Double-Dip in Divorce: A Double Misconception

June 29, 2015 Leave a comment

In the July 2015 Edition of BVUpdate, Robert Levis, CPA/ABV, ASA, CFE authored an article, entitled “The Double-Dip Concept is Often a Misconception,” that argued that the concept of the economic “double-dip” in divorce is a “delusion.” In the context of divorce, the double-dip refers to simultaneously requiring the propertied spouse to equitably divide an asset and pay alimony based upon the income derived from the same asset. Mr. Levis argues that the double-dip is a misconception because the double-dip only occurs when the income is paid from a depreciating asset. As he states, “As long as the future income assumed to be generated by the business asset is a return on investment (i.e. “capital”), there is no double counting of the income for property and spousal support purposes.” This argument is completely incorrect. The double-dip concept is not dependent upon whether an asset represents a “return on” or a “return of” the principal of an asset or whether the asset is a depreciation or appreciating. His article fails to consider present value mechanics and fails to consider that the propertied spouse must pay out the present worth of the asset today, which impairs the future ability of the propertied spouse to maintain the level of “return on” the investment.

Mr. Levis provides an example utilizing a bond stating the following:

“Let’s take an example of a one-year bond to illustrate the return of capital/investment. Assume the marital estate owns a $100,000, one-year bond that generates 10% annual interest at the time of the divorce. Assume further that its face value and market value are the same.

At the end of the year following the divorce, the spouse who received the bond will receive $110,000 ($100,000 principal and $10,000 interest). It is doubtful that anyone would say that the $100,000 of the bond principal the spouse receives at the end of Year 1 is income available for spousal maintenance. Most judges or divorce professional would consider it a double count to do so.”

What his example fails to realize, however, is the spouse who received the bond would have to pay the other spouse half of the $100,000 bonds value (i.e. $50,000) in equitable distribution payment on the date of divorce. In order to generate those proceeds, the spouse keeping the bond post-divorce would have to sell half of the $100,000 bond, thereby reducing the available income for distribution from $10,000 to $5,000 on a post-divorce basis. The spouse keeping the bond, therefore, would have to sell $50,000, reducing his portfolio from $100,000 to $50,000, and payout his entire post-equitable distribution payment income of $5,000 in alimony. As such, the propertied spouse receive no income from his post-divorce bonds.

To understand the issue in the context of a business valuation, let’s consider the following example. Suppose the business generates $100,000 in after-tax income and faces a cost of equity capital of 10%. Assume the business has no growth. For simplicity, further assume the only marital asset is the business. Accordingly, the value of the company is $1,000,000 (i.e. $100,000/10% = $1,000,000) and the propertied spouse must pay the non-propertied spouse $500,000 in equitable distribution. However, since the parties have no other assets, the only way in which the propertied spouse can provide the equitable distribution payment to the wife is to monetize the business interest (i.e. either from selling half of the interest in the open market or leveraging half of the business value). For simplicity assume the propertied spouse sells half of the business in order to pay the equitable distribution payment (note that the results would be the same if the business was leveraged). Therefore, on a post-divorce basis the propertied spouse, has only a 50% interest in the company worth $500,000, and is generating income of only $50,000 (i.e. 10% of $500,000). However, the alimony payment is 50% of the pre-divorce income of $100,000. Therefore, the propertied spouse must pay too the wife ALL of the $50,000 in post-divorce business income generated from the asset.

The table below illustrates the outcome:

Pre-Divorce (both parties)
Busines Value      1,000,000
Income          100,000
Post-Divorce (Propertied Spouse) Post-Divorce (Non Propertied Spouse)
Pre-divorce value      1,000,000 Pre-Divorce Value          –
Less: Sale of 50% of Value (or loan) to generate equitable distirbution payment        (500,000) Equitable Distribution Payment 500,000
Post-Divorce Equity Interest in Company:          500,000 Post-Divorce Net Worht 500,000
Income of Remaining 50% Interest in Company            50,000 Pre-Divorce Income          –
Less: 50% of Pre-Divorce Income (Alimony)          (50,000) Alimony Payment    50,000
Net Cash Flow to Propertied Spouse                  – Net Cash flow to Non-Propertied Spouse    50,000

Therefore, on a post-divorce basis, the propertied spouse is left with no income, while the non-propertied spouse receives $50,000. Moreover, in order to pay the equitable distribution payment to the the non-propertied the propertied spouse must keep the the $500,000 in post marital value “locked-up” in an business or another investment of similar risk, otherwise there is no income to provide the alimony payment. The non-propertied spouse, however, may invest the $500,000 in cash received as equitable distribution payment to further increase income available to the non-propertied spouse. In fact, if you eliminate the annual income and simply accrue the economic equivalent of the alimony payments as an asset for lump sum alimony (using the business cost of capital as the discount) rate, the following shows that the propertied spouse has no economic assets remaining:

Propertied Spouse Non-Propertied Spouse
Present Value of 50% Interest in Business $500,000 Cash/Equitable Distribution Payment $500,000
Present Value of Alimony Payments (Assuming Perpetual Alimony)        (500,000) Present Value of Alimony Payments (Assuming Perpetual Alimony)      500,000
Economic Assets                  – Economic Assets $1,000,000

As shown, the propertied spouse has no economic assets, while the non-propertied spouse is allocated the entire marital net worth. It’s important to note that I have assumed (for illustration) perpetual alimony payments in the above schedule. In reality, alimony payments are no perpetual. Therefore, there would not be a full transference of the entire marital net worth from the propertied spouse to the non-propertied spouse. This can be easily adjusted, however, by reducing the alimony payment by an appropriate probability weighted actuarial adjustment factor to reflect the length of alimony and probability of termination.

In any event, the non-propertied spouse receives a substantial economic windfall when alimony is awarded and the asset generating the income is equitably divided. We can understand this further in terms of future values. For example, assume the non-propertied spouse invests the $500,000 cash equitable distribution payment in the stock market to generate a return of 10% (same as business cost of capital). Further assume, the non-propertied spouse invests the $50,000/year alimony in the market at the same rate. The propertied spouse, however, has no income to invest. Furthermore, the asset is no growth so it has no appreciate. Assume alimony continues for 10 years, at which time alimony stops, and the propertied spouse can sell the business asset supporting the alimony payments (recall this asset will not appreciate because it all of the return is being paid to the non-propertied spouse in terms of alimony). At the end of 10 years, the future net worth of the parties would look as follows:

Propertied Spouse Non-Propertied Spouse
Future Value of 50% Business Interest          500,000 Future Value of $500,000 Invested @ 10%    1,296,871.23
Future Value of Income Invested (no income available for investment)                  – Future Value of $50,000 alimony invested @ 10%        796,871.23
Total Future Net Worth          500,000 Total Future Net Worth    2,093,742.46

As shown above, a significant economic injustice is created because, with the non-propertied spouse amassing over $2 million within 10 years. The propertied spouse holds an asset worth $500,000. Now contrasts this economic result to the case with no alimony and simply an award of half the business value. In this case, the propertied spouse retains a business worth $500,000 (recall the propertied spouse must sell or leverage 50% of the business value of $1 million to finance the equitable distribution payment) and can reinvest business cash flows at the business cost of capital of 10%. The non-propertied spouse holds cash of $500,000 which can be reinvested in an asset of comparable risk. Thus, in 10 years time, both parties have the same equitable value of $1,296,871.23. Assets have been equitably divided.

As such, the “Double-Dip” does not depend upon the nature of the asset or whether the asset is amortizing or appreciating. Simply put, requiring the propertied spouse to pay alimony AND pay out the present worth of the asset that is generating the income to support alimony payments, results in a significant economic injustice. Mr. Robert Levis Misconception is really the double misconception.

JA

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Update on Inflation and Inflation Expectations

April 3, 2013 1 comment

Inflation is an important variable that investors and valuation professionals follow on a frequent basis. Inflation is important because it not only influences interest rates, but it also impacts the pricing of assets and the terms of long-term contracts. A high rate of inflation can extract a hefty toll on investments, eroding the purchasing power of the future dollars received. Low or moderate rates of inflation, particularly moderate expected rates, can help stimulate investment and economic activity.

With the U.S. Federal Reserve drastically expanding the monetary base, individuals, investors, and governments alike have become concerned with the prospect of higher inflation. Despite concerns, however, inflation expectations have remained within reasonable levels, and, although inflation expectations have risen in recent quarters, the projected rates of inflation remain relatively low.

In this article, I provide a brief overview of current inflation and inflation expectations. I will also decompose the current inflation rate to illustrate the primary sources driving the current inflation.

Current Inflation

During the 1st Quarter of 2013, the year-over-year increase in the U.S. Consumer Price Index (CPI-U) was 1.98 percent. Core-CPI inflation, which removes the volatile food and energy, increased at a year-over-year rate of 2.0 percent during the 1st Quarter of 2013. Exhibit 1 (below) presents recent trends in CPI and Core-CPI inflation.

Inflation #1

 

Prior to the collapse of Lehman Brothers (near the 3rd quarter of 2008) CPI inflation was about 5 percent, with core-CPI inflation hovering around the 2.3-2.5 percent range. The financial crisis brought about a precipitous drop in the rate of non-core CPI inflation (largely a result of a wide-scale drops in asset prices), with core-CPI inflation exhibiting a modest downward trend from the Q3 2008 through Q4 2010.

