Use or Not Use : RAND and RANDBETWEEN function in Excel


In financial modelling especially in showing the application of monte-carlo simulation using Excel to the capital budgeting, we use RAND and RANDBETWEEN function.

According to this article, RAND and RANDBETWEEN in Excel doesn’t really generate truly random values. So it is safe to say that those 2 functions in Excel are adequate for casual use and approximately correct, which implicate that we don’t use such function for analysis involving  billion-dollar projects.

Microsoft Excel’s ‘Not The Wichmann-Hill’ random number generators” by B.D. McCullough. Computational Statistics and Data Analysis. (accessed on 13 September 2020)

Excel (and almost any other program) are generating pseudo-random numbers. RANDBETWEEN is only generating discrete numbers.

Diversification : Investor Level or Project Level?


Recently I read a book on capital budgeting and the book will say that the company needs  to diversify its project investment portfolio similar to what is suggested by Modern Portfolio Theory (MPT). It is said that

…. In a world without uncertainty, an investor would choose to own the investment that will provide the highest return over their holding period. In the real world, fraught with uncertainty, investors cannot possibly know which investment will provide the highest return, or even if an investment will earn a profit. Therefore, it makes sense to spread one’s funds across several investments in the hope that some will be profitable enough to more than offset the losses of others. This is known as diversification

I don’t really concur with the way that book tries to convince to use or apply MPT to project investment portfolio, yet my personal opinion:

I see that project portfolio and stock portfolio is certainly different stuff. Reducing risk in stock portfolio by applying diversification is well documented, pioneered by Dr. Markowitz in his ground-breaking paper, leading to the birth of MPT.

My question, can we bring the MPT concept to Project Investment Portfolio?

My personal opinion, no, we can’t. Though logically, it makes sense to not put all your eggs in one basket, yet, project is different in many fronts compared to stock. The specific risk is more dominant, and in many cases, the company might not have a chance to remove it at all by investing in many more projects both in the same industry and/or different industries. It might even make the company’s business risk higher, if it invests its money in the project that is beyond its expertise.

Additionally, project investment is difficult to be compared to stock investment. The specific project risk under different sectors or industries (if this is what the diversification under MPT means) might even make the company’s risk higher, for example, investing in project that is beyond the company’s expertise or industry. Additionally, switching is almost impossible to do once the company puts in the money. The issue with this project portfolio, the specific risk plays much bigger role compared to the stock portfolio, and in many cases, probably is not easy to remove at all.

Respondent 1 to Karnen:

The risk diversification assumes that you are constraint in one dimension: money.

However, in reality, companies are constraint in money and other aspects like time, management attention, specialization etc.

So I think that diversifying projects like portfolios is not a practical exercise since the company will not be able to manage it. However all else equal – it is better to invest in low correlation projects than in high correlation.

Karnen :

Statement : The company invested in various projects with lower or no correlation (or covariance) to each other, the risk will be lower. Though I accept that statement but the risk reduction itself doesn’t create value. This is two different thing. The idiosyncratic risk in overall might be lower, for example, under the acquisition setting, yet if the market is perfect and CAPM holds up, then the cost of capital will remain the same under the combined company.

To be more precise, diversification by the company itself doesn’t create or destroy value under perfect market assumption. Which means diversification per se can never expand the opportunity set on the investor’s side given perfect security markets. Value will only be created via this diversification if there is a new Positive Net Present Value (NPV) Project being created from that diversification, something that is not explored before.

Let me put that in the example to explain the above, what I meant.

Acquirer company A data: Market value US$ 900 million, beta of 2 and risk of 25%

Target company B data : Market value US$ 100 million, beta of 1 and risk of 25%


  • both companies have the same total risks of 25%.
  • both companies are not in the same business line.

The market value of the combined company AB will be US$ 900 million + US$ 100 million => US$ 1,000 million.

