Let’s say one investor A after long negotiation with you as a founder of a early stage startup, offer you a TERM SHEET, which they will give you US$ 2 million for 10% equity in your startup.

This will mean that the POSTMONEY valuation is:

US$2mio/10% = US$ 20 mio

And the PREMONEY valuation is

US$ 20 mio – US$ 2 mio = US$ 18mio

The next question is how many shares that need to be issued to that investor?

Answering this question, will necessitate the founder to look at his/her Capitalization Table, or called CAP TABLE.

Cap Table will be pretty much a table showing all securities (common share, preferred share for each Series, Warrants, etc.) that you have issued so far, plus the reserve for Employee or Talent Option Pool

As we have known the Premoney valuation, then we could :

(1) Calculate the current share price, and

(2) Then dividing the Investor’s investment against the current share price to come up with the total number of shares being issued to that investor.

Let’s put the above into the Cap Table, as demonstrated below.

So, we could see from the above Cap Table, that founders need to issue 666,667 shares to Series A investors.

Here we noted that the Cap Table should consider the shares that will be issued as well under Employee/Talent Option Pool that will be issued later.

Some notes from the above Cap Table:

First, though it is Preferred Shares that are issued to Series A Investors, and not Common Shares, however in the Cap Table, such Preferred Shares will always be assumed as “as-converted basis”, which means that when determining the right or benefit of preferred stock, it is assumed that the Preferred Share has been converted into some number of common shares.

Second, the Employee or Talent Pool Option that have not been issued yet, will be incorporated into the Cap Table as “fully diluted” basis. Here, the Series A investors, want to know all parties that will have a claim on the startup exit value.

Why in the Cap Table we need to use “fully diluted” assumption and “as-converted basis”, from the perspective of Investor, regardless whether the shares have been issued or not yet, or whether the options has been executed or not?

First, it must be related to risk of the deal.

Rob Johnson, in his paper under the title : Valuing Early-Stage Business: The Venture Capital Method, April 2020) said that such assumptions are necessary in view of the risk of the deal.

He said:

What is important to understand is that the use of such instruments does not actual change the risk of the deal – the capital invested is still at risk; rather these instruments are used to achieve other objectives. The investor will have invested in preferred shares or debt (a) first and foremost to secure simultaneously an equity position in the company (b) while putting most of his/her capital in a senior instrument that achieves the other objective described above (Note: that is to ensure that their investment is in a senior position). For this reason, one must always use the total amount invested – irrespective of what instruments the capital is invested in – to calculate the post-financing and pre-financing valuations.

Second, it must be related to the Exit scenario.

Though risk is definitely one element of this deal, yet, another equally important element to include all instruments (such as non-vested and all non-issued options and shares), I believe it is because all those investors focus on the valuation at a successful exit (otherwise, what is the point to sit there and do such exercise?), and when that times come, logically, all these shares will certainly be issued, vested and valuable. Here Venture Capital method assumes that all equity classes will effectively have the equal claims on the company’s value, although their respective interests might typically have different rights and privileges, which might again translate into differences in exit proceeds per share unit. One thing, which I read, quite common in practice, upon exit (for example, IPO), the terms might require “qualified IPO” meaning that all outstanding preferred stock will be automatically converted to common stock. From the perspective of the buyer at exit, they want to remove all those interests with special privileges and rights.

Third, it must be related to investor’s protection of their interest.

Investor will manage to anticipated dilution in the next rounds of financing. Any unanticipated dilution is shared among all shareholders, whereas anticipated dilution is borne by premoney investors. Thus the insertion of a hypothetical future round is an attempt by current-round investors to hedge future dilution and shift it to the current round’s premoney investors. The same intuition applies to refreshing the incentive stock plan pool in a way that shifts the dilution from shared to premoney investors.

Anticipated (claimed) additional funding necessary to reach a fixed exit value is also that the insertion of an additional future round is actually the insertion of additional costs not currently in the business plan. This indirect insertion is due to using a financing-flows valuation rather than an operating flows valuation like that most common in corporate finance.

