Valuing Securities Using the Option Pricing Method by Frank Kiepura

DEPARTMENTS | Business Valuation

Valuing Securities Using the Option Pricing Method By Frank Kiepura

he option pricing model (OPM) is a popular and results are analyzed with respect to the Black-Scholes option commonly used model to allocate equity value pricing model and how changes to its parameters can affect to securities in the complex capital structures of allocations of value. privately held companies. Given the absence of active markets for privately issued securities, one The OPM Tof the challenges that valuation specialists face is deter- Many venture capital and private equity backed compa- mining how to allocate value to each specific security in nies are financed through a combination of different equity the capital structure. Although the OPM is one of the more securities, each of which provides its holders with unique common methods, choosing it as the optimal allocation tech- rights. Because of the high risks associated with investments nique is dependent on certain characteristics of the company. in privately held companies, investors often require different The OPM is most appropriate for companies with longer classes of equity that convey various rights to their holders. liquidity event timelines and access to a variety of exit In many capital structures of privately held companies, options; the model also works well in valuing option-like these classes of equity consist of preferred shares, common payoffs (e.g., common stock options and warrants). Using shares, options, and other securities, all of which provide an example of a complex privately held company, the author different sets of preferences and privileges. The purpose of constructed a hypothetical capital structure that incorporates granting different rights to different groups of investors is the characteristics associated with choosing the OPM. The to facilitate financing through capital investments at various

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07-2020-BusinessVal-Kiepura.indd 50 8/10/20 1:00 PM stages of enterprise development. By or control rights. Economic rights are ■ Risk-free rate. The risk-free rate is issuing different classes of securities designed to provide preferred share- derived from the yield on U.S. Treasury with different rights, companies create holders with superior economic returns instruments with a term equal to the time a situation where some securities have as compared to common shareholders. to liquidity utilized in the OPM. more economic privileges than others, These rights include preferred dividends, Step 4: Calculate each breakpoint and thus more value. redemption rights, conversion rights, value. The call option value for each The OPM is a method for allocating and participation rights; these are key breakpoint can be determined using the equity value to multiple classes of secu- to understanding liquidation preferences. Black-Scholes model. The call option rities in a company’s capital structure— Step 2: Calculate each breakpoint. value of the range between adjacent break- it is not a method for estimating the A breakpoint is the point at which each points is allocated to each security class. enterprise value for the entire company. class of equity reaches in-the-money Step 5: Allocate incremental option An overall equity value must first be status. The value in excess of any given values. Allocate the incremental option estimated using the valuation methods breakpoint is equal to a call option on values ownership interests to arrive at the under the asset, income, and market the total equity value of the company. per-share value of the equity securities. approaches before the OPM can be Step 3: Determine Black-Scholes applied. The OPM is a tool to allocate parameters. The OPM typically employs Comprehensive Illustration equity value; therefore, the value of the Black-Scholes option pricing model The following illustration is an exam- any debt should be removed from the to treat the different classes of securities ple of how to utilize the OPM to value enterprise value. as call options on the company's equity the equity securities of a privately held The OPM works by treating each value. The following are the inputs and company with a complex capital struc- class of security as a call option on the assumptions, some of which require a ture. To provide an example of a com- total equity value of the company. To significant degree of judgment: pany that a valuation specialist would accomplish this, OPM typically employs ■ Stock price. The stock price in the encounter in practice, securities with the Black-Scholes model to value the OPM is the total equity value of the sub- different issue dates and their own par- call options. The Black-Scholes model ject. The total equity value is estimated ticular set of economic rights are exam- incorporates the parameters of stock from the application of traditional valu- ined. The selection of a specific method price, exercise price, time, volatility, ation methods under the asset, income, to allocate value among different equity and risk-free rate to determine the price and market approaches. If the company's securities depends upon various factors of a European call option. In the OPM, enterprise value contains debt, the fair related to the specific facts and circum- the stock price becomes the company’s value of the debt should be subtracted stances of the company and its various total equity value, and the exercise price to derive the equity value. classes of equity securities. becomes the liquidation preference of ■ Exercise price. The exercise price Consider the hypothetical example of the security. When the OPM utilizes represents the equity value break point Cotopaxi Tech, a venture capital-backed the Black-Scholes model, it becomes determined in Step 2. startup, initially funded with the owners’ highly sensitive to estimates for time to ■ Time to liquidity. A single point is esti- capital in 2010. The company’s owners liquidity and volatility. The following mated for the liquidity event. Estimates currently hold 2 million common shares, is a detailed step-by-step process for for the time to liquidity can generally be which is the total amount of common applying the OPM. derived by reference to management’s shares issued and outstanding. Step 1: Analyze the capital structure. expected time to exit scenarios. From 2010 to 2016, Cotopaxi Tech Identify and understand the subject ■ Volatility. Volatility can be derived issued several financing rounds consist- company's equity interests outstanding from observed volatilities of guide- ing of Series A preferred stock. As of as of the valuation date. In addition to line public companies, benchmarks, the valuation date, there were 15 million common equity, a company's capital and other sources. For early stage Series A shares outstanding with a liq- structure can include preferred shares, companies, volatility will often uidation preference of $1.30 per share. restrictive shares, options, warrants, and approach the upper end of observed Series A shareholders are entitled to other dilutive securities. Estimating the volatilities for guideline public com- either participate in its liquidation pref- value of each of these different classes of panies. The range of guideline public erence or convert into common stock on equity requires an analysis of the rights company stock returns should equal a one-for-one basis. Series A preferred associated with each class. Generally, the same period as the estimated time shareholders earn a cumulative dividend preferred shareholders have economic to liquidity. of 10% per annum, payable upon the