It is possible for the investors in the market to hold 90% Company A and 10% Company B in their well diversified portfolio, without having for the Company to acquire Company B.

Will the acquisition of Company B by Company A ADD VALUE TO THE WELL-DIVERSIFIED INVESTORS?

We need to check whether the cost of capital  (or investor’s expected rate of return) will remain the same after the acquisition. If yes, then no value is created or destroyed from this diversification by Company A to Company B under different business line. Meaning there is no diversification benefits coming from this corporate action.

Let us check from CAPM world.

Let’s say we have risk-free rate of 3% and the market equity premium of 5%.

CAPM-based expected rate of return of Company A = 3% + (5% x 2) = 13%

CAPM-based expected rate of return of Company B = 3% + (5% x 1) = 8%

The combined Company AB will have:

  • the expected rate of return = (90% x 13%) + (10% x 8%) = 12.5%
  • the market beta = (90% x 2) + (10% x 1) = 1.9

Let’s now check what the CAPM-based expected rate of return will say.

CAPM-based expected rate of return = 3% + (5% x 1.9) = 12.5%

From the above calculation, we could see that the expected cost of capital from investors will remain the SAME, not increased or not declined.

We might ask about what happens to the risk of 25% above of each company? Since both companies are not operating in the same business line, or in other words, they are not perfectly correlated (or said to be lower co-variance), then it will result to the non-systematic (idiosyncratic) risk of the combined company AB be lower or reduced. However, under CAPM world, only the reduction of the market (systematic) risk that will be compensated by the well-diversified investors.

We might have another question, what happens to the all the news we read about the acquisition, that there is synergy benefits arising from such corporate action? If there is really synergy benefits coming (which I might have a big doubt. For interested readers, they might want to read the book  Synergy Trap : How Companies Lose the Acquisition Game by by Mark L. Sirower  (1997, Simon and Schuster)), this benefits are not coming from the diversification per se.

In the pursuit of diversification, we might be trapped to put so much focus on the gain ONLY:

Gain = PV(AB) − {PV(A) + PV(B)} = ΔPV(AB)

PV = Present Value

As that many things in life, if this gain is positive, then we need to think the other side of the coin:

How come the selling shareholders of the acquired company will give the acquiring company’s shareholders THOSE BENEFITS FOR FREE? There is certainly COST TO PAY. THERE IS NO FREE LUNCH IN THIS WORLD!

Cost = cash paid − PV(B)

Let’s say we get the NET POSITIVE GAIN from the above calculation, we need to put some grain of salt into this analysis, and ask ourselves, whether this NET POSITIVE GAIN is there not because of the acquisition, but simply we have put too optimistic expectation (over-confidence) cash flow projections?

Respondent 2 to Karnen:

More generally, diversification matters at the investor level.  The firm doesn’t need to diversify, since its investors can. The potential gain from internal diversification is that it may support more leverage.

Respondent 3 to Karnen:

Well, there are mixed feelings and special considerations….

A firm could diversify if it sells/produces products/services that have “inverse” behaviors… For instance, imagine a firm with different products that sell in different seasons of the year. It’s a kind of diversification… Or a firm that produces the same product BUT has customers in different areas of the same countries with peaks at different dates. In fact, I remember I worked in a printing firm that used to sell notebooks for schools. In Colombia we have two calendars: schools that start classes at February and schools that start say in July. The firm had diversified according to markets. Moreover, this firm diversified its products/services because it doesn’t only manufactured notebooks but (in that time 1960’s/1970’s) there were no computers and the firms used to keep the bookkeeping in physical accounting books. Yes, the firm manufactured those accounting books. But also, it printed calendars with coloured photos, and printed poster according to its customers’ needs… and so on. The firm was diversified….

Agree, investor in equities are more prone to diversify, yes. And Markovitz theory was developed for those investors. No discussion on that. And no discussion on flexibility to change portfolios when it is based on stocks, say…. However, for a firm it is good to have some degree of diversification and yes, it is not easy to switch from one given portfolio to another….

