Discounted Cash Flow - Part 2

Part 2 – The Basics Continued

I left off the last discussion by pointing out some of the advantages and disadvantages of DCF analysis. It’s reproduced below for a quick reference –

Theoretically, the DCF is arguably the most  sound method of valuation. The method is forward-looking and depends on more future expectations rather than historical results. It is more inward-looking, relying on the fundamental expectations of the business or asset, and is influenced to a lesser extent by volatile external factors. DCF analysis is focused on cash flow generation and is less affected by accounting practices and assumptions. The method allows expected (and different) operating strategies to be factored into the valuation. It also allows different components of a business or synergies to be valued separately. However; it also has certain disadvantages. The accuracy of the valuation determined using the DCF method is highly dependent on the quality of the assumptions regarding FCF, TV, and discount rate. As a result, DCF valuations are usually expressed as a range of values rather than a single value by using a range of values for key inputs. It is also common to run the DCF analysis for different scenarios, such as a base case, an optimistic case, and a pessimistic case to gauge the sensitivity of the valuation to various operating assumptions. While the inputs come from a variety of sources, they must be viewed objectively in the aggregate before finalizing the DCF valuation.

One thing that immediately catches my attention is that the DCF method has its shortcomings. For starters, the DCF model is only as good as its inputs. If the inputs - free cash flow forecasts, discount rates and other factors - are wide off the mark, the fair value generated for the asset won't be accurate.

Here I would like to quote Tim Koller, Marc Goedhart, and David Wessels from  their book “Valuation: Measuring and Managing the Value of Companies” –

“Discounted cash flow analysis is the most accurate and flexible method for valuing projects, divisions, and companies. Any analysis, however, is only as accurate as the forecasts it relies on. Errors in estimating the key ingredients of corporate value . . . can lead to mistakes in valuation.”

Let’s take the case of FCF, the first and most important factor in calculating the DCF value of an asset. There are a number of inherent problems with earnings and cash flow forecasting that can generate problems with DCF analysis. Notice here that I said “earnings and cash flow”. From our previous discussion we know that both are different. However, we also know that cash flows are dependent on earnings; earnings being the super-set. Earnings can be volatile both as a result of the normal ebb and flow of business and as a result of accounting transactions. As a corollary, uncertainty in cash flow projection increases for each year in the forecast. We  may have a good idea of what cash flows will be for the current year and the following year, but beyond that, the ability to project earnings and cash flow diminishes rapidly.  In my opinion anything beyond a couple of years is suspect. Also cash flow projections in any given year will most likely be based largely on results for the preceding years. Small, erroneous assumptions in the first couple years of a model can amplify variances in cash flow projections in the later years of the model.

The next input (and a very important one) is the discount rate that will be used.  What should be the appropriate discount rate? Emphasis is on the word “appropriate”. Warren Buffett has been frequently quoted to use the long-term U.S. Bonds as his discount rate for doing his DCF calculations. At the 1997 Berkshire Hathaway meeting, Buffett was quoted to have said:

“We use the risk-free rate merely to equate one item to another. In other words, we’re looking for whatever is the most attractive. In order to estimate the present value of anything, we’re going to use a number. And, obviously, we can always buy government bonds. Therefore, that becomes the yardstick rate.”

At the 1998 Berkshire Hathaway meeting, Buffett was quoted to have said:

“We don’t discount the future cash flows at 9% or 10%; we use the U.S. treasury rate. We try to deal with things about which we are quite certain. You can’t compensate for risk by using a high discount rate.”

Fine. After all Warren Buffett is a successful man. So the question is do we do what  he does or is his method not the best for our purpose. After all there are some more avenues of deciding on the discount rate. As Ben McClure (a regular contributor to investopedia.com) says “A wide variety of methods can be used to determine discount rates, but in most cases, these calculations resemble art more than science. Still, it is better to be generally correct than precisely incorrect”.

Determining a reasonable rate is the most difficult part of DCF analysis. There are many ways to choose the discount rate to use in DCF analysis. The choice of the rate is the most critical input into the analysis; the choice of an unreasonable rate can skew the investment decision. Some popular discount rates adopted are – the Weighted Average Cost of Capital (WACC), the Internal Rate of Return (IRR), and treasury or government bond rate. The choice differs from person to person doing the analysis. And even from different sectors of businesses.