Non-Core CPI inflation hit a record low (over the period analyzed) in Q2 2009, declining by 1.43 percent on a year-over-year basis. Non-core CPI inflation began a strong rebound from about Q3 2009 (deflation of 1.29 percent year-over-year) through Q3 2011, with year-over-year inflation at 3.87 percent. The rebound in price can, in part, be attributed to the fiscal and monetary stimulus of the U.S. Government and U.S. Federal Reserve, respectively.

After peaking in Q3-2011, non-core CPI inflation declined each quarter before reaching a low of 1.66 percent in Q2 2012. Since then, non-core CPI inflation has increased slightly, but has stayed within a range of about 1.75 to 2.0 percent.

Core-CPI inflation witnessed a pickup after Q3-2010 through Q1-2012, with the year-over-year rate of core inflation rising from 0.8 percent to 2.26 percent. The rate of core inflation has fluctuated within a range of about 1.90 to 2.25 percent since Q1 – 2012, in-line with the Fed’s target.

Since Q3 2013, both core and non-core CPI have been modest and relatively in line with each other. The data suggest that current rates of inflation have moderated at about the 2.0 percent level.

Also, a noticeable observation in the data is the dramatic reduction in inflation volatility. Generally speaking, the Federal Reserve has, in my opinion, done a good job of stabilizing the price level, despite the rapid growth in the monetary aggregates over this period.

Short-Term Inflation Expectations

Short-term inflation expectations, defined as the rate of expected inflation in the U.S. CPI over the next 1-5 years, have appeared to increase during Q1 2013 vs. Q4 2012 (based upon breakeven estimates from the U.S. Treasury Market).

As of Q1 2013, market participants are pricing in a break-even inflation rate of about 2.3 percent over the next year, slightly above the 1.98 percent rate realized on a year-over year basis during Q1 2013.  One year breakeven inflation was about 1.3 percent during Q4 2012 and 2.18 percent one year ago. Five year break-even expectations have also blipped up slightly during the Q1 2013, with the breakeven inflation rate rising from 2.07 percent to 2.33 percent.

Overall, short-term inflation expectations have increased during the Q1 2013, but are generally in-line with recent realized rates of inflation. Market participants are currently pricing in moderate rates of inflation over the next 1-5 years, with the expected rate remaining fairly stable at about 2.3 percent over the entire period. These rates are slightly higher than the Fed’s target of 2.0 percent.

Long-Term Inflation Expectations

Long term inflation expectations, defined as inflation expectations over the next 10 and 20 years have also crept up during Q1 2013, but only slightly. The 10 year breakeven inflation rate is approximately 2.52 percent, compared to 2.45 percent during Q4-2012, and 2.34 percent one-year ago. The 20 year break-even inflation rate is about 2.49 percent, compared to 2.45 percent during Q4-2012 and 2.42 percent one year-ago. Overall, long-term inflation estimates are moderate.

The chart below illustrates the term structure of U.S. breakeven inflation expectations, as measured by the U.S. Treasury Market.

Inflation #2

As shown, inflation expectations have crept up at the short-end of the term structure relative to Q4 2012. Today, however, the term structure is relatively flat, suggesting that the market anticipates a moderate rate of inflation in the 2.3-2.5 percent range over the next 20 years. In my opinion, the stability in inflation, as evidenced by the flatness of the inflation term structure, is a positive development; that is, long-term inflation expectations are stabilizing, suggesting less inflation uncertainty over the long-term.

Decomposition of Inflation Using the Quantity Theory of Money

The quantity theory of money is a mathematical identify that relates the realized inflation rate to the growth rate in the money supply, the growth rate in the velocity of money, and the growth rate in real GDP. The theory states the following

Inflation Rate = %Chg. Money Supply + %Chg. Velocity of Money – % Chg. In Real GDP

The velocity of money is the average frequency with which a unit of money is spent on new goods and services produced domestically in a specific period of time. It can be viewed as the amount of economic activity associated with a given money supply. It is defined as the ratio of nominal GDP divided by the money supply. The table below decomposes recent inflation rate using the quantity theory of money. Please note that the inflation rates reported below differ from the CPI inflation rates reported above because these rates are based upon the GDP Deflator (an alternative measure of inflation) and are reflected as a continuous rate of inflation. The rates are also seasonally adjusted rates. The differences, however, do not affect the purpose of the analysis, which is to understand the drivers of recent inflation.

Inflation #3

In 2008, the year-over-year change in inflation (as measured by the continuously compounded change in the GDP deflator) ranged about 2.0-2.5 percent. Most of the monetary inflation during this period was fueled by a rapid expansion in the money supply, which was heavily offset, however, by significant declines in the velocity of money. In the early parts of 2008, real GDP growth also helped reduce inflation fueled by the expansion of money, but as real GDP declined in Q3 and Q4 it actually boosted the inflation rate slightly.

In 2009, inflation dropped significantly, with the year-over-year inflation rate ranging from about 1.8 percent to 0.27 percent. The drop in inflation during this period was largely a function of a slower expansion in the U.S. money supply, followed by large drops in the velocity money. In other words, as the financial crisis unfolded, the amount of economic activity declined considerably, thereby placing pressure on the price level given the outstanding stock of money. The decline in real GDP also contributed to the inflation rate.

Inflation picked up strongly in 2010 relative to 2009. The pick-up in inflation was a surge in the velocity of money during this period. Real GDP growth also expanded sharply, which held the inflation rate back somewhat. Monetary growth over this period was not significant.

In 2011, the money supply started to expand again, primarily as a result of QE2, which boosted the rate of inflation. The monetary expansion, however, was partially offset by the velocity of money, which remained negative all throughout 2011. Real GDP growth was also somewhat sluggish during this period, which contributed to inflation over the period.

In 2012, inflation declined relative to 2011. This decline was primarily a result of strong real GDP growth, slower monetary expansion, and continued declines in the velocity of money.

It should be evident that over this entire period that the money supply increased rapidly, but this monetary expansion did not result in a high level of inflation because of a precipitous decline in the velocity of money. Indeed, although the average year-over-year change in the money supply was about 7.4 percent since Q1 2008, velocity declined on average at about 5.1 percent. Real GDP expanded at an average year-over-year rate of only about 0.60 percent, which also reduced the inflation rate.

Reconciling & Projecting Inflation: Quantity Theory of Money,

U.S. Economists project that real GDP will increase at an annualized rate of 1.9 percent, 2.8 percent, and 2.9 percent, and 3.0 percent for the years 2013, 2014, 2015, and 2016, respectively (based upon the Q1-2013 Survey of Professional Forecasters). The Federal Reserve is currently purchasing approximately $1 trillion in mortgage backed securities and treasury bonds per annum, representing about an 8-9 percent growth rate in the money supply. These factors would suggest about a 6-7 percent inflation rate, ignoring changes in the velocity of money. Since current inflation expectations, however, are settled around 2-2.5 percent, market participants are expecting continued declines in the velocity of money at the rate of about 4-5 percent per annum. Currently, MZM velocity, much like the 10 year treasury yield, is at historically lows levels (see chart below).

Inflation #4

It should be evident that an unexpected rise in the velocity of money could result in an unexpected increase in the rate of inflation.  However, so long as MZM velocity continues to decline, we should expect the inflation rate to remain low.

Conclusions

Overall, U.S. inflation expectations remain within reasonable levels. Although the U.S. monetary supply has increased rapidly over this period, declines in MZM velocity have largely offset the increase in the money supply, a situation that has kept the inflation rate low. Both short-term and long-term inflation expectations are strongly grounded at the 2-2.5 percent level. Much of the low inflation appears to a result of continuing expectations regarding a continued decline in the velocity of money. So long as MZM velocity does not unexpectedly pick up significantly, inflation should remain within reasonable levels over the short-run.

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Historical Risk Premium of Real Estate Investment Trusts (REITs)

January 28, 2013 2 comments

What is the historical risk premium for investing in publicly traded equity real estate investment trusts (REITs)? The chart below illustrates the historical realized risk premium of equity REITs (as measured by the FTSE NAREIT Equity Total Return Index) in excess of the 10 year treasury from February 1, 1972 (inception) to December 31, 2012.

REIT Risk Premium

The chart illustrates that REITs have historically earned a sizable risk premium in excess of the 10 year treasury of approximately 6% over the entire period (4.0% during the 1st half and 8.1% during the second half).

 

 

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Do European Put Options Measure Marketability?

September 7, 2012 1 comment

Proponents of the option pricing methodologies commonly argue that an At-the-Money (ATM) European Put Option provides a lower bound estimate of the discount for lack of marketability. According to the argument, if an investor purchases an ATM European Put Option with a strike price equal to the marketable price, the investor has, in effect, purchased the ability to sell the restricted shares at the marketable price. In the words of Chaffe, the investor has purchased marketability.  If this is the case, then we should expect a portfolio of a ATM European Put Option and a restricted security to provide the same economic payoffs at expiration as that of owning liquid stock directly. However, working through the math, one will discover that a portfolio comprised of a European put option and a restricted security actually provides better economic payoffs at expiration than owning liquid stock directly. To illustrate this important economic concept, consider the following hypothetical example:

Suppose the marketable value of the common stock is currently $10 per share and the underlying stock price volatility is 40%. The stock pays no dividends, and the risk free rate is currently 5%. The illiquid stock is subject to a 1 year holding period restriction. Therefore, under these assumptions, the cost of a 1 year European ATM Put Option (strike of $10) is $1.32 per share (computed using the Black Scholes Option Pricing Model) and the theoretical value of the illiquid stock is $8.68 per share (i.e. $10 – $1.32 = $8.68). Now, assume an investor purchases the illiquid stock at its theoretical value of $8.68 and simultaneously purchases the ATM Put Option at its theoretical value of $1.32. In effect, this investor has just purchased marketability.[1] Therefore, we should expect the portfolio to have the same economic payoffs as that of owning the liquid stock directly. However, consider the economic payoffs at expiration in all states of the world (i.e. when the liquid stock expires in the money, at the money, and out of the money). For this illustration we will assume the liquid stock trades at $8, $10, and $12 at expiration. Keep in mind that at expiration the illiquid stock is now liquid and, therefore, trades at its liquid value. The table below illustrates the payoffs at expiration for all three scenarios:

Price of Liquid Stock at Expiration

Profit on European Put Option At Expiration[2]

Profit on Illiquid Stock at Expiration[3]

Profit on Portfolio of Option and Illiquid Stock[4]

Profit on Liquid Stock at Expiration[5]

$8

$0.68

-$0.68

$0

-$2

$10

-$1.32

$1.32

$0

$0

$12

-$1.32

$3.32

$2

$2


[1] This assumption is presumed to be true according to Chaffe’s theory.

