Guidance on cash flow and dicounted cash flow calculations

Cash flow is the movement of money into or out of a business, project, or financial product. It is usually measured during a specified, limited period of time. Measurement of cash flow can be used for calculating other parameters that give information on a company's value and situation.

1. To determine a project's rate of return or value. The time of cash flows into and out of projects are used as inputs in financial models such as internal rate of return and net present value.

2. To determine problems with a business's liquidity. Being profitable does not necessarily mean being liquid. A company can fail because of a shortage of cash even while profitable.

3. As an alternative measure of a business's profits when it is believed that accrual accounting concepts do not represent economic realities. For instance, a company may be notionally profitable but generating little operational cash (as may be the case for a company that barters its products rather than selling for cash). In such a case, the company may be deriving additional operating cash by issuing shares or raising additional debt finance.

4. Cash flow can be used to evaluate the 'quality' of income generated by accrual accounting. When net income is composed of large non-cash items it is considered low quality.

5. To evaluate the risks within a financial product, e.g., matching cash requirements, evaluating default risk, re-investment requirements, etc.

Cash flow notion is based loosely on cash flow statement accounting standards. the term is flexible and can refer to time intervals spanning over past-future. It can refer to the total of all flows involved or a subset of those flows. Subset terms include net cash flow, operating cash flow and free cash flow.

Business' financials

The (total) net cash flow of a company over a period (typically a quarter, half year, or a full year) is equal to the change in cash balance over this period: positive if the cash balance increases (more cash becomes available), negative if the cash balance decreases. The total net cash flow is the sum of cash flows that are classified in three areas:

1. Operational cash flows: Cash received or expended as a result of the company's internal business activities. It includes cash earnings plus changes to working capital. Over the medium term this must be net positive if the company is to remain solvent.

2. Investment cash flows: Cash received from the sale of long-life assets, or spent on capital expenditure (investments, acquisitions and long-life assets).

3. Financing cash flows: Cash received from the issue of debt and equity, or paid out as dividends, share repurchases or debt repayments.

Examples

Description	Amount ($)	totals ($)
Cash flow from operations		+10
Sales (paid in cash)	+30
Incoming loan	+50
Loan repayment	-5
Taxes	-5
Cash flow from investments		-10
Purchased capital	-10
Total		0

The net cash flow only provides a limited amount of information. Compare, for instance, the cash flows over three years of two companies:

	Company A			Company B
	Year 1	Year 2	year 3	Year 1	Year 2	year 3
Cash flow from operations	+20M	+21M	+22M	+10M	+11M	+12M
Cash flow from financing	+5M	+5M	+5M	+5M	+5M	+5M
Cash flow from investment	-15M	-15M	-15M	0M	0M	0M
Net cash flow	+9M	+10M	+11M	+13M	+14M	+15M

Company B has a higher yearly cash flow. However, Company A is actually earning more cash by its core activities and has already spent 45M in long term investments, of which the revenues will only show up after three years.

DISCOUNTED CASH FLOW

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 or price 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.

Contents

Discount rate

Main article: Discounting

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. 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

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.

$DCF = \frac{CF_1}{(1+r)^1} + \frac{CF_2}{(1+r)^2} + \dotsb + \frac{CF_n}{(1+r)^n}$

$FV = DCF \cdot (1+r)^n$

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

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

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:

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

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 integration:

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

Where

is now the rate of cash flow, and $\lambda = log(1+r)$ .

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 (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, 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 (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.

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.

3. Fisher, Irving. "The theory of interest." New York 43 (1930).

Friday, 4 September 2015