aluatTRATEGIES ıo NOVEMBERn/DECEMBER 2010 vS

MEASURING THE INCREMENTAL DISCOUNT FOR LACK OF MARKETABILITY

Steven D. Kam and Darren Wan STEVEN D. KAM AND DARREN WAN

Recent developments in determining the discount for lack of marketability rely on quantifiable parameters of a specific asset rather than empirical data on historical transactions.

4 VALUATION STRATEGIES MEASURING THE INCREMENTAL DISCOUNT FOR LACK OF MARKETABILITY

VALUATION STRATEGIES 5 Thecurrent thinking regarding the adjustment or discount ue of $100 and is restricted from sell- the underlying asset price gives the for the lack of marketability (DLOM) ing the security for five years. The investor the right, at the maturity of has been based, in part, on empirical investor purchases a five-year put the , to sell the asset at the studies of private placements involv- for $35, providing her the right (but price determined at the applicable val- ing securities restricted under Rule 144 not the obligation) to sell the asset for uation date. The of the under- (known as “letter stock”) that compare $100 at the end of five years. By paying lying asset can be measured by the stock’s private placement price to its $35, or 35% of the marketable security calculating the implied volatilities of freely-traded price. The FMV Restrict- price, the investor has assured herself traded options on that asset or volatil- ed Stock Study (“FMV Study”) and liquidity2 at the end of the option term ities calibrated from traded options on Valuation Advisors’ Pre-IPO Study and has eliminated all downward pric- comparable assets. Alternatively, the (“Pre-IPO Study”) are examples of ing risk. The investor’s overall invest- volatility can be measured by calcu- these empirical studies. More recent ment position at the time of the option lating the asset’s historical trading insight, however, has led to the devel- purchase is reduced by 35% from $100 volatility, or, if no trading data exists, opment of approaches that do not rely to $65, but the entire risk attendant to by calculating the historical trading on empirical data from historical trans- identifying a buyer to purchase the volatility of comparable assets. The actions. Instead, these approaches focus security has been neutralized. term of the protective put option is on quantifiable parameters that are spe- As a proxy for DLOM, the rationale the holding period for the security. cific to the asset being valued. One such in this example assumes that an investor Last, the risk-free rate corresponds approach applies the Black-Scholes- who has the choice between purchasing with the term of the put option. Exhib- Merton (BSM) option pricing method- two identical securities—except that it 2 compares the relationship of the ology to value protective put options. one is completely marketable and the inputs to the BSM model to the price other is completely nonmarketable— of the protective put option and the would pay 35% less for the nonmar- corresponding DLOM. Use of Protective Put Options to ketable security than she would for the Quantify Lack of Marketability marketable security. The investor would In a protective put option calculation, pay 35% less for the nonmarketable Application to the cost of purchasing a European put security because he would pay 35% of Nonmarketable Securities option to protect against downward the security’s price to purchase a pro- Protective put option methodology is price changes in the underlying asset tective put option to assure liquidity particularly useful for investors with is considered a proxy for the cost of and downside price protection at the assets that are restricted from sale, marketability. For private entity valu- end of her investment holding period. whether by legal restrictions or mar- ation purposes, the cost of a put option ket conditions. For example, assume is the price an investor would pay, on that a valuation has been performed the valuation date, for the right to sell Using the on a nonmarketable security (Security the underlying asset at a guaranteed Black-Scholes-Merton Model A) using the comparative analysis of price at the maturity of the option One way to calculate the cost of a pro- publicly traded companies. The value of (thus assuring its marketability). The tective put option is through the BSM Security A was determined to be $100 investor’s net position1 as of the valu- model (see Exhibit 1). This is com- at the marketable, minority level of val- ation date is the value of the underly- prised of six inputs: ue. To account for the differences in ing asset minus the price of the put 1. Security price. option. From this calculation, the 2. . 1 “Position” is an investor’s stake in a particular DLOM can be observed as the price 3. Volatility. security or market. A long position equals the number of shares owned; a short position of the put option divided by the mar- 4. Term. equals the number of shares owed by a dealer ketable value of the underlying asset. 5. Risk-free rate. or an individual. Dictionary of Finance and Investment Terms, 4th ed. (Barron’s Educational For example, assume an investor 6. Dividend yield. Series, 1995). owns a security with a marketable val- In a protective put option, the strike 2 Discussion of the differences between mar- price is equal to the value of the under- ketability and liquidity is outside of the scope STEVEN D. KAM, ASA, is a managing director and of this article. The terms will be used inter- DARREN WAN, CFA, is a vice-president at Cogent lying asset. In a business valuation con- changeably without distinction for the purpos- Valuation, San Francisco, California. text, setting the strike price to equal es of this article.

