The basic DuPont model starts by decomposing return on assets into two parts, net income and asset turnover. The breakdown is as follows;

From there, a leverage factor is included to find return on equity;

This is the basic DuPont formula. It's simply the product of net income, asset turnover, and leverage. The breakdown can be useful for understanding the driving factors of ROE. It acts as a window into a company's business model and helps answer questions such as;

- Is a company's ROE high because of high profit margins, large amounts of leverage, efficient use of assets, or some proportion of all three?
- How do the ROE drivers of one company compare with those of another in that industry?

**Expansion for Retail Operations**

Fortunately, the Dupont breakdown can be expanded to allow for a more granular analysis of the company's operations. There are many possibilities depending on one's focus. The specific expansion presented below was derived by the author to analyze retail operations, and is one that works well for companies which hold inventory. The complete derivation can be found at the end of this article.

__Notes:__

- Many of the inputs used in this analysis are accrual accounting values and are subject to manipulation by the presenting company. The Dupont analysis does a poor job spotting accrual accounting manipulations, and should be used in conjunction with other cash flow analysis techniques.
- The above equation uses NET PPE to represent fixed capital invested. One can include off-balance sheet capital by replacing NET PPE with [GROSS PPE + CAPITALIZED OPERATING LEASES]. Additionally, ASSETS can be replaced by (ASSETS + COL) in the denominator of both (NET PPE/ASSETS) as well as the numerator of the leverage ratio, (ASSETS/EQUITY). This allows one to view the adjusted leverage ratio corrected for operating leases.

This ROE breakdown is designed to illuminate critical characteristics of a company's business operations. The equation is written in such a way to allow for quick analysis of popular metrics such as inventory turns or interest coverage. Each term represents an element in the operations story. In a retail environment, inventory is purchased from a supplier and then stocked on shelves in a layout and density designated by the company. The product is then marked up and turned. Each square foot of that retail space is supported by property plant and equipment. Once the inventory is sold, funds from the inventory markups are combined. Out of that pool, SG&A expenses are paid, then interest, then taxes. The return to shareholders is leveraged in this entire process by using assets that the equity holders do not own.

Since ROE is simply a product of all terms (excluding net PPE per sqft, in this case), one may have an initial instinct to simply maximize each term, thus increasing ROE. Unfortunately, there's no free lunch. Changing the conditions of one term tends to affect a neighboring term in an opposing manner. For instance, increasing the resale markup or inventory-per-sqft will also have the effect of decreasing inventory turns. The problem is not one of maximization, but optimization. The retailer must decide what type of model works best for that company. Is the business one of low margins and high volumes, or one of higher margins and lower volumes? Understanding ones market, and the company's place in it, will help management find the optimal solution to their unique ROE equation. It will also give analysts a deeper understanding of management's strategic approach.

]]>The basic DuPont model starts by decomposing return on assets into two parts, net income and asset turnover. The breakdown is as follows;

From there, a leverage factor is included to find return on equity;

This is the basic DuPont formula. It's simply the product of net income, asset turnover, and leverage. The breakdown can be useful for understanding the driving factors of ROE. It acts as a window into a company's business model and helps answer questions such as;

- Is a company's ROE high because of high profit margins, large amounts of leverage, efficient use of assets, or some proportion of all three?
- How do the ROE drivers of one company compare with those of another in that industry?

**Expansion for Retail Operations**

Fortunately, the Dupont breakdown can be expanded to allow for a more granular analysis of the company's operations. There are many possibilities depending on one's focus. The specific expansion presented below was derived by the author to analyze retail operations, and is one that works well for companies which hold inventory. The complete derivation can be found at the end of this article.

__Notes:__

- Many of the inputs used in this analysis are accrual accounting values and are subject to manipulation by the presenting company. The Dupont analysis does a poor job spotting accrual accounting manipulations, and should be used in conjunction with other cash flow analysis techniques.
- The above equation uses NET PPE to represent fixed capital invested. One can include off-balance sheet capital by replacing NET PPE with [GROSS PPE + CAPITALIZED OPERATING LEASES]. Additionally, ASSETS can be replaced by (ASSETS + COL) in the denominator of both (NET PPE/ASSETS) as well as the numerator of the leverage ratio, (ASSETS/EQUITY). This allows one to view the adjusted leverage ratio corrected for operating leases.

This ROE breakdown is designed to illuminate critical characteristics of a company's business operations. The equation is written in such a way to allow for quick analysis of popular metrics such as inventory turns or interest coverage. Each term represents an element in the operations story. In a retail environment, inventory is purchased from a supplier and then stocked on shelves in a layout and density designated by the company. The product is then marked up and turned. Each square foot of that retail space is supported by property plant and equipment. Once the inventory is sold, funds from the inventory markups are combined. Out of that pool, SG&A expenses are paid, then interest, then taxes. The return to shareholders is leveraged in this entire process by using assets that the equity holders do not own.

Since ROE is simply a product of all terms (excluding net PPE per sqft, in this case), one may have an initial instinct to simply maximize each term, thus increasing ROE. Unfortunately, there's no free lunch. Changing the conditions of one term tends to affect a neighboring term in an opposing manner. For instance, increasing the resale markup or inventory-per-sqft will also have the effect of decreasing inventory turns. The problem is not one of maximization, but optimization. The retailer must decide what type of model works best for that company. Is the business one of low margins and high volumes, or one of higher margins and lower volumes? Understanding ones market, and the company's place in it, will help management find the optimal solution to their unique ROE equation. It will also give analysts a deeper understanding of management's strategic approach.

]]>This model of reality, one where a nice result is somehow a mixture of luck and skill, is simple and intuitive. In my view, it's also incomplete. I want to propose an alternative framework for thinking about luck and skill, one that is more consistent with our current understanding of a probabilistic world.

__Luck__

Let's first define luck as follows;

A lucky event is one that is;

- Not likely to occur
- Favorable
- Actually does occur

This definition of luck is probabilistic. If the prior probability of any given event is rare, the outcome is favorable, and that event occurs, then you got lucky. Examples of this could include winning the lotto, surviving a dangerous cancer, or simply finding a good parking spot at the mall during the holidays. The level of luck involved is proportional to the rarity of the circumstance. Powerball winners are much much much luckier than a family with a short walk to the Macy's entrance.

What's important is that the event is actually rare. At a minimum, the odds of occurrence must be less than 50%. Favorable things happen to people all the time. If the prior probability was high for that event to occur in the first place, it would be a mistake to talk about luck. Unfortunately, this is exactly the error we tend to make in our common conversations. Talk to any new mother about their healthy baby and they will tell you how lucky they were. The religiously minded might even say they are "blessed". The statistical truth is that having a healthy baby in the U.S. is a much more likely result than not. If something good was likely to happen, and it did, then luck was not part of the picture. (BTW, You won't grow your facebook friend list by telling new moms that their kid is no big deal, statistically speaking.) In fact, it seems that there's a black hole in our vocabulary with respect to matters of normality. Maybe things that are likely to happen, and do happen, just make for a lousy story. In my view, this absence of conversation has a strange effect on the way we perceive our reality and the probabilities that govern it.

__Unluck__

On the other end of the luck spectrum is unluck. (We'll get to skill in a minute.) I'll define unluck as follows;

- Not likely to occur
favorable__Un__- Actually does occur

This is the same definition as before, with one difference. The outcome that occurs is unfavorable. Most of us can dream up unlucky situations at the drop of the hat. The news media knows we feed off of this fear. Plane crashes, murders, shark attacks, lighting strikes, terrorist bombings, you name it. That's not to say these things don't exist. They're just really rare, which makes the victims of one of these events very very unlucky.

Just as before, it's important to weed out all the events that are more likely than not to occur in the first place. Losing the pick six is NOT unlucky. It's almost a certainty that any given player will lose, regardless of one's lucky numbers, how the stars are aligned, or what "system" one uses to play. Losing the lotto is the default outcome, one that borders on the inevitable. It has nothing to do with being unlucky. It's simply the norm.

Luck and unluck are depicted graphically below.

As you can see, as the prior probability of an event decreases, the amount of luck (or unluck) increases exponentially, approaching infinity as p goes to 0. If the prior probability is above 50%, and the event happens, luck is nil.

__Skill__

Once luck and unluck are defined, it's a simple matter to describe skill. Skill is simply the ability to change the prior probabilities in a manner favorable to the "player". The odds of a hole-in-one on a par 3 might be 1-in-5,000 for your buddy Marty, but more like 1-in-900 for a skilled professional. The range of error (the distance from the hole) will also be smaller for the pro. It's the years of disciplined training that allowed for the shift in the professional's prior probabilities. If one can increase the odds of success relative to others by way of repetition, training, and practice, then that person possess skill. The greater the shift in probabilities relative to the average, the higher the skill level.

Notice also that some games are designed to be altogether immune to skill. That is to say, skill cannot affect the players' results over time. Roulette is a great example. It is impossible to be a skilled roulette player. Thankfully for those downing free vodka tonics at Harrah's, it's also impossible to be an *unskilled* roulette player. The expected return will always be the same for every player making a "number" bet. (This assumes the amount of "zero's" on the wheel stays constant.) As with every game of pure chance, each roll is totally random. No amount of strategy will ever change one's expected return, and the laws of statistics are always silent on the outcome of the next roll. The casinos know this, but the folks searching for patterns in the brightly lit spin history display do not.

__Talent__

This probabilistic model can be stretched even further. If skill is the ability to change ones prior probability of success, then maybe the *rate* of that change is simply talent. Youtube is filled with young virtuosos showing off their talents. From music, to chess, to cup stacking, it's clear that some people are just hard-wired to learn certain things faster than others. If you don't believe me, go watch a 13-year-old Magnus Carlsen play Garry Kasparov. It doesn't matter how long I live or how hard I try, I'll never sing like a twenty-something Edele, or arrest a crowd like a 17-year-old Malala. Their talents are undeniable, and almost certainly a function of their makeup. I'm not suggesting hard work isn't involved, but the speed of learning is what sets them apart. That's what makes raw talent.