Jakarta, 9 January 2021
Rainy day

DERIVATION OF STOCK PRICE WITH NPVGO (Net Present Value of Growth Opportunities)

Dear Readers,

Please find my trying to get the well-known formula of

Stock Price equal to EPS/r + NPVGO

NPVGO, the one I see here is positive NPVGO, meaning that the company will only invest in the projects that will generate return on investment higher that its cost of capital.

If you have any comments or inputs about this formula derivation, feel free to share.

I have not yet finished up with this thinking.

This is only for educational purposes.

It comes up to me that we could solve this issue of its derivation, by having two equations that will lead us to the same results.

(1) Dividend Constant Growth Model =>
Stock Price = Dividends_1/(r – g)
Here we are discounting in growth perpetuity of Dividends (and not Earnings)

(2) Growth Opportunities (GO) Model ==>

Stock Price = Earnings_1/r + NPV of first year investment/(r-g)

Here we are discounting first without growth for Earnings, and then add it the NPV of first year investments discounted at cap rate of (r-g)

Though mathematically I could get that, what is the insight that we could take from the above 2 formula?

GO Model will make more sense since we could link it to the return on investment higher than cost of capital, or the other way around, if the NPV is negative.

Next, will show you how to have both formula mathematically be proven and the example, to see whether they are the same indeed.

So we could see from the above hand-written proof, that both Models, either we are discounting the DIVIDENDS WITH GROWTH, or EARNINGS WITHOUT GROWTH and then add on it the NPV of the GROWTH OPPORTUNITIES (discounted WITH GROWTH), then the results will be the same.

Respondend 1 to Karnen:

The growth opportunities model has the right concepts in mind, but is too vague on the metric to capture the NPV. That will be the (discounted) growth in earnings over the no-growth forecast expressed by Earnings/r. See Chapter 6 : Accrual Accounting and Valuation: Pricing Earnings of the textbook : Financial Statement Analysis and Security Valuation (5th Edition, McGraw-Hill) by Stephen Penman for the full expression.

Respondent 2 to Karnen:

Yes your derivation is correct.
As for insight, it is mainly used to assess the whether the stock price is mainly from current operations or future growth.

Respondent 3 to Karnen:

I think that your problem with this issue stems from the presentation of Gordon formula. The one you are using is ok but not the full one.

If you use the following one: PV = DIV_1/(r-g) = [EPS_1 * (1-b)]/[(r – b*ROE)]

Now, the only step needed is to plug in b =0 for the no growth (and get EPS1/r) and positive b for the growth value.

Karnen to Respondent 3

Noted for that formula, but not the one I am trying to solve.

I have been able to match the Dividends Constant Growth Model against NPV Growth Opportunities Model, under (i) growth situation and(2) return on investment is higher than cost of capital. I attach my (very bad) hand-written doodle to match both models.
The interesting insight during this exercise, I noted that that the cost of capital will be the function of growth rate, as growth is a risky thing, then the higher growth rate, then this will push up the cost of capital as well. The concept that growth is a risky one, came up to me upon reading the books by Prof. Stephen Penman (Financial Statement Analysis and Security Valuation) and Douglas R Emery, John D. Finnerty and John D. Stowe (Corporate Financial Management).

Respondent 4 to Karnen:

It is just a version of decomposing an observable stock price into COV (current operations value) and FGV (future growth value), which assumes COV being measured by capitalising current no growth EPS. Then p – COV gives an estimate of FGV. Typical approach with unrealistic simplifying assumptions regarding COV.



