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.