This model of reality, one where a nice result is somehow a mixture of luck and skill, is simple and intuitive. In my view, it's also incomplete. I want to propose an alternative framework for thinking about luck and skill, one that is more consistent with our current understanding of a probabilistic world.

__Luck__

Let's first define luck as follows;

A lucky event is one that is;

- Not likely to occur
- Favorable
- Actually does occur

This definition of luck is probabilistic. If the prior probability of any given event is rare, the outcome is favorable, and that event occurs, then you got lucky. Examples of this could include winning the lotto, surviving a dangerous cancer, or simply finding a good parking spot at the mall during the holidays. The level of luck involved is proportional to the rarity of the circumstance. Powerball winners are much much much luckier than a family with a short walk to the Macy's entrance.

What's important is that the event is actually rare. At a minimum, the odds of occurrence must be less than 50%. Favorable things happen to people all the time. If the prior probability was high for that event to occur in the first place, it would be a mistake to talk about luck. Unfortunately, this is exactly the error we tend to make in our common conversations. Talk to any new mother about their healthy baby and they will tell you how lucky they were. The religiously minded might even say they are "blessed". The statistical truth is that having a healthy baby in the U.S. is a much more likely result than not. If something good was likely to happen, and it did, then luck was not part of the picture. (BTW, You won't grow your facebook friend list by telling new moms that their kid is no big deal, statistically speaking.) In fact, it seems that there's a black hole in our vocabulary with respect to matters of normality. Maybe things that are likely to happen, and do happen, just make for a lousy story. In my view, this absence of conversation has a strange effect on the way we perceive our reality and the probabilities that govern it.

__Unluck__

On the other end of the luck spectrum is unluck. (We'll get to skill in a minute.) I'll define unluck as follows;

- Not likely to occur
favorable__Un__- Actually does occur

This is the same definition as before, with one difference. The outcome that occurs is unfavorable. Most of us can dream up unlucky situations at the drop of the hat. The news media knows we feed off of this fear. Plane crashes, murders, shark attacks, lighting strikes, terrorist bombings, you name it. That's not to say these things don't exist. They're just really rare, which makes the victims of one of these events very very unlucky.

Just as before, it's important to weed out all the events that are more likely than not to occur in the first place. Losing the pick six is NOT unlucky. It's almost a certainty that any given player will lose, regardless of one's lucky numbers, how the stars are aligned, or what "system" one uses to play. Losing the lotto is the default outcome, one that borders on the inevitable. It has nothing to do with being unlucky. It's simply the norm.

Luck and unluck are depicted graphically below.

As you can see, as the prior probability of an event decreases, the amount of luck (or unluck) increases exponentially, approaching infinity as p goes to 0. If the prior probability is above 50%, and the event happens, luck is nil.

__Skill__

Once luck and unluck are defined, it's a simple matter to describe skill. Skill is simply the ability to change the prior probabilities in a manner favorable to the "player". The odds of a hole-in-one on a par 3 might be 1-in-5,000 for your buddy Marty, but more like 1-in-900 for a skilled professional. The range of error (the distance from the hole) will also be smaller for the pro. It's the years of disciplined training that allowed for the shift in the professional's prior probabilities. If one can increase the odds of success relative to others by way of repetition, training, and practice, then that person possess skill. The greater the shift in probabilities relative to the average, the higher the skill level.

Notice also that some games are designed to be altogether immune to skill. That is to say, skill cannot affect the players' results over time. Roulette is a great example. It is impossible to be a skilled roulette player. Thankfully for those downing free vodka tonics at Harrah's, it's also impossible to be an *unskilled* roulette player. The expected return will always be the same for every player making a "number" bet. (This assumes the amount of "zero's" on the wheel stays constant.) As with every game of pure chance, each roll is totally random. No amount of strategy will ever change one's expected return, and the laws of statistics are always silent on the outcome of the next roll. The casinos know this, but the folks searching for patterns in the brightly lit spin history display do not.

__Talent__

This probabilistic model can be stretched even further. If skill is the ability to change ones prior probability of success, then maybe the *rate* of that change is simply talent. Youtube is filled with young virtuosos showing off their talents. From music, to chess, to cup stacking, it's clear that some people are just hard-wired to learn certain things faster than others. If you don't believe me, go watch a 13-year-old Magnus Carlsen play Garry Kasparov. It doesn't matter how long I live or how hard I try, I'll never sing like a twenty-something Edele, or arrest a crowd like a 17-year-old Malala. Their talents are undeniable, and almost certainly a function of their makeup. I'm not suggesting hard work isn't involved, but the speed of learning is what sets them apart. That's what makes raw talent.

Now assume that a second security A2 can be purchased. Let's say A2 is in every way the same as A1, yielding a $10 annual payout that lasts forever, ** except** for one difference. If you purchase security A2, it also comes with an additional $100. The $100 bill is yours to keep, with no restrictions. What should a knowledgeable market participant pay for security A2? Easy. The fair value of security A2 is simply A1+$100, or $300.

Now let's take a look at the P/E of A2. Since you are buying the security for $300 and getting $10/year in earnings, the P/E is $300/$10, or **30x**. Notice this is a much __higher__ multiple than what was paid for A1. A simple relative comparison based *solely* on the P/E ratio would suggest that A2 is overpriced at 30x earnings relative to A1 at 20x. We know that isn't true, however, and both A1 and A2 are priced appropriately. So what's going on? P/E is supposed to allow for an apples-to-apples comparison. What went wrong?

In this example, the price of security A1 was based solely on the present value of the earnings stream. Thus, it was appropriate to say that one would be paying 20x the annual earnings. However, in the case of security A2, the price was a composite of both the present value of earning *and* the excess cash. Using an unadjusted P/E ratio to evaluate A2 just isn't appropriate because the resulting price is due to more than just earnings. This condition is depicted below;

As you can see, the excess cash causes a distortion in the P/E measure. For a P/E ratio to be comparable across the board, it should exclude the market value of excess capital. This means that any capital not required for maintaining or growing earnings must be deducted from the security's price. In the case of A2, backing out the $100 in excess capital results in an adjusted P/E of ($300-$100)/$10, or 20x. The adjusted value is now consistent with A1's P/E.

Adjusting P/E in this manner will have a bigger effect for companies that hoard cash. For instance, Apple's P/E looks much lower when adjusted for its excess capital, even after that capital has been discounted for repatriation taxes. This adjustment is also useful for market indices through different time periods. During the recession, many of the S&P 500 companies were holding cash they really didn't need. An adjusted P/E ratio would do a better job comparing market levels from time periods of higher excess capital with the market levels of more cash lean times. For those of you out there analyzing cash rich companies, you might be surprised by what you find. Good luck and happy investing.

]]>Now assume that a second security A2 can be purchased. Let's say A2 is in every way the same as A1, yielding a $10 annual payout that lasts forever, ** except** for one difference. If you purchase security A2, it also comes with an additional $100. The $100 bill is yours to keep, with no restrictions. What should a knowledgeable market participant pay for security A2? Easy. The fair value of security A2 is simply A1+$100, or $300.

Now let's take a look at the P/E of A2. Since you are buying the security for $300 and getting $10/year in earnings, the P/E is $300/$10, or **30x**. Notice this is a much __higher__ multiple than what was paid for A1. A simple relative comparison based *solely* on the P/E ratio would suggest that A2 is overpriced at 30x earnings relative to A1 at 20x. We know that isn't true, however, and both A1 and A2 are priced appropriately. So what's going on? P/E is supposed to allow for an apples-to-apples comparison. What went wrong?

In this example, the price of security A1 was based solely on the present value of the earnings stream. Thus, it was appropriate to say that one would be paying 20x the annual earnings. However, in the case of security A2, the price was a composite of both the present value of earning *and* the excess cash. Using an unadjusted P/E ratio to evaluate A2 just isn't appropriate because the resulting price is due to more than just earnings. This condition is depicted below;

As you can see, the excess cash causes a distortion in the P/E measure. For a P/E ratio to be comparable across the board, it should exclude the market value of excess capital. This means that any capital not required for maintaining or growing earnings must be deducted from the security's price. In the case of A2, backing out the $100 in excess capital results in an adjusted P/E of ($300-$100)/$10, or 20x. The adjusted value is now consistent with A1's P/E.

Adjusting P/E in this manner will have a bigger effect for companies that hoard cash. For instance, Apple's P/E looks much lower when adjusted for its excess capital, even after that capital has been discounted for repatriation taxes. This adjustment is also useful for market indices through different time periods. During the recession, many of the S&P 500 companies were holding cash they really didn't need. An adjusted P/E ratio would do a better job comparing market levels from time periods of higher excess capital with the market levels of more cash lean times. For those of you out there analyzing cash rich companies, you might be surprised by what you find. Good luck and happy investing.

]]>"Thus our first lesson: businesses logically are worth far more than net tangible assets when they can be expected to produce earnings on such assets considerably in excess of market rates of return. The capitalized value of the excess return is economic goodwill."

Buffett goes on to talk about the valuation of See's and the idea of what he calls economic goodwill. He defined economic goodwill as the amount in excess of book value that one should be willing to pay if a company is purchased at fair price. He said that since See's ROE was higher than the market rate, it should sell at a multiple of book value. On the other hand, if a business is earning rates of return lower than the market, then paying full price for a company's equity is most likely a mistake.