Karnen to Respondent 3:

I guess, diversification is an old stuff, long before MPT was introduced (mathematically) by Harry Markowitz. This diversification makes sense since we don’t want to put all eggs in one basket. No discussion about that.

Yet, my big question: can we use MPT to apply practically to project diversification. I believe even Mr. Markowitz doesn’t want to bring his proposition to project level. Mr. Markowitz uses stock portfolios, and built that efficient stock portfolio frontier. Will that be possible again to bring that MPT and its efficient frontier to project diversification?

I just copied below from Principles of Corporate Finance textbook by Brealey, Myers and Allen (12th Edition, McGraw-Hill Education, 2017, page 184). This brings me home to this understanding.

We have seen that diversification reduces risk and, therefore, makes sense for investors. But does it also make sense for the firm? Is a diversified firm more attractive to investors than an undiversified one? If it is, we have an extremely disturbing result. If diversification is an appropriate corporate objective, each project has to be analyzed as a potential addition to the firm’s portfolio of assets. The value of the diversified package would be greater than the sum of the parts. So present values would no longer add.

Diversification is undoubtedly a good thing, but that does not mean that firms should practiceit. If investors were not able to hold a large number of securities, then they might want firms to diversify for them. But investors can diversify. In many ways they can do so more easily than firms. Individuals can invest in the steel industry this week and pull out next week. A firm cannot do that. To be sure, the individual would have to pay brokerage fees on the purchase and sale of steel company shares, but think of the time and expense for a firm to acquire a steel company or to start up a new steel-making operation. You can probably see where we are heading. If investors can diversify on their own account, they will not pay any extra for firms that diversify. And if they have a sufficiently wide choice of securities, they will not pay any less because they are unable to invest separately in each factory. Therefore, in countries like the United States, which have large and competitive capital markets, diversification does not add to a firm’s value or subtract from it. The total value is the sum of its parts.

This conclusion is important for corporate finance, because it justifies adding present values.

The concept of value additivity is so important that we will give a formal definition of it. If the capital market establishes a value PV(A) for asset A and PV(B) for B, the market value of a firm that holds only these two assets is

PV(AB) = PV(A) + PV(B)

A three-asset firm combining assets A, B, and C would be worth PV(ABC) = PV(A) + PV(B) + PV(C), and so on for any number of assets.

We have relied on intuitive arguments for value additivity. But the concept is a general one that can be proved formally by several different routes. The concept seems to be widely accepted, for thousands of managers add thousands of present values daily, usually without thinking about it.

As Brealey, Myers and Allen said above, if shareholders could do the diversification by themselves, then they will only pay for the market risk of the company’s business that they can’t diversify away. Any idiosyncratic nature from the unsystematic risk being taken by the company in any projects will be not be compensated by the investors.

The company of course could put their money into many projects that, is expected to have low correlation between one project to another project, or even put their projects in different geography or even different segment of business. However though logically by doing the aforementioned, we expect that the risk will be reduced from the diversification, yet behind this way of thinking, the company might have another risk coming in to counter that expected lower risk from diversification. The more projects in different segment, geography, etc. might give the company a challenge to manage such business and whether the management has such skill and business knowledge to deal with the conglomeration of the business. Not many business could have such skills and expertise. However, it might be whatever the risk adding to the project portfolio, it could be specific project risk, which in nature, non-systematic risk, and the investor could easily diversify it away using the portfolio diversification.

Respondent 4 to Karnen

But sometimes I really like the idea of Markowitz.  In the extreme case you can invest in every asset in the whole economy and then you should earn the overall growth of the economy plus the amount transferred from non asset owners to owners.

Brealey and Myers should simply go back to Merton Miller.  Miller said that leverage did not matter because investors could create leverage themselves.  Of course investors can diversify themselves so what value is added by companies doing something that investors can do anyway by themselves.