For those of us who are not familiar with some of the terms used in the preceding paragraph there’s no cause for worry. We’ll quickly run through what goes into the makings of WACC and IRR.

Let’s first look at the Internal Rate of Return (IRR). In simple terms it can be defined as the discount rate at which the present value of all future cash flow is equal to the initial investment or in other words the rate at which an investment breaks even. What it means in terms of NPV is that the discount rate should be such that when all future cash flows are discounted the NPV should be equal to zero.

Noticed something? Here the discount rate is not known; it needs to be calculated. In NPV the discount rate is known or rather the discount rate is  optimised. However, since IRR is just an offshoot of NPV (where NPV = 0, remember?) it is prone to the same estimation errors that can creep in – in this case the projected cash flows.

The formula for Internal Rate of Return is given below.

             n

NPV = ∑         FV_t  = 0

            t=0      (1+i)^t

Here, all parameters are given except ‘i’ which can be found by trial and error if  a financial calculator or appropriate software is not available. The simplest way to calculate IRR is to find the two discount rates at which the NPVs are positive and negative and then interpolate till we find the appropriate rate at which NPV = 0. I actually found the interpolation part to be tedious till my friend, Cmdr. Arun Nair of the Indian Navy, re-introduced me to high school geometry on the concept of similar triangles.

I suggest a quick visit to the internet to refresh on the concept of similar triangles will be worth your while. You could either google it or visit “http://www.mathopenref.com/similartriangles.html”. By definition “Triangles are similar if they have the same shape, but can be different sizes”. 

Consider the two triangles ΔPQR & ΔP’Q’R’of the same shape but with different sizes.   Similar triangles have two very important properties -

Corresponding angles are congruent i.e. of the same measure. So, the angle P=P',  Q=Q',  and R=R'; and

Corresponding side are all in the same proportion. So if PQ is twice the length of P'Q' the other pairs of sides are also in that proportion. PR is twice P'R' and RQ is twice R'Q'.  Formally, in two similar triangles PQR and P'Q'R' :

PQ     QR     RP

----- = ----- = ----

P’Q’   Q’R’   R’P’

Also similar triangles can have shared parts i.e. both can have common sides.

Now consider a set of investment given by the sequence of cash flows as

Year (n)	Cash Flow (CFn)
0	-123400
1	36200
2	54800
3	48100

We first work out the two discount rates which gives us one positive and one negative NPV. Incidentally the values are for i = 4%, NPV = 4834.10 and for i = 10%, NPV = (-) 9063.41. If we then plot the graph for the positive and negative NPVs using the co-ordinates as (i₁ = 4%, NPV₁ = 4834.10) and (i₂ = 10%, NPV₂ = (-) 9063.41) and join them with a straight line then we notice that the point where the line cuts the X-axis has values (IRR, 0) i.e. to say this is the point where the value of “i” is such that NPV = 0. A visual inspection will also show that the line cuts the X-axis at 6 i.e. for NPV = 0, i = 6%. However, we’ll solve it mathematically to try and arrive at a more accurate answer.

We now need to construct triangles by joining the points. There are two triangles which are similar as per the SAS theorem for similar triangles. We’ll not go into the theorem; suffice to say that two sides are in the same proportion and one angle is equal. If the larger triangle is named ΔPQR and the smaller triangle is named ΔP’Q’R’ then we have PR and P’R’ in some proportion. Using their co-ordinate points we have -

NPV₁ – NPV₂          i₁ – i₂

--------------------  =  ----------

NPV₁ – 0                 i₁ - IRR

Substituting the values in the above formula

4834.10 – (- 9063.41)          4% – 10%

---------------------------   =  --------------

      4834.10 – 0                    4% - IRR

Solving the equation we have IRR = 0.06087 or 6.08%

Just follow the above steps and with a few seconds of practice you’ll get the hang of it. It’s that simple. Certainly better then the interpolation that one is required to do.

We’ll discuss WACC, TV, etc. in the next session and then go on to solve a simple problem. In the coming sessions we’ll try to cover as many aspects of DCF analysis as possible.

