# Discounted cash flow

In finance, the discounted cash flow (or DCF) approach describes 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. The discount rate used is generally the appropriate cost of capital and may incorporate judgments of the uncertainty (riskiness) of the future cash flows.

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

Very similar is the net present value.

## 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), ie 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: $\mbox{DPV} = \sum_{t=0}^{N} \frac{FV_t}{(1+i)^{t}}$

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.

(1+i)^(-t) can of course also be expressed as exp(-it).

### Continuous cash flows

With continuous cash flows, the summation in the above formula is replaced by an integration - nothing else changes:

DPV= integral over the required time period of FV(t) * (1-exp(-it)) dt

where FV(t) is now the rate of cash flow.

## 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, that means that the value of $150,000 received in three years actually has a present value of$129,576 (rounded off). Those future dollars aren't worth the same as the dollars we have 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). (As a check, 1.050 x 1.090 = 1.145 approximately.)

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, of course). There is of course a lot of uncertainty about house prices, and the outturn 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 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, 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.

## 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.