I think about the diversification issue in terms of venture capital and high tech companies.  They try to earn about 30% on start-ups that have demonstrated proven concepts and about 50% with no proof of concept (this is what private equity apparently wants).  Maybe on average they earn something like 10% or even less.  This means they are making a whole bunch of bets to get a few successes.  This is a kind of diversification that seems to promote innovation and maybe without this diversification there would be less investment in start-ups.

Karnen to Respondent 4:

Yes, I heard that 10% success rate from one of the articles or books about start-ups, this is why the expected [required} rate for start-up or any seed stage of entrepreneurship is quite high, even reaches 50%. However, this makes finance sense.

About the diversification, I guess, if the company’s owners could do the diversification by themselves, then logically (at least from MPT), no point for the company to do the diversification at its project level. If the owners do not have such opportunity for diversification, for instance, many privately owned corporation, then diversification at project level might make sense. Having said that, it doesn’t mean that diversification at project level is not needed, it is still necessary to  reduce the risk level, leading to higher leverage capacity for the company overall, in this case, the company could assume higher debt-to-equity ratio.

Another thing about this diversification at investor level vs project level, I guess, related to the size of the money being invested and agency issue.

Let’s say we flip a coin 100 times for a dollar bet each time. Investor might accept this. However at project level, the management of the company might not have such thing, they might flip a coin once for $100, so either they get that money back or lose it all (note: this assumed bell-shape probability distribution, though in reality, we should have this distribution is skewed to the right).

Agency issue also comes up since disproportionate share of their potential future (compensation) is tied up to that project’s success as well.

What I am trying to say, MPT cannot be brought to project level diversification discussion. Stock portfolio and project is again two different stuffs.

Respondent 5 to Karnen:

I disagree. An investment is an investment. The idea of diversification and optimization is what is important. Stocks were used in the chapter simply because they are familiar and we have relatively continuous data on returns which makes it easy to measure risk. Would you argue that a property insurance company should concentrate all of its policies in one small geographic area? I wouldn’t as they could be bankrupted by one natural disaster in that area. Geographic diversification seems like a good idea. Just a couple of weeks ago I saw an academic paper using Markowitz’s ideas in marketing. Years ago, I recall seeing an application in information technology projects. A quick Google search turned up this article:

You might also do a search on “project portfolio optimization,” and keep in mind that it is the general principal, not the exact details that matter.


Neil Steiz and Mitch Ellison (Capital Budgeting and Long-Term Financing Decisions. 3rd Edition. The Dryden Press. 1999. Page 417) said that the problems encountered by managers in applying mean-variance MPT include the following:

  • Indivisibility of Assets

Full mean-variance portfolio analysis is based on the assumption that securities are infinitely divisible, while capital investments often come in very large, indivisible units…. Fortunately, the lack of divisibility can be handled with integer quadratic programming, which forces the selection of only whole units of specified investments. Constraints can also be used to limit the choice between zero and one unit.

  • Holding Period Choice

We could use mean-variance analysis to find the portfolio of capital investments that has the highest net present value for each level of risk, where risk is measured as the standard deviation of the net present value. We could do the same thing with the profitability index or the internal rate of return. Unfortunately, though, the efficient frontiers that result are difficult to interpret because asset lives are not all identical. (Note from Karnen: MPT is about stock that has not limited life. The stock of that company could be delisted, yet, the company itself might still exist, via merger, acquisition, etc., unless it goes bankrupt or be liquidated by the owners)….To avoid ambiguity caused by unequal lives, we usually measure return over some holding period. One alternative is to use a long holding period, such as the life of the longest-life asset being considered. To measure portfolio risk and return over this holding period, though, it is necessary to identify all assets that will be selected between now and the end of the holding period. In addition, there is no date at which the lives of all assets to be acquired both now and later will end, unless the company plans to liquidate. Therefore, some terminal value estimates will be required as well. Beyond these difficulties of application, there is a serious conceptual problem. Risk analysis based on the results over a multiyear holding period does not give a complete picture of risk because it ignores fluctuations during the holding period. Problems like the risk of not being able to meet financial obligations during a bad year are missed when a long holding period is used, so the long holding period gives an incomplete picture of risk. …..These problems are avoided by using a short holding period, such as a year, and using the present value of remaining cash flows as the value of each asset at the end of the holding period. Present value of remaining cash flows would be used instead of market value because many profitable capital investments cannot really be sold after such a short holding period, without suffering a large loss. Unfortunately, the lack of opportunity to sell an asset for the present value of tis cash flows limits our ability to modify the portfolio at the end of the holding period. This disadvantage is not assigned a specific value by the model, so the short holding period also gives an incomplete picture of risk and return.