6 VALUATION STRATEGIES November/December 2010 INCREMENTAL DLOM EXHIBIT 1 Black-Scholes-Merton Model

p = c – S + Xe-rT c = Se-qTN(d1) − Ke-rTN(d2) d1 = ln(S/K) + (r − q + σ 2/2)T / σ√T d2 = d1– σ√T Where: p=price of a written put c=price of a written call S=value of the underlying asset at the applicable valuation date K= price or strike price σ = annualized volatility of the underlying stock T=time to (in years) r=continuously compounded risk-free rate q=continuously compounded dividend yield N= cumulative probability function for a standardized normal distribution e=a mathematical constant; the base of the natural logarithm

EXHIBIT 2 Effect of Inputs to the BSM Model on Put Option Prices and DLOM

Put Option DLOM Value ᭝ No Effect No Effect Price of the underlying securitya (1) S ᭞ No Effect No Effect ᭝ No Effect No Effect Exercise pricea (2) K ᭞ No Effect No Effect

Annualized volatility ᭝ Increases Increases (3) σ of the underlying stock ᭞ Decreases Decreases ᭝ Increases Increases Time to expiration (in years) (4) T ᭞ Decreases Decreases

Continuously compounded ᭝ Decreases Decreases (5) r risk-free rate ᭞ Increases Increases

Continuously compounded ᭝ Increases Increases (6) q dividend yield ᭞ Decreases Decreases

a The asset and exercise price are set equal to each other because the asset price is the value the investor desires to protect. Since S and X maintain a constant 1:1 relationship, changes to S and X do not affect the relationship between the price of the underlying security and the put option value.

EXHIBIT 3 Lack of Marketability Calculation for an LP Interest

Price of the underlying asset at the valuation date /a/ $100.00 Strike price $100.00 Annualized volatility of the underlying asset /b/ 0.45 Option term (in years) 5.00 Continuously compounded risk-free rate /c/ 4.00% Continuously compounded dividend yield 0.00% Value of the Protective Put Option $26.60 Put Option Divided by Underlying Asset Price 26.60% Marketable, Minority Value of LP Interest $100.00 Less: Cost of Marketability ($100.00 * 26.6%) ($26.60) Non-Marketable, Minority Value of LP Interest $73.40 /a/Marketable, minority value of the LP or LLC interest, not net asset value. /b/Volatility matched to option term. /c/Risk-free rate to maturity is matched to option term.