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07-2020-BusinessVal-Kiepura.indd 51 8/10/20 1:01 PM DEPARTMENTS I Business Valuation

EXHIBIT 1 - Capital Structure implementing the OPM. Information regarding the rights, privileges, and LiquidaonExercise Shares Conversion Fully DilutedPercentage of preferences of securities can be found Preference PriceOutstanding Rao Shares (FDS)FDS in the company’s purchase agreements, Series B $7,500,000 - 5,000,000 1.0 5,000,000 13.9% Series A Dividend $6,000,000 - - - - 0.0% shareholder agreements, and discussions Series A $19,500,000 - 15,000,000 1.0 15,000,000 41.7% with management. Common Shares - - 2,000,000 1.0 2,000,000 5.6% Series B holders have preference over Opons - 0.75 1,000,000 1.0 1,000,000 2.8% Warrants I- 2.00 10,000,000 1.0 10,000,000 27.8% all other securities and receive a return Warrants II - 3.00 3,000,000 1.0 3,000,000 8.3% on their original investment before distri- Total $33,000,000 36,000,000 36,000,000 100.0% butions can be made to any of the other equity securities. Upon the receipt of their initial investment, the shares con- EXHIBITEXHIBIT 2 - Breakpoint2 - Break-point Calc Calculationsulaons vert to common shares on a one-for-one Break-points #1 #2 #3 #4 #5 #6 #7 #8 basis. Series A cumulative dividends are Preferred Securies Series B$0 $7,500,000 $7,500,000$7,500,000$7,500,000 $7,500,000 $7,500,000 $7,500,000 due upon a qualifying liquidation event Series A Dividend - - 6,000,000 6,000,000 6,000,000 6,000,000 6,000,000 6,000,000 Series A- - - 19,500,000 19,500,000 - - - and are payable prior to the liquidation Liquidaon Preference A$0 $7,500,000 $13,500,000 $33,000,000 $33,000,000 $13,500,000 $13,500,000 $13,500,000 preference or conversion of series A to Equity Shares Series B- 5,000,000 5,000,000 5,000,000 5,000,000 5,000,000 5,000,000 5,000,000 common shares. Series A holders have Common Stock- - - 2,000,000 2,000,000 2,000,000 2,000,000 2,000,000 Opons - - - - 1,000,000 1,000,000 1,000,000 1,000,000 the option to either participate in the Series A- - - - - 15,000,000 15,000,000 15,000,000 Warrants I------10,000,000 10,000,000 proceeds of a liquidation event or convert Warrants II ------3,000,000 Total Outstanding Shares B- 5,000,000 5,000,000 7,000,000 8,000,000 23,000,000 33,000,000 36,000,000 into common shares on a one-for-one Price Per ShareC - - - - 0.75 1.30 2.00 3.00 basis upon the closing of such an event. Incremental Equity BXC- - - - $6,000,000 $29,900,000 $66,000,000 $108,000,000 Less Proceeds From Opons - - - - (750,000) (750,000) (750,000) (750,000) Common shares receive distributions pari Less Proceeds From Warrants I------(20,000,000) (20,000,000) Less Proceeds From Warrants II ------(9,000,000) passu with other converted and dilutive Incremental Equity, NetD - - - - $5,250,000 $29,150,000 $45,250,000 $78,250,000

