# Discounted cash flow

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Discounted cash flow
Excel spreadsheet uses Free cash flows to estimate stock's Fair Value and measure the sensibility of WACC and Perpetual growth

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 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 or price of the cash flows in question.

Using DCF analysis to compute the NPV takes as input cash flows and a discount rate and gives as output a price; the opposite process — taking cash flows and a price and inferring a discount rate, is called the yield.

Discounted cash flow analysis is widely used in investment finance, real estate development, and corporate financial management.

## Discount rate

The most widely used method of discounting is exponential discounting, which values future cash flows as "how much money would have to be invested currently, at a given rate of return, to yield the cash flow in future." 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.

The discount rate used is generally the appropriate Weighted average cost of capital (WACC), that reflects the risk of the cashflows. The discount rate reflects two things:

1. The 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. A 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.

An alternative to including the risk in the discount rate is to use the risk free rate, but multiply the future cash flows by the estimated probability that they will occur (the success rate). This method, widely used in drug development, is referred to as rNPV (risk-adjusted NPV), and similar methods are used to incorporate credit risk in the probability model of CDS valuation.

Oxera (2011) [1] reviews the selection of a discount rate suitable for the assessment of new and emerging energy technologies.

## History

Discounted cash flow calculations have been used in some form since money was first lent at interest in ancient times. As a method of asset valuation it has often been opposed to 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

### Discrete cash flows

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

$FV = DPV \cdot (1+i)^n$

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

$DPV = \frac{FV}{(1+i)^n} = {FV} {(1-d)^n}$

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;
• i is the interest 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;
• d is the discount rate, which is i/(1+i), i.e. the interest rate expressed as a deduction at the beginning of the year instead of an addition at the end of the year;
• 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:

$DPV = \sum_{t=0}^{N} \frac{FV_t}{(1+i)^{t}}$

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 i, that is the internal rate of return.

All the above assumes that the interest rate remains constant throughout the whole period.

### Continuous cash flows

For continuous cash flows, the summation in the above formula is replaced by an integration:

$DPV= \int_0^T FV(t) \, e^{-\lambda t} dt \,,$

where FV(t) is now the rate of cash flow, and λ = log(1+i).

## Example DCF

To show how discounted cash flow analysis is performed, consider the following simplified 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 100000 = 150000 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 (14.5%-5.0%)/(100%+5%) or approximately 9.0% (still very respectable).

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, of course). There is of course 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 (9.0%-5.0%)/(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.

## Methods of appraisal of a company or project

This is necessarily a simple treatment of a complex subject: more detail is beyond the scope of this article.

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.

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

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

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.

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.[2]

## Shortcomings

Commercial banks have widely used discounted cash flow as a method of valuing commercial real estate construction projects. This practice has two substantial shortcomings. 1) The discount rate assumption relies on the market for competing investments at the time of the analysis, which would likely change, perhaps dramatically, over time, and 2) straight line assumptions about income increasing over ten years are generally based upon historic increases in market rent but never factors in the cyclical nature of many real estate markets. Most loans are made during boom real estate markets and these markets usually last less than ten years. Using DCF to analyze commercial real estate during any but the early years of a boom market will lead to overvaluation of the asset.

Discounted cash flow models are powerful, but they do have shortcomings. 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. Instead of trying to project the cash flows to infinity, terminal value techniques are often used. A simple annuity is used to estimate the terminal value past 10 years, for example. This is done because it is harder to come to a realistic estimate of the cash flows as time goes on[3] involves calculating the period of time likely to recoup the initial outlay.

## References

1. ^ Oxera (April 2011). Discount rate for low-carbon and renewable generation technologies. Oxford, United Kingdom: Oxera Consulting Ltd.
2. ^ Pratt, Shannon; Robert F. Reilly, Robert P. Schweihs (2000). Valuing a Business. McGraw-Hill Professional. McGraw Hill. ISBN 0071356150.
3. ^ "Discounted Cash Flow - DCF". investopedia.com. Retrieved 22 November 2010.

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