Discounted Cash Flow - Part 1

Part 1 – The Basics

Preamble

One of the primary roles of financial analysis is to determine the monetary value of an asset. In part, this value is determined by the income generated over the lifetime of the asset. This can make it difficult to compare the values of different assets since the monies might be paid at different times. Let’s start with a simple case. Would you rather have an asset that paid you `1,000 today, or one that paid you `1,000 a year from now? It turns out that money paid today is better than money paid in the future (we will see why in a moment). This idea is called the time value of money. The time value of money is at the centre of a wide variety of financial calculations, particularly those involving value. What if you had the choice of `1,000 today or `1,100 a year from now? The second option pays you more (which is good) but it pays you in the future (which is bad). So, on net, is the second better or worse? In this section we will see how investors make that comparison.

The financial and economic analysis is typically carried out using the technique of Discounted Cash Flow (DCF) analysis. This module introduces concepts of discounting and DCF analysis for the derivation of criteria such as net present value (NPV) and internal rate of return (IRR). The concepts and criteria are introduced with simple examples.

Wikipedia defines DCF as -

“..........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. Present value may also be expressed as a number of years' purchase of the future undiscounted annual cash flows expected to arise.”

The basic underlying principles of DCF are -

Ø  Time Value of Money

Ø  Present/Future Value

Ø  Opportunity Cost

Let’s look at what “Time Value of Money” means. Simply put; a rupee today is worth more than a rupee tomorrow. In other words a rupee today can be invested to earn a rate of return or interest.

Reasons for time value of money

Money has time value because of the following reasons :

1)      Risk and Uncertainty : Future is always uncertain and risky. Outflow of cash is in our control as payments to parties are made by us. There is no certainty for future cash inflows. Cash inflows are dependent out on our Creditor, Bank etc. As an individual or firm is not certain about future cash receipts, it prefers receiving cash now.

2)      Inflation : In an inflationary economy, the money received today, has more purchasing power than the money to be received in future. In other words, a rupee today represents a greater real purchasing power than a rupee a year hence.

3)      Consumption : Individuals generally prefer current consumption to future consumption.

4)      Investment opportunities : An investor can profitably employ a rupee received today, to give him a higher value to be received tomorrow or after a certain period of time.

Thus, the fundamental principle behind the concept of time value of money is that, a sum of money received today, is worth more than if the same is received after a certain period of time.

Which brings us to the concept of the present and future value of money. What is today’s rupee worth tomorrow (future value)? Conversely, what is tomorrow’s rupee worth today (present value)?

Please recall that a rupee today can be invested to earn a rate of return or interest. Since we live in an inflationary economy (as most economies tend to be!!) the return on the invested rupee is compounded over a period. The future value that can be earned is then a function of the present value and the interest rate. Mathematically it can be exhibited as –

FV = PV (1+i)ⁿ

Where “FV” is the Future Value

“PV” is the Present Value

“i”is the interest rate and

“n” is the number of periods

This is also known as the compounding method as the interest earned on the initial principal amount becomes a part of the principal at the end of the compounding period.

Conversely, the present value of future sums of money can also be worked out. This method, whereby, the interests are reversed is known as the discounting method. Mathematically it can be depicted as –

PV = FV/(1+i)ⁿ

Where all the notations have the same meaning as previously discussed.

Let’s look at an example.

You are given `5,000 and decide to invest it in the stock market for 10 years and expect an average annual rate of return of 10%.  What is that `5,000 worth 10 years from now?

FV = `5,000 X (1+10%)^{10 years}

                                                               = `12,969

Likewise…

PV = `12,969/(1+10%)^{10 years}

                                                               = `5,000

That was easy. Now let’s now have a look at opportunity cost.  Investopedia  defines it as –

“The cost of an alternative that must be forgone in order to pursue a certain action. Put another way, the benefits you could have received by taking an alternative action.

The difference in return between a chosen investment and one that is necessarily passed up. Say you invest in a stock and it returns a paltry 2% over the year. In placing your money in the stock, you gave up the opportunity of another investment - say, a risk-free government bond yielding 6%. In this situation, your opportunity costs are 4% (6% - 2%).”

To put it simply a choice between two or more mutually exclusive options must be made given limited resources It would be an easy decision if you knew the end outcome; however, the risk that you could achieve greater "benefits" (be they monetary or otherwise) with another option is the opportunity cost.

Now that we have a good understanding of the three principles that are at the heart of the discounted cash flow method let us look at the three key components of a DCF which are -

Ø  Free cash flow (FCF) – Cash generated by the assets of the business (tangible and intangible) available for distribution to all providers of capital. FCF is often referred to as unlevered free cash flow, as it represents cash flow available to all providers of capital and is not affected by the capital structure of the business.