[2] European put profit = MAX(K – S, 0) – $1.32 (i.e. cost of option); where K = $10 and S = Price of liquid stock

[3] Profit on illiquid stock = S – $8.68 (cost of illiquid stock); where S = price of liquid stock (recall that the illiquid stock is liquid at expiration of the option)

[4] Profit on portfolio = sum of profit on option and profit on illiquid stock.

[5] Profit on liquid stock = S – $10; where S = price of liquid stock at expiration.

As one can see, the portfolio provides the holder with superior economic payoffs when the put options expires in-the-money. Accordingly, a portfolio comprised of restricted stock and a put is more valuable than liquid stock and, therefore, we should expect a European Put Option to actually overstate the cost of lack of marketability.

Does this Mean that European Put Options are Not Useful?

No, not necessarily. Although a European Put Option does not directly measure (and likely overstates) the cost of illiquidity, the option does eliminate a large portion of pricing fluctuation over the restriction period. A rationale market participant can, therefore, use a European Put Option to eliminate price risk and guarantee a specific economic payoff, or price, at expiration of the option. For example,  suppose an investor owns a restricted security and simultaneously purchases a 1 year ATM put option at $10 per share. This technique will guarantee $10 per share for the investor at expiration of the option. And, while this economic payoff is slightly different than the real option of liquidity (i.e. liquidity allows the purchaser to receive $10 today), and can actually be constructed in a less costly fashion (i.e. with a costless collar), the strategy, nevertheless, is a viable hedging option for an investor owning illiquid securities.

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The Impact of Warrants on Restricted Stock Discounts in the Pluris DLOM Database

June 11, 2012 1 comment

The Pluris DLOM Database is a new restricted stock study database that is available to appraisers and can be utilized to quantify the discount for lack of marketability. The database is much larger than other restricted stock studies (i.e. as of December 31, 2011 the database had well over 3,000 transactions vs. approximately 600 in the FMV Opinions Database), primarily because Pluris includes restricted stock transactions issued with warrants. A warrant is a non-tradable option that is often issued in the private placement of restricted stock. Most databases exclude warrant transactions because warrants are difficult to value. Pluris includes them to increase the size of their samples. To my knowledge, very little, if any resesarch has been published on the impact of warrants on the restricted stock discounts in Pluris. The purpose of this post is to report my preliminary findings on the issue. In particular, I will demonstrate the following:

1. The warrant transactions have considerably higher discounts than the non-warrant transactions.

2. The magnitude of the discounts on warrant transactions is affected by the fraction of warrants placed in the transaction;  in particular, the greater the fraction of warrants placed in the transaction, the larger the reported restricted stock discount.

3. The moneyness of the warrant (i.e. whether the warrant is in-the-money, out-of-the-money, or at-the-money) affects the discounts on warrant transactions. In-the-money and at-the-money transactions command significantly higher discounts than out-of-the-money transactions. In addition, I find a strong relationship between the relative moneyness of the warrant (defined later) and the restricted stock study discounts on warrant transactions.

4. Other factors, such as whether the warrants are issued with a cashless exercise or call-cap feature, impact the discounts on the warrant transactions.

5. After controlling for the aforementioned variables, among other factors known to impact the restricted stock discounts, I am able to explain approximately 57% of the variation in discounts on restricted stock transactions issued with warrants.

These issues/findings are discussed further below.

Higher Discounts Observed on Warrant Transactions vs. Non-Warrant Transactions

First, I find that transactions issued with warrants (i.e. 1,655 of the 3,169 transactions) have considerably higher discounts than transactions issued without warrants. This is clearly demonstrated in the graphic below, which summarizes the median discounts on both types of transactions.

As one can see, the median discount on warrant transactions is 29.0% vs. 12.20% for non-warrant transactions. The differential is both economically and statistically significant. Unfortunately, it is unclear why the warrant transactions command considerably higher discounts than the non-warrant transactions (in fact, my initial hypothesis was for the discounts to be comparable). To my knowledge, there is no significant difference in the liquidity of restricted stock issued with or without warrants. Nonetheless, the data suggests that either (a) market participants view warrant transactions to be considerably less attractive than non-warrant transactions or (b) that the warrants are being systematically overvalued by Pluris (overvaluation of the warrants would result in a higher implied discount).

Relationship of Discounts and Fraction of Warrants Placed

In addition to higher discounts, I find a strong, positive, linear relationship between the fraction of warrants placed (i.e. total warrant value expressed as a percentage of total gross proceeds raised) and the size of the restricted stock discount. To demonstrate this concept, I grouped the warrant transactions into 25 equally (or approximately equal) sized portfolio baskets based upon the fraction of warrants placed in the transaction, and computed the median discount for each portfolio basket. The results are summarized in the chart below:

As one can see, there is a strong relationship between the fraction of warrants placed and the observed discounts; that is, as the fraction of warrants placed increases the restricted stock discounts increase. Again, it is unclear what specific economic factor is contributing to this relationship. However, in my opinion, illiquidity is unlikely to be a valid explanation because the fraction of warrants placed should not affect the degree of restricted stock liquidity.

Relationship of Discounts and Moneyness of Warrants

I also find a strong relationship between the discounts on restricted stock transactions issued with warrants and the monenyess of the warrant. The moneyness of the warrant refers to whether the warrant is in-the-money, at-the-money, or out-of-the-money.  The chart below summarizes the discounts on transactions based upon moneyness:

As one can see, at-the-money and in-the-money warrants command significantly higher discounts than out-of-the-money warrants. In addition, I find a very powerful, positive relationship between the discounts and the relative moneyness of the warrants (I define relative moneyness as the stock price less the strike price expressed as a percentage of the stock price). To demonstrate this concept, I grouped the warrant transactions into 25 equally (or approximately equal) sized portfolio baskets based upon relative moneyness, and computed the median discounts for each portfolio basket. The chart below summarizes the relationship:

As one can see, the transactions that have warrants that are deep out-of-the-money (left side of chart) command much lower discounts that transactions that are slightly-out-of-the-money. Moreover, transactions at-the-money command higher discounts than transactions out-of-the-money, and the discounts increase the further the warrants are in-the-money. Again, it is unclear what economic factor is causing this relationship, although illiquidity does not appear to be the answer.

Impact of Other Options Provisions/Features on Discounts

I also find that call caps and cashless exercise features impact the discounts. A call cap refers to a special right that allows the issuer of the warrant to force the holder, under specified conditions, to exercise the warrant once the stock reaches a trigger price. A cashless exercise feature allows the purchaser of the warrant to exercise the warrant without actually paying the strike price in cash.  The table below summarizes the discounts on call cap and non-call cap warrant transactions.

The table below summarizes the discounts on cashless and non-cashless exercise warrant transactions:

As one can see, there is a slight difference in median discounts on transactions issued with call cap and cashless exercise features. Again, it is unclear why these warrant features impact the illiquidity discounts on restricted stock, as neither of the features directly affect the illiquidity restrictions.

Regression Analysis

Based upon the relationships above, and after controlling for other factors known to impact the restricted stock discounts, I formulated a regression equation to explain the variation in discounts on the warrant transactions (1,655 warrants transactions vs. 3,169 total transactions).  The regression equation is summarized below:

DLOM = a +b1*Warrant% + b2*Moneyness + b3*Cashless + b4*CallCap + b5*vol + b6*block +b7*exchange + b8*reg + b9*hp

Where:

Warrant%:   This variable is for the fraction of warrants placed (i.e. warrant proceeds divided by total gross proceeds). This variable is included to control for the observed relationships between discounts and the fraction of warrants placed in the restricted stock transactions.

Moneyness:  This variable is the relative moneynes of the warrant (i.e. stock price less strike price divided by stock price). This variable is included to control for the observed relationship between the moneyness of the warrant and the discounts on restricted stock transactions.

Cashless:   This is a dummy variable to control for the impact of the cashless exercise feature. The variable is set to 1 for warrant transactions with a cashless exercise and 0 for all other warrant transactions.

CallCap:   This is a dummy variable to control for the impact of the call cap feature. The variable is set to 1 for warrant transactions with a call cap feature and 0 for all other warrant transactions.

Vol:   This variable is for volatility of the issuer. This variable controls for differences in the risk of the issuer, which is known to impact the restricted stock discounts.

Block:  This variable is for the fraction of shares placed expressed as a percentage of total shares outstanding. This variable controls for block size, which is known to increase discounts for large block transactions in restricted stock (presumably because the dribble-out provision of Rule 144 make large block sales more difficult).