  • Defining the Borrowing-Lending Rate

[Karnen: Once a risk-free asset is introduced and assuming that investors can borrow and lend at the risk-free rate , the conclusion of Markowitz MPT,  and every combination of
the risk-free asset and the Markowitz efficient portfolio M is shown on the capital market line (CML) below.

Source: Equity Portfolio Management. Frank J. Fabozzi and James L. Grant. 2001. Wiley. Page 27.

The interest rate depends on the maturity of the debt, and the only risk-free interest rate for a holding period is that for debt with a maturity equal to the holding period. For a long holding period, the rate on long-term bonds can be used. For a short holding period, a short-term rate would be appropriate. However, the possibility of entering into long-term loan agreements must be recognized, even if a short holding is used for the analysis. One way to recognize the long-term borrowing is to treat long-term debt as a risky asset in which a negative position can be taken. The long-term debt is risky over a short holding period because the terminal value of that asset will depend on the interest rate prevailing at the end of the holding period.

  • Data Needs

Another problem with the use of mean-variance portfolio analysis for capital budgeting is the tremendous amount of data required. Covariance estimates are needed for each asset in relation to every other asset. The number of covariance terms needed (excluding variances which are covariances of an asset with itself) is 0.5N^2 – 0.5N where N is the number of assets being considered. Including expected return and standard deviation for each asset, the total number of terms needed is 0.5N^2+1.5N. If 1,000 capital investments are being considered, 501,500 risk and return measures must be estimated. The estimation is made more difficult by the fact that covariances and correlation coefficients are not the types of measures for which the average manager will be prepared to make judgmental estimates.


Most of the problems mentioned above are related to the assumptions of MPT.

MPT is built from the perspective of well-diversified shareholders. If shareholders are well diversified in their own stock portfolios, then following MPT, their main concern is only about that asset’s contribution to non-diversifiable or systematic risk for a broadly based portfolio, and in this case mean-variance analysis of the company’s portfolio is not needed for that purpose.

Applying MPT to capital budgeting or project level, this will mean that we analyze the risk from the perspective of the company and its manager.

Respondent 3:

The concept of diversification might be applied to everything (projects and stocks) BUT the mathematical concept and development of it, by Harry Markowitz, is for stocks.

No debate on this.

Karnen to Respondent 3:

We do know that MPT can’t be or not that easy to be applied to corporate or project level.

That’s one thing.

Another thing is whether there is benefit (read : Value Creation) for the company doing this diversification at corporate level or project level? By having low correlation among the projects being taken up, this will reduce the risk. However, will the risk reduction lead to the value creation? that’s the question. This is a slippery question. According to Corporate Finance fundamentals, such risk reduction doesn’t give rise to value creation, for example, in the merger and acquisition cases.

Respondent 3:

Not sure…

If the investor is rational, he/she will perceive that projects have different risks in different projects and might/should change his/her expectations of returns accordingly….

That might be a good subject for research.

Respondent 6:

Diversifying into new business a firm may mitigate the volatility of its aggregate cash flow. On the other hand, it exposes itself to all risks of doing business where it’s competences are not strong enough ending up with a loss in cash flows and value destruction. There are lost of examples of inefficient conglomerates moving to restructuring  and refocusing on the core.