Unfortunately, the language in the appendix was a bit too vague. How much was "considerably in excess of"? How much would the valuation increase with increasing ROE? Was it a linear increase or was it exponential? How did this model square with the free cash flow models I have in my excel spreadsheets at the moment? To answer these questions, one needs equations. This article is meant to fill the missing mathematics gap left by that 1983 chairman's letter. I believe the following model is the framework for which Buffett's (and Munger's) thinking is built upon.

__Economic Goodwill__

In the 1983 letter, Buffett defined economic goodwill as "the capitalized value of the excess return". This idea is known today as the Residual Income Model. ^{1} The General Residual Income model is as follows;

**Eq.1** may look a bit daunting, but it's not that bad. It says that the value of a company is worth the company's book value, B_{0}, plus some additional value. The term (ROE-r) is the company's return in excess of the market (discount) rate. If you multiply it by last period's book value, B_{t-1}, you find the income earned by the company (in dollars) **above and beyond** what can be earned in a market index. This is the residual income, and it's the driver for economic goodwill.

The factor in the denominator of **Eq.1** will discount each chunk of residual income from now until eternity. (Picking a proper market/discount rate gets a little hairy, but the total return rate on the S&P 500 might be a good starting point.) The big sigma sign just means that each discounted residual income from next period (t=1), until eternity (t=∞), are added together. Thus, the entire term on the right side of book value B_{0} ** is** economic goodwill!!!

Here is Buffett's quote from before, but with some emphasis added…

"…businesses logically are worth far more than net tangible assets when they can be expected to produce earnings on such assets considerably

in excessof market rates of return.The capitalized value of the excess return is economic goodwill."

__No Excel Required__

As it turns out, the model is simply a variation of the dividend discount model, and is far more familiar to all of us than meets the eye. Like the dividend discount model, it can also be simplified into a single-stage discount model, assuming the discount rate and ROE stay constant.^{3} Here are the results of that derivation;

For those of you who are familiar with the single-stage dividend discount model, this will feel familiar. No spreadsheets are needed. The model suggests the same results found in the first equation. The value of a company will be equal to the company's book value plus the capitalized value of any additional earnings received in excess of a diversified market return.^{4} If a company really does have a "sustainable competitive advantage", it should be evident by a permanently elevated ROE rate. Under those conditions, the company is worth more than book value. Conversely, if the company's ROE is less than the market rate, a rational buyer should pay less than its book value.

__Owner Earnings__

I wanted to briefly touch on 'owner earnings'. Plenty has been written on the topic, so I won't reinvent the wheel. However, I would like to make a definitional distinction between owner earnings and free cash flow. The residual income model uses accounting earnings to find fair value. It could certainly be adjusted to use a cash flow measure instead. However, many free cash flow measures add back non-cash items while deducting __ all__ current capex. Often times no distinction is made between replacement capex and growth capex. Owner earnings is essentially a free cash flow measure that does not deduct the growth capex. The logic is simple. Although growth capex is not available to the shareholder, it is still earned. (It's just immediately turned around and reinvested by management.) Thus, growth capex should be counted, whether the capital is subsequently reinvested for growth, paid out to shareholders, or reinvested in shares of one's own company.

__Price-to-Book__

Using some algebra, one can divide both sides of the single-stage residual income model by B_{0}, and replace the stock value V_{0}, with price P_{0.} What results is a justified price-to-book ratio;

The equation finds the fair multiple of book value for the company. If ROE is greater than the market rate, P/B will be greater than 1. The following chart shows how the justified P/B changes with different growth rates. The discount rate in this case has been set to 10%.

**Capital Allocation**

If you watch popular business news, then you know that Berkshire has been questioned for years about its dividend policy. The answer has been consistent every year. No dividend. What's motivating that decision? Leaving taxes aside for now, I think **Eq.2** and **Eq.3** give us a good guide for the dividend/no dividend decision. As you can see from **Eq.3**, the growth rate is directly proportional to the retained earnings. Over time, lower dividends mean higher growth. When a higher growth rate is plugged into equation **Eq.2**, the denominator gets smaller. The results are as follows;

- Retaining earnings will increase company value
the company's ROE is higher than r.__if and only if__ - Conversely, if the company's ROE is less than r, (ROE-r) is negative and retaining earnings becomes
for shareholders.__worse__ - If ROE is equal to r, the effect is neutral.

The point is that the decision to retain earnings has a leveraging effect on economic goodwill. It only makes sense to hold capital when economic goodwill is positive. Companies earning subpar returns on equity while simultaneously retaining earnings for growth are most likely destroying shareholder value. (Maybe you can think of one or two companies guilty of this.) Things get worse if management is buying back overpriced stock and/or making foolish acquisitions. In the case of Berkshire, they will retain their earnings until their overall ROE drops near the rate of the general market, where ROE ≈ r. Until then, don't expect a dividend.

__Footnotes:__

- All of the content in this article comes from the CFA Level II reading "Residual Income Valuation" by Pinto, Henry, Robinson, Stowe. The residual income model is also known as the discounted abnormal earnings model and the Edwards-Bell-Ohlson model, after the names of researchers in this field.
- There are a number of assumptions in the model, so I recommend anyone using it to familiarize yourself with them. One critical assumption states that any changes in equity from period to period flow through the income statement. This so-called 'clean surplus' accounting does not hold when other comprehensive income is present. Interestingly, due to the large amount of unrealized capital gains, Berkshire Hathaway itself is one of these companies. Adjustments can be made to correct for the distortions of OCI.
- This assumes clean surplus accounting, a constant payout ratio, and a constant ROE.
- The simplified model only converges when r > g.

"Thus our first lesson: businesses logically are worth far more than net tangible assets when they can be expected to produce earnings on such assets considerably in excess of market rates of return. The capitalized value of the excess return is economic goodwill."

Buffett goes on to talk about the valuation of See's and the idea of what he calls economic goodwill. He defined economic goodwill as the amount in excess of book value that one should be willing to pay if a company is purchased at fair price. He said that since See's ROE was higher than the market rate, it should sell at a multiple of book value. On the other hand, if a business is earning rates of return lower than the market, then paying full price for a company's equity is most likely a mistake.

Unfortunately, the language in the appendix was a bit too vague. How much was "considerably in excess of"? How much would the valuation increase with increasing ROE? Was it a linear increase or was it exponential? How did this model square with the free cash flow models I have in my excel spreadsheets at the moment? To answer these questions, one needs equations. This article is meant to fill the missing mathematics gap left by that 1983 chairman's letter. I believe the following model is the framework for which Buffett's (and Munger's) thinking is built upon.

__Economic Goodwill__

In the 1983 letter, Buffett defined economic goodwill as "the capitalized value of the excess return". This idea is known today as the Residual Income Model. ^{1} The General Residual Income model is as follows;

**Eq.1** may look a bit daunting, but it's not that bad. It says that the value of a company is worth the company's book value, B_{0}, plus some additional value. The term (ROE-r) is the company's return in excess of the market (discount) rate. If you multiply it by last period's book value, B_{t-1}, you find the income earned by the company (in dollars) **above and beyond** what can be earned in a market index. This is the residual income, and it's the driver for economic goodwill.

The factor in the denominator of **Eq.1** will discount each chunk of residual income from now until eternity. (Picking a proper market/discount rate gets a little hairy, but the total return rate on the S&P 500 might be a good starting point.) The big sigma sign just means that each discounted residual income from next period (t=1), until eternity (t=∞), are added together. Thus, the entire term on the right side of book value B_{0} ** is** economic goodwill!!!

Here is Buffett's quote from before, but with some emphasis added…

"…businesses logically are worth far more than net tangible assets when they can be expected to produce earnings on such assets considerably

in excessof market rates of return.The capitalized value of the excess return is economic goodwill."

__No Excel Required__

As it turns out, the model is simply a variation of the dividend discount model, and is far more familiar to all of us than meets the eye. Like the dividend discount model, it can also be simplified into a single-stage discount model, assuming the discount rate and ROE stay constant.^{3} Here are the results of that derivation;

For those of you who are familiar with the single-stage dividend discount model, this will feel familiar. No spreadsheets are needed. The model suggests the same results found in the first equation. The value of a company will be equal to the company's book value plus the capitalized value of any additional earnings received in excess of a diversified market return.^{4} If a company really does have a "sustainable competitive advantage", it should be evident by a permanently elevated ROE rate. Under those conditions, the company is worth more than book value. Conversely, if the company's ROE is less than the market rate, a rational buyer should pay less than its book value.

__Owner Earnings__

I wanted to briefly touch on 'owner earnings'. Plenty has been written on the topic, so I won't reinvent the wheel. However, I would like to make a definitional distinction between owner earnings and free cash flow. The residual income model uses accounting earnings to find fair value. It could certainly be adjusted to use a cash flow measure instead. However, many free cash flow measures add back non-cash items while deducting __ all__ current capex. Often times no distinction is made between replacement capex and growth capex. Owner earnings is essentially a free cash flow measure that does not deduct the growth capex. The logic is simple. Although growth capex is not available to the shareholder, it is still earned. (It's just immediately turned around and reinvested by management.) Thus, growth capex should be counted, whether the capital is subsequently reinvested for growth, paid out to shareholders, or reinvested in shares of one's own company.

__Price-to-Book__

Using some algebra, one can divide both sides of the single-stage residual income model by B_{0}, and replace the stock value V_{0}, with price P_{0.} What results is a justified price-to-book ratio;

The equation finds the fair multiple of book value for the company. If ROE is greater than the market rate, P/B will be greater than 1. The following chart shows how the justified P/B changes with different growth rates. The discount rate in this case has been set to 10%.

