Discounted Cash Flow - Part 3

Part 3 – The Basics Continued

Today we continue with our discussion. We’ll cover Terminal Value, WACC and solve a simple problem based on the discussions that have been done so far.

Terminal Value

Most projects require an initial investment and then eventually they give off positive net cash flows for a given period then stop. An appropriate example could be a mine which produces some ore or gold or coal only for a finite time. Other projects or assets could produce positive cash flows in perpetuity e.g. a piece of land which can be utilised for anything. If you keep the cash flows going for years (including Year 0) and discount them, you’ll notice they get smaller and smaller as you go further in time until eventually they are a fraction of a rupee. You could keep going another 5,000 years but the discounted cash flows are already so small that it wouldn’t really make a difference. Eventually by adding up all the discounted cash flows we get the NPV.

However, it would not be practical to forecast the cash flows beyond a certain time horizon (probably 5 to 10 years). Please recall what I had said in the part 2 of this article –

“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.”

Now, having estimated the cash flows for the horizon period we need to come up with a reasonable value of the cash flows beyond this period. Forecasting results beyond the horizon period is impractical and exposes such projections to a variety of risks limiting their validity, primarily the great uncertainty involved in predicting industry and macroeconomic conditions beyond a few years. Instead of an attempt to forecast the cash flow for each individual year beyond the horizon period, one can use a single value representing the discounted value of all subsequent cash flows. This single value is referred to as the terminal value (TV). Thus, the terminal value allows for the inclusion of the value of future cash flows occurring beyond a several-year projection period while satisfactorily mitigating many of the problems of valuing such cash flows. Depending on what is being valued the terminal value can be calculated either based on the value if liquidated or based on the value of the firm as an ongoing concern. A mine, for example, will have a small terminal value vis-à-vis a business which leases a piece of land.

There are three methods in vogue in which the TV can be calculated –

Liquidation method: By assuming a firm will cease operations and liquidate its assets at the end of the discounted cash flow analysis, you can simply calculate the value of the firm's existing assets and adjust for inflation. However, this approach is limited as it doesn't reflect the earning power of the firm's assets.

Multiple approach: The value of the firm is estimated by applying a multiple to the firm's earnings or revenues. For example, one might multiply an appropriate industry price to earnings ratio to the estimated earnings in order to arrive at a terminal value for the firm.

Stable growth model: This model assumes that the company will grow at a constant rate forever. To calculate this, use the formula –

TV_t =  Cash Flow _t+1

            -------------------

              r - g_stable

Cash Flow t+1 represents the first year beyond the discounted cash flow analysis, r is the interest rate and g is the stable growth rate. If the company is assumed to disappear at some point in the future, a negative growth rate can be used. Please also refer to the web page available on terminal value Estimating Terminal Value from NYU Stern.

When applied to the discounted cash flow, the terminal value should be discounted by dividing the terminal value by (1 + r)^t.

It would be prudent to mention here that a detailed study of stable growth model is beyond the scope of this article. We’ll cover this topic at a later stage. However, for those of you interested at this stage I suggest a reference to Professor Aswath Damodaran’s notes titled  Valuation: Estimating Terminal Value.

Please appreciate that as with any forecast or prediction, the further out it is, the greater the chance of error. Keep in mind that terminal value is typically forecasted for some X periods into the future, for an indefinite amount of time beyond. A number of assumptions must hold true to obtain even a modestly accurate terminal value.

Because of this, many analysts may opt to instead use a 'base case' terminal value, with as conservative assumptions as possible - while this has the potential benefit of limiting downside and maximizing upside, it does not necessarily imply accuracy (which is typically how the greatest returns are generated).

All this goes to say is that while terminal value is a useful and sometimes necessary metric for valuation, it should be subject to significant scrutiny, as even small changes in the underlying assumptions can have meaningful overall impact.

Weighted Average Cost of Capital

Companies raise money from a number of sources: common equity, preferred stock, straight debt, convertible debt, exchangeable debt, warrants, options, pension liabilities, executive stock options, governmental subsidies, and so on. Different securities, which represent different sources of finance, are expected to generate different returns. The WACC is the minimum acceptable return that a company must earn on an existing asset base to satisfy its creditors, owners, and other providers of capital, or they will invest elsewhere. The WACC is calculated taking into account the relative weights of each component of the capital structure. The more complex the company's capital structure, the more laborious it is to calculate the WACC.

The rate used to discount future cash flows and the terminal value (TV) to their present values should reflect the blended after-tax returns expected by the various providers of capital. The discount rate is a weighted-average of the returns expected by the different classes of capital providers (holders of different types of equity and debt), and must reflect the long-term targeted capital structure as opposed to the current capital structure. While a separate discount rate can be developed for each projection interval to reflect the changing capital structure, the discount rate is usually assumed to remain constant throughout the projection period.