Break-points A +D $0 $7,500,000 $13,500,000 $33,000,000$38,250,000 $42,650,000 $58,750,000$91,750,000 securities. Options, and warrants, will be exercised based on their lowest to highest exercise prices. The fully diluted number sale or liquidation of the company; as basis. Cotopaxi Tech granted options of common shares in this illustration is 36 of the valuation date, the cumulative to officers and employees to purchase million. Exhibit 1 summarizes Cotopaxi dividends are $6 million. The company common shares under its stock option Tech’s capital structure. issued warrants in connection with the plan. The options have an exercise price sale of Series A that entitles the holder of $0.75 per option and expire 10 years Step 2: Calculate Each to purchase the company’s preferred from the grant date. As of the valuation Breakpoint shares; as of the valuation date, Cotopaxi date, there are 1 million options out- When allocating a company’s total Tech had 10 million warrants outstand- standing and convert to common shares equity value to multiple classes of secu- ing at an exercise price of $2.00 per on a one-for-one basis. rities, a valuation analyst should con- share (Warrants I) and 3 million war- Cotopaxi Tech retained the services sider ownership in each security as a rants outstanding at an exercise price of of a valuation specialist to estimate right to receive a portion of the proceeds $3.00 per share (Warrants II). the fair value of its common stock from a liquidity event. A liquidity event In July 2016, the company issued as of June 30, 2018. The company’s can be achieved through dissolution, 5 million Series B preferred shares, enterprise value was estimated using strategic sale, initial public offering with a liquidation preference of $1.50 the asset, income, and market approach (IPO), or other transaction. In a privately per share. According to the sharehold- methods, resulting in a total equity held company, there can be many differ- er’s agreement, Series B investors are value for the company of approxi- ent classes of equity. To allocate value to receive a return on their invest- mately $40 million. to each one of these classes of equity, ment before distributions are made to a breakpoint needs to be determined all other shareholders. After an event Step 1: Analyze the Capital for each security. A breakpoint can be resulting in a return on their invest- Structure thought of as the point where a security ment, Series B automatically convert Understanding the company’s cap- reaches in-the-money status through a to common shares on a one-for-one ital structure is key to successfully hypothetical liquidity event. In other