**Capital Allocation**

If you watch popular business news, then you know that Berkshire has been questioned for years about its dividend policy. The answer has been consistent every year. No dividend. What's motivating that decision? Leaving taxes aside for now, I think **Eq.2** and **Eq.3** give us a good guide for the dividend/no dividend decision. As you can see from **Eq.3**, the growth rate is directly proportional to the retained earnings. Over time, lower dividends mean higher growth. When a higher growth rate is plugged into equation **Eq.2**, the denominator gets smaller. The results are as follows;

- Retaining earnings will increase company value
the company's ROE is higher than r.__if and only if__ - Conversely, if the company's ROE is less than r, (ROE-r) is negative and retaining earnings becomes
for shareholders.__worse__ - If ROE is equal to r, the effect is neutral.

The point is that the decision to retain earnings has a leveraging effect on economic goodwill. It only makes sense to hold capital when economic goodwill is positive. Companies earning subpar returns on equity while simultaneously retaining earnings for growth are most likely destroying shareholder value. (Maybe you can think of one or two companies guilty of this.) Things get worse if management is buying back overpriced stock and/or making foolish acquisitions. In the case of Berkshire, they will retain their earnings until their overall ROE drops near the rate of the general market, where ROE ≈ r. Until then, don't expect a dividend.

__Footnotes:__

- All of the content in this article comes from the CFA Level II reading "Residual Income Valuation" by Pinto, Henry, Robinson, Stowe. The residual income model is also known as the discounted abnormal earnings model and the Edwards-Bell-Ohlson model, after the names of researchers in this field.
- There are a number of assumptions in the model, so I recommend anyone using it to familiarize yourself with them. One critical assumption states that any changes in equity from period to period flow through the income statement. This so-called 'clean surplus' accounting does not hold when other comprehensive income is present. Interestingly, due to the large amount of unrealized capital gains, Berkshire Hathaway itself is one of these companies. Adjustments can be made to correct for the distortions of OCI.
- This assumes clean surplus accounting, a constant payout ratio, and a constant ROE.
- The simplified model only converges when r > g.

For decades banks have hired armies of MBAs and CFAs who have spent countless hours digging through SEC documents, talking with company management, and working tirelessly on DCF models that incorporate factors like company market share, gross margins, tax rates, earnings estimates, free cash flow, and everything else an analyst might use to estimate the current value or the expected future price of a security. Reports are published with price targets or buy and sell recommendations. A typical conclusion may well read something like…

"After careful study of company So-and-So's current and projected financial condition, market position, strategic alternatives, and previous execution track record, we believe that the current stock price of $15.00/share does not reflect the true value of the company. We have set a price target of $20.00/share for company So-And-So, with a time horizon of 9 months from today."

Yet at the same time on a different floor at the same bank, physics and applied science PhDs are busily pricing the options of these same stocks, with seemingly little regard for what their research analyst colleagues have to say about the company. These wizards of financial alchemy might care more about a security's historic or implied return volatility than the company's earnings or cash flow.

One can't help noticing just how divorced these two worlds appear. I find myself asking;

- If an analyst uses fundamental analysis to find a stock's value, and he/she expects the price to revert to this expected value in the future, why shouldn't the PhDs use that same analysis to calculate the option value?
- Why do options pricing models focus so acutely on consistency with other liquid assets in the market without ensuring consistency with the economic fundamentals of the underlying?
- If a research analyst has spent time forming an opinion about where the future price of a stock might be, and has an idea of how long it might take to get to that target, then why not use that information to price the option?
- Shouldn't risk be defined as the uncertainty in ones analysis, not how much stock prices tend to bounce around?
- Why must we assume that current market prices are correct in the first place?

Many modern option pricing models start by assuming that the current market price is correct. The binomial model for instance, sends you on a trip into the future (a bunch of futures actually), tabulating option payouts along the way by constructing the probabilities of potential parallel future realities. Oddly, these so-called "risk-neutral" probabilities envisioned by the model have more to do with ensuring consistency with current bond returns than with trying to predict future events. When the trip is over, the models come back to the present in a DeLorean with Marty McFly carrying an option price that's in every way consistent with today's stock price, current bond yields, and all possible future scenarios imagined.

Unfortunately, many models such as the binomial model share similar drawbacks. Mainly, the future realities projected are rigidly constrained by what is happening today or in the recent past. These expectations are hinged to current stock or interest rate levels, and assume them to be "correct." If they weren't, then armies of arbitragers would make a bunch of money until prices came back into line, so the story goes. The models also assume that risk is defined by stock price fluctuations, or the volatility of stock price returns. In other words, how vigorously any random day's return bounces around its mean return.

But what if you were certain that the market is __wrong__, and you want to make an unhedged bet to capitalize on it? For some time this idea was considered a heresy in the academic community. It's clear now that the academics had latched onto unfounded memes like the efficient market hypotheses for way too long, despite the indisputable track record of so many successful value investors. It takes a certain collective tenacity by the academic community to ignore the significance of writings like "Security Analysis" or "The Super Investors of Graham and Doddsville". I believe that this can only be justified by a fatal flaw in the constitutions of some of our oldest and most respected institutions. Ivory tower hubris, perhaps. Only now has the consensus finally started to drift toward ideas that have been common understanding since the days of Graham himself. Mainly that, markets can be very wrong, and a keen observer can profit from it.

__A Model for the Buy-Side__

This option pricing model is geared towards research analysts with a view of the future. As such, it starts from the future, and works its way back to the present. Not only does the model price an option based on ones assumptions of the future, but it also provides an implied current fair value of the underlying based on that future.

WARNING: Option critics would often profess that it's hard enough to value a stock, let alone get the timing right on an option. They would argue that since the expiration of an option acts like a fuse on a ticking bomb, options can be much more risky than being long or short the underlying. I believe the critics are absolutely right and their warnings should be heeded. Models such as these probably work best for long dated options, where prices have ample time to correct to ones expectations and the correction time window can be predicted with some level of certainty. If the timing aspect of your position is too uncertain, you're probably better off just buying the underlying.

To start, let's assume that, upon completion of extensive due diligence, the analyst __can__ predict with some level of certainty where the stock price will be on the expiration date of the option. Certainly, this is no easy task. It's never possible to predict the future with exacting precision. What we *can* attempt to do is form an expectation of the future by our knowledge of the present, then estimate how uncertain we are about it. If we can be honest with ourselves about our uncertainty, we can form a conservative probabilistic view of future events. Once a reasonable expectation of the future is established, it's a simple task to find the expected option payout and discount that back to the present.

One solution is to assume that the future price of a stock will fall within a range of equally probable prices, with the average of that range being the expected value. For instance, if a European option expires in a year from now and we expect the stock price to be $10.00 at expiration, we can make an assumption on the range of possible prices at that date based on our uncertainty with our own analysis and timing expectations. Let's define the range as $10.00 ± $5.00 and assume that all prices in the expected range are equally probable. We can also assume that the exercise price of the European option is $12.00. What we end up with is a probability density function that looks like Figure 1.

__Figure 1__-Probability Density Function

The function shows an equally likely chance that the future stock price will fall between $5.00 and $15.00, with an expected price of $10.00. The area under the curve must always sum to one, thus explaining all possible future events.

If we then multiply the probability density function, P(S_{T}), by the call (put) option pay0ff function *f*_{c}_{(}S_{T)} (*f*_{p}_{(}S_{T)}) at each interval , we get;

The results of our example are shown graphically in Figure 2.

*(click to enlarge)*

__Figure 2__ - Payoff Distribution of a Call and Put Option with Strike Price of $12.00.

To find the option value at expiration, simply integrate (SUM) the area under both the green and red curves from 0 to +∞ to get;

The expected payoff in our example yields;

c_{T}= $0.45

p_{T}= $2.45

Finally, we can discount the expected option values to the present using the current risk free rate.

In our example, the current European option values assuming a $12.00 strike price, a 3.0% risk free rate, and one year until expiration are as follows;

c_{0}= __$0.43__

p_{0}= __$2.38__

Notice that these values were found without knowledge of the current stock price. When pricing American options, one can simplify the problem by taking the higher of the European option value, or the difference between the current market price and the strike price. This assumption implies a smooth price convergence toward the expected future price over the time of the option.

__Honing the Model__

The previous example assumed that there were equal probabilities for each potential future outcome inside the range chosen. Since the probability density function represents the analyst's subjective or *a posteriori* probabilities, there are no restrictions on the *shape* of the function used in the model, as long as the total area under the curve sums to 1. Since the analyst typically expects a higher probability for a certain expected future value, and a lower probability for values farther from the expected value, a bell shaped curve may better reflect the analyst's expectations. Instead of equal probabilities, we can use a normal distribution, where the analyst's expected future stock price at expiration is equal to mean µ, and the uncertainty inherent in the analyst's valuation and timing expectations is equal to standard deviation σ. Notice here that σ has less to do with historic or implied return volatility, and more to do with uncertainty of future prices. The model assumptions are as follows;

- The analyst has an opinion about the current stock value.
- The analyst has formed an expectation of the future stock price at expiration.
- Stock prices tend to revert to the expected price over time.
- The analyst has an opinion about the uncertainty inherent in both the valuation process and the expected price target timing.
- The probabilities of future stock prices at option expiration follow one or more normal distributions, defined by the analyst's expected future stock price at expiration µ, and the uncertainty inherent in the analyst's valuation and timing expectations σ.
- The sum of all probabilities is equal to 1.

As in the previous example we have;

But in this case;

P(S_{T})= One or more normal distributions defined by the expected future stock price at expiration µ, and the uncertainty inherent in the analyst's valuation and timing expectations σ.

To proceed with our example, if we define the expected return µ=$10.00, with a standard deviation σ= ±$3.00, we get the following probability and payout distribution shown in Figure 3. (Remember here that σ represents our uncertainty of the future, not return volatility.)

*(click to enlarge)*

__Figure 3__- Expiration Payoff of European Option with $12.00 Strike Price and Expectation Defined By Normal DIstribution of µ =$10.00, and σ=$3.00.

