# Discounted cash flow

(Redirected from Required rate of return)

In finance, discounted cash flow (DCF) analysis is a method of valuing a project, company, or asset using the concepts of the time value of money. 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.[1]

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.

Discounted cash flow analysis is widely used in investment finance, real estate development, corporate financial management and patent valuation. It was used in industry as early as the 1700s or 1800s, widely discussed in financial economics in the 1960s, and became widely used in U.S. Courts in the 1980s and 1990s.

 Example DCF calculation To show how discounted cash flow analysis is performed, consider the following example. John Doe buys a house for $100,000. Three years later, he expects to be able to sell this house for$150,000. 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 over the three years, his implied annual return (known as the 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 3-year US Treasury Note 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 market for the purchase and sale of T-Notes. If he hadn't 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 for the relevant period (3 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 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 3 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 3 years isn't 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 results in the net 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 3 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, 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 for a further discussion of this. For the sake of the example (and this is a gross simplification), let's 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 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. See #Methods of appraisal of a company or project for cases where multiple periods are considered.

## Discount rate

The act of discounting future cash flows answers "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; see further under John Burr Williams#Theory.

This "required return" thus incorporates:

1. Time value of money (risk-free rate) – according to the theory of time preference, investors would rather have cash immediately than having to wait and must therefore be compensated by paying for the delay
2. Risk premium – reflects the extra return investors demand because they want to be compensated for the risk that the cash flow might not materialize after all.

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 Capital asset pricing model#Asset-specific required return and Asset pricing#General Equilibrium Asset Pricing.

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".)

Note that "expected return", although formally the mathematical expected value, is often used interchangeably with the above wording, where "expected" means "required" in the corresponding sense.

## 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 of the future cash flows. This method of asset valuation differentiated between the accounting book value, which is based on the amount paid for the asset.[2] 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.[3]

## Mathematics

### Discounted cash flows

The discounted cash flow formula is derived from the future value formula for calculating the time value of money and compounding returns.

${\displaystyle DCF={\frac {CF_{1}}{(1+r)^{1}}}+{\frac {CF_{2}}{(1+r)^{2}}}+\dotsb +{\frac {CF_{n}}{(1+r)^{n}}}}$
${\displaystyle FV=DCF\cdot (1+r)^{n}}$

Thus the discounted present value (for one cash flow in one future period) is expressed as:

${\displaystyle DPV={\frac {FV}{(1+r)^{n}}}}$

where

• DPV is the discounted present value of the future cash flow (FV), or FV adjusted for the delay in receipt;
• FV is the nominal value of a cash flow amount in a future period;
• r is the interest rate or discount rate, which reflects the cost of tying up capital and may also allow for the risk that the payment may not be received in full;[4]
• n is the time in years before the future cash flow occurs.

Where multiple cash flows in multiple time periods are discounted, it is necessary to sum them as follows:

${\displaystyle DPV=\sum _{t=0}^{N}{\frac {FV_{t}}{(1+r)^{t}}}}$

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:

${\displaystyle DPV=\int _{0}^{T}FV(t)\,e^{-\lambda t}dt=\int _{0}^{T}{\frac {FV(t)}{(1+r)^{t}}}\,dt\,,}$

where ${\displaystyle FV(t)}$ is now the rate of cash flow, and ${\displaystyle \lambda =\log(1+r)}$.

## Methods of appraisal of a company or project

Here, a spreadsheetvaluation, uses Free cash flows to estimate stock's Fair Value and measure the sensibility of WACC and Perpetual growth

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 simple treatment; for the components / steps of business modeling here, see the list for "Equity valuation" under Outline of finance#Discounted cash flow valuation.

### Equity-approach

• Flows to equity approach (FTE)
• Discount the cash flows available to the holders of equity capital, after allowing for cost of servicing debt capital
• Advantages: Makes explicit allowance for the cost of debt capital
• Disadvantages: Requires judgement on choice of discount rate

### Entity-approach

• Adjusted present value approach (APV)
• Discount the cash flows before allowing for the debt capital (but allowing for the tax relief obtained on the debt capital)
• Advantages: Simpler to apply if a specific project is being valued which does not have earmarked debt capital finance
• Disadvantages: Requires judgement on choice of discount rate; no explicit allowance for cost of debt capital, which may be much higher than a risk-free rate
• Weighted average cost of capital approach (WACC)
• Derive a weighted cost of the capital obtained from the various sources and use that discount rate to discount the cash flows from the project
• Advantages: Overcomes the requirement for debt capital finance to be earmarked to particular projects
• Disadvantages: Care must be exercised in the selection of the appropriate income stream. The net cash flow to total invested capital is the generally accepted choice.
• Total cash flow approach (TCF)[clarification needed]
• This distinction illustrates that the Discounted Cash Flow method can be used to determine the value of various business ownership interests. These can include equity or debt holders.
• Alternatively, the method can be used to value the company based on the value of total invested capital. In each case, the differences lie in the choice of the income stream and discount rate. For example, the net cash flow to total invested capital and WACC are appropriate when valuing a company based on the market value of all invested capital.[5]