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07-2020-BusinessVal-Kiepura.indd 52 8/10/20 1:01 PM words, each security reaches certain EXHIBIT 3 - Allocaon Matrix economic milestones at different levels Series A Common of proceeds; in the OPM, these levels Break-points Series B Dividend Series A Shares Opons Warrants IWarrants II of proceeds from a liquidity event form $0.0to $7,500,000 100.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% $7,500,000 to $13,500,000 0.0% 100.0% 0.0% 0.0% 0.0% 0.0% 0.0% the breakpoints. Exhibit 2 calculates the $13,500,000 to $33,000,000 0.0% 0.0% 100.0% 0.0% 0.0% 0.0% 0.0% breakpoints for each class of security $33,000,000 to $38,250,000 71.4% 0.0% 0.0% 28.6% 0.0% 0.0% 0.0% $38,250,000 to $42,650,000 62.5% 0.0% 0.0% 25.0% 12.5%0.0%0.0% after the liquidation preferences have $42,650,000 to $58,750,000 21.7% 0.0% 65.2% 8.7% 4.3%0.0%0.0% been settled in full and the equity secu- $58,750,000 to $91,750,000 15.2% 0.0% 45.5% 6.1% 3.0%30.3%0.0% rities begin to participate in the proceeds. $91,750,000 to < 13.9%0.0% 41.7% 5.5% 2.8% 27.8%8.3% Breakpoint #1. In this example, proceeds from a liquidity event are paid tions owed to the Series B and Series A than their exercise price of $750,000. first to the holders of Series B, to the preferred shareholders have been paid. The Therefore, the breakpoint #3 floor of extent of each holders’ unreturned cap- fourth breakpoint to the common share- $33 million plus $5.25 million gives us ital investment. Series B shareholders holders begins at $3 million, which is the the next breakpoint at $38.25 million. have a liquidation preference of $1.50 amount after the $7.5 million payoff for Breakpoint #6. Series A can benefit per share, with 5 million shares out- Series B, $6 million payoff for the Series economically from its liquidation pref- standing; as a result, Series B holders A cumulative dividend, and $19.5 million erence or its conversion to common will receive proceeds up to $7.5 million payoff for Series A. Series B also benefits shares. Breakpoint #6 is the incremental before any of the other equity securities from its conversion to common shares amount of liquidation proceeds where can participate. Thus, the first breakpoint once the shareholders have received all Series A shareholders will find it eco- is $0.0 to $7.5 million. their unreturned capital investment. After nomically beneficial to forgo its liquida- Breakpoint #2. After the Series B Series B converts to common shares, there tion preference and convert to common shareholders receive their liquidation will be 7 million total shares outstanding; shares. For Series A shareholders to find preference, additional proceeds above the therefore, all proceeds from $33 million it more beneficial to convert to common $7.5 million breakpoint will be paid to the to the next breakpoint will be distributed shares, they must receive more than their holders of Series A cumulative preferred to the common and Series B shareholders liquidation preference of $19.5 million. dividends. Any cumulative dividends due on a pro rata basis. The conversion of Series A will add 15 at the liquidation date are to be paid prior Breakpoint #5. The next breakpoint million shares, for a total of 23 million to the liquidation preference of the Series corresponds to the exercise of out- shares outstanding. When Series A con- A shareholders. As of the valuation date, standing options. Under the OPM, it verts to common shares, Cotopaxi Tech there were $6 million in unpaid cumu- is assumed that all options and other will no longer be liable for the original lative dividends; therefore, the payoff to dilutive securities as of the valuation liquidation preference of $19.5 mil- the Series A cumulative dividend will date will be exercised upon a liquidity lion and the liability prior to the equity constitute the second breakpoint from event. Cotopaxi Tech’s options have participation will decrease to $13.5 $7.5 million to $13.5 million. an exercise price of $0.75 per share, so million (i.e., the Series B and Series A Breakpoint #3. The third payoff will the exercise of the options will add 1 dividend). For Series A to receive a pro be to the holders of Series A preferred million shares to the total outstanding rata amount of $19.5 million, additional shares, to the extent of each holders’ shares and provide cash proceeds of net incremental equity of $29.15 million unreturned capital investment. Series A $750,000. The cash proceeds from the will need to be added to a new floor shareholders have a liquidation preference exercise will reduce the incremental preference of $13.5 million. At break- of $19.5 million, and any proceeds above equity by $750,000, to $5.25 million, as point #6, Series A preferred stock will the second breakpoint will go to Series A calculated in Exhibit 2. Options will be constitute 65.2% of shares outstanding shareholders. The third breakpoint will be exercised only when the 8 million total and will share proceeds pari passu with from $13.5 million to $33 million. outstanding shares have a per-share price the common shares and option holders. Breakpoint #4. Common equity will of $0.75. For the option holders to find it Breakpoint #7. Like options, Warrants begin to participate in liquidation pro- economically beneficial to exercise, the I have an exercise price, and therefore ceeds once liquidation preferences of all liquidity event would have to provide will dilute the company’s equity and the preferred shares have been fulfilled. an additional $5.25 million in proceeds. provide cash proceeds. There are 10 The common shareholders will receive Before that point, the option’s pro rata million Warrants I with an exercise proceeds when the company’s distribu- allocation of proceeds would be less price of $2.00. Warrants I holders will

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EXHIBIT 4 - Call Opon Value

Break-points #1 #2 #3 #4 #5 #6 #7 #8 Total Firm Value $40,000,000 $40,000,000 $40,000,000 $40,000,000$40,000,000 $40,000,000 $40,000,000 $40,000,000

Break-points 0 7,500,000 13,500,000 33,000,000 38,250,000 42,650,000 58,750,000 91,750,000