(to be continued with more arguments to say that diversification at the project level could reduce risk but create no VALUE to investors that could diversify by themselves.)

IRR (Internal Rate of Return) Rule and Mutually Exclusive Investments : TIMING of CASH FLOWS issue

One of the problems with using IRR for making decision to select which project or investment among two mutually exclusive projects has a better financial feasibility is faced when two projects have different cash flow patterns, or referred to as the cash flow timing issue. This timing issue will also give rise to the conflicting results of NPV and IRR, which might lead potentially to the ranking problem.

Example 1

In the very extreme example, let’s say we have two mutually exclusive Project A (short-term) and Project B (long-term) with the timing of the cash flows displayed below.

Both projects have the SAME IRR, that is 50%, however, NPV of both projects have an extreme different gulf, which long-term project have 9.3x higher NPV compared to short-term project, under the assumption of 10% discount rate.

If we look carefully long-term project, then we see that that project doesn’t bring any cash flows in year 1 to year 4, but that project has HIGHER long-run cash flows.

Additionally, even if we pocket 150 in year 1 from short-term project, net of the investment being made -100, net of 50 and compounding that at IRR 50% a year, this will only bring 50 * (1+50%)^4 = 253, which is only 1/3 of the cash flow generated by the long-term project.

Using the above simple example, we could see that using IRR singly (without comparing it with NPV) might potentially lead to less-than-optimal decisions. In this case, NPV calculation should come first.

We might challenge the above example, since we are comparing two projects that are not apple-to-apple, meaning that those two projects do not have the same or equal project duration, one is 1 year and another one is 5 years.

However, this problematic IRR with the cash flow timing will still persist even we compare two projects with the same duration. Let’s move to Example 3 which might give more apple-to-apple comparison.

Example 2

Let’s say we have two mutually-exclusive projects with 5-year duration, with the pattern of the cash flow timing, the calculation of NPV and IRR as depicted below. The discount rate for both projects are 10%.

Here we see that we have conflicting results between NPV and IRR.

  • Using NPV as the main decision criteria, Project A will be chosen.
  • Using IRR as the main decision criteria, Project B will be chosen.

This conflicting results come from the fact that the cash flow pattern or timing is different between Project A and Project B.

Let me show the differential cash flow between Project A and Project B, and calculate its NPV and IRR as well, as demostrated below.

Then we can see here that the differential cash flows will give us POSITIVE NPV of 42.32 and IRR of 12.48%.

Since the differential cash flow has POSITIVE NPV, then selecting Project A over Project B will give us the optimal decision. In other words, the cash flow pattern of Project A is equivalent to Project B cash flow PLUS the differential cash flow.

By rejecting Project A, and selecting Project B, then it means we have rejected the POSITIVE NPV generated by the differential cash flows.

If we reflect the NPV profile of Project A and Project B, then we could see that as long as the discount rate is lower than IRR of the Differential Cash Flow (= 12.48%) then Project A will have higher NPV over Project B. However, if the discount rate is higher than 12.48% then Project B will have higher NPV over Project A, or this case, Project B will be preferred over Project A.

The above slope of NPV profile of Project A and Project B gives us another insight. Project A’s NPV profile has steeper slope compared to that of Project B, which means that when the discount rate is creeping northwest, then the NPV of the Project A will drop quicker than that of Project B. This comes as no surprise, since most of the cash flows from Project A comes later than that of Project B, in other words, those higher long-run cash flows will get bigger “punished” by higher discount rate.

Here, I show again the NPV and IRR if we change the discount rate to be the IRR of the Differential Cash Flows, that is 12.48%, and here we can find that at the cross-over of 12.48%, then the NPV of the Project A will be the same with that of Project B. Yet, IRR of Project B will be higher and thus, will be preferred over Project A.