Exchange: This is a dummy variable to control for the trading exchange of the issuer. The variable is set to 1 for issuers that trade OTC or OTC BB and 0 for all other trading exchanges (i.e. Nasdaq and other National Exchanges)

Reg:  This is a dummy variable to control for registration rights. The variable is set to 1 for transactions issued without registration rights and 0 for all other transactions. We note that certain transactions did not report whether registration rights were included. In those cases, we assumed that the transactions did not have registration rights (primarily because the discounts are comparable to transactions issued without registration rights).

Hp:  This is a dummy variable for the holding period rules applicable to the restricted stock. The variable is set to 1 for transactions occurring after February 15, 2008 (i.e. when the holding period restriction was reduced to 6 months) and 0 for all other transactions (we note that all other transactions were subject to a 1 year holding period).

The table below summarizes the regression output:

As one can see, the regression equation does an excellent job of explaining the variation in discounts, explaining approximately 57% of the variation in discounts on transactions issued with warrants. More importantly, all of the economic variables, except for trading exchange and call cap features, are statistically significant at the 99% level. The call cap feature is statistically significant at the 95% level. The Exchange variable is not statistically significant. The equation provides important perspective on the impact of certain variables on the discounts. In particular, the following:

1. For each 1% increase in the fraction of warrants placed, the discounts increase by 0.8776%. Therefore, a transactions that has 30% of its value attributable to a warrant is expected to have a discount 17.552% higher than a transaction with only 10% of its value attributable to a warrant. This is statistically significant at the 99% level.

2. For each 1% increase in the relative moneyness of the warrant, the discount increases by 0.3177%. Therefore, a transactions that is 100% in-the-money will have a discount 31.77% higher than a transaction in the money. This is statistically significant at the 99% level.

3. Warrant transactions issued with a cashless exercise feature have discounts 2.97% lower than warrant transactions issued without a cashless exercise feature. This is statistically significant at the 99% level.

4. Warrant transactions issued with a call cap feature have discounts 2.01% higher than warrant transactions issued without a call cap feature. This is statistically significant at the 95% level.

5. For each 1% increase in volatility, the discounts increase by 0.0202%. This is statistically significant at the 99% level.

6. For each 1% increase in block-size, the discounts increase by 0.1143%. This is statistically significant at the 99% level.

7. Firms that trade OTC or OTC-BB have discounts 3.37% higher than firms that trade on other major trading exchanges. This variable is not statistically significnat.

8. Transactions issued without registration rights command discounts 5.24% higher than transactions with registration rights. This variable is statistically significant at the 99% level.

9. Transactions subject to a 6 month holding period command discounts 3.34% lower than transactions subject to a 1 year holding period. This is statistically significant at the 99% level.

The most interesting observation, in my opinion, is the significant contribution that the warrant% and moneyness of the warrants have on the discounts. In particular, appraisers could considerably alter the median discounts by “titling” the sample of transactions towards transactions issued with certain option characteristics. For example, if an appraiser wanted to artificially increase the discounts, the appraiser could simply focus on transactions wherein a large fraction of in-the-money warrants was issued. Again, it is unclear whether the relationships are related to illiquidity. In fact, many of the variables (such as warrant% and cashless exercise, among others) appear completely unrelated to illiquidity. Nonetheless, these factors have a large impact on the discounts.

Conclusion

This post briefly examined the impact of warrants in the Pluris DLOM Database. I demonstrate that warrant transactions have considerably higher discounts than non-warrant transactions. More importantly, the magnitude of discounts on warrant transactions is affected by the fraction of warrants placed, the relative moneyness of the warrant, and other warrant features (i.e. cashless exercise and call caps). In particular, both the fraction of warrants and the moneyness of the warrants have a positive and statistically significant relationship to the discounts, even after controlling for many other factors that are known to impact discounts, such as volatility and block size.  In fact, after controlling for these variables, I can develop regression equations that explain approximately 57% of the variation in the restricted stock discount on transactions issued with warrants, and show that many warrant variables, which appear unrelated to illiquidity, impact the magnitude of the discounts. Appraisers should carefully consider the impact of warrants when selecting transactions from the Pluris DLOM Database. It is currently unclear why certain variables impact the discounts and whether these variables are related to illiquidity. Therefore, evidence from the warrant transactions may bias the discounts considerably, and should be approached with caution until valid explanations are found.

Categories: Uncategorized

A Quick Theory on the Pass-Through Premium at the Level of Control

April 14, 2012 3 comments

By Joshua B. Angell

Introduction

There is a general consensus among appraisers that a controlling ownership interest in a pass-through firm should not be worth much more than a controlling ownership interest in an otherwise identical double-tax firm. This “general consensus” primarily stems from empirical studies that have compared the pricing multiples of pass-through firms to that of double-tax firms in the market for control. For example, in a recent study of the Pratt’s Stats Database, the author of this post compared the pricing multiples of over 7,000 market transactions and found very weak, if any, statistical evidence in support of a pass-through premium in the market for controlling ownership interests. As such, many appraisers have concluded that a pass-through premium should not apply when determining the fair market value of a controlling ownership interest in a pass-through firm.

Despite the empirical evidence, however, appraisers should understand that a controlling ownership interest in a pass-through firm can theoretically command (and investors should be willing to pay) a price premium, price discount, or even no pricing differential relative to an otherwise identical controlling ownership interest in a double tax firm. The purpose of this article is to demonstrate this concept and highlight the primary  economic factors that contribute to the pass-through premium at the enterprise level.

A Theoretical Model for the Pass-Through Premium

To understand why the pass-through premium can be positive, negative, or non-existent, we begin with a basic formula for valuing the after-tax cash flows of a pass-through firm from the perspective of  the investor (we assume the investor maintains the pass-through election):

ValuePT = EBTPT*(1-RPT-Tp) / (KatPT-GatPT)

Where:

EBTPT = Earnings before tax of pass-through firm, next year

Tc        = Corporate income tax rate

RPT      = Reinvestment ratio expressed as a percentage of EBTPT

Td         = Dividend tax rate

KatPT    = After-shareholder level tax cost of capital

GatPT    = Long-term sustainable growth rate in free cash flow

This formula is effectively the Gordon Growth Model for a pass-through firm. The numerator, EBTPT*(1-RPT-Tp), represents the after-tax cash flow available to the investor after all taxes, including personal taxes, have been paid. The denominator is the after-tax capitalization rate applicable to this income stream. Therefore, KatPT, is the after-shareholder level tax discount rate. This rate differs from the rate that an appraiser would obtain from a source such as Ibbotson. Nonetheless, this equation values the perpetual income stream of a pass-through entity from the perspective of an investor who plans to maintain the pass-through election into perpetuity.

Next, we extend this same underlying concept to a double tax firm:

ValueDT = EBTDT*(1-Tc-RDT)*(1-Td) / (KatDT-GatDT)

Where:

EBTDT = Earnings before tax of double-tax firm, next year

Tc        = Corporate income tax rate

RDT      = Reinvestment ratio expressed as a percentage of EBTDT

Td         = Dividend tax rate

KatDT    = After-shareholder level tax cost of capital

GatDT    = Long-term sustainable growth rate in free cash flow

This formula is effectively the Gordon Growth Model for a double tax firm (as customarily defined in most investment textbooks), except we deduct dividend taxes, (i.e. 1-Td) from corporate free cash flow (i.e. EBTDT*(1-Tc-RDT). We deduct dividend taxes from corporate free cash flow in order to obtain the actual cash flows available to the investor after all taxes, including personal taxes, have been paid. Therefore, the discount rate, KatDT, is, again, the after-shareholder level tax discount rate, and not the pre-shareholder level tax discount rate that one would obtain from a source such as Ibbotson. In our opinion, this method is more theoretically appealing than using pre-tax rates. Nonetheless, this equation values the perpetual income stream available to an investor in a double-tax firm, assuming the investor maintains the double-tax status of the firm into perpetuity.

Finally, to determine theoretical pass-through premium percentage, (PTP %), we express equation 1 (i.e. the value of pass-through firm) as a percentage of equation 2 (i.e. the value of a double-tax firm) and subtract 1 as follows:

PTP (%) = [EBTPT*(1-RPT-Tp)/(KatPT-GatPT) *  (KatDT-GatDT)/EBTDT*(1-Tc-RDT)*(1-Td)] -1

Where:

EBTPT = Earnings before tax of pass-through firm, next year

EBTDT  = Earnings before tax of double-tax firm, next year

RPT      = Reinvestment rate of pass-through firm, expressed as a % of EBT

RDT        = Reinvestment rate of double-tax firm, expressed as a % of EBT

Tp         = Effective personal income tax rate

Tc         = Effective corporate income tax rate

Td         = Effective dividend tax rate

KatPT    = After-shareholder level tax return of pass-through firm

KatDT     = After-shareholder level tax return of double-tax firm

GatPT    = Long-term sustainable growth rate of pass-through firm

GatDT     = Long-term sustainable growth rate of double-tax firm

This equation is a mathematical identity that relates the value of a pass-through firm to the value of a double-tax firm. If we assume that the pass-through firm is identical to the double tax firm in every way except for incorporation status – that is, EBTPT, EBTDT, RPT, RDT, KatPT, KatDT, GatPT, and GatDT, are identical – the equation above simplifies to following mathematical identity:

PTP (%) = (1-RPT-Tp)/[(1-RDT-Tc)*(1-DT)] – 1

Where:

RPT      = Reinvestment rate of pass-through, expressed as a % of EBTPT

RDT        = Reinvestment arte of double-tax firm, expressed as a % of EBTDT

Tp         = Effective personal income tax rate on flow-through income from pass-through

Tc         = Effective dividend tax rate on corporate distributions

This equation tells us that the relative pricing of a pass-through firm and an otherwise identical double tax firm depends on only four economic variables:

  1. The reinvestment rates, RPT and RDT (which equal for identical firms)
  2. The corporate income tax rate, Tc
  3. The personal income tax rate, Tp; and
  4. The dividend tax rate, Td

Exploring the Model in Various Scenarios

The above formula can be utilized to evaluate the economic characteristics of the pass-through premium under various different scenarios. For example, consider the situation in which the corporate income tax rate is equal to the personal income tax rate; that is, Tp equals Tc. In that case, (1-RPT-Tp) and (1-RDT-Tc) cancel, and the PTP (%) simplifies to the following equation:

PTP (%) = 1/ (1-Td) – 1

Where:

Td = Dividend tax rate

Thus, when personal and corporate income tax rates are identical, the pass-through premium is simply equal to 1 dividend by 1 minus the dividend tax rate, Td, less 1.  The table below demonstrates the theoretical pass-through premium under various different dividend tax and reinvestment rate scenarios:

Image

As one can see, when the corporate and personal income tax rate are the same, the pass-through premium is always positive, unless the dividend tax rate is equal to 0%. In addition, the pass-through premium does not depend on the reinvestment rate. Therefore, when corporate and personal tax rates are the same, an investor should be willing to pay a premium for a pass-through firm equal to 1/(1-Td) -1.

Next, consider the situation in which the personal income tax rate exceeds the corporate income tax rate; that is, Tp > Tc. In this case, the PTP (%) equation cannot simplify any further. However, since RPT and RDT are identical, the numerator will always be smaller than the denominator if Td < 1 – (1-RPT-Tp)/(1-RDT-Tc). Therefore, the PTP (%) can be positive, negative, or zero depending on the corporate income tax rate, the personal income tax rate, the reinvestment ratio and the dividend tax rate. The table below demonstrates this concept assuming a personal income tax rate of 40% and a corporate income tax rate of 35%.

Image

As one can see, the PTP (%) takes several different values depending upon the dividend tax rate and the reinvestment ratio. Several notable observations can be made from the table above. First, notice that the PTP (%) is always positive when the dividend tax rate is greater than 1 – (1-Rpt-Tp)/(1-RDT-Tc). Conversely, the PTP (%) is always negative when the dividend tax rate is less than 1 – (1-Rpt-Tp)/(1-RDT-Tc).  Also, notice that the PTP (%) always increases as the dividend tax rate increases. More importantly, notice that the PTP (%) decreases as the reinvestment rate increases; that is, as a firm increases its reinvestment rate the differential in value between a pass-through firm and a double-tax firm declines (assuming the corporate tax rate is below the personal income tax rate). In fact, with the right combination of dividend tax rates and reinvestment ratios a double-tax firm can be worth more than a pass-through firm. The reason is not obvious, but relates to shifting the composition of returns from dividends to capital gains, thereby increasing the after-tax cash flows available to the investor in a double tax firm (*note our model assumes a perpetual holding period, therefore, the present value of capital gains taxes is minimized. For shorter holding periods, however, we would need to consider the capital gains liability. This is outside of the scope of this article). Nonetheless, this shows that when the corporate tax rate is below the personal income tax rate, that a pass-through firm can theoretically command a price premium, price discount, or no-pricing differential relative to an otherwise identical double tax firm.

Finally, consider the situation in which the personal income tax rate is below the corporate income tax rate; that is, Tp < Tc. In this scenario, the numerator will always be greater than the denominator. Therefore, the PTP (%) will always be positive, but the magnitude of the PTP (%) will depend upon the dividend tax rate and the reinvestment ratio. This concept is demonstrated in the table below assuming the personal income tax rate is 35% and the corporate income tax rate is 40%.

Image

As one can see, the PTP (%) is positive in all scenarios and is dependent upon the dividend tax rate and the reinvestment ratio. In particular, the PTP (%) increases as both the dividend tax rate and the reinvestment ratio increase. Therefore, if the personal income tax rate is expected to be lower than the corporate income tax rate, an investor should be willing to pay a premium for a pass-through firm, and that premium will vary directly with the dividend tax rate and the reinvestment rate into the firm.

The Break-Even Personal Income Tax Rate

We can also use the PTP (%) formula to determine when an investor will be indifferent between owning a double-tax firm and a pass-through firm (i.e. will pay the same price). For example setting the PTP (%) to zero and solving for the personal tax rate Tp, we discover the following:

Tp = Tc + Td*(1-RDT-Tc)

Where:

Tp         = Personal Tax Rate

Tc         = Effective Corporate Tax Rate

Td         = Effective Dividend Tax Rate

RDT      = Reinvestment of pre-tax earnings

Notice that (1-RDT-Tc) is an alternative way of expressing the free cash flow payout ratio, PDT, as as percentage of earnings before tax (EBT). Therefore, an investor will be indifferent from owning a pass-through firm and a double-tax firm when the personal tax rate equals the (a) corporate tax rate plus (b) the dividend tax rate times the free cash flow payout ratio (expressed as a % of pre-tax earnigns). More formally, we can state the following:

  1. When Tp = Tc + Td*(PDT) the pass-through commands the same price as an otherwise identical double-tax firm
  2. When Tp > Tc + Td*(PDT) the pass-through commands a discount to an otherwise identical double-tax firm.
  3. When Tp < Tc + Td*(PDT) the pass-through commands a premium to an otherwise identical double-tax firm.

This demonstrate that the pass-through premium can be positive, negative, or non-existence, depending upon the personal tax rate, the corporate income tax rate, the dividend tax rate, and the reinvestment ratio. Thus, investors purchasing a pass-through firm (an appraisers valuing them) should consider these factors in determining the appropriate price for a pass-through firm.

Conclusions

In this post, I developed a quick theory on the pass-through premium. Using basic financial theory, I show that an investor in a pass-through firm should be willing to pay a price premium, price discount, or no pricing differential relative to an otherwise identical double-tax firm. In particular, the pricing differential will depend upon the corporate income tax rate, the personal income tax rate, the dividend tax rate, and the reinvestment ratio of the firm. I show that when the personal income tax rate is equal to the corporate income tax rate, that an investor should be willing to pay a price premium equal to 1/(1-Td) -1. In addition, I show that when the personal income tax rate exceeds the corporate income tax rate, that the investor should be willing to pay a price premium, price discount, or no-pricing differential depending upon the corporate income tax rate, the personal income tax rate, the dividend tax rate, and the reinvestment ratio. In particular, the investor will pay a premium when the dividend tax rate is greater than 1 – (1-Rpt-Tp)/(1-RDT-Tc). The premium will increase with the dividend tax rate and decline with the reinvestment ratio. In addition, I show that when the personal tax rate is below the corporate tax rate, that the investor should always be willing to pay a premium for a pass-through firm. This premium is positively related to the dividend tax rate and the reinvestment ratio. Finally, I show that investors will be indifferent between owning a pass-through firm and a double-tax firm when. Tp = Tc + Td*(1-RDT-Tc). Appraisers should consider this information when determining the value of a pass-through firm at the level of control.

Author’s Note:

The above analysis rests upon the following assumptions:

1.  The pass-through firm is identical to the double-tax firm in every way but for incorporation status

2. The investor maintains the tax status (i.e. pass-through vs. double-tax) of the firm into perpetuity.

3. The investor holds the investment into perpetuity

4. The investor does not take advantage of any favorable tax provisions, such as 338(h)(10) election

5. The corporate income tax rate, personal income tax rate, dividend tax rate, and reinvestment ratio remain constant, forever.

These assumptions are in addition to the assumptions of the Gordon Growth Model that was utilized to develop the underlying formulas. We acknowledge that these assumptions will not always hold up in practice. For example, the buyer may not always be able to maintain the pass-through status (i.e. C-Corp acquiring the S-Corp). Therefore, the buyer may not always pay the full PTP (%). In addition, if investors do not plan to hold an investment into perpetuity, then capital gains taxes would need to be introduced into the equation. This would alter some of the conclusions of my analysis. Nonetheless, the formulas derived in this post demonstrate the complexity in valuing pass-through firms, and show that rational investors can be willing to pay meaningful price premiums and/or price discounts for pass-through firms depending upon the facts of the investment.

Categories: Uncategorized

Thoughts on Total Beta, Idiosyncratic Risk, and Valuation

March 6, 2012 3 comments

I just finished reading the two articles regarding Total Beta, or private company beta, in the January/February 2012 issue of the Value Examiner. The two articles highlight a serious intellectual debate within the valuation community regarding the application of financial concepts, principally Beta. Larry Kasper, MBA, Mac, CPA, CVA, CBA, authored the first article, entitled “Portfolio Theory and Total Beta – A Fairy Tale of Two Betas.”, which argues that Butler-Pinkerton’s Total Beta concept violates the underlying assumptions of modern portfolio theory. The second article written by Peter Butler, CFA, ASA, argues that Total Beta is a relevant concept for private companies, as individuals in this marketplace cannot diversify (an assumption of modern portfolio theory) and, therefore, require compensation for both systematic and idiosyncratic, or company specific risk. The authors of both articles make very compelling points. However, both authors also make some statements that require further scrutiny. The purpose of this post is as follows:

  1. First, I would like to simplify Larry Kasper’s compelling mathematical arguments into laymen terms (at least simplify them the best that I can) and explain why his arguments hold, assuming a very strict interpretation of modern portfolio theory (MPT). I will also briefly describe some important concepts from modern portfolio theory, namely the Efficient Frontier, the Capital Market Line, and the Security Market Line
  2. Second, I would like to briefly review Butler-Pinkerton’s arguments about Total Beta highlight some of the important assumptions, and raise some theoretically perplexing questions regarding Total Beta, again under a strict interpretation of MPT.
  3. Third, I would like to briefly review some empirical research from the academic community on traditional CAPM Beta and Total Beta and demonstrate that the assumptions of traditional CAPM Beta and MPT are often violated in practice, while some academic research supports the concept of Total Beta. I then raise questions about the implications for asset pricing, Larry’s arguments, and the use of Total Beta in the context of recent empirical research
  4. Finally, I would like to summarize these points and suggest areas for further research.