I just finished up reading Appendix 14-A Present Value of Risky Outflows of the textbook Capital Budgeting and Long-Term Financing Decisions by Neil Seitz and Mitch Ellison (Third Edition, Harcourt Brace College Publishers, 1999, page 470 – 473). That Appendix’s example makes a reference to Laurence D. Booth’s article: Correct Procedures for the Evaluation of Risky Cash Outflows, Journal of Financial and Quantitative Analysis 27 (June 1982), page 287-300. Prof. Laurence D. Booth is a name that will be familiar to many finance students (

I am a bit intrigued to read further that article since that Appendix said that the present value of a series of cash outflows is the same as the present value of a series of cash inflows with identical characteristics. This is interesting, since generally speaking, when we do the present value, we just jump to the NET cash flows, but this Appendix shows that underlying this general practice, it is assumed that both cash outflows and cash inflows have the same characteristics, though the book doesn’t really elaborate more about what it means with “having the same characteristics”.

The summary of that Appendix said:

Note that a positive beta on a cash outflow means that the outflow tends to move with revenues, and the variability of the outflow, therefore, decreases risk. A risk-averse decision maker would prefer variable outflows that decrease risk over a known cost with the same expected amount. The higher discount rate applied to risky outflows means a smaller present value of the outflow, which is consistent with the risky outflow being preferred over a riskless outflow.

Likewise, a cash outflow with a negative beta increases risk and, therefore, is less desirable than a riskless outflow. The low discount rate and resulting high net present value reflect the undesirability of the increased risk.

I will be using the example taken from Neil and Mitch’s demonstration of the validity and application of the basic principle of valuation for risky outflows is that the present value of a series of cash outflows is the same as that of the present value of a series of cash inflows with identical characteristics. The intuitive behind this, is that someone’s cash outflow will be another person’s cash inflow. In an equilibrium capital market, the present value of a series of outflows is the present value of those flows to the person receiving them. The above article by Prof. Laurence D. Booth provides a CAPM-based proof and a state-preference-based proof as well as a more general example.

Example 1


  • One-period capital investment (Note: this assumption rings familiar since one of the assumptions underlying the CAPM is that all investors make decisions for a single-period horizon and can revise their portfolios at the end of that horizon, albeit CAPM itself is silent about the length of the holding period. Generally speaking, it is treated as a short period. This assumption might be challenged when we are talking about capital investment, since such investment (i) will yield return over a number of years, and (ii) not always be marketable compared to stock portfolios, and (iii) even it is marketable, will the project owner be able to capture the market value of that projects?)
  • There is only two possible outcomes, State I and State II, each of which with a probability of 50%.
  • The current market value of the market portfolio is $100.
  • The cash flows for the capital investment is identical to the ending value of the market portfolio [Note: in this example, both market portfolio and cash inflows will have the same value of $100 and $130 under State I and State II.]
  • The risk free rate is 10%.

The table below shows the cash flows in each State for the investment and the market portfolio:

With the expected value of market portfolio at the end of the period is $115, then the expected return on the market portfolio is $115/$100 (current market value) – 1 = 15%.

Because the cash INflows from the capital investment are identical or the same to the ending market portfolio value, we could say that the equilibrium present value of the risky INflow stream will be equal to the value of the market portfolio, that is $100, its current market value.

Concerning NET cash flows, since both will have the same value either in State I and State II, that is $20, then it is risk free, and we could discount the NET cash stream using the risk free rate of 10%.

The present value (PV) of a risk-free of NET cash flows = $20/(1+10%) = $18.18.

With PV-out = equilibrium present value of cash OUTflows, then,

PV_out = PV_in (= current market value) – PV_net

PV_out = $100 – $18.18 = $81.82.

Once we have PV_out and the expected value of cash OUTflows, we could get the implied risk-adjusted discount rate (k_out) by discounting it:

PV_out = expected value_out/(1+k_out)

k_out = expected value_out/PV_out – 1 = $95/$81.82 – 1 = 16.11%

As we get 16.11%, we could now confirm that this will be the same risk-adjusted discount rate that would be applied to a series of cash INflows with the same characteristics. We could use CAPM to prove it.