When using the normal distribution, we can define a confidence interval to be used as a mental litmus test. If a 90% confidence interval is defined, the analyst would expect that 9 out of 10 expected future price scenarios would lie between µ ± 1.64 σ, or in our case between $5.07 and $14.93. With the future probabilities defined as they are, and a risk free rate of 3.0%, a one year European call and put option with a strike price of $12.00 is worth;

c_{0}= __$0.44__

p_{0}= __$2.37__

By varying the exercise price, a model generated option pricing table is produced as follows;

*(click to enlarge)*

(Here we assume a current stock price of $11.50 to determine American option values and ITM status.)

There are a number of observable differences between this model, which I'll call the expected value model, and others such as Black-Scholes-Merton. In the BSM model, call values increase and put values decrease as the risk free rate increases. This is due mainly to the expectation that the stock prices drift upward from the current price, allowing the holder of the underlying to earn a return that is at least equivalent to that of a risk free bond. The expected value model on the other hand starts at expiration and works back to the present. Thus, any return expectations of the underlying are already implicit in the probability density function defined by the analyst. If the analyst believes that a stock price will grow at the risk free rate (or more likely at the company's sustainable ROE in the event that no dividends are paid out), she will reflect that in her expectations of the future. The only explicit effect of increasing the risk free rate assumption is to decrease both call and put values as a result of the discounting process.

On the surface, it may also appear that the expected value model is much less sensitive to the time-to-expiration than BSM. As this time is varied, the risk free rate appears to be the only influencing variable affecting the option value, which may have relatively little impact when rates are low. However, a proper evaluation made by the analyst of the uncertainty inherent in greater lengths of time would likely result in the modeling of a larger expected standard deviation as time to expiration increases. This results in a larger confidence interval, and acts as an additional indirect factor affecting option value.

Finally there is volatility. Unlike the BSM, the expected value model cares nothing about the volatility of returns. The "σ" of the expected value model embodies the risks one inherits each time one tries to predict the future (without a DeLorian and a flux capacitor). The information available to us is imperfect, the past is only a so-so predictor of the future, and our understanding of markets is nothing like our understanding of our planetary orbits. We just can never know the future for certain, but we might be able to estimate the bounds of our uncertainty.

__Stretching the Model__

Occasions often arise where two distinct future outcomes are possible. This might happen following a merger announcement where the closing is less than certain, prior to a material litigation ruling, or before a critical earnings announcement. The expected return model can account for this by defining a probability density function with two separately defined probability curves. The following example will explore this concept.

__Example- Post Merger Announcement__

A tender offer has just been announced for the XYZ company at $15.00/share. It is being bought by a private equity firm with a checkered history of closing deals. After extensive due diligence, an analyst determines that the probability for the deal closing is about 70%. For simplicity, we can assume that the deal is expected to close on the option expiration date. He believes that it is reasonable to expect a $15.00 future price within a narrow band of expectations. However it is also expected that the stock price might fall to $9.00 if the deal falls through. The analyst is less certain about the magnitude of the price drop if this happens. He constructs the following probability distribution based on his expectation;

*(click to enlarge)*__Figure 4__ - XYZ Company Expected Probability Density Function

The distribution shows two curves, one with an expected price of $15.00 and standard deviation of $0.50, and the other with an expected price of $9.00 and standard deviation of $2.00. The area under the left curve is 7/3 times the area of the left, reflecting the analyst's expectation of the deal closing. The cumulative probability of the entire function must always equal one. If the analyst is interested in pricing a call and put option with a strike price of $15.00, the following payoff distribution will result;

*(click to enlarge)*

__Figure 5__ - XYZ Merger Payoff Distribution with $15.00 Strike Price

The model-generated European option values assuming 6 months until expiration, a 3% risk free rate, and a $15.00 strike price are as follows;

c_{0}= __$0.14__

p_{0}= __$1.91__

With the current price at $14.20, the model-generated pricing table is;

*(click to enlarge)*

__Put-Call Parity__

Put-Call parity states that a fiduciary call (C_{0} +PV(Exercise Price)) should remain equal to a protective put (S_{0}+P_{0}), or an arbitrage opportunity is possible. Typically it assumes that today's current market price S_{0} is as it should be, and the option prices must adjust to maintain the balance. However, the expected return model assumes that S_{0} may be incorrect. In our case, put-call parity is maintained if it assumed that one's expectation about the future is correct, and the *implied* spot price that results from this view is substituted for S_{0}. The implied spot price is simply the present value of the future expected stock price. Revisiting the previous example, the put-call parity equation is as follows;

If anyone is interested in the copy of Excel file, I expect to post a link in the comments sometime in the near future. Here is a screen shot of what it looks like.

*(click to enlarge)*

-Joe Gradzki

]]>For decades banks have hired armies of MBAs and CFAs who have spent countless hours digging through SEC documents, talking with company management, and working tirelessly on DCF models that incorporate factors like company market share, gross margins, tax rates, earnings estimates, free cash flow, and everything else an analyst might use to estimate the current value or the expected future price of a security. Reports are published with price targets or buy and sell recommendations. A typical conclusion may well read something like…

"After careful study of company So-and-So's current and projected financial condition, market position, strategic alternatives, and previous execution track record, we believe that the current stock price of $15.00/share does not reflect the true value of the company. We have set a price target of $20.00/share for company So-And-So, with a time horizon of 9 months from today."

Yet at the same time on a different floor at the same bank, physics and applied science PhDs are busily pricing the options of these same stocks, with seemingly little regard for what their research analyst colleagues have to say about the company. These wizards of financial alchemy might care more about a security's historic or implied return volatility than the company's earnings or cash flow.

One can't help noticing just how divorced these two worlds appear. I find myself asking;

- If an analyst uses fundamental analysis to find a stock's value, and he/she expects the price to revert to this expected value in the future, why shouldn't the PhDs use that same analysis to calculate the option value?
- Why do options pricing models focus so acutely on consistency with other liquid assets in the market without ensuring consistency with the economic fundamentals of the underlying?
- If a research analyst has spent time forming an opinion about where the future price of a stock might be, and has an idea of how long it might take to get to that target, then why not use that information to price the option?
- Shouldn't risk be defined as the uncertainty in ones analysis, not how much stock prices tend to bounce around?
- Why must we assume that current market prices are correct in the first place?

Many modern option pricing models start by assuming that the current market price is correct. The binomial model for instance, sends you on a trip into the future (a bunch of futures actually), tabulating option payouts along the way by constructing the probabilities of potential parallel future realities. Oddly, these so-called "risk-neutral" probabilities envisioned by the model have more to do with ensuring consistency with current bond returns than with trying to predict future events. When the trip is over, the models come back to the present in a DeLorean with Marty McFly carrying an option price that's in every way consistent with today's stock price, current bond yields, and all possible future scenarios imagined.

Unfortunately, many models such as the binomial model share similar drawbacks. Mainly, the future realities projected are rigidly constrained by what is happening today or in the recent past. These expectations are hinged to current stock or interest rate levels, and assume them to be "correct." If they weren't, then armies of arbitragers would make a bunch of money until prices came back into line, so the story goes. The models also assume that risk is defined by stock price fluctuations, or the volatility of stock price returns. In other words, how vigorously any random day's return bounces around its mean return.

But what if you were certain that the market is __wrong__, and you want to make an unhedged bet to capitalize on it? For some time this idea was considered a heresy in the academic community. It's clear now that the academics had latched onto unfounded memes like the efficient market hypotheses for way too long, despite the indisputable track record of so many successful value investors. It takes a certain collective tenacity by the academic community to ignore the significance of writings like "Security Analysis" or "The Super Investors of Graham and Doddsville". I believe that this can only be justified by a fatal flaw in the constitutions of some of our oldest and most respected institutions. Ivory tower hubris, perhaps. Only now has the consensus finally started to drift toward ideas that have been common understanding since the days of Graham himself. Mainly that, markets can be very wrong, and a keen observer can profit from it.

__A Model for the Buy-Side__

This option pricing model is geared towards research analysts with a view of the future. As such, it starts from the future, and works its way back to the present. Not only does the model price an option based on ones assumptions of the future, but it also provides an implied current fair value of the underlying based on that future.

WARNING: Option critics would often profess that it's hard enough to value a stock, let alone get the timing right on an option. They would argue that since the expiration of an option acts like a fuse on a ticking bomb, options can be much more risky than being long or short the underlying. I believe the critics are absolutely right and their warnings should be heeded. Models such as these probably work best for long dated options, where prices have ample time to correct to ones expectations and the correction time window can be predicted with some level of certainty. If the timing aspect of your position is too uncertain, you're probably better off just buying the underlying.

To start, let's assume that, upon completion of extensive due diligence, the analyst __can__ predict with some level of certainty where the stock price will be on the expiration date of the option. Certainly, this is no easy task. It's never possible to predict the future with exacting precision. What we *can* attempt to do is form an expectation of the future by our knowledge of the present, then estimate how uncertain we are about it. If we can be honest with ourselves about our uncertainty, we can form a conservative probabilistic view of future events. Once a reasonable expectation of the future is established, it's a simple task to find the expected option payout and discount that back to the present.

One solution is to assume that the future price of a stock will fall within a range of equally probable prices, with the average of that range being the expected value. For instance, if a European option expires in a year from now and we expect the stock price to be $10.00 at expiration, we can make an assumption on the range of possible prices at that date based on our uncertainty with our own analysis and timing expectations. Let's define the range as $10.00 ± $5.00 and assume that all prices in the expected range are equally probable. We can also assume that the exercise price of the European option is $12.00. What we end up with is a probability density function that looks like Figure 1.

__Figure 1__-Probability Density Function

The function shows an equally likely chance that the future stock price will fall between $5.00 and $15.00, with an expected price of $10.00. The area under the curve must always sum to one, thus explaining all possible future events.