Understanding Larry Kasper’s Arguments

First, in order to understand Larry Kasper’s arguments, we must first review three very important concepts from modern portfolio theory: the Markowitz efficient frontier, the capital market line, and the security market line. These are discussed below.

The Markowitz Efficient Frontier

According to modern portfolio theory, investors only care about risk and return and, therefore, seek to maximize their utility by constructing efficient portfolios that generate the highest rate of return for a given level of risk. Investors construct these efficient portfolios by combining individual risky securities into portfolios and minimizing total risk for a given level of return through diversification. The possible combinations of these risky assets are represented by the investment opportunity set, which reflects all risky assets and combinations of risky assets in the marketplace. The chart below depicts a hypothetical Markowitz efficient frontier based upon available risky assets in the marketplace

The line represents the different combinations of risky assets (or portfolios of risky assets) that achieve the lowest variance, or risk, for a given rate of return. The individual risky assets that comprise these portfolios fall to the right of this line. The global minimum variance portfolio represents the lowest variance portfolio achievable in the marketplace given the available opportunity set of risky investments in the marketplace. The portfolios that fall on the line and above the minimum variance portfolio refer to the Markowitz Efficient Frontier. In an economy with no risk free asset, investors only select portfolios that fall along the Markowitz efficient frontier because these portfolios dominate all other risky portfolios and individual risky assets in terms of risk and return; that is, the portfolios on the efficient frontier earn the highest rate of return available in the marketplace for a given level of risk. Alternatively stated, this line represents those portfolios that have the lowest level of risk for a given level of return. In the Markowitz framework, all rational investors should select portfolios on this efficient frontier.  These investors are referred to as mean-variance efficient investors.

The Capital Market Line (CML)

When the risk-free rate is introduced into the Markowitz framework, however, investors can simplify their portfolio allocation decision and improve their risk/return profile by borrowing and lending at the risk free rate. In particular, investors no longer select any portfolio along the efficient frontier; instead, rational investors purchase one “ideal” risky portfolio, often referred to as the “market portfolio,” and adjust their risk and return profile by allocating between this ideal risky portfolio (i.e. the market portfolio) and the risk free asset. Investors prefer this methodology because the portfolios that are created through the combination of the risk free asset and the market portfolio dominate all other portfolios on the efficient frontier (other than the market portfolio) in terms of risk and return. This concept is illustrated in the chart below.

As one can see, the line connecting the risk free asset (i.e. line intersecting the vertical axis) to the “ideal” market portfolio (i.e. tangency point on the efficient frontier) has the highest slope and, therefore, dominates all other portfolio choices in the market in terms of risk and return. This line is referred to as the capital market line. According to modern portfolio theory, all investors should plot on this line; that is, all rational investors choose some combination of the “ideal” market portfolio and the risk free asset (i.e. portfolios represented by the line) because these portfolio combinations dominate all other portfolio combinations available in the marketplace in terms of risk and return. The equation for this line is expressed as follows:

Rp = Rf + Op/Om *(Rm – Rf)

Where:

Rp          = Expected Return on Portfolio

Rf         = Risk Free Rate of Return

Op        = Standard Deviation, or total risk, of Portfolio Returns

Om       = Standard Deviation, or total risk, of Market Returns

Rm       = Expected Return on Market

The capital market line is a critical concept of modern portfolio theory because it forms the basis of many pricing models in modern finance and capital budgeting, including the capital asset pricing model (which will be proven below). There are several important concepts about the capital market line, however, that must be understood.

  1. First, the “market portfolio,” represented by the tangent point in this model, does not refer to the S&P 500 (although the S&P 500 is often used as a proxy), but to a very specific market portfolio that is comprised of all risky assets in the economy (i.e. bonds, public stock, private stock, art, commodities, real estate, labor capital, etc.) held in proportion to their actual market weights. This market portfolio, by definition, is highly diversified, and, because of its diversification, has eliminated all idiosyncratic, or company specific risk. Therefore, the total risk of the market portfolio, represented by the term (Om), only reflects market, or systematic risk. No company specific risk is priced.
  2. Second, the individual portfolios (i.e. the portfolios on the line), do not refer to any portfolio, such as a portfolio of bonds, but to a very specific portfolio combination comprised of the risk free asset and the market portfolio. At one extreme, represented by the intersection at the vertical axis, is a portfolio 100% allocated to the risk-free asset. At the other extreme, represented by the tangent point on the Markowitz frontier, is a portfolio 100% allocated to the market portfolio as previously defined. Investors can also leverage the market portfolio by borrowing money at the risk-free rate and investing the proceeds into the market, thereby extending the line beyond the market portfolio.
  3. Third, since the individual portfolios in the CML simply represent combinations of the risk-free asset and the market portfolio, these individual portfolio combinations, by definition, are perfectly positively correlated with the market return (i.e. have a correlation coefficient equal to 1.0). Therefore, the “beta” of these portfolios is simply represented by the relative standard deviation of the portfolio and the market (i.e. Op/Om = Beta). Furthermore, these individual portfolios must, by definition, be fully diversified portfolios. Therefore, the total risk of the individual portfolios, represented by the term Op, only reflects market, or systematic risk. No company specific risk is priced.

This last point is particularly important because it allows us to reconcile the capital market line with the single index model (a pricing model in finance), which forms the basis of the capital asset pricing model. To demonstrate this concept (this is the most complicated math in this article), we begin by rearranging the terms from the equation above and generalizing the subscript from p (a portfolio) to i (a security), thereby rewriting the equation of the CML for any security in terms of realized returns as follows:

Ri – Rf = Oi/Om*(Rm-Rf) + ei

Where:

Ri         = Return of Security i

Rf         = Risk Free Rate

Oi         = Standard deviation of security i

Om       = Standard deviation of market portfolio

Rm       = Return of Market portfolio

ei          = Error term of returns (i.e. idiosyncratic or company specific risk)

By definition, the variance, or risk, of this equation can be expressed as follows (note that Rf drops out because the risk-free asset is constant and has no risk):

Oi^2 =(Oi/Om*pi,m)^2*(Om^2) + Oe^2 + 2Cov(Rm,ei)

Further notice that the non-systematic risk component, represented by ei, of this portfolio is, by definition, zero because the portfolios in the CML are fully diversified. Accordingly, the 2Cov(Rm,ei) term and the Oe^2 term drop-out from this equation, and the standard deviation of the security simplifies to the following equation:

Oi = (Oi/Om*pi,m)*Om

Substituting the above equation into capital market line, we discover the following:

Ri = Rf + ((Oi/Om*pi,m)*Om)/Om)*(Rm-Rf)

First, notice that Oi/Om*pi,m is simply the definition of Beta. Therefore, the capital market line can be alternatively expressed as follows:

Ri = Rf + Bi*Om/Om*(Rm-Rf)

Finally, the Om terms cancel and the final equation simplifies to the capital asset pricing model:

Ri = Rf + Bs * ( Rm – Rf)

This demonstrates that the capital market line is fully consistent with capital asset pricing model. However, further, notice that the correlation coefficient between the security (which plots on the CML) and the market is, by definition, 1.0 (see point 3 from above). Therefore, if we replace Bs, with Oi/Om*pi,m (i.e. the alternative formulation for beta), and substitute the correlation coefficient with 1.0, we obtain the capital market line:

Ri = Rf + Oi/Om*(Rm-Rf)

Again, this demonstrates that the capital market line is fully consistent with the capital asset pricing models. More importantly, however, we can equate these two relationships for fully diversified portfolios that plot on the CML, and derive the following mathematical relationship:

Rf + Bs*(Rm-Rf) = Rf + Os/Om*(Rm-Rf)

Solving for Bs, we discover that

Bs = Os/Om

Therefore, for fully diversified portfolios that plot on the CML, the Beta is equal to the standard deviation of the portfolio divided by the standard deviation of the market. This is Butler-Pinkerton’s Total Beta. This shows that Total Beta is consistent with CML when the correlation coefficient is 1.0 (the thrust of Larry’s argument). However, this concept does not apply to individual securities. To understand this concept, let us review a related concept in modern finance: the security market line.