Black Scholes Inputs Years to Liquidity Event 3.03.0 3.03.0 3.03.0 3.03.0 Volality 80.0% 80.0% 80.0% 80.0% 80.0% 80.0%80.0% 80.0% Risk-Free Rate 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0%

Call Opon Value 40,000,000 33,935,184 30,393,433 22,895,835 21,481,600 20,422,584 17,279,014 13,029,043 Incremental Opon Value $6,064,816 $3,541,751 $7,497,598$1,414,236$1,059,016 $3,143,570 $4,249,970 $13,029,043

therefore pay $20 million to convert to Step 3: Determine Black-Scholes value. The calculation of value at each 10 million common shares. The con- Parameters breakpoint can be determined using the version of Warrants I will increase the Once the capital structure has been Black-Scholes model. In our illustration, total to 33 million shares outstanding. analyzed and the breakpoints for each the option to purchase the company for Consequently, a liquidation event would class of equity have been calculated, $0 has a value of $40 million, the option have to add $66 million in proceeds, less the next step is to derive the value of to purchase the company for $7.5 mil- the proceeds from the option exercise of the call options using the Black-Scholes lion has a value of $33,935,184, and so $750,000 and less the proceeds from the model. The Black-Scholes model has the on. The Black-Scholes model calculates Warrants I exercise of $20 million for following critical inputs: the total equity the value of each breakpoint in its entire a net incremental increase of $45.25. value for the company; the exercise price amount. However, liquidation prefer- Warrants I holders will not exercise for each breakpoint; the expected time to ences are incremental, as they increase their option to convert to common shares liquidity; the volatility of the equity for from one liquidity event to the next. For until liquidation proceeds reach $58.75 the company; and the risk-free rate. example, proceeds from $0.0 to $7.5 million and above. Below that amount, In this illustration, Cotopaxi Tech’s million from a liquidity event are allo- their pro rata share of proceeds would equity value is estimated to be $40 mil- cated 100% to series B, yielding break- not equal their cost to exercise. lion using the asset, market, and income point #1. Any proceeds between $7.5 Breakpoint #8. Warrants II will be approaches to valuation. In the OPM, million and $13.5 million go to Series the final security to exercise, at $3.00 the exercise prices in the Black-Scholes A cumulative dividends, which yields a per warrant. Warrants II holders will model correspond to the breakpoints range of $6 million between breakpoints need to pay $9 million to convert to calculated in Exhibit 2. Management #2 and #3. In Exhibit 4, a range is the common shares; at this point, equity estimated that a liquidity event will difference between two adjacent break- will be fully diluted at 36 million shares. occur in three years. A volatility of points and it is the value of these ranges Additional proceeds of $108 million, net 80% for the equity of the company was that are allocated to the securities in contributions from option and warrants, estimated using observed volatilities of order to determine their per-share value. will produce net incremental equity of comparable guideline public companies The option value of the ranges can be $78.25 million, at which point it will be and a risk-free rate of 2.0% was derived derived by calculating the incremental economically viable for the holders of from the yield on U.S. Treasury Bonds value of the option. As shown in Exhibit Warrants II to exercise. as of the valuation date. 4, the incremental option value for Exhibit 3 summarizes the break- breakpoint #1 is $6,064,816 (40,000,000 points and illustrates the ownership Step 4: Calculate Each Break- – 33,935,184) and $3,541,751 for break- interest of each security. The owner- point Value point #2 (33,935,184 – 30,393,433). ship interest is calculated by dividing The Black-Scholes model can now The same approach is used to deter- an individual security’s shares by the be applied to determine the call option mine the value of all the ranges up to fully diluted shares outstanding at value for each breakpoint. The results of breakpoint #7. The entire amount of each breakpoint. These percentages the OPM are summarized in Exhibit 4. the call option value for breakpoint are used to allocate value to each The OPM treats each breakpoint as #8 is used, because this is the point security in Step 5. a call option on the company’s equity where all securities in the capital struc-