For a series of cash OUTflows, under two possible end-of-period State I and State II, the returns on each State will be as follows:

State I = $80/$81.82 – 1 = -0.0222 (Minus sign), or -2.22%

State II = $110/$81.82 – 1 = +0.3444 (Positive sign), or 34.44%

The market portfolio return for each State will be as follows:

State I = $100/$100 – 1 = 0%

State II = $130/$100 – 1 = 30%

As the beta is the slope coefficient for the relationship between returns for some assets and return for the market portfolio, then the beta for the cash OUTflows is:

beta_out,m = [34.44% – (-2.22%)]/[30% – 0%] = 1.222 (Positive sign)

We now could calculate the required return using this beta_out,m and CAPM standard formulation, as follows:

K_out = Risk free + Beta_out,m (market portfolio return – risk free rate)

K_out = 10% + 1.222 (15% – 10%) = 16.11%

So we could see here that the equilibrium risk-adjusted discount rate is the same, regardless the cash flows is COSTs (for cash OUTflows) or Revenue stream (for cash INflows).

Under Example 1 here, the readers are requested to note that :

A positive beta on a cash OUTflow (in this case, +1.222), means that the OUTflows tend to move IN LINE with

  • the Revenues, in which from State I to State II, revenue is up from $100 to $130, and the cash OUTflows is also moving up from $80 to $110;
  • the variability of the cash OUTflows.

This IN-LINE movement, therefore, DECREASES the risk of the project.

A risk-averse decision maker, when faced with two options: (A) Variable cash OUTflows that decrease risks, or (B) A KNOWN cost with the same expected amount, might prefer Option A.

How do we know that Option A is preferable over Option B?

This is crystal clear if we compare the discount rate of Option A and Option B.

Option A have the discount rate of 16.11%, while Option B have the risk-free discount rate of 10%.

With higher discount rate being applied to the risky cash OUTflows, resulting to a lower PV of the cash OUTflows. The lower the cost (or cash OUTflows), this option will be preferred over a risk-less cash OUTflow.

Example 2


  • We will carry all assumptions from Example 1, but we just change now the direction of the cash OUTflows which will move in the opposite direction from the general market returns, which implicitly this will INCREASE the risk. So when the market returns are up from State 1 to State II, along with the cash INflows going up as well, yet, the cash OUTflows are going down.

Table below displays the cash flows in each State:

Here we need to solve jointly :

  • The present value of the NET cash flows (= PV_net);
  • The beta of the net cash flows (beta_net,m)
  • The required rate of return for the NET cash flows (k_net)

The calculation is as follows:

PV_net = Expected Value of NET cash flows / (1 + k_net) = $20/(1+k_net)

K_net = Risk free rate + beta_net,m (market return – risk free rate) = 5% + beta_net,m (15% – 10%)

Beta_net,m = [($50 – PV_net)/PV_net – (-$10 – PV_net)/PV_net] divided by (30% – 0%) = $200/PV_net

By substituting k_net and beta_net,m above to PV_net, then

PV_net = $20/(1 + (10% + (200/PV_net) x (15% – 10%))

PV_net = $9.09

Once we have the PV_net of $9.09, PV of cash OUTflows will be:

PV_out = PV_in – PV_net = $100 (its current market value) – $9.09 = $90.91.

The implied risk-adjusted discount rate of the cash OUTflows =

PV_out = expected amount / (1 + k_out)

k_out = expected amount/PV_out – 1 = $95/$90.91 – 1 = 4.5%

The beta of this cash OUTflows will be :

State I : ($110/$90.91) – 1 = +20.9988%

State II : ($80/$90.91) – 1 = – 12.0009%

Beta_out,m = (-12.0009% – 20.9988%)/(30% – 0%) = – 1.1 (Minus sign)

The CAPM required rate of return = risk free + beta_out,m (market return – risk free)

K_out = 10% + (-1.1) x (15% – 10%) = 4.5%

Again, we have here the risk-adjusted present value for the risky cash OUTflows that is the same present value for risky cash INflows with identical characteristics.

The readers are requested to note that under Example 2, which we have NEGATIVE Beta for the risky cash OUTflows, this has given rise to increase the risk. When the market is going down, from State II to State I, followed by the decline in the cash INflows, the cash OUTflows shows it has higher costs. This is of course, less desirable compared to a risk-less OUTflows.