Security Market Line (SML)

A related concept to the CML is the security market line, also referred to as the capital asset pricing model. The SML is a pricing equation for individual securities that is based upon the mathematical relationships derived from the CML (see above). In particular, the equation for the SML is expressed as follows:

E(R)s = Rf + Bs * ( Rm – Rf)

Where:

E(R)s   = Expected Return on the Stock

Rf         = Risk Free Rate of Returns

Bs        = Beta of the Stock

Rm       = Expected Return on the Market

The equation tells us that the expected return on an individual stock is simply equal to the risk free rate (first term of equation) plus the equity risk premium multiplied by the stock’s beta (second term of equation), or systematic risk factor. This line is effectively the index model derived from the CML. Again, recall that the beta of a stock can be expressed as follows:

Bs = Os/Om * ps,m

Where:

Bs        = Beta of Stock

Os        = Standard deviation, or total risk, of stock returns

Om       = Standard deviation, or total risk, of market returns

ps,m      = Correlation of stock and market returns

Therefore, substituting this equation into the original equation, we obtain the following:

 E(R)s = Rf + Os/Om * ps,m * ( Rm – Rf)

Where:

E(R)s   = Expected Return on the Stock

Rf         = Risk Free Rate of Returns

Os        = Standard deviation, or total risk, of stock returns

Om       = Standard deviation, or total risk, of market returns

ps,m      = Correlation of stock and market returns

Notice that this equation is essentially the capital market line, except that the correlation coefficient does not drop from the equation. The correlation coefficient does not drop from the equation because unlike a diversified portfolio on the CML, an individual security does not necessarily have perfect positive correlation with the market. More importantly, since investors in the Markowitz framework are presumed to be mean-variance efficient and hold the market portfolio (or some combination of the market portfolio and the risk free asset as predicted and described by the capital market line), these investors, by definition, only care about how an individual security’s total risk contributes to the total risk of their portfolio on the CML. Therefore, they price investments using economic relationships derived or related to the CML (namely the index model/capital asset pricing model). This model, as derived from the CML, includes the correlation coefficient. Investors require it because they only care about their systematic risk.

Understanding Larry Kasper’s Argument

If one accepts the premises of Modern Portfolio theory, then Larry Kasper’s arguments regarding Total Beta should become very clear. In particular, modern portfolio mandates that all investors are mean-variance efficient. Therefore, these investors diversify and develop portfolios that plot on the Markowitz Efficient Frontier. More importantly, since a risk-free asset exists, these investors further optimize by holding only one risky portfolio (i.e. the market portfolio), and allocate between this portfolio and the risk-free asset based upon their risk preferences (as demonstrated in the capital market line). Since investors plot on the capital market line, are mean-variance efficient, and well diversified, they should price individual risky assets on the basis of the economic relationships that exist on the capital market line. The capital market line tells us that the relevant economic relationship is the single-index model, or capital asset pricing model, which is defined as follows:

E(R)s = Rf + Os/Om * ps,m * ( Rm – Rf)

The term Os/Om*p,m is also referred to as Beta.  In the case of a diversified portfolio on the CML, the correlation coefficient of the portfolio and the market is, by definition, 1.0, and, therefore, the Beta of a diversified portfolio on the CML simplifies to Os/Om, or Butler-Pinkerton’s Total Beta. However, in the case of an individual risky asset, such as a privately owned business, the correlation coefficient does not always drop-out from the equation because the correlation coefficient for a private business is not always 1.0. Therefore, in the case of an individual risky asset, the pricing equation from the CML cannot simplify to Total Beta, but remains beta, unless the correlation coefficient of the security and the market is 1.0. Accordingly, Beta is the relevant risk metric within the context of modern portfolio theory because it quantifies the amount of risk that will actually contribute to the risk of the portfolio that investors hold on the CML. If investors priced individual risky assets using any metric other than Beta (i.e. such as Total Beta), then these securities, by definition, would be incorrectly priced unless their correlation coefficient with the market portfolio was equal to 1.0. This is the thrust of Larry Kasper’s argument, and is very compelling if we assume modern portfolio theory applies to all markets.

Understanding Butler-Pinkerton’s Argument

Peter Butler primarily argues against Larry Kasper’s arguments by suggesting that individual investors in the marketplace for privately owned businesses are not fully diversified, and, therefore, require compensation for all risk (both systematic and idiosyncratic). In particular, Peter Butler states that individual investors in the private market are undiversified price setters (a direct violation of the primary assumptions of CAPM and modern portfolio theory) and, therefore, require compensation for their total risk due to their inability (or conscious decision not to) diversify. Peter Butler then concludes that due to their lack of diversification that these investors require compensation equal to Total Beta, which is defined as follows:

TB = Os/Om

Where:

TB       = Total Beta

Os        = Standard deviation of security

Om       = Standard deviation of market

Therefore, in the private market, the relevant pricing equation is no longer traditional CAPM, but an “improved” version of CAPM using Total Beta, which is expressed as follows:

Ri = Rf + Os/Om*(Rm-Rf)

Notice that this equation is effectively the equation for the capital market line. Therefore, in effect, Peter Butler is suggesting that non-diversified investors in the private market (due to their lack of diversification) require compensation above traditional CAPM, In particular, these investors will require their excess return (i.e. return in excess of the risk-free rate) per unit of total risk (i.e. standard deviation), also known as the Sharpe Ratio, to equal that of the market, irrespective of the securities correlation with the market. In modern portfolio theory, investors only require an excess return per unit of Beta (i.e. systematic risk) equal to that of the market. Consequently, Total Beta results in a higher required rate of return than conventional beta, thereby lowering the price of these investments and causing them to plot above the security market line as contemplated in modern portfolio theory (which should not happen if the assumptions of modern portfolio theory hold). Appraisers should understand that Total Beta, by definition, makes the following assumptions regarding the private marketplace;

  1. Investors in the private marketplace do not diversify, even though empirical research demonstrates that diversification is highly advantageous, and, therefore, price investments outside of the framework of modern portfolio theory (i.e. since modern portfolio theory presumes that all investors are diversified).
  2. Investors who are diversified cannot or do not enter the market for private businesses and, therefore, compete away the idiosyncratic risk that is, according to Peter Butler, being priced by this market.
  3. Investors in this marketplace, because of their lack of diversification, therefore, price investments such that their excess return per unit of total risk equals that of the market portfolio as contemplated in the capital market line, which presumes that all investors are fully diversified and hold the market portfolio.

These assumptions may or may not be unreasonable (discussed later). However, some interesting questions are raised if these assumptions are accurate. First, why do investors in the marketplace elect not to diversify when there are clear economic benefits from doing so? Secondarily, if individual investors in the private market do not diversify, what is preventing outside investors from entering this market and extracting the “free-lunch” that exists due to the pricing of idiosyncratic risk. In fact, if idiosyncratic risk is fully priced, there would seem to be a wonderful investment opportunity for large individual investors: namely, enter the market for private businesses, purchase, say, 30 private businesses with low correlation that are being priced for “full” risk, create a diversified portfolio, eliminate the non-systematic risk component, and earn an excess return per unit of total risk that far exceeds that achievable in the marketplace. Clearly, something is preventing this from happening, or Total Beta is incorrect.

More importantly, if assumptions 1 and 2 are accurate, then we still need to demonstrate empirically whether assumption 3 is valid; that is, do undiversified investors really demand a return premium per unit of total risk equal to that of the market portfolio? If this is not the case, then we cannot really use Total Beta confidently. This problem is further compounded because we suffer a big weakness when we infer Total Beta from the market; that is, unlike the coefficient on regular beta, the coefficient term on Total Beta cannot be tested for statistical significance..

The Real Debate & Empirical Support

We can debate the points of Total Beta and Beta all day, but the “real” debate really centers around whether the assumptions of modern portfolio theory or Total Beta are accurate. If we accept, for example, a strict interpretation of modern portfolio theory, then Larry Kasper’s arguments should govern this debate; that is, all investors are Markowitz Efficient investors, they form diversified portfolios, hold the market portfolio, plot on the CML and price individual securities using the capital asset pricing model that is derived from the CML. If this is the case, then investors cannot price investments using Total Beta because Total Beta, by definition, presumes that investors are undiversified, which is a violation of CAPM. Alternatively, if modern portfolio does not hold (i.e. real world investors do not hold market portfolio, plot on CML, etc., etc.) then perhaps an alternative pricing equation exists. Perhaps investors do price idiosyncratic risk. Perhaps investors even use a formulation of Total Beta. To evaluate theses questions, let us consider what empirical research has actually demonstrated. First, let’s start with the assumptions of CAPM beta and modern portfolio theory, and, then, we will proceed to Total Beta and idiosyncratic risk.

The Assumptions and Empirical Support of CAPM Beta

First, despite the ubiquitous use of CAPM Beta and modern portfolio theory, the empirical record in support of these models is very weak (see The Capital Asset Pricing Model: Theory and Evidence, Eugene F. Fama and Kenneth R. French). For example, early research of traditional CAPM beta statistically rejected the model by demonstrating consistent “alphas” in the market model, suggesting that CAPM was not capturing all risk factors. In addition, financial researches have widely documented that CAPM beta is too “flat” in terms of predicting returns; that is, CAPM beta overstates the expected return of high beta stocks and understates the return of low beta stocks (see the Ibbotson year book, for example). Moreover, later research, including the work conducted by Fama and French and others, has demonstrated that other factors including size, value, and momentum are all important factors (in addition to the traditional market beta) in terms of pricing securities.