If we then multiply the probability density function, P(S_{T}), by the call (put) option pay0ff function *f*_{c}_{(}S_{T)} (*f*_{p}_{(}S_{T)}) at each interval , we get;

The results of our example are shown graphically in Figure 2.

*(click to enlarge)*

__Figure 2__ - Payoff Distribution of a Call and Put Option with Strike Price of $12.00.

To find the option value at expiration, simply integrate (SUM) the area under both the green and red curves from 0 to +∞ to get;

The expected payoff in our example yields;

c_{T}= $0.45

p_{T}= $2.45

Finally, we can discount the expected option values to the present using the current risk free rate.

In our example, the current European option values assuming a $12.00 strike price, a 3.0% risk free rate, and one year until expiration are as follows;

c_{0}= __$0.43__

p_{0}= __$2.38__

Notice that these values were found without knowledge of the current stock price. When pricing American options, one can simplify the problem by taking the higher of the European option value, or the difference between the current market price and the strike price. This assumption implies a smooth price convergence toward the expected future price over the time of the option.

__Honing the Model__

The previous example assumed that there were equal probabilities for each potential future outcome inside the range chosen. Since the probability density function represents the analyst's subjective or *a posteriori* probabilities, there are no restrictions on the *shape* of the function used in the model, as long as the total area under the curve sums to 1. Since the analyst typically expects a higher probability for a certain expected future value, and a lower probability for values farther from the expected value, a bell shaped curve may better reflect the analyst's expectations. Instead of equal probabilities, we can use a normal distribution, where the analyst's expected future stock price at expiration is equal to mean µ, and the uncertainty inherent in the analyst's valuation and timing expectations is equal to standard deviation σ. Notice here that σ has less to do with historic or implied return volatility, and more to do with uncertainty of future prices. The model assumptions are as follows;

- The analyst has an opinion about the current stock value.
- The analyst has formed an expectation of the future stock price at expiration.
- Stock prices tend to revert to the expected price over time.
- The analyst has an opinion about the uncertainty inherent in both the valuation process and the expected price target timing.
- The probabilities of future stock prices at option expiration follow one or more normal distributions, defined by the analyst's expected future stock price at expiration µ, and the uncertainty inherent in the analyst's valuation and timing expectations σ.
- The sum of all probabilities is equal to 1.

As in the previous example we have;

But in this case;

P(S_{T})= One or more normal distributions defined by the expected future stock price at expiration µ, and the uncertainty inherent in the analyst's valuation and timing expectations σ.

To proceed with our example, if we define the expected return µ=$10.00, with a standard deviation σ= ±$3.00, we get the following probability and payout distribution shown in Figure 3. (Remember here that σ represents our uncertainty of the future, not return volatility.)

*(click to enlarge)*

__Figure 3__- Expiration Payoff of European Option with $12.00 Strike Price and Expectation Defined By Normal DIstribution of µ =$10.00, and σ=$3.00.

When using the normal distribution, we can define a confidence interval to be used as a mental litmus test. If a 90% confidence interval is defined, the analyst would expect that 9 out of 10 expected future price scenarios would lie between µ ± 1.64 σ, or in our case between $5.07 and $14.93. With the future probabilities defined as they are, and a risk free rate of 3.0%, a one year European call and put option with a strike price of $12.00 is worth;

c_{0}= __$0.44__

p_{0}= __$2.37__

By varying the exercise price, a model generated option pricing table is produced as follows;

*(click to enlarge)*

(Here we assume a current stock price of $11.50 to determine American option values and ITM status.)

There are a number of observable differences between this model, which I'll call the expected value model, and others such as Black-Scholes-Merton. In the BSM model, call values increase and put values decrease as the risk free rate increases. This is due mainly to the expectation that the stock prices drift upward from the current price, allowing the holder of the underlying to earn a return that is at least equivalent to that of a risk free bond. The expected value model on the other hand starts at expiration and works back to the present. Thus, any return expectations of the underlying are already implicit in the probability density function defined by the analyst. If the analyst believes that a stock price will grow at the risk free rate (or more likely at the company's sustainable ROE in the event that no dividends are paid out), she will reflect that in her expectations of the future. The only explicit effect of increasing the risk free rate assumption is to decrease both call and put values as a result of the discounting process.

On the surface, it may also appear that the expected value model is much less sensitive to the time-to-expiration than BSM. As this time is varied, the risk free rate appears to be the only influencing variable affecting the option value, which may have relatively little impact when rates are low. However, a proper evaluation made by the analyst of the uncertainty inherent in greater lengths of time would likely result in the modeling of a larger expected standard deviation as time to expiration increases. This results in a larger confidence interval, and acts as an additional indirect factor affecting option value.

Finally there is volatility. Unlike the BSM, the expected value model cares nothing about the volatility of returns. The "σ" of the expected value model embodies the risks one inherits each time one tries to predict the future (without a DeLorian and a flux capacitor). The information available to us is imperfect, the past is only a so-so predictor of the future, and our understanding of markets is nothing like our understanding of our planetary orbits. We just can never know the future for certain, but we might be able to estimate the bounds of our uncertainty.

__Stretching the Model__

Occasions often arise where two distinct future outcomes are possible. This might happen following a merger announcement where the closing is less than certain, prior to a material litigation ruling, or before a critical earnings announcement. The expected return model can account for this by defining a probability density function with two separately defined probability curves. The following example will explore this concept.

__Example- Post Merger Announcement__

A tender offer has just been announced for the XYZ company at $15.00/share. It is being bought by a private equity firm with a checkered history of closing deals. After extensive due diligence, an analyst determines that the probability for the deal closing is about 70%. For simplicity, we can assume that the deal is expected to close on the option expiration date. He believes that it is reasonable to expect a $15.00 future price within a narrow band of expectations. However it is also expected that the stock price might fall to $9.00 if the deal falls through. The analyst is less certain about the magnitude of the price drop if this happens. He constructs the following probability distribution based on his expectation;

*(click to enlarge)*__Figure 4__ - XYZ Company Expected Probability Density Function

The distribution shows two curves, one with an expected price of $15.00 and standard deviation of $0.50, and the other with an expected price of $9.00 and standard deviation of $2.00. The area under the left curve is 7/3 times the area of the left, reflecting the analyst's expectation of the deal closing. The cumulative probability of the entire function must always equal one. If the analyst is interested in pricing a call and put option with a strike price of $15.00, the following payoff distribution will result;

*(click to enlarge)*

__Figure 5__ - XYZ Merger Payoff Distribution with $15.00 Strike Price

The model-generated European option values assuming 6 months until expiration, a 3% risk free rate, and a $15.00 strike price are as follows;

c_{0}= __$0.14__

p_{0}= __$1.91__

With the current price at $14.20, the model-generated pricing table is;

*(click to enlarge)*

__Put-Call Parity__

Put-Call parity states that a fiduciary call (C_{0} +PV(Exercise Price)) should remain equal to a protective put (S_{0}+P_{0}), or an arbitrage opportunity is possible. Typically it assumes that today's current market price S_{0} is as it should be, and the option prices must adjust to maintain the balance. However, the expected return model assumes that S_{0} may be incorrect. In our case, put-call parity is maintained if it assumed that one's expectation about the future is correct, and the *implied* spot price that results from this view is substituted for S_{0}. The implied spot price is simply the present value of the future expected stock price. Revisiting the previous example, the put-call parity equation is as follows;

If anyone is interested in the copy of Excel file, I expect to post a link in the comments sometime in the near future. Here is a screen shot of what it looks like.

*(click to enlarge)*

-Joe Gradzki

]]>The mechanics of a share repurchase are straight forward. Typically, the management of a company uses excess cash on the company's balance sheet to purchase outstanding shares of its own company. Often times, the shares are bought directly from the secondary market at the prevailing market price. These shares are effectively taken out of circulation, creating a greater concentration of equity ownership in the hands of the remaining shareholders. Although it's natural to assume that this will benefit shareholders in all instances, it's not always the case. The following examples will illustrate why this is;

__Example1: Share Repurchases During a Period of Undervaluation__

Company XYZ is a hypothetical company with a total of only 5 common shares outstanding. This simple company does not own any property, plant, or equipment, doesn't provide a service, has no debt, doesn't pay any salaries, and doesn't do much of anything for that matter. It does, however, hold two types of assets:

- The company keeps a total of $50 in a bank account. The money earns interest at the prevailing risk-free interest rate at all times.
- The company owns a perpetual security yielding $10/year. The cashflows (earnings) are paid out to the shareholders as an annual dividend.

Let's also assume that the tax rate on all company dividends and earnings is 0%, and the discount rate used to value the earnings is 10%. (These assumptions, although unrealistic, will affect only the company valuation, not the underlying principles discussed here).

Under these conditions, the $50 in the company's bank account should generate at least $50 dollar of intrinsic business value, or $10/share. The $10/year of perpetual interest payments would add a present value of $100, or $20/share (The value of a perpetuity = cashflow/discount rate). During "normal" times, the market value of the stock will most likely be aligned with the intrinsic value of the company. The shares will trade close to their fair value of $30/share. This scenario is illustrated below:

*(click to enlarge)**(click to enlarge)*Each of the 5 shares of company XYZ have an equal claim on 1/5th of the $50 in the bank account, as well as 1/5th of the earnings value from the perpetuity. The different colors represent the two different types of assets, the green blocks being cash, and purple blocks, earnings power. The relative length of each block depicts the dollar amount of the asset's value.