INCREMENTAL DLOM November/December 2010 VALUATION STRATEGIES 7 Security A’s marketability relative to Application to LP or LLC Interests marketability, albeit partial mar- the marketable securities to which it Protective put option methodology can ketability. For example, unlike a secu- was compared, a complete DLOM must also be used to determine DLOMs for rity listed on the NYSE, a RELP trades be taken on Security A to develop a limited partner interests (“LP inter- periodically (not daily), privately, and nonmarketable, minority indication. In ests”) and nonmanaging, member in small, irregular volumes. Further, Exhibit 3, a protective put option is cal- interests (“LLC interests”) in closely unlike a security listed on the NYSE, culated for Security A, the underlying held entities that own real estate, secu- the purchase or sale of an LP or LLC asset. An investor in Security A has no rities, or a combination of both. First, interest is a customized transaction public or recognized market to sell the the marketable, minority value of the between parties without the benefits asset and faces restrictions on sale or LP or LLC interest is determined, typ- of an established marketplace. There- transfer. A term of five years represents ically using comparable publicly trad- fore, RELP interests are neither com- the holding period for the investment. ed interests. Then, depending on the pletely marketable (because they do For a security with a five-year hold- degree of marketability of the compa- not trade in meaningful volume over ing period, the investor can purchase rable publicly traded interests, an public exchanges such as the NYSE), a five-year put option for 26.6% of the appropriate DLOM is applied to the nor completely nonmarketable underlying asset price, thereby reduc- LP or LLC interest. (because a secondary market exists for ing his or her overall net position by LP and LLC interests share several their resale). Here, trades are made 26.6% but eliminating illiquidity risk common attributes with real estate lim- with far less frequency and with a and downward pricing risk. By imple- ited partnership (RELP) interests trad- greater level of uncertainty as to par- menting this investment strategy, it is ed in the secondary market.3 Over ticipant interest and pricing than assumed that the investor would pay up time, the pricing of RELP interests in demonstrated over public exchanges. to 73.4% of the marketable price, or the secondary market has been con- No identifiable market exists for close- $73.40, for the nonmarketable LP inter- sidered an appropriate proxy for the ly held LP or LLC interests. est. Thus, the investor would pay 26.6% pricing of LP and LLC interests. There Investors’ pricing of RELP interests less for the nonmarketable security, are two methods to value LP and LLC in the secondary market includes their because it would cost 26.6% of the interests: the net asset value method partial marketability. In contrast, LP marketable price to purchase a pro- and the price-to-yield method. or LLC interests have no marketabili- tective put option to assure liquidity Net Asset Method. The net asset val- ty, because they are interests in close- and downside price protection at the ue method derives an indication of ly held entities that do not trade in any end of the five-year holding period. value from the application of a price- secondary market. Therefore, any val- Effect of Option Term on Option Val- to-NAV multiple (“NAV Multiple”) as uation of closely held LP or LLC inter- ue. In Exhibit 4, the values of two pro- determined from a secondary market ests based on RELPs must be adjusted tective put options are calculated on analysis of RELP interests, with cor- to compensate investors for the com- the same underlying asset to demon- roborating data from other sources, plete lack of marketability of these strate the effect of the term on the val- including traders, investors, and asset interests relative to the partial mar- ue of the protective put option, managers. ketability of the traded RELP interests. assuming that all other inputs are held The Price-to-Yield Method. The constant. The first protective put option price-to-yield method considers mar- has a term of six months; the second ket-based yields (“Required Yield”) of Application to Securities has a term of five years. The put option traded partnerships as a source of pric- with Partial Marketability with the six-month term is purchased ing data for LP and LLC interests. The When developing a DLOM for valua- so that after six months the owner of price-to-yield method capitalizes tion indications that reflect partial the underlying asset is assured liquid- expected distributions by dividing the marketability, the DLOM derived from ity for the asset at the stated strike price. market-required yield into the repre- methods such as the FMV Study or Similarly, the owner of the five-year sentative distributions. Market pricing the Pre-IPO Study must be reduced. put option is assured liquidity at the information for the traded limited DLOMs derived from the FMV Study stated strike price for the underlying partnerships is deemed to be a useful and the Pre-IPO Study represent com- asset after five years has passed. Risk- proxy for LP and LLC interests, since plete lack of marketability discounts free rates and volatilities corresponding LP, LLC, and RELP interests have many that are used to adjust from the (com- to the terms of the put options are also attributes in common, including: pletely) marketable level of value to presented in Exhibit 4. 1. Rights and restrictions provided in the (completely) nonmarketable level Clearly, the difference between option their governing documents. of value. The multiples derived from terms has a material effect on the values 2. Underlying assets (primarily real RELPs already include partial lack of of the protective puts (assuming that all estate properties). marketability discounting. Protective other inputs are held constant). As 3. Investment opportunities (income- put option methodology can be used shown in Exhibit 2, the five-year put producing investments). to quantify the difference between option costs more than the six-month 4. Operating performance. 3 Kam, Schroeder, and Smith, “The Market option because the investor is protect- RELP interests that are traded in Pricing of Syndicated LPs and the Valuation of ed over a longer period of time. the secondary market exhibit some FLPs,” Trusts & Estates 35 (February 1996).