In its simplest form the WACC formula is -

WACC = (E/V) × R_e + [(D/V) × R_d] × (1 - T_c)

where: 
R_e = cost of equity 
R_d = cost of debt 
E = market value of the firm's equity 
D = market value of the firm's debt 
V = E + D = firm value 
E/V = percentage of financing that is equity 
D/V = percentage of financing that is debt 
T_c = corporate tax rate

Calculating the Cost of Equity

The cost of equity is usually calculated using the Capital Asset Pricing Model (CAPM), which defines the cost of equity as follows:

R_e = R_f + β X (R_m – R_f)

Where:

R_{f =} Risk free rate (normally the government bond rate)

β = Predicted equity beta

(R_m – R_f) = Market risk premium

A short commentary on Beta will be appropriate at this point. Beta is a measure of the volatility of a stock's returns relative to the equity returns of the overall market. It is determined by plotting the stock's and market's returns at discrete intervals over a period of time and fitting (regressing) a line through the resulting data points. The slope of that line is the levered equity beta. When the slope of the line is 1.00, the returns of the stock are no more or less volatile than returns on the market. When the slope exceeds 1.00, the stock's returns are more volatile than the market's returns.

This is the simplest way in which the Beta can be worked out. However, Professor Aswath Damodaran says Avoid regression betas. Regression betas, commonly used in calculating the cost of equity, generally have large standard errors. Betas should reflect the business the firm operates in, its operating leverage, and its debt level. Damodaran calls for the use of sector betas as a way to eliminate the noise that comes with regression betas calculated on individual firms. A reading of chapter 2 in  Aswath Damodaran’s book “Damodaran on valuation” and chapter 6 in I M Pandey’s book “Financial Management” will be helpful.

Simply put, R_d reflects the current market rates the firm pays for debt. Interest paid on debt reduces the Net Income and therefore, reduces the tax payments for the firm. This interest tax shield depends on the tax rate.

We have covered everything on the basics of DCF. Advanced topics in DCF will be discussed later. For now the discussions till date should suffice for our purpose.

Let us now try to solve a few problems taking into account all that has been discussed.

Problem 01:  Determine the NPV (at discount rate of 30%) and IRR for two mutually exclusive projects that cost ` 5,00,000 each and would yield after-tax cash flows as given in the chart below.

	Cash Flows
Year	Project A	Project B
0	-500000	-500000
1	400000	100000
2	300000	200000
3	200000	300000
4	100000	400000

Solution

Project A
Year	Cash Flows	PV @ 30%	PV @ 40%	PV @ 50%
0	-500000	-500000	-500000.00	-500000.00
1	400000	307692.3	285714.29	266666.67
2	300000	177514.8	153061.22	133333.33
3	200000	91033.23	72886.30	59259.26
4	100000	35012.78	26030.82	19753.09
NPV		111253.11	37692.63	-20987.65
IRR (Using Excel function)				46.17%
IRR (Calculated)				46.42%

Project B
Year	Cash Flows	PV @ 20%	PV @ 30%
0	-500000	-500000.00	-500000.00
1	100000	83333.33	76923.08
2	200000	138888.89	118343.20
3	300000	173611.11	136549.84
4	400000	192901.23	140051.12
NPV		88734.57	-28132.77
IRR (Using Excel function)			27.27%
IRR (Calculated)			27.59%

I have used the method of graphing and constructing similar triangles and then solving for IRR which we had discussed in part 2 of this series.

I would like to comment some more on the IRR at this point. IRR allows managers to rank projects by their overall rates of return rather than their net present values, and the investment with the highest IRR is usually preferred.  Also, IRR does not measure the absolute size of the investment or the return. This means that IRR can favour investments with high rates of return even if the rupee amount of the return is very small. For example, a `1 investment returning `3 will have a higher IRR than a `1 million investment returning `2 million. Finally, IRR does not consider cost of capital and can’t compare projects with different durations.

Problem 02: The ABC Company has consulted with its investment bankers and determined that they could issue new debt with a yield of 8%. If the firm’s ' marginal tax rate is 39%, what is the after-tax cost of debt to the firm?

Solution

r_d = 0.08 (1 – 0.39) = 0.0488 or 4.88%

Problem 03: The XYZ Company has common stock outstanding that has a current price of  `20 per share and a `0.5 dividend. XYZ’s dividends are expected to grow at a rate of 3% per year, forever. The expected risk-free rate of interest is 2.5%, whereas the expected market premium is 5%. The beta on XYZ’s stock is 1.2. What is the cost of equity for XYZ using the Capital Asset Pricing Model (CAPM)?

Solution

r_e = 0.025 + (0.05) 1.2 = 0.025 + 0.06 = 8.5%

Problem no. 3 is the kind where examiners try to confuse the student. The additional information given regarding the current price of the stock, the dividend and the growth rate are not utilized in CAPM but only in the Dividend Valuation Model. We’ll discuss this topic when we do valuation later.

I end this discussion on DCF at this point. We’ll try to cover other topics in subsequent posts.

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.