# Discounted cash flow

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In finance, the discounted cash flow (or DCF) approach describes a method to value a project or an entire company. The DCF methods determine the present value of future cash flows by discounting them using the appropriate cost of capital. This is necessary because cash flows in different time periods cannot be directly compared since most people prefer money sooner rather than later (put simply: a dollar in your hand today is worth more than a dollar you may receive at some point in the future). The same logic applies to the difference between certain cash flows and uncertain ones, or "a bird in the hand is worth two in the bush". This is due to opportunity cost and risk over time.

DCF procedure involve three problems

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

Depending on the financing schedule of the company four different DCF methods are distinguished today. Since the underlying financing assumptions are different they do not need to arrive at the same value of the project or company:

## Math

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

${\displaystyle FV_{n}=CF_{0}\cdot (1+r)^{n}}$

Where ${\displaystyle CF_{0}}$ is the Cash Flow received at time ${\displaystyle t=0}$.

The simplified version of the Discounted cash flow equation (for one cash flow in one future period) is expressed as:

${\displaystyle PV=\left({\frac {CF_{t}}{(1+r)^{n}}}\right)}$

where

• PV is the present value of the future cash flow (FV), or FV adjusted for the opportunity cost of future receipts and risk linked to the uncertainty of the expected value.;
• ${\displaystyle ''CF_{t}''}$ is the nominal value of a cash flow amount in a future period;
• r is the discount rate, which is the risk-free rate plus an adjustement for the riskiness of the cash flow (or the [[time value of money);
• n is the number of discounting periods used (the period in which the future cash flow occurs). I.e. if the receipts occur at the end of year 1, n will be equal to 1; at the end of year 2, 2—likewise, if the cash flow happens instantly, n becomes 0, rendering the expression an identity (PV=FV).

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

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

For each future cash flow (FV) at any time period (t) for all time periods. The sum can then be used as a net present value figure or used to further calculate the internal rate of return for a cash flow pattern over time.

## 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 13.6%. Looking at those figures, he might be justified in thinking that the purchase looked like a good idea.

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 Bill rate is 5%. Treasury Bills are generally considered to be inherently less risky than real estate, since the value of the Bill is guaranteed by the US Government and there is a liquid market for the purchase and sale of T-Bills.

So, calculating exclusively for opportunity cost, we get a discount rate of 5% per year. Using the DCF formula above, that means that the net present value of $150,000 received in three years is actually$129,146 (rounded off). Those future dollars aren't worth the same as the dollars we have now.

Using simple subtraction again, the present-value profit on the sale would then be $29,146 or a little more than 29%. Amortized over the three years, that implies a discounted annual return of 8.6% (still very respectable, but only 63% of the profit he previously thought he would have). Note that the original internal rate of return (13.6%) minus the discount rate (5%) equals the discounted internal rate of return (8.6%). The discount rate directly modifies the annual rate of return. But what about risk? • 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.

For the sake of the example, let's then estimate his risk factor is about 5% (we could perform a more precise probabilistic analysis of the risk, but that is beyond the scope of this article). Therefore, this analysis should now include both opportunity cost (5%) and risk (5%), for a total discount rate of 10% per year.

Going back to the DCF formula, $150,000 received three years from now and discounted at a rate of 10% is only worth$111,261 (rounded off) in present-day dollars. The present-value profit on the sale is now down to $11,261 discounted dollars from$50,000 nominal dollars. The implied annual rate of return on that discounted profit is now 3.6% per year.

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 opportunity cost and 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 it appear 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 buying the house would actually cause John to lose money in present-value terms (about$6,000 in the first case, and about $9,000 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 make a profit in discounted terms even if 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, DCF analysis must be performed on each cash flow in each period and summed into a single net present value.

## 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.