8 VALUATION STRATEGIES November/December 2010 INCREMENTAL DLOM EXHIBIT 4 Value of Protective Put Options

Six-Month Put Five-Year Put Price of the underlying asset/a/ $100.00 $100.00 Strike price $100.00 $100.00 Annualized volatility of the underlying asset 0.45 0.45 Option term (in years) 0.50 5.00 Continuously compounded risk-free rate 4.00% 4.00% Continuously compounded dividend yield 0.00% 0.00% Value of the Protective Put Option $11.60 $26.60 Put Option Divided by Underlying Asset Price 11.60% 26.60%

/a/ Marketable minority value of the LP or LLC interest, not net asset value

complete lack of marketability and par- traded partnership or other security ously, however, the indications of val- tial lack of marketability. This method- with partial marketability seeking ue developed from RELPs already ology can be useful for adjusting cash would not have the certainty of include a partial lack of marketability DLOMs that reflect partial mar- a readily available market or even an discount. Thus, applying the full lack ketability when the situation requires estimate of the timing of the sale. of marketability discount of 30% from incrementally more DLOM for a secu- Consequently, there is greater uncer- the FMV and Pre-IPO Studies would rity that should have a full DLOM built tainty as to the timing of the sales overestimate the appropriate discount into its price. transaction and sales price. The to the LP interest. Instead of applying As discussed, an investor who expected timeframe to achieve liq- the complete 30% discount, it would be wants to assure liquidity and price uidity of an asset with partial mar- more appropriate to compute a dis- protection could purchase a protec- ketability will vary according to the count that quantifies the incremental tive put option. Moreover, the length specific characteristics of each asset lack of marketability between a par- of time required to effectuate the sale and the depth of market activity. tially marketable RELP interest and of an asset corresponds to the price Assume an investor owns a LP inter- the completely nonmarketable private an investor will pay to assure liquid- est with a marketable value of $100 LP interest. ity at a future date. With a complete- determined through a RELP compar- To calculate the incremental dis- ly marketable security, the investment ison analysis. Comparing the under- count, the values of two protective put can be quickly liquidated for cash. lying business of the LP interest to options on the LP interest are calcu- For example, an investor holding companies within the FMV Study and lated. An investor in the closely held LP shares of Johnson and Johnson, Inc. Pre IPO Study yields a DLOM of 30%, has no public or established market to stock could call her broker and which reflects the difference between sell his assets and faces restrictions on receive cash within three days. An complete marketability and complete the sale or transfer contained in the investor holding shares in a thinly nonmarketability. As discussed previ- LP governing documents. A term of

INCREMENTAL DLOM November/December 2010 VALUATION STRATEGIES 9 five years is assumed to represent a tive put option of 26.6%, which reasonable holding period. To reflect yielded 0.39, representing the pro- this inherent difference between a com- portion of DLOM not incorporated pletely marketable security and a secu- within the value that was based on rity with partial marketability, a RELP pricing in the secondary mar- proportion is developed that quantifies ket (“Proportion”). the difference between an asset with 4. Fourth, the Proportion (0.39) was partial marketability and a complete- multiplied by the complete DLOM ly nonmarketable asset. developed from the FMV Study, Pre In Exhibit 4, the protective put IPO Study, and protective put option used to quantify partial mar- option of 30.0% to yield an incre- ketability is given a term of six months. mental DLOM of (12.0%). This The protective put option with a term incremental DLOM was then of five years is used to quantify com- applied to the value of the LP inter- plete lack of marketability. The put est ($100.00) that included partial option with the six-month term pro- marketability. The product that vides assurance to the owner of the includes the additional incremen- underlying asset that in six months he tal DLOM ($88) is the nonmar- will receive the stated strike price for ketable, minority value of the his asset if exercised. Similar logic private LP Interest. applies to the five-year option. Use of Protective Put Options to Unlike the example in Exhibit 4, Determine Lack of Marketability. In a which illustrated the effect on option protective put option calculation, the value of varying the term of the option cost of purchasing a put option to (other inputs held constant), the exam- assure against downward price changes ple in Exhibit 5 calculates a real-world of the underlying asset is a proxy for example using varied inputs—vari- the cost of marketability. This down- ables that change with the term of the side price protection represents a mea- option. surement for DLOM, because an The implied lack of marketability investor who was given the choice discounts was computed as follows: between purchasing two identical secu- 1. First, the put option values ($16.20 rities would pay less for the nonmar- and $26.60) were divided by the ketable security than for the partially marketable, minority value of the marketable security. Factors that influ- LP or LLC interest ($100.00). These ence the value of downside price pro- proportions (16.2% and 26.6%) tection (DLOM) include: represents the cost to assure liq- 1. The volatility of the underlying FMV Study, Pre-IPO Study, and pro- uidity and price protection for six asset’s price. tective put option is applied without months and five years respectively. 2. The holding period or term of the adjustment. If, however, the private LP The five-year put option costs more security. or LLC interest is valued using a par- than the six-month option because 3. The risk-free rate corresponding to tially marketability security, such as a the investor is protected for a longer the expected holding period. RELP, the complete DLOM must be period of time. Differences between Partial and Com- reduced because the discount that is 2. Second, the discount for incremen- plete Lack of Marketability. Protective derived would reflect the complete lack tal lack of marketability was com- put option methodology can also be of marketability discounts, whereas puted. The six-month put option used to determine a DLOM for LP and RELP pricing includes a partial lack of DLOM (16.2%) was subtracted LLC interests in closely held entities marketability discount. That is, apply- from the five-year DLOM (26.6%) that own real estate, securities, or a ing a complete lack of marketability for an incremental lack of mar- combination of both. After the value discount from the FMV and Pre-IPO ketability of 10.40%. This incre- of the LP or LLC interest is determined, Studies would overestimate the appro- mental DLOM represents the and depending on the level of mar- priate discount to the LP or LLC inter- discount for lack of marketability ketability of the comparable traded est value that is based on RELP pricing. not incorporated within the par- interests, a complete or incremental tially liquid value that is based on lack of marketability discount is 4 Longstaff, “How Much Can Marketability Affect the RELP pricing in the secondary applied. If the LP or LLC interest is val- Security Value?,” 50 J. Finance 1767 (December market. ued using a completely marketable 1995). 3. Third, the incremental DLOM security, such as a REIT that can be 5 Finnerty, “The Impact of Transfer Restrictions on Stock Prices,” November 2007, available at (10.40%) was then divided by the exchanged for cash in three days, a http://www.fma.org/Prague/Papers/TheImpacto DLOM from the five-year protec- complete DLOM developed from the fTransferRestrictions.pdf.