## Shortcomings

1) Traditional DCF models assume we can accurately forecast revenue and earnings 3–5 years into the future. But studies have shown that growth is neither predictable nor persistent.[6]

In other terms, using DCF models is problematic due to the problem of induction, or presupposing that a sequence of events in the future will occur as it always has in the past. Colloquially, in the world of finance, the problem of induction is often simplified with the common phrase: past returns are not indicative of future results. In fact, the SEC demands that all mutual funds use this sentence to warn their investors.[7]

This observation has led some to conclude that DCF models should only be used to value companies with steady cash flows. For example, DCF models are widely used to value mature companies in stable industry sectors, such as utilities. For industries that are especially unpredictable and thus harder to forecast, DCF models can prove especially challenging.

• Industry Examples:
• Real Estate- Investors use DCF models to value commercial real estate development projects. This practice has two main shortcomings. First, the discount rate assumption relies on the market for competing investments at the time of the analysis, which may not persist into the future. Second, assumptions about ten-year income increases are usually based on historic increases in the market rent. Yet the cyclical nature of most real estate markets is not factored in. Most real estate loans are made during boom real estate markets and these markets usually last fewer than ten years. In this case, due to the problem of induction, using a DCF model to value commercial real estate during any but the early years of a boom market can lead to overvaluation.[8]
• Early-stage Technology Companies- In valuing startups, the DCF method can be applied a number of times, with differing assumptions, to assess a range of possible future outcomes—such as the best, worst and mostly likely case scenarios. Even so, the lack of historical company data and uncertainty about factors that can affect the company's development make DCF models especially difficult for valuing startups. There is a lack of credibility regarding future cash flows, future cost of capital, and the company's growth rate. By forecasting limited data into an unpredictable future, the problem of induction is especially pronounced.[9]

2) Traditional DCF models assume that the capital asset pricing model can be used to assess the riskiness of an investment and set an appropriate discount rate. But, according to some economists, the capital asset pricing model has been empirically invalidated.[10]

3) Input-output problem

• DCF is merely a mechanical valuation tool, which makes it subject to the principle "garbage in, garbage out." Small changes in inputs can result in large changes in the value of a company. This is especially the case with terminal values, which make up a large proportion of the Discounted Cash Flow's final value.

4) Doesn't account for all variables

• Traditional DCF calculations only consider the financial costs and benefits of a decision. They do not include the environmental, social and governance performance of an organization.[11]

## Integrated Future Value (IntFV)

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. 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.[12] 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.[13]

## References

1. ^ "Wall Street Oasis (DCF)". Wall Street Oasis. Retrieved 5 February 2015.
2. ^ O.E.H. Neugebaner, The Exact Sciences in Antiquity (Copenhagen :Ejnar Mukaguard, 1951) p.33 (1969). O.E.H. Neugebaner, The Exact Sciences in Antiquity (Copenhagen :Ejnar Mukaguard, 1951) p.33. US: Dover Publications. p. 33. ISBN 0486223329.CS1 maint: Multiple names: authors list (link)
3. ^ Fisher, Irving. "The theory of interest." New York 43 (1930).
4. ^ "Discount rates and net present value". Centre for Social Impact Bonds. Retrieved 28 February 2014.
5. ^ Pratt, Shannon; Robert F. Reilly; Robert P. Schweihs (2000). Valuing a Business. McGraw-Hill Professional. McGraw Hill. ISBN 0-07-135615-0.
6. ^ Chan, Louis K.C.; Karceski, Jason; Lakonishok, Josef (May 2001). "The Level and Persistence of Growth Rates". Cambridge, MA.
7. ^ "SEC.gov | Mutual Funds, Past Performance". sec.gov. Retrieved 8 May 2019.
8. ^ Reilly, Robert F.; Schweihs, Robert P. (28 October 2016). "Guide to Intangible Asset Valuation". doi:10.1002/9781119448402.
9. ^ "Measuring and Managing Value in High-Tech Start-ups", Valuation for M&A, John Wiley & Sons, Inc., pp. 285–311, 12 September 2015, ISBN 9781119200154, retrieved 8 May 2019
10. ^ Fama, Eugene F.; French, Kenneth R. (2003). "The Capital Asset Pricing Model: Theory and Evidence". SSRN Electronic Journal. doi:10.2139/ssrn.440920. ISSN 1556-5068.
11. ^ Sroufe, Robert, author. Integrated management : how sustainability creates value for any business. ISBN 178714562X. OCLC 1059620526.CS1 maint: Multiple names: authors list (link)
12. ^ Eccles, Robert; Krzus, Michael (2010). One Report: Integrated Reporting for a Sustainable Strategy. Wiley.
13. ^ Sroufe, Robert (July 2017). "Integration and Organizational Change Towards Sustainability". Journal of Cleaner Production. 162: 315–329 – via Research Gate.