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.

A Short Note on Key Drivers behind Price-Earnings (PE) Ratio at Terminal Period

Hi Readers,

PE ratio (or earnings-multiple) is very common being used in the valuation (including in the start-up valuation) regardless many finance scholars will suggest the use of Discounted Cash Flow (DCF) approach, but as you probably already know, that building a full set of projected Financial Statements is easier said than done. Additionally, discounting projected cash flows is about:


I bet you know as well that EXPECTED is not the same with REALIZED Cash Flows.

For example, if you put the card no. 1, 2, 3, 4 and 5 into a box, and doing thousands of drawings from that box, the EXPECTED card will be :

the Mean = (1+2+3+4+5)/5 = 15/5 = 3

But your reality will be either 1, 2, 3, 4, or 5, so 3 is just of them.

Ok, back to PE ratio.

The hard time for getting the Price of a stock at the end of Terminal period, let’s put it _t (t= terminal).

We could use (i) Direct Comparison or (ii) Direct Capitalization approach.

I guess, the most important is to understand the key drivers or factors that will impact the PE ratio, which I am trying to hand-write it as depicted below.

The conclusion, the P/E at terminal period is other things being equal, the investors should logically pay more for a stock with

  • a higher potential growth (g); and
  • lower required rate of return (r), and
  • lower plowback ratio (b).

Discount rate for Tax Shield : Unanswered Question?


I would like to get your views on my thoughts about the discount rate for Tax Shield (TS). I know this is a classic discussion but important.

According to Taggart, Jr, R. A. (1991): Consistent Valuation Cost of Capital Expressions with Corporate and Personal Taxes. Financial Management, Autumn, pp. 8–20, the CORRECT formula to unlever and lever beta or cost of capital will really depend on the assumption of the Tax Shield discount rate.

I have checked Taggart’s formula being given in that paper, and I found that his paper and understanding is correct.

If that’s the case, then discount rate for Tax Shield is an important topic, though in many finance classes, this topic is not really being emphasized intensely.

Respondent 1 to Karnen:

The behavior of TS depends on EBIT. See (UO = EBIT, OI = Other Income, AI = TS)


As TS depends on EBIT shall we assume that the discount rate for TS should be Ku (cost of unlevered equity)?

On the other hand, if we see the Cash Flow conservation equation, we have




Note :

FCF = Free Cash Flows

TS = Tax Shield (flow)
CFE = Cash Flow to Equityholders
CFD = Cash Flow to Debtholders

Clearly the TS is “owned” by the equity holders. Hence, the Discount Rate for TS should be Ke (cost of levered equity)?

When you assume Ke as a discount rate for TS you might be able to obtain an optimal leverage and optimal VTS (Value of Tax Shield). That doesn’t happen with Kd and Ku as Discount Rate for TS. (see paper written by Felipe Mejia-Pelaez, Ignacio Velez-Pareja and James W. Kolari (2011) : Optimal Capital Structure for Finite Cash Flows, downloadable from

The idea of rebalancing or keeping constant debt is unrealistic. You need debt when you have a deficit. That’s all. Show me any case that tries to keep debt at some level or another one that adjusts debt to keep D% constant… (and don’t forget he should have the value of firm first and apply D%)  etc.

Karnen to Respondent 1:

My current understanding in my financial modelling:

The discount rate for Tax Shield will depend whether in the financial model, we are going to use predetermined debt or debt will keep rebalancing. In other words, the value of the TS will depend how certain they will be in the future and again this will sit on our modelling on debt (predetermined or rebalancing).

If we follow predetermined debt modelling, Kd will – in my opinion – be used to discount the TS. There is no difference with Kd being  used to discount and to get the value of debt. There might be a challenge that even the debt is sure but TS is not sure since the EBIT is not always big enough to absorb all interest burden. To such challenge, I will respond by using banker’s hat: will the bank extend the company with a predetemined debt in the first place if they knew that the project would not have big enough EBIT in the future to ensure interest will be able to be serviced during the debt term? So in this case, there is at least a high probability for that company to have sufficient EBIT in the future during the debt term, otherwise the bank would not give the predetermined loan to that company.

Under the debt rebalancing modelling, the value of TS will logically tie to the project value in the future. Somehow this TS could be certain or not certain hinging upon the success rate of the project…which means tax shields somehow be affected by the business risk. Under this logic, I would say Ku could be used as a discount rate for TS.

However, this logic though sounds good, still not quite satisfactory to me.

Discount rate is all about Opportunity Cost of Capital (OCC), the alternative use of money and ultimately as the required rate of return.

Tax Shield is something previously from the portion that should belong to the government, however, via the tax regime privilege, the tax authority is willing to have the interest expense be deductible on calculating the company’s corporate tax, reducing their tax liability and this government’s portion then goes to shareholders.

So, in other words, Tax Shield, sounds like a “bonus” to the shareholders. The money is not coming from the shareholders, but the shareholders enjoy it.


How to relate this Tax Shield concept to the OCC, alternative use of money, if the money itself is not coming from the shareholder, but government’s portion? Can we “penalize” this Tax Shield using Kd (cost of debt), Ku (cost of unlevered equity), Ke (cost of levered equity) or in between?

About the debt, I don’t think we could use one model to fit all sizes. It is really dependent on the company, meaning:

1. Permanent debt = this is possible, if the company keeps rolling over the loan. The bank will also enjoy this, as they keep receiving interest. I have seen this practice in some companies. Mutual benefits for both parties.

2. On-off loan depends on the company’s needs for cash. This will refer as working capital loan. this is also I have seen in some companies.

3. Loan taken by linking it to the market value of the project. This one-time loan is taken at certain point of the project finance life, by linking it to the value of the project. It will be locked for example, max 80% to the project value.

Constant D% or target leverage sometimes come up in the Valuation Analysis, but I have no idea whether it is really applied in reality. The analyst might just follow finance textbook approach, that is using constant leverage ratio (called target ratio). Again, this in reality, I don’t think it will be applied in corporate life.

Let me put in the Decision Tree to help a better understandign about what I meant above.

Prof. PDM’s Comments:

It doesn’t matter if it is a bonus, or if it comes from shareholders.  The valuation is the same, and the discount rate should depend on risk.  The two extremes are r_d (no adjustment) and r_u (continuous adjustment) and reality is somewhere between the two.  I prefer to use r_u as the main example pedagogically, since firms that use leverage for the tax shield do tend to adjust it over time.

Tax shields should be discounted at rate r_u if leverage is adjusted continuously to a target level.  (Yes, r_u is presumed stable.)
Karnen’s responses to Prof. PDM:

Once we apply r_u ad the discount rate for tax shield, unlever and relever will be quite simple :

Ke_t = Ku_t + (Ku_t – Kd_t) D_t-1/V_t-1

The same above formula will be used both applied to:

(i) CCF = FCF + TS = CFD  + CFE; and
(i) FCF (without TS)

Combining r_u (or Ku) as discount rate for tax shield and Capital Cash Flows (as suggested by RS Ruback in his paper (2000) : Capital Cash Flows: A Simple Approach to Valuing Risky Cash Flows, downloadable from, then this is the simplest approach to valuation which the WACC will be compressed to Ku.

Other readings:

  • Valuing the Debt Tax Shield by Ian Cooper and Kjell G. Nyborg (2011), downloadable at
  • Corporate Income Taxes and the Cost of Capital : Revision by James W. Kolari and Ignacio Velez-Pareja (2013), downloadable from
  • Cost of Capital with Levered Cost of Equity as the Risk of Tax Shields by Joseph Tham, Ignacio Velez-Pareja, and James W. Kolari (2011), downloadable from