More importantly, the models are subject to many unreasonable assumptions that do not hold up in practice. For example, CAPM Beta is based upon the following assumptions:

  1. All investors aim to maximize economic utilities
  2. All investors are rational and risk-adverse
  3. All investors are broadly diversified across a range of investments
  4. All investors are price takers, i.e. they cannot influence prices
  5. All investors can lean and borrow unlimited amounts at the risk free rate
  6. All investors trade without transaction or taxation costs
  7. All investors deal with securities that are highly divisible into small parcels
  8. All investors assume all information is available at the same time to all investors

In addition, modern portfolio theory prescribes the following additional assumptions:

  1. All investors are only interested in maximizing mean for a given variance
  2. Asset returns are jointly normally distributed random variables
  3. Correlations between assets are fixed and constant forever
  4. All investors aim to maximize economic utility
  5. All investors believes about probability of returns match the true distribution of returns
  6. There are no taxes or transactions costs

However, it is has been widely documented, for example, that individual investors do not own broadly diversified portfolios as contemplated by CAPM Beta. More importantly, investors rarely ever own the market portfolio has described by the capital market line. In addition, common sense dictates that individuals do not only care about mean and variance, but also care about other variables such as liquidity and taxes. Moreover, academic researches know that asset returns are not normally distributed about the mean; stocks, for example, exhibit the “fat-tails” problem, wherein large negative returns occur much more frequently than predicted by a normal distribution. Furthermore, investors do not have homogenous expectations about the future return distributions, and, often, focus on long-term returns instead of their single period return.

The point is that if (a) many of the underlying assumptions related to traditional CAPM beta and modern portfolio theory are violated in practice and if (b) the empirical record generally does not support CAPM beta, then is CAPM beta really the proper metric for pricing risk? More importantly, if these assumptions are violated in practice can we really make the claim that Total Beta is wrong using modern portfolio theory to prove that it is wrong (as Larry Kasper does)? Or, are investors in the marketplace using a different “tool” to price risk? Moreover, if investors are not using CAPM to price investments in the marketplace (i.e. given all the violations in the marketplace), then what metric are they using? Perhaps total risk is important to investors as the concept of Total Beta suggests. Let us evaluate this concept next.

The Assumptions and Empirical Support for Total Beta

Perhaps interestingly, and despite all the criticism of Total Beta, there is actually a wide body of financial research that provides both empirical and theoretical support for the notion that investors not only price systematic risk, but idiosyncratic risk as well, especially when they are non-diversified. For example, in a recent working paper published by the National Bureau of Economic Research, entitled “Entrepreneurial Finance and Non-Diversifiable Risk,” Hui Chen, Jianjun Mio, and Neng Wang demonstrate that “non-diversified entrepreneurs demand both systematic and idiosyncratic risk premium.” In the concluding remarks of the paper, these authors assert

“Entrepreneurial investment opportunities are often illiquid and non-tradeable. Entrepreneurs cannot completely diversify away project-specific risks for reasons such as incentives and information asymmetry. Therefore, the standard law-of-one-price based valuation/capital structure paradigm in corporate finance cannot be directly applied to entrepreneurial finance…In addition to compensation for systematic risks, the entrepreneur also demands sizable premium for bearing idiosyncratic risks, which increase with his risk aversion, his equilibrium inside ownership, and the project’s idiosyncratic variance.”

Furthermore, research in even the public market may suggest that investors consider idiosyncratic risk when pricing investments. For example, in a paper entitled Idiosyncratic Risk Matters (Journal of Finance Volume 58, Issue 3, page 975-1008, June 2003) Amit Goyal and Pedro Santa-Clara find “a significant positive relation between average stock variance (largely idiosyncratic) and the return on the market.” Furthermore, Burton G. Malkiel and Yexio Xu develop theoretical and empirical work that supports the concept that idiosyncratic risk is priced. As the authors describe in the concluding remarks of their paper:

“Other things being equal, idiosyncratic risk will affect asset returns when not every investor is able to hold the market portfolio. Even after controlling for factors such as size, book-to-market, and liquidity, evidence from both individual US stocks and a sample of Japanese equities supports the predictions of our model. Most importantly, the cross sectional results demonstrate that idiosyncratic volatility variable is more powerful than either beta or size measures in explaining the cross section of returns [italics added]”

At the same time, however, the empirical research is still unclear as to whether total risk is priced. For example, in another National Bureau of Economic Research working paper, entitled “The Returns to Entrepreneurial Investment: A Private Equity Premium Puzzle?,” Tobias J. Moskowitz and Annette Vissing-Jorgensen, find that the returns to private equity are no higher than returns to public equity, even though entrepreneurial investment is extremely concentrated (i.e. poor diversification). Moreover, even if idiosyncratic risk is priced, none of these articles directly test whether this premium is directly proportional to the market price of risk (i.e. Rm-Rf/Om) from the capital market line, as Total Beta presumes.

The point here is that we really cannot determine whether idiosyncratic risk is priced (as Butler-Pinkerton suggest) or whether systematic risk is only priced (as Larry Kasper suggest); that is, research provides support for pricing idiosyncratic risk, but other research does not. Moreover, none of the empirical research indicates that investors price idiosyncratic risk using Total Beta, or at least test whether Total Beta has any predictive power in terms of pricing idiosyncratic risk. Therefore, while Butler-Pinkerton may have uncovered a useful pricing tool, they have not exactly supported whether this pricing equation is used by investors.

Concluding Remarks & Areas for Further Research

This article has presented the underlying arguments for and against Total Beta. In particular, this article demonstrates that under a very strict interpretation of modern portfolio theory that Total Beta should not hold; that is, if investors truly optimize portfolios pursuant to Markowitz, hold the market portfolio, and plot of the CML, that these investors should rationally price investments using traditional CAPM Beta and not Total Beta. Furthermore, the only time in which Total Beta will price risk properly in the context of modern portfolio theory is when the correlation of the security and the market portfolio as defined by the capital market line is equal to 1.0. This rarely happens. However, if modern portfolio theory does not hold (i.e. if investors do not diversify), then Total Beta may be a useful concept. However, if this is the case, investors must assume that non-diversified investors require an excess return per unit of risk equal to that of the market portfolio. This article further suggests that the “real” debate concerning Total Beta is related to whether one accepts the premises of modern portfolio theory. If modern portfolio holds, then Total Beta does not hold. However, if modern portfolio theory does not hold, then Total Beta may hold. This article presented empirical research that challenges the assumptions of modern portfolio theory and presented other research in support of pricing models that consider idiosyncratic risk. These articles provide mixed results, but generally support the notion that idiosyncratic risk is priced, especially for non-diversified investors. These articles, however, do not verify whether these investors require compensation equal to Total Beta, as suggested by Peter Butler.

Appraisers should be aware of the factors discussed in this article before ardently supporting either model (i.e. CAPM Beta or Total Beta). The reality is that the financial community is still debating the merits of these models; that is the empirical record generally does not support traditional CAPM Beta under a strict interpretation of modern portfolio theory, and there is no empirical basis to support the assumption that Total Beta properly compensates investors for lack of diversification. Therefore, as appraisers, we really need to develop answers to many unresolved questions before using either these models blindly. These include the following questions:

  1. Why do investors in the private marketplace elect not to diversify, especially when there are clear economic benefits from do so?
  2. What is preventing outside investors from entering the private market an competing away idiosyncratic risk being priced as Butler-Pinkerton claim?
  3. Why do investors in the public marketplace not follow modern portfolio theory or adhere to the assumption of the CAPM if these pricing models are accurate?
  4. Do investors actually price idiosyncratic risk, and, if so, how do they price it?
  5. Is Total Beta measuring idiosyncratic risk and is Total Beta able to reasonably predict market returns for undiversified investors?

Until these questions are answered, I believe that Total Beta and even traditional CAPM Beta should be used with relative caution by appraisers; that is, appraisers, in my opinion, should not automatically rely upon the company specific risk premiums suggested by Total Beta or automatically default to the cost of capital estimates obtained from traditional CAPM Beta because these equations are likely to provide false estimates of risk under the façade of mathematical accuracy. Unfortunately (or perhaps fortunately), our “reasoned judgment” is a better predictor of future required returns than the mathematical “precision” that these models provide. As Warren Buffet states in his essays

“Academics… like to define investment “risk” differently, averring that the relative volatility of a stock or portfolio of stocks-that is, their volatility as compared to that of a large universe of stocks. Employing data bases and statistical skills, these academics compute with precision the “beta” of a stock-its relative volatility in the past-and then build arcane investment and capital-allocation theories around this calculation. In their hunger for a single statistic to measure risk, however, they forget a fundamental principle: it is better to be approximately right than precisely wrong….

For owners of a business-and that’s the way we think of shareholders-the academics’ definition of risk is far off the mark, so much so that it produces absurdities….

In our opinion, the real risk an investor must assess is whether his aggregate after-tax receipts from an investments (including those he receives on sale) will, over his prospective holding period, give him at least as much purchasing power as he had to begin with, plus a modest rate of interest on that initial stake. Though this risk cannot be calculated with engineering precision, it can in some cases be judged with a degree of accuracy that is useful. The primary factors bearing upon this evaluation are:

  1. The certainty with which the long-term economic characteristics of the business can be evaluated
  2. The certainty with which management can be evaluated, both as to its ability to realize the full potential of the business and to wisely employ its cash flows
  3. The certainty with which management can be counted on to channel the reward from the business to the shareholders rather than to itself;
  4. The purchase price of the business;
  5. The levels of taxation and inflation that will be experienced and that will determine the degree by which an investor’s purchasing-power return is reduced from his gross return

These factors will probably strike many analysts as unbearably fuzzy since they cannot be extracted from a data base of any kind. But the difficulty of precisely quantifying these matters does not negate their importance nor is it insuperable. Just as Justice Steward found it impossible to formulate a test for obscenity but nevertheless asserted, “I know it when I see it.” So also can investors-in an inexact but useful way –“see” the risks inherent in certain investments without reference to complex equations or price histories.”

Perhaps we should follow his advice.

Categories: Cost of Capital