10 VALUATION STRATEGIES November/December 2010 INCREMENTAL DLOM EXHIBIT 5 Lack of Marketability Calculation for a Private LP Interest Partial Complete Marketability Non-Marketability Underlying Security Price (S) $100.00 $100.00 Strike Price (X) $100.00 $100.00 Equity Volatility Factor (σ) 0.60 0.45 Continuous Risk Free Rate (r) 2.00% 4.00% Continuous Dividend Yield (q) 0.00% 0.00% Holding Period or Term (T) 0.50 5.00 Put Option Value (p) $16.20 $26.60 Put Option Value divided by Underlying Security Price 16.20% 26.60% Implied Incremental Lack of Marketability (26.60% - 16.20%) 10.40% Proportion (10.40% / 26.60%) 0.39 Complete Lack of Marketability Discount (FMV Study, Pre-IPO Study, and Protective Put Option) 30.00% Discount for Incremental Lack of Marketability (0.39 * 30.0%) (rounded) 12.00% The Value of the LP Interest with Partial Marketability $100.00 Less: Discount for Incremental Lack of Marketability ($100.00 * 12.0%) ($12.00) Non-Marketable, Minority Value of LP Interest $88.00

Instead of applying the complete private LP or LLC interest is an this discount, two other options-based discount, the applicable discount is approach that does not rely on empir- models exist to determine the discount: computed that captures the incremen- ical data from historical transactions, 1. A lookback put option model devel- tal lack of marketability between a par- but on quantifiable parameters that are oped by Francis A. Longstaff (the tially marketable RELP interest and a specific to the asset being valued. In “Longstaff model”).4 completely nonmarketable private LP addition, this incremental methodolo- 2. An average strike put option mod- or LLC interest. This computation gy corrects the potential for overesti- el developed by John D. Finnerty adjusts for the difference in mar- mation of the discount for lack of (the “ASP model”).5 ketability between a security with par- marketability when valuing an LP or The Longstaff Model. The Longstaff tial marketability (six months to sale) LLC interest using partially liquid model considers a hypothetical and a completely nonmarketable secu- RELPs. investor with perfect market-timing rity (five-year holding period) using ability who is restricted from selling a put options calculated with corre- security for a fixed period. The value sponding six-month and five-year Alternative Methods of marketability is the present value of terms. The proportionate difference for Completely the incremental cash flow an investor between these put options of varied Nonmarketable Securities would receive if she were to the terms represents the incremental dis- Earlier in this article, the BSM option value of the security at the end of the count for lack of marketability not pricing methodology was applied to restricted period for the maximum val- included within the minority value of determine the lack of marketability ue of the security during the restrict- the private LP or LLC interest. discount on a security that was ed period. Using put options to quantify the assumed to be completely nonmar- Unlike the BSM model, the incremental DLOM that is not includ- ketable. While BSM is considered an Longstaff model captures the oppor- ed within the minority value of the appropriate methodology to determine tunity cost associated with the

