylonian mathematics suggest that they used techniques similar to discounting future cash flows. Modern discounted cash flow analysis has been used since at least the early 1700s in the UK coal industry. Discounted cash flow valuation is differentiated from the accounting book value, which is based on the amount paid for the asset. Following the stock market crash of 1929, discounted cash flow analysis gained popularity as a valuation method for stocks. Irving Fisher in his 1930 book The Theory of Interest and John Burr Williams's 1938 text The Theory of Investment Value first formally expressed the DCF method in modern economic terms.

Mathematics

Discounted cash flows

The discounted cash flow formula is derived from the present value formula for calculating the time value of money and compounding returns: Thus the discounted present value (for one cash flow in one future period) is expressed as: where Where multiple cash flows in multiple time periods are discounted, it is necessary to sum them as follows: for each future cash flow (FV) at any time period (t) in years from the present time, summed over all time periods. The sum can then be used as a net present value figure. If the amount to be paid at time 0 (now) for all the future cash flows is known, then that amount can be substituted for DPV and the equation can be solved for r, that is the internal rate of return. All the above assumes that the interest rate remains constant throughout the whole period. If the cash flow stream is assumed to continue indefinitely, the finite forecast is usually combined with the assumption of constant cash flow growth beyond the discrete projection period. The total value of such cash flow stream is the sum of the finite discounted cash flow forecast and the Terminal value (finance).

Continuous cash flows

For continuous cash flows, the summation in the above formula is replaced by an integration: where FV(t) is now the rate of cash flow, and.

Discount rate

The act of discounting future cash flows asks "how much money would have to be invested currently, at a given rate of return, to yield the forecast cash flow, at its future date?" In other words, discounting returns the present value of future cash flows, where the rate used is the cost of capital that appropriately reflects the risk, and timing, of the cash flows. This "required return" thus incorporates: For the latter, various models have been developed, where the premium is (typically) calculated as a function of the asset's performance with reference to some macroeconomic variable - for example, the CAPM compares the asset's historical returns to the "overall market's"; see and. An alternate, although less common approach, is to apply a "fundamental valuation" method, such as the "T-model", which instead relies on accounting information. Other methods of discounting, such as hyperbolic discounting, are studied in academia and said to reflect intuitive decision-making, but are not generally used in industry. In this context the above is referred to as "exponential discounting". The terminology "expected return", although formally the mathematical expected value, is often used interchangeably with the above, where "expected" means "required" or "demanded" by investors. The method may also be modified by industry, for example various formulae have been proposed when choosing a discount rate in a healthcare setting; similarly in a mining setting, where risk-characteristics can differ (dramatically) by property.

Methods of appraisal of a company or project

For these valuation purposes, a number of different DCF methods are distinguished today, some of which are outlined below. The details are likely to vary depending on the capital structure of the company. However the assumptions used in the appraisal (especially the equity discount rate and the projection of the cash flows to be achieved) are likely to be at least as important as the precise model used. Both the income stream selected and the associated cost of capital model determine the valuation result obtained with each method. (This is one reason these valuation methods are formally referred to as the Discounted Future Economic Income methods.) The below is offered as a high-level treatment; for the components / steps of business modeling here, see.

Equity-approach

Entity-approach

Shortcomings

The following difficulties are identified with the application of DCF in valuation:

<!-- == Example == TOO MUCH To show how discounted cash flow analysis is performed, consider the following example. Simple subtraction suggests that the value of his profit on such a transaction would be $150,000 − $100,000 = $50,000, or 50%. If that $50,000 is [amortized](https://bliptext.com/articles/amortization-business) over the three years, his implied annual return (known as the [internal rate of return](https://bliptext.com/articles/internal-rate-of-return)) would be about 14.5%. Looking at those figures, he might be justified in thinking that the purchase looked like a good idea. 1.1453 x $100,000 = $150,000, approximately. However, since three years have passed between the purchase and the sale, any cash flow from the sale must be discounted accordingly. At the time John Doe buys the house, the three-year [US Treasury Note](https://bliptext.com/articles/united-states-treasury-security) rate is 5% per annum. Treasury notes are generally considered to be inherently less risky than real estate, since the value of the note is guaranteed by the US government and there is a [liquid](https://bliptext.com/articles/liquidity) market for the purchase and sale of T-notes. If he had not put his money into buying the house, he could have invested it in the relatively safe T-Notes instead. This 5% per annum can, therefore, be regarded as the [risk-free interest rate](https://bliptext.com/articles/risk-free-interest-rate) for the relevant period (three years). Using the DPV formula above (FV=$150,000, i=0.05, n=3), that means that the value of $150,000 received in three years actually has a [present value](https://bliptext.com/articles/present-value) of $129,576 (rounded off). In other words, we would need to invest $129,576 in a T-bond now to get $150,000 in three years almost risk-free. This is a quantitative way of showing that money in the future is not as valuable as money in the present ($150,000 in three years is not worth the same as $150,000 now; it is worth $129,576 now). Subtracting the purchase price of the house ($100,000) from the [present value](https://bliptext.com/articles/present-value) results in the net [present value](https://bliptext.com/articles/present-value) of the whole transaction, which would be $29,576 or a little more than 29% of the purchase price. Another way of looking at the deal as the excess return achieved (over the risk-free rate) is (114.5 - 105)/(100 + 5) or approximately 9.0% (still very respectable). But what about risk? We assume that the $150,000 is John's best estimate of the sale price that he will be able to achieve in three years time (after deducting all expenses). There is a lot of uncertainty about house prices, and the outcome may end up higher or lower than this estimate. (The house John is buying is in a "good neighborhood", but market values have been rising quite a lot lately and the real estate market analysts in the media are talking about a slow-down and higher interest rates. There is a probability that John might not be able to get the full $150,000 he is expecting in three years due to a slowing of price appreciation, or that loss of liquidity in the real estate market might make it very hard for him to sell at all. Under normal circumstances, people entering into such transactions are [risk-averse](https://bliptext.com/articles/risk-averse), that is to say that they are prepared to accept a lower expected return for the sake of avoiding risk. See [Capital asset pricing model](https://bliptext.com/articles/capital-asset-pricing-model) for a further discussion of this. For the sake of the example (and this is a gross simplification), let us assume that he values this particular risk at 5% per annum (we could perform a more precise probabilistic analysis of the risk, but that is beyond the scope of this article). Therefore, allowing for this risk, his expected return is now 9.0% per annum (the arithmetic is the same as above). And the excess return over the risk-free rate is now (109 - 105)/(100 + 5) which comes to approximately 3.8% per annum. That return rate may seem low, but it is still positive after all of our discounting, suggesting that the investment decision is probably a good one: it produces enough profit to compensate for tying up capital and incurring risk with a little extra left over. When investors and managers perform DCF analysis, the important thing is that the net present value of the decision after discounting all future cash flows at least be positive (more than zero). If it is negative, that means that the investment decision would actually lose money even if it appears to generate a nominal profit. For instance, if the expected sale price of John Doe's house in the example above was not $150,000 in three years, but $130,000 in three years or $150,000 in five years, then on the above assumptions buying the house would actually cause John to lose money in present-value terms (about $3,000 in the first case, and about $8,000 in the second). Similarly, if the house was located in an undesirable neighborhood and the [Federal Reserve Bank](https://bliptext.com/articles/federal-reserve-bank) was about to raise interest rates by five percentage points, then the risk factor would be a lot higher than 5%: it might not be possible for him to predict a profit in discounted terms even if he thinks he could sell the house for $200,000 in three years. In this example, only one future cash flow was considered. For a decision which generates multiple cash flows in multiple time periods, all the cash flows must be discounted and then summed into a single [net present value](https://bliptext.com/articles/net-present-value). 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Integrated future value

To address the lack of integration of the short and long term importance, value and risks associated with natural and social capital into the traditional DCF calculation, companies are valuing their environmental, social and governance (ESG) performance through an Integrated Management approach to reporting, that expands DCF or Net Present Value to Integrated Future Value (IntFV). This allows companies to value their investments not just for their financial return but also the long term environmental and social return of their investments. By highlighting environmental, social and governance performance in reporting, decision makers have the opportunity to identify new areas for value creation that are not revealed through traditional financial reporting. As an example, the social cost of carbon is one value that can be incorporated into Integrated Future Value calculations to encompass the damage to society from greenhouse gas emissions that result from an investment. This is an integrated approach to reporting that supports Integrated Bottom Line (IBL) decision making, which takes triple bottom line (TBL) a step further and combines financial, environmental and social performance reporting into one balance sheet. This approach provides decision makers with the insight to identify opportunities for value creation that promote growth and change within an organization.