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07-2020-BusinessVal-Kiepura.indd 54 8/10/20 1:02 PM ture participate in the EXHIBIT 5 - Allocate Incremental Opon Value

proceeds of a liquida- Break-points tion event. The sum of Ownership Interests #1 #2 #3 #4 #5 #6 #7 #8 Series B 100.0% 0.0% 0.0% 71.4%62.5% 21.7%15.2% 13.9% the incremental option Series A Dividend 0.0% 100.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% values should equal the Series A 0.0% 0.0% 100.0% 0.0% 0.0% 65.2%45.5% 41.7% Common Shares 0.0% 0.0% 0.0% 28.6%25.0% 8.7% 6.1% 5.6% company’s total equity Opons 0.0% 0.0% 0.0% 0.0% 12.5%4.3%3.0%2.8% Warrants I0.0%0.0%0.0%0.0%0.0%0.0%30.3% 27.8% value of $40 million. Warrants II 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 8.3% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%

Step 5: Allocate Incremental Opon Value Series B $6,064,816 $0 $0 $1,010,168 $661,885 $683,385 $643,935 $1,809,589 Incremental Option Series A Dividend - 3,541,751 ------Values Series A - - 7,497,598 - - 2,050,154 1,931,805 5,428,768 Common Shares - - - 404,067 264,754 273,354 257,574 723,836 The final step is to Opons - - - - 132,377 136,677 128,787 361,918 allocate the incremen- Warrants I------1,287,870 3,619,179 Warrants II ------1,085,754 tal option values to Total $6,064,816 $3,541,751 $7,497,598$1,414,236$1,059,016 $3,143,570 $4,249,970 $13,029,043 each security, based upon their ownership interests at each breakpoint. These security with respect to EXHIBIT 6 - Per Share Values ownership interests were calculated the Black-Scholes model. and summarized in Exhibit 3. Exhibit 5 The most influential fac- AggregatePer Share Equity ValueFDS Value demonstrates the process of allocating tor on option price is the Series B $10,873,779 5,000,000 2. 17 value to each security. current market value of Series A Dividend $3,541,751 - - As shown in Exhibit 5, series B is the underlying asset. As Series A $16,908,325 15,000,000 1. 13 allocated the entire amount of the first can be observed with the Common Shares $1,923,585 2,000,000 0. 96 Opons $759,759 1,000,000 0. 76 breakpoint, Series A dividends are then per-share values, securi- Warrants I $4,907,048 10,000,000 0. 49 allocated 100% of breakpoint #2, Series ties with senior ranking Warrants II $1,085,754 3,000,000 0. 36 A preferred shares receive 100% of liquidation preferences Total $40,000,000 36,000,000 breakpoint #3 and so forth. The total are the most valuable. allocated amounts should equal the incre- For example, the high- mental option values in Exhibit 4. Adding est-ranking liquidation preference of because the loss is limited to the price the allocated values across for each Series B results in those shares having of the option. This means that the junior security will yield their total aggregate the highest value at $2.17 per share. This ranking liquidity preference sharehold- values. The aggregate values can be con- makes sense, because the call option’s ers will benefit more from the possi- verted to per-share amounts, as shown strike price is the lowest relative to the bility that the company’s value will in Exhibit 6. market value of the underlying asset. converge to their breakpoints. Similarly, There are several key metrics in In other words, Series B’s breakpoint call options become more valuable as assessing the OPM output of Exhibit 6. is below the company’s total equity the time to expiration increases, because First, the sum of the allocated vales to value, putting it in-the-money. As the there is more time for the value of the each security must equal the company’s company’s securities move from in-the- underlying asset to increase. In both total equity value of $40 million. Second, money to out-of-the-money relative to situations, per-share value shifts from the sum of the fully diluted shares (FDS) the company’s equity value, their per senior-ranking securities to junior-rank- equals 36 million, which corresponds to share prices decrease accordingly. ing securities. As mentioned before, the the amount from the capital structure The Black-Scholes model is also OPM is an allocation model and thus a table. Finally, the rank of the per-share highly sensitive to volatility and time. zero-sum game. The only impact that values is consistent with each security’s The reason for this shift is that, as changes to volatility and time can have liquidation preferences, exercise prices, volatility increases, there is a greater is on the allocation of value between and conversion rights. probability that the company’s value the securities. ■ will increase or decrease. The owner Market Value, Volatility, and Time of a call option benefits from increases Frank Kiepura, CPA/ABV, is a senior The results in Exhibit 6 confirm in stock price, but has limited downside manager in the valuation advisory prac- expectations regarding the value of each risk in the event of price decreases, tice at Lieberman LLP, New York, N.Y.

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