In a nutshell, as long as the discount rate is 12.48% or higher, then Project B will be selected over Project A.

Source: Financial Analysis with Microsoft Excel. 7th Edition. Timothy R. Mayes and Todd M. Shank. Cengage Learning. 2015.

Thoughts on Corporate Valuation : Which One to Use?


Trying to share my thoughts on Corporate Valuation:

The first and foremost equation:

FCF + Tax Shield (TS) = CFD + CFE = CCF (Capital Cash Flows)

FCF = Free Cash Flows (=unlevered cash flows)
CFD = Cash Flows accruing to Debt holders
CFE = Cash Flows accruing to Equity holders

From my experience, tackling the valuation from the CASH FLOWS perspective, is easier to understand. Cash flow is the simple stuff to explain away. How much money will you get at the end of the day…I guess, this is the big concern, before bringing the topic of discount rate, or either the company will finance the project with debt (with its attendant TS) or not (just removed out the TS from the above equation).

Once the equation is brought into the audience, usually, there will be a question about what is Tax Shield cash flows. Then it needs to explain away that the Tax Shield will only show up when the debt is being used to finance a project or a business, and a big note that TS will only have a value if the company’s business has enough EBIT to cover it, or if the fiscal loss could be carried forward (or carried backward like in US tax system).

The next question, who is going to receive the Tax Shield Cash Flow? Equity holders!

Upon discussing the Cash Flow, then it is important to know whether the cash flow is flowing to the FIRM, or the cash flow to the debt holder and/or equityholder. Under the GAAP Statement of Cash Flows, it is cash flows to the Firm, though we could still identify, which cash flows accruing to debtholders and equityholders.

Then, I am always back to the above equation:  FCF   + TS = CFD + CFE

The whole point about the Valuation and Financing is about how the analyst to take account of the tax benefits of the interest deduction from the debt financing. Which we might put under the Cash Flow or we might put under the Discount Rate. I guess, this is what your slide shown on page 11.

I don’t think the FTE (Flow to Equity) method is even comparable to WACC and APV (Adjusted Present Value). FTE is only talking about CFE, however WACC and APV is about FCF + TS or CFD + CFE.

FTE Method is not much used if we are talking about corporate finance, it is mostly spoken in terms of project finance (with or without debt recourse).

WACC and APV is more relevant for corporate finance discussion.

The next question, then what is the difference between WACC and APV.

WACC will treat the tax benefits of interest deduction in the discount rate, by lowering the discount rate, and have higher value, accordingly, as we deal with FCF ONLY = CFD + (CFE – TS)

However, this is easier said than done, since there is a STRONG assumptions by lumping the tax benefits into the WACC formula. First, we assume that the tax benefits will be realized in the year it is incurred (the company has enough EBIT to absorb it or loss carryforward is possible to do and there is enough future EBIT to cover that loss carry forward). Second, the leverage is assumed constant over the life of the cash flow profile, which might be challenging for growth company. Third, classic circularity issue. Fourth, the formula of traditional WACC, though intuitive, but misleading.

Traditional WACC = Kd (1-Tax) Debt/Value + Ke Equity/Value

The correct one:

WACC = Kd Debt/Value + (Ke Equity/Value – Kd x Tax x Debt/Value); or

After_tax WACC = Before_tax WACC – (Kd x Tax x Debt/Value)

The last term in the right hand of equation is the tax benefits from debt financing and this cash flows GOING to the equityholder, and not the debt holder.

APV, as introduced by Steward Myers (1974), then will take care of tax shield separately from WACC, and address it directly as part of the cash flows, which means, then we don’t have stand-alone FCF anymore, but needs to have the Value of the project (FCF) plus the value of TS from the project financing.

The last one, which is most easier one to use is CCF by lumping FCF altogether with TS and discount it with Ku (as suggested by Richard S. Ruback, 1995 and revised in 2000, see, being accessed on 17 August 2020).