Let's now assume that the entire stock market suddenly loses 1/3 of its value as the result of a large widespread market panic. The fundamentals of XYZ remain intact (and interest rates remain unchanged), but the share price of the stock plummets to $20/share from its original $30. Each share can now be acquired for a discount to its intrinsic value of $10/share. The situation is represented below;

*(click to enlarge)**(click to enlarge)*

Aware of this large undervaluation, management decides to repurchase its own stock. Since the company does not need the cash to make operating or capital expenditures, the management considers all of the $50.00 as "excess" capital, and is free to spend it buying back stock. The next trading day the company buys back a single share of XYZ (20% of the outstanding shares) for $20/share. The result of the share repurchase is shown below.

*(click to enlarge)**(click to enlarge)*

Since the company has one less shareholder after the repurchase, the $20 earnings value of the repurchased share has now been transferred equally to the remaining shareholders. This will materialize in an annual earnings increase of $0.50/share, or an increased present value of $5.00/share. It's critical to note that the EPS increase is only half of the story. Since the stock was repurchased with $20.00 from the company's balance sheet, the company's new cash position has decreased to $30.00, or $7.50/share. To properly evaluate the economic effect of the repurchase, both the benefit and the cost to remaining shareholders must be considered. In this example, each remaining shareholder paid 2.50/share in cash to receive $5.00/share in earnings value. The net effect of management's decision to buy back shares resulted in an increase per-share intrinsic value of $2.50, to $32.50, or 8.3% (relative to intrinsic value). Put another way, for every $1 the continuing shareholders gave up in cash, they received $2 in earnings value from the folks leaving. This example demonstrates a repurchase situation that created value for the remaining shareholders.

__Example 2: Share Repurchases During a Period of Share Overvaluation__

Now let's assume the share repurchase in Example 1 never happened and XYZ is back to 5 shares. This time, instead of a depression, the market surges into a period of euphoria. As a result, market values become detached from underlying business values and market prices skyrocket. Instead of the company selling in the market at a discount to intrinsic value, as in the previous example, it now sells for a $10 premium, or $40/share. XYZ's financial position is shown below;*(click to enlarge)**(click to enlarge)*

Despite the share premium, the management decides to repurchase one share of the company's stock. XYZ's financial position after the repurchase is as follows;

*(click to enlarge)**(click to enlarge)*

This time the economic effect of the repurchase yields a quite different result. As in the last example, management's decision to repurchase shares has led to an increase in EPS of $0.50 per annum, or an added earnings value of $5/share. (The management team will always be quick to advertise this seemingly pleasant result). The unfortunate truth is that the economic benefit to the remaining shareholder did not outweigh the cost. As one can see, the company spent $7.50/share for only $5 of increased earnings value. The net result of the repurchase is a total per-share intrinsic value decrease of $2.50. In this case, the management effectively took $10.00 of cash from the balance sheet and donated it to the shareholders dumping the stock. So much for fiduciary duty!

It becomes clear that the increase in per-share intrinsic value resulting from share repurchases originates *exclusively* from the undervaluation of the repurchased stock. In Example 1, the $10.00/share discount was effectively transferred pro-rata from the single exiting shareholder to the four remaining shareholders. During a well executed share repurchase, capital is not so much returned to shareholders as it is transferred from active ones to the less active ones. (More on this later). In Example 2 however, that same $10.00 flowed out of the company to the selling shareholder.

__A Bit of Math__

We can mathematically express the value added (or lost) as a result of the repurchases. The following formula shows the economic per-share value gain or loss (in dollars) as a result of repurchases in a given period;

*(click to enlarge)**(click to enlarge)*

The formula simply expresses what was already observed in the previous examples. It shows that the per-share value created (or destroyed) as a result of share repurchases is simply the average per-share discount to intrinsic value (V_{i}-P_{avg}), times the number of shares repurchased S_{rp}, distributed evenly across the remaining shareholder base S_{o}.

It's just as useful to express this result as a percentage of intrinsic value;

*(click to enlarge)**(click to enlarge)*

Close investigation into these formulas yields the following conclusions;

- The per-share value created, V
_{c}, is proportional to the discount to intrinsic value, D_{avg}. The more the market discount to intrinsic value widens, the greater the economic benefit to remaining shareholders. Conversely, if share repurchases are executed when the company is selling at a premium to intrinsic value, D_{avg}will be negative, and the result is a transfer of business value from the continuing shareholder to the exiting shareholder. When the stock is fairly valued, average market price and business value are equal (D_{avg}is zero), and nobody benefits. - In addition to the market discount, the per-share value created, V
_{c}, is also proportional to the number of shares repurchased during the period, S_{rp}, and inversely proportional to the amount of shares remaining after the repurchase, S_{o}. As the numerator increases, the denominator decreases (excluding the effects of executed options, warrants, share issuance, etc.) This effect, taken alone, creates an increasing marginal benefit to the remaining shareholder. Figure 1 illustrates this pleasant result under different market discounts;

*(click to enlarge)**(click to enlarge)*__Figure 1__- Economic Benefit of Repurchases at Different Market Discounts

The graph uses Equation 1c to plot the value added per-share as a result of repurchases, V_{c}/V_{i}, against the percentage of the company repurchased in a given period, S_{rp}/S_{tot}. Each curve represents a different discount to intrinsic value. As discussed above, a greater discount will result in a proportionally greater added value. In this undervalued condition, the value added to each share increases exponentially as more if the company is repurchased.

In practice, the amount of shares repurchased will be limited by the amount of excess cash on a company's balance sheet. A greater market discount will not only magnify the repurchase benefit, but will also allow for a greater number of shares to be repurchased in a given period. This will thus maximize the increasing marginal benefit effect of the repurchase. The benefit to shareholders becomes self-reinforcing the more the stock price falls relative to value. The dotted lines in Figure 1 represent the maximum amount of shares available for repurchase with a given amount of excess cash. The excess cash is represented as a proportion of intrinsic value. It's clear that the optimum benefit to shareholders will result from the repurchase of the maximum amount of shares possible during a period of maximum undervaluation. In the case of share repurchases, an opportunistic management is the friend of the long term investor.

__Example 3: Entering Shareholders__

So far, the discussion has focused on the benefits of share repurchases to existing shareholders. However, shareholders buying into a company during a period of both undervaluation and share repurchases can receive substantially more value than the existing shareholders of that same company. Both groups receive the benefit of the share repurchase previously discussed, but the new shareholder also receives the value of the current market discount. This is because the initial investment is made at the depressed market price. The outcome can be illustrated using the post-repurchase results of Example 1 presented earlier;

*(click to enlarge)**(click to enlarge)*

Since a new shareholder can buy a share at the depressed market price of $20.00, the total benefit from this well timed share acquisition is $2.50/share due to the repurchase, plus $10/share from the market discount. The savvy investor can buy $32.50 of value for only $20.00. This "dual benefit" for the new shareholder is expressed mathematically below;

*(click to enlarge)**(click to enlarge)*

Equation 2b is also presented below as a percent of intrinsic value;*(click to enlarge)**(click to enlarge)*

The expression sums both the value created from the ongoing repurchase and value obtained from the initial purchase of the undervalued share. It also assumes that the average purchase price of the entering investor is the same price the management is paying for their shares. If the company is indeed undervalued, the total economic benefit received by the new shareholder will be, at a minimum, equal to the discount to intrinsic value. The benefit will multiply as the company repurchases greater amounts of stock.

__Management Responsibility and the Return of Capital Myth__

Managements that choose to repurchase shares must first acknowledge that, unlike a dividend, their judgment alone determines the benefit to shareholders. As shown above, the optimum benefit arises when repurchases are executed shrewdly. Thus, it is managements' responsibility to perform the following tasks when confronted with the decision of share repurchases;

- Formulate a proper valuation of the company before shares are repurchased. (An outside opinion by one or more independent parties may be better for shareholders than the "instinct" of an optimistic CEO.)
- Compare the conservatively estimated share repurchase benefit to all other capital allocation alternatives and execute the best option.
- Act at
__all__times in the best interest of the long term shareholder. This includes displaying restraint during periods of fair or overvaluation, even when the exercise of performance stock options start to dilute the EPS figure.

Incidentally, these guidelines are nearly identical to the due diligence performed before any typical business investment or acquisition. The main difference for a repurchase evaluation is that management already knows everything there is to know about their potential target. Although share repurchases are often labeled as a "return" of capital, and can be shown as mathematically equivalent under certain circumstances, it may be more appropriate to classify them as a *reinvestment* of capital. Beware of managements that hide behind the pretext of the former definition to avoid the accountability inherent in the latter. Whether investing in whole companies, real estate, oil wells, or the earnings of exiting shareholders, the benefit to long term shareholders originates from management paying less for an asset than what that asset is intrinsically worth. As in any investment, it's management's judgment and skill, or lack of, that will be the deciding factor. When evaluating companies that regularly repurchases their own stock, choosing a company with the right management will be just as important as choosing the right company. Be sure to choose wisely.

The mechanics of a share repurchase are straight forward. Typically, the management of a company uses excess cash on the company's balance sheet to purchase outstanding shares of its own company. Often times, the shares are bought directly from the secondary market at the prevailing market price. These shares are effectively taken out of circulation, creating a greater concentration of equity ownership in the hands of the remaining shareholders. Although it's natural to assume that this will benefit shareholders in all instances, it's not always the case. The following examples will illustrate why this is;

__Example1: Share Repurchases During a Period of Undervaluation__

Company XYZ is a hypothetical company with a total of only 5 common shares outstanding. This simple company does not own any property, plant, or equipment, doesn't provide a service, has no debt, doesn't pay any salaries, and doesn't do much of anything for that matter. It does, however, hold two types of assets:

- The company keeps a total of $50 in a bank account. The money earns interest at the prevailing risk-free interest rate at all times.
- The company owns a perpetual security yielding $10/year. The cashflows (earnings) are paid out to the shareholders as an annual dividend.

Let's also assume that the tax rate on all company dividends and earnings is 0%, and the discount rate used to value the earnings is 10%. (These assumptions, although unrealistic, will affect only the company valuation, not the underlying principles discussed here).