Ø  Terminal value (TV) – Value at the end of the FCF projection period (horizon period).

Ø  Discount rate – The rate used to discount projected FCFs and terminal value to their present values.

We’ll not be considering the terminal value for this discussion and only employ the FCF and the discount rate for a little while.

Please have a look at the formula given below –

       Operating Profit

         (-)      Adjusted taxes_______________________

          =   Net Operating Profit After Taxes (NOPAT)

         (+)     Depreciation

         (+/-) Working Capital

         (-)      Capital expenditure_____________________

            =   Free Cash Flows (FCF)_________________   

In the above formula we started with operating profit (EBITA - Earnings Before Interest, Taxes and Amortization) and deducted the taxes that need to be paid thereby arriving at NOPAT. So far so good. Subsequently we added back depreciation. Why you may ask. What was the purpose of deducting it from EBITDA (Earnings Before Interest, Taxes, Depreciation and Amortization) in the first place to arrive at the operating profit (EBITA) and then adding it back. It may please be appreciated that depreciation is only a book entry which has no effect on the cash flows. To arrive at the correct amount of cash flow; depreciation has to be added back into the system. Remember; we need to discount the cash flow and not the operating profit or NOPAT.

Putting DCF into Action

Now that we have covered the workings of the discounted cash flow (DCF) model in general, we'll dig a little deeper into how to determine fair values for assets. We'll walk  through a step-by-step sample DCF model that uses the "free cash flow" method. Here are the main steps to generating fair value estimate with this method :

Step 1. Project free cash flow for the forecast period.

Step 2. Determine a discount rate.

Step 3. Discount the projected free cash flows to the present, and sum.

Step 4. Calculate the perpetuity value and discount it to the present.

For the present discussion we will go only upto step 3.

We are now ready to build the DCF formula. Recall Wikipedia’s definition; a part of which says “....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”. The operative part of the definition for our purpose of building the DCF formula will be “....the sum of all future cash flows, both incoming and outgoing.....”.

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

DCF = FCF₁ + FCF₂ + ..........+ FCF_n

             (1+i)¹   (1+i)²                 (1+i)ⁿ

and FV = DCF X (1+i)ⁿ

The discounted present value (DPV) for one period will be

DPV = FV                      Where n = 1

                                                               (1+i)ⁿ

As a corollary where multiple cash flows in multiple time periods are discounted, it is necessary to sum them as follows :

                                                                         n

DPV = ∑         FV_t

             t=0     (1+i)^t

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

Let’s now look at an example

	Year 0	Year 1	Year 2	Year 3
Initial Cost	(-) 50,000
Operating Profit		75,000	84,000	1,00,000
Less taxes @34%		25,500	28,560	34,000
NOPAT		49,500	55,440	66,000
Add Depreciation		3,750	4,200	5,000
Less CapEx		4,500	5,040	6,000
FCF		48,750	54,600	65,000

Discount Rate = 10%

DPV	(-) 50,000	44,318	45,123	48,835
NPV	88,276

The DPV for each year when summed up gives the Net Present Value (NPV).

Theoretically, the DCF is arguably the most sound method of valuation. The method is forward-looking and depends on more future expectations rather than historical results. It is more inward-looking, relying on the fundamental expectations of the business or asset, and is influenced to a lesser extent by volatile external factors. DCF analysis is focused on cash flow generation and is less affected by accounting practices and assumptions. The method allows expected (and different) operating strategies to be factored into the valuation. It also allows different components of a business or synergies to be valued separately. However; it also has certain disadvantages. The accuracy of the valuation determined using the DCF method is highly dependent on the quality of the assumptions regarding FCF, TV, and discount rate. As a result, DCF valuations are usually expressed as a range of values rather than a single value by using a range of values for key inputs. It is also common to run the DCF analysis for different scenarios, such as a base case, an optimistic case, and a pessimistic case to gauge the sensitivity of the valuation to various operating assumptions. While the inputs come from a variety of sources, they must be viewed objectively in the aggregate before finalizing the DCF valuation.

We’ll discuss more of discount rate and other concepts like TV (terminal value), sensitivity, NPV and IRR in the next session.