With CCF, we don’t have issue with the following:

(i) classic circularity and

(ii) no need to calculate Ke (Cost of levered equity, which I believe nobody could come up with a good guess about Ke) and

(iii) no need to come up with the value of Tax Shield (the big problematic questions that nobody could give satisfying answer up to now, the question that we need to address separately under Adjusted Present Value method).

Showing FCF + TS = CCF = CFD + CFE, in my opinion, is very important to readers.

There is equally important to show (which much not found in many good textbook when they discussed about NPV)

NPV (Owners or Equity holders) = NPV (Project) + NPV (Financing), which is under Perfect Markets assumption => NPV (Financing) = ZERO.

Respondent 1 to Karnen: 

However, since in most cases you assume that the finance institution is sophisticated and that the financial industry is a competitive one (i.e. perfect markets), we usually get that NPV(financing) is zero.

Respondent 2 to Karnen:

When I read the last sentence and earlier thoughts you have, I think at the inside, you respect Miller and nobody else.  I really believe finance is where medicine was before Louis Pasteur or where physics was before Faraday. The establishment believes silly things.  I really like the way you study things — you study the existing theory carefully and now I hope you are seeing that a lot of it is not making sense.  This non-senses comes to some extent from using integral formulas rather than making proofs with simple examples.

Here is the way I think you should think about the debt shield.  Think about it as a government grant.  The interest deduction does not change the cost of funds for the debt holders.  Lets say debt holders get 5% as the interest rate.  They may have different personal tax rates on their funds.  That is the cost of debt.  This is the same for equity.  When you measure the cost of equity (most methods are absurd including the CAPM), you do not think about what kind of tax rate the investor pays; you measure the pre-tax cost of equity for the investors, not their after-tax proceeds.

So if the company gets any kind of gift from the government, customers or anybody else, the traditional thing to do is to use the formula:

EV (from FCF without accounting for tax gift and using cost of funds for debt holders)

Less: Debt
Plus: Cash
Add: Customer contributions
Add: GOVERNMENT GIFT — from tax deduction

Common Equity Value

The government gift occurs on an on-going basis so you cannot just add back the government gift as above.  When you work through the math you can prove what method works best.  I know that changing the amount of debt — including the reduction in debt from the gift — is different from changing the interest rate as in the traditional WACC.  I think it is essential and easily possible to make a proof of what is the correct value.

My suggestion which is different from everything else is:

                    %                                     Cost                           Weighted Average
Debt         Debt x (1-t)/Total             Nominal Cost
Equity          Equity/Total                  Nominal Cost

June 2020 update: U.S. Normalized Risk-Free Rate to 2.5%

D&P lowers U.S. normalized risk-free rate to 2.5%

Duff & Phelps has decreased its recommended U.S. normalized risk-free rate from 3.0% to 2.5% for use as of June 30, 2020, according to

This new rate, used in conjunction with a (reaffirmed) recommended equity risk premium of 6.0%, implies a “base” U.S. cost of equity capital estimate of 8.5% (6.0% + 2.5%).

Personal note: Market risk premium of 6% is not really a big gap from what many analysts believe that 5% has been used in many sources as a reliable estimate of the MRP, based on historical data and forward-looking market data. Since that the recently outbreak of Covid-19 pandemic, has shifted upward the MRP to be 6% as mentioned by D&P above.

Quoted from Business Valuation Resources (BVR):

Most use spot yield: The concept of normalizing the risk-free rate emerged around the time of the 2008 financial crisis and is generally based on historical rates. A number of thought leaders disagree with the use of a normalized rate, including Professor Aswath Damodaran of the New York University Stern School of Business, who wrote that “you should be using today’s risk free rates and risk premiums, rather than normalized values, when valuing companies or making investment assessments.” In a BVWire survey from last year, most respondents said they use the 20-year spot yield on Treasury bonds for their risk-free rate. A quarter of respondents said they use the D&P normalized risk-free rate.