When we compare the discount rate of 4.5% vs risk free rate of 10%, then with lower discount rate, this will result in higher net present value (= higher cost), reflecting the un-desirability of the increased risk.

One closing note on Example 1 and Example 2 is that if components of NET cash flows are discounted separately, the sum of the present values (for cash INflows and cash OUTflows) will equal the present value of the NET cash flows.

My understanding, that the discount rate for cash inflows and cash outflows, theoretically, could be different. Only if both shares the same risk factors, then we can use the same discount rate.

Under CAPM world, seems market risk, this single risk, is claimed to impact almost everything of the company’s cash flows, including INflows and OUTflows, and this has implicated that we should use one discount rate for both. When we are talking about cash outflows, this could occur horizontally or vertically. Horizontally, what I meant, is those cash OUTflows during the initial stages of the project, or in many cases, put as Io (Initial Investment) which could take years or months before the project will generate positive cashflows. The risk factors during this initial phases of the project could be said will be quite different from the risk factors impacting the later stages of the project life.

Additionally, cash OUTflows could occur vertically, that is during the period when the project has booked cash INflows, and the project will have costs of manufacturing, production, buying inventory, labor cost, overhead cost, marketing cost, project administration, etc. It is possible that the risk factors around this vertical cash OUTflows will be different from those leading to the cash INflows (which might more market oriented).

This will be logically bringing us to be able:

1. using different discount rate for cash INflows and vertical cash OUTflows

2. using different discount rate for horizontal cash OUTflows (early stage or seed phase of the project)

Respondent 1:

I agree to the general idea that the PV of cash inflow = cash outflow. However, there is an underlying assumption and that is same risk for inflows and outflows.

However, in reality capital outflow will have my risk rating and capital inflow has the payer’s risk rating (I would rather get an inflow from the US government than from a guy at the street). This should be reflected by the risk factors, though.

To make things short: I agree to the statement GIVEN same risk factors, however I do not agree that the risk factors ARE the same.

Karnen to Respondent 1:

In the practical leve, it is not that easy, to say that the risk factors impacting cash inflows and cash outflows are not the same, and this will require different discount rate.

In many cases, the analysts just assume away both cash flows (out and in) are having the same risk factors, and this is why, the NPV formula is applied to the NET cash flows, using one single discount rate.

Respondent 1 to Karnen:

Agree. That is my assumption as well.

Respondent 2:

Well, I would say that there are some generalizations that oversimplify the situation.

They say that an outflow has an equivalent inflow to other person. Yes, in general, but inflows and outflows usually are composed of outflows and inflows of MANY other people. 

For me, a cashflow (in or out) poses a given risk as perceived by me, the investor. This is the risk that is relevant: the ones I perceive. What other parties perceive about the risk of the contrary sign cashflow, is irrelevant to me.

Respondent 3:

A couple of marginal notes.

“a positive beta on a cash outflow means that the outflow tends to move with revenues”. This sounds counterintuitive. Generally, a positive beta means that an asset’s returns (let them be outflows) tend to move with the MARKET. Perhaps, the “beta” should be replaced with “correlation”, shouldn’t it?

“when we do the present value, we just jump to the NET cash flows, but this Appendix shows that underlying this general practice, it is assumed that both cash outflows and cash inflows have the same characteristics” Not sure that the latter statement is correct. We discount an expected net cash flow (a mix of inflows and outflows) at a rate corresponding to its risk, and that does not mean that we assume all components of the net cash flow have the same characteristics. Formally one obtains the same result when splitting the cash flow into parts and discounting them all at, say, the project’s (firm’s) cost of capital, but, in fact, according to corporate finance fundamentals each component should be discounted at it’s specific discount rate, and the weighted average if these rates aggregates into the discount rate  applied to the net cash flow.

Respondent 3:

Yes, that is an interesting issue that you mention and it’s something
I’ve worked out my own explanation for based on microeconomic theory.

Respondent 4:

I totally agree with this.

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.)