Under these conditions, the $50 in the company's bank account should generate at least $50 dollar of intrinsic business value, or $10/share. The $10/year of perpetual interest payments would add a present value of $100, or $20/share (The value of a perpetuity = cashflow/discount rate). During "normal" times, the market value of the stock will most likely be aligned with the intrinsic value of the company. The shares will trade close to their fair value of $30/share. This scenario is illustrated below:

*(click to enlarge)**(click to enlarge)*Each of the 5 shares of company XYZ have an equal claim on 1/5th of the $50 in the bank account, as well as 1/5th of the earnings value from the perpetuity. The different colors represent the two different types of assets, the green blocks being cash, and purple blocks, earnings power. The relative length of each block depicts the dollar amount of the asset's value.

Let's now assume that the entire stock market suddenly loses 1/3 of its value as the result of a large widespread market panic. The fundamentals of XYZ remain intact (and interest rates remain unchanged), but the share price of the stock plummets to $20/share from its original $30. Each share can now be acquired for a discount to its intrinsic value of $10/share. The situation is represented below;

*(click to enlarge)**(click to enlarge)*

Aware of this large undervaluation, management decides to repurchase its own stock. Since the company does not need the cash to make operating or capital expenditures, the management considers all of the $50.00 as "excess" capital, and is free to spend it buying back stock. The next trading day the company buys back a single share of XYZ (20% of the outstanding shares) for $20/share. The result of the share repurchase is shown below.

*(click to enlarge)**(click to enlarge)*

Since the company has one less shareholder after the repurchase, the $20 earnings value of the repurchased share has now been transferred equally to the remaining shareholders. This will materialize in an annual earnings increase of $0.50/share, or an increased present value of $5.00/share. It's critical to note that the EPS increase is only half of the story. Since the stock was repurchased with $20.00 from the company's balance sheet, the company's new cash position has decreased to $30.00, or $7.50/share. To properly evaluate the economic effect of the repurchase, both the benefit and the cost to remaining shareholders must be considered. In this example, each remaining shareholder paid 2.50/share in cash to receive $5.00/share in earnings value. The net effect of management's decision to buy back shares resulted in an increase per-share intrinsic value of $2.50, to $32.50, or 8.3% (relative to intrinsic value). Put another way, for every $1 the continuing shareholders gave up in cash, they received $2 in earnings value from the folks leaving. This example demonstrates a repurchase situation that created value for the remaining shareholders.

__Example 2: Share Repurchases During a Period of Share Overvaluation__

Now let's assume the share repurchase in Example 1 never happened and XYZ is back to 5 shares. This time, instead of a depression, the market surges into a period of euphoria. As a result, market values become detached from underlying business values and market prices skyrocket. Instead of the company selling in the market at a discount to intrinsic value, as in the previous example, it now sells for a $10 premium, or $40/share. XYZ's financial position is shown below;*(click to enlarge)**(click to enlarge)*

Despite the share premium, the management decides to repurchase one share of the company's stock. XYZ's financial position after the repurchase is as follows;

*(click to enlarge)**(click to enlarge)*

This time the economic effect of the repurchase yields a quite different result. As in the last example, management's decision to repurchase shares has led to an increase in EPS of $0.50 per annum, or an added earnings value of $5/share. (The management team will always be quick to advertise this seemingly pleasant result). The unfortunate truth is that the economic benefit to the remaining shareholder did not outweigh the cost. As one can see, the company spent $7.50/share for only $5 of increased earnings value. The net result of the repurchase is a total per-share intrinsic value decrease of $2.50. In this case, the management effectively took $10.00 of cash from the balance sheet and donated it to the shareholders dumping the stock. So much for fiduciary duty!

It becomes clear that the increase in per-share intrinsic value resulting from share repurchases originates *exclusively* from the undervaluation of the repurchased stock. In Example 1, the $10.00/share discount was effectively transferred pro-rata from the single exiting shareholder to the four remaining shareholders. During a well executed share repurchase, capital is not so much returned to shareholders as it is transferred from active ones to the less active ones. (More on this later). In Example 2 however, that same $10.00 flowed out of the company to the selling shareholder.

__A Bit of Math__

We can mathematically express the value added (or lost) as a result of the repurchases. The following formula shows the economic per-share value gain or loss (in dollars) as a result of repurchases in a given period;

*(click to enlarge)**(click to enlarge)*

The formula simply expresses what was already observed in the previous examples. It shows that the per-share value created (or destroyed) as a result of share repurchases is simply the average per-share discount to intrinsic value (V_{i}-P_{avg}), times the number of shares repurchased S_{rp}, distributed evenly across the remaining shareholder base S_{o}.

It's just as useful to express this result as a percentage of intrinsic value;

*(click to enlarge)**(click to enlarge)*

Close investigation into these formulas yields the following conclusions;

- The per-share value created, V
_{c}, is proportional to the discount to intrinsic value, D_{avg}. The more the market discount to intrinsic value widens, the greater the economic benefit to remaining shareholders. Conversely, if share repurchases are executed when the company is selling at a premium to intrinsic value, D_{avg}will be negative, and the result is a transfer of business value from the continuing shareholder to the exiting shareholder. When the stock is fairly valued, average market price and business value are equal (D_{avg}is zero), and nobody benefits. - In addition to the market discount, the per-share value created, V
_{c}, is also proportional to the number of shares repurchased during the period, S_{rp}, and inversely proportional to the amount of shares remaining after the repurchase, S_{o}. As the numerator increases, the denominator decreases (excluding the effects of executed options, warrants, share issuance, etc.) This effect, taken alone, creates an increasing marginal benefit to the remaining shareholder. Figure 1 illustrates this pleasant result under different market discounts;

*(click to enlarge)**(click to enlarge)*__Figure 1__- Economic Benefit of Repurchases at Different Market Discounts

The graph uses Equation 1c to plot the value added per-share as a result of repurchases, V_{c}/V_{i}, against the percentage of the company repurchased in a given period, S_{rp}/S_{tot}. Each curve represents a different discount to intrinsic value. As discussed above, a greater discount will result in a proportionally greater added value. In this undervalued condition, the value added to each share increases exponentially as more if the company is repurchased.

In practice, the amount of shares repurchased will be limited by the amount of excess cash on a company's balance sheet. A greater market discount will not only magnify the repurchase benefit, but will also allow for a greater number of shares to be repurchased in a given period. This will thus maximize the increasing marginal benefit effect of the repurchase. The benefit to shareholders becomes self-reinforcing the more the stock price falls relative to value. The dotted lines in Figure 1 represent the maximum amount of shares available for repurchase with a given amount of excess cash. The excess cash is represented as a proportion of intrinsic value. It's clear that the optimum benefit to shareholders will result from the repurchase of the maximum amount of shares possible during a period of maximum undervaluation. In the case of share repurchases, an opportunistic management is the friend of the long term investor.

__Example 3: Entering Shareholders__

So far, the discussion has focused on the benefits of share repurchases to existing shareholders. However, shareholders buying into a company during a period of both undervaluation and share repurchases can receive substantially more value than the existing shareholders of that same company. Both groups receive the benefit of the share repurchase previously discussed, but the new shareholder also receives the value of the current market discount. This is because the initial investment is made at the depressed market price. The outcome can be illustrated using the post-repurchase results of Example 1 presented earlier;

*(click to enlarge)**(click to enlarge)*

Since a new shareholder can buy a share at the depressed market price of $20.00, the total benefit from this well timed share acquisition is $2.50/share due to the repurchase, plus $10/share from the market discount. The savvy investor can buy $32.50 of value for only $20.00. This "dual benefit" for the new shareholder is expressed mathematically below;

*(click to enlarge)**(click to enlarge)*

Equation 2b is also presented below as a percent of intrinsic value;*(click to enlarge)**(click to enlarge)*

The expression sums both the value created from the ongoing repurchase and value obtained from the initial purchase of the undervalued share. It also assumes that the average purchase price of the entering investor is the same price the management is paying for their shares. If the company is indeed undervalued, the total economic benefit received by the new shareholder will be, at a minimum, equal to the discount to intrinsic value. The benefit will multiply as the company repurchases greater amounts of stock.

__Management Responsibility and the Return of Capital Myth__

Managements that choose to repurchase shares must first acknowledge that, unlike a dividend, their judgment alone determines the benefit to shareholders. As shown above, the optimum benefit arises when repurchases are executed shrewdly. Thus, it is managements' responsibility to perform the following tasks when confronted with the decision of share repurchases;

- Formulate a proper valuation of the company before shares are repurchased. (An outside opinion by one or more independent parties may be better for shareholders than the "instinct" of an optimistic CEO.)
- Compare the conservatively estimated share repurchase benefit to all other capital allocation alternatives and execute the best option.
- Act at
__all__times in the best interest of the long term shareholder. This includes displaying restraint during periods of fair or overvaluation, even when the exercise of performance stock options start to dilute the EPS figure.

Incidentally, these guidelines are nearly identical to the due diligence performed before any typical business investment or acquisition. The main difference for a repurchase evaluation is that management already knows everything there is to know about their potential target. Although share repurchases are often labeled as a "return" of capital, and can be shown as mathematically equivalent under certain circumstances, it may be more appropriate to classify them as a *reinvestment* of capital. Beware of managements that hide behind the pretext of the former definition to avoid the accountability inherent in the latter. Whether investing in whole companies, real estate, oil wells, or the earnings of exiting shareholders, the benefit to long term shareholders originates from management paying less for an asset than what that asset is intrinsically worth. As in any investment, it's management's judgment and skill, or lack of, that will be the deciding factor. When evaluating companies that regularly repurchases their own stock, choosing a company with the right management will be just as important as choosing the right company. Be sure to choose wisely.