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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.

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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.

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Understanding the Fundamentals of Price-to-Revenue Multipliers

February 15, 2012 1 comment

The price-to-revenue multiplier is a popular valuation multiple, especially for service firms such as accounting practices or insurance companies. Mechanically, valuing a company using a price-to-revenue multiple is relatively straightforward (i.e. multiply the underlying revenue of the subject company by the revenue multiplier to derive an indication of value). However, proper application of the price-to-revenue multiplier is more complicated than the simple mathematics suggest, as profit margins have a significant theoretical impact on the size of the price-to-revenue multiplier.

To understand this concept, let’s first begin with the basic formula for valuing a business:

Price = EBIT*(1-T)*P / (Kw-G)

Where:

EBIT = Earnings before interest and tax (next year)

T        = Effective corporate tax rate

P       = Free cash flow payout ratio (i.e. after-tax free cash flow as % of after-tax EBIT)

Kw   = Weighted average cost of capital

G      = Long-term sustainable growth rate

This formula is effectively the Gordon Growth Model using after-tax free cash flow instead of dividends. The valuation formula essentially equates the value of a business to the present value of its expected after-tax cash flows. Dividing both sides by Revenue we discover that the price-to-revenue multiplier is simply:

Price / Revenue = EBIT*(1-t)*p/(Revenue*(Kw-g))

Notice, that EBIT*(1-t)/Revenue is the after-tax profit margin of a business. Therefore, defining this variable as M, the equation above simplifies to the following:

Price / Revenue = M*p/(Kw-G)

Where:

M = After-tax profit margin (next year)

p = Free cash flow payout ratio

Kw = Weighted Average Cost of Capital

G = Long-Term Growth

This formula indicates that the price-to-revenue multiplier is influenced by the following factors:

1. After-Tax Profit Margin

2. Free cash flow payout ratio

3. Weighted Average Cost of Capital

4. Long-Term Expected Growth

Factors 2-4 essentially impact every valuation multiple. Factor 1 (i.e. after-tax profit margin), however, is a variable that is unique to the price-to-revenue multiplier. In effect, the after-tax profit margin is the theoretical fundamental variable that drives the variation in price-to-revenue multiplies, holding all else constant. The formula shows that a firm with a high after-tax profit margin will command a higher price-to-revenue multiplier than that of a firm with a low-after-tax profit margin.

By way of example, consider a firm that is expected to generate an after-tax profit margin of 15%. The free cash flow payout ratio is 50%. The cost of capital is 10% and the expected long-term growth rate is 5%. Under these assumptions, a price-to-revenue multiplier is computed as follows:

Price-to-revenue = 15%*50%/(10%-5%) = 1.5x

Now, consider a firm that generates 10% after-tax profit margin. Under these assumptions, the price-to revenue multiplier is computed as follows:*

Price-to-Revenue = 10%*50%/(10%-5%) = 1.0x

As one can see, the price-to-revenue multiple is lower in the second example due to the company’s lower after-tax profit margin. In fact, since risk and growth are the same, the entire differential can be explained by the ratio of profit margins (i.e. 1.5x * ( 10%/15%) = 1.0x)

Since after-tax profit margins influence the price-to-revenue multiplier, appraisers should be cognizant of differences in after-tax profit margins when relying upon them in the market approach. For example, suppose the appraiser generates a sample of market transactions with a median price-to-revenue multiplier of 2.0x. The appraiser also notes that the median after-tax profit margin of the underlying companies is 10%. The appraiser is now valuing a similar business, whose profit margin is only 5%.  This business generates $10 million in sales. If the appraiser relied upon the sample of transactions without adjustment the appraiser would value the company at $20 million (i.e. 2.0x * $10 million). However, this would overvalue the subject company due to differences in after-tax profit margins. In fact, assuming the risk and long-term growth of the businesses was comparable, the appropriate price-to-revenue multiple for the subject business is actually 1.0x sales (i.e. 2.0x*5%/10%) due to its lower profit margin. Therefore, the correct value for this business is $10 million vs. the $20 million that one would obtain relying upon the unadjusted price-to-revenue multiples of the guidelines. The discrepancy is entirely attributable to differences in the after-tax profit margin.

This article demonstrates that the price-to-revenue multiple is influenced by the after-tax profit margin. In particular, a firm with a high after-tax profit margin will command a higher price-to-revenue multiple than that of a firm with a low after-tax profit margin, holding all else constant. If appraisers utilize price-to-revenue multipliers in the valuation of a business, they should focus on differences in the profit margin. If significant differences exist, then appropriate adjustment should be made to the multiples in order to derive a proper estimate of value, as failure to adjust the multiple for differences in profit-margins can lead to an erroneous conclusions of value.

*Theoretically speaking, the adjustment would be even greater than that described above because a lower-after profit margin, holding all else constant, would lower the return on equity and, therefore, the long-term sustainable growth rate of the business. I have assumed for illustrative purposes that the reduction in profit margin would be offset by improvements in asset efficiency such that the long-term growth rate would remain unchanged. If this assumption is violated than the appraiser should adjust the multiple for differences in growth as well.

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