INCREMENTAL DLOM November/December 2010 VALUATION STRATEGIES 11

EXHIBIT 6

The ASP Model

( (

(

(r-q)T (r-q)T 1 ( (r-q)T 1

D(T) = exp N ϩ ( v√T ϪN Ϫ (v√T ( v√T 2 ( v√T 2

( ␴ ( ␴ v√T = √ ␴ 2TT + ln 2( exp 2 Ϫ ␴ 2 T – 1 – 2 ln (exp T 2 Ϫ 1

exp = a mathematical constant; the base of the natural logarithm q = annualized dividend yield of security r = risk-free rate T=time to expiration of option (in years) N=cumulative probability function for a standardized normal distribution σ=annualized volatility of the underlying stock D(T) = percentage discount for lack of marketability on the security

EXHIBIT 7 Comparison of Lack of Marketability Indications

BSM ASP Model Price of the underlying asset/a/ $100.00 $100.00 Strike price $100.00 N/A /b/ Annualized volatility of the underlying asset 0.45 0.45 Option term (in years) 5.00 5.00 Risk-free rate 4.00% 4.00% Dividend yield 0.00% 0.00% Resulting D(T) $26.60 $35.93 Marketable, Minority Value of Security $100.00 $100.00 Less: Discount for Lack of Marketability ($26.60) ($35.93) Non-Marketable, Minority Value of Security $73.40 $64.07 /a/ Marketable minority value of the LP or LLC interest, not net asset value. /b/ The ASP Model is an average strike price model. Therefore, no explicit strike price is assumed in the model.

EXHIBIT 8 Lack of Marketability Calculation for Private LP Interest Using the ASP Model

Partial Complete Marketability Non-marketability Underlying Security Price (S) $100.00 $100.00 Equity Volatility Factor (σ) 0.60 0.45 investor’s ability to sell her security at Risk Free Rate (r) 2.00% 4.00% the maximum attainable price during Dividend Yield (q) 0.00% 0.00% the restricted period rather than sell Holding Period or Term (T) 0.50 5.00 Put Option Value (p) $10.16 $35.93 her security at the price at the end of Put Option Value divided the restricted period. The Longstaff by Underlying Security Price 10.16% 35.93% model is a “lookback model,” in that Implied Incremental Lack the investor is assumed to have perfect of Marketability (35.93% - 10.16%) 25.77% hindsight to observe the maximum Proportion (10.16% / 35.93%) 0.28 price of the security during the Complete Lack of Marketability Discount (FMV Study, Pre-IPO Study, and Protective Put Option) 30.00% restricted period and to sell the secu- Discount for Incremental Lack of Marketability rity at the precise moment the securi- (0.28 * 30.0%) 8.40% ty reaches its maximum price in the The Value of the LP Interest with Partial Marketability $100.00 Less: Discount for Incremental Lack of Marketability market. ($100.00 * 12.0%) ($8.40) The Longstaff model is valuable in Nonmarketable, Minority Value of LP Interest $91.60 that it yields an upper-bound for the value of marketability that can be

12 VALUATION STRATEGIES November/December 2010 INCREMENTAL DLOM tant to consider the opportunity cost point difference can be viewed as a of the value foregone by holding an measure of the opportunity cost investment until option maturity associated with the investor’s inabil- rather than selling at some point with- ity to sell the security during the in the option term, the Longstaff mod- restricted period. It reflects value that el will likely lead to an overestimation is foregone assuming that the investor of the DLOM if applied directly to a must hold on to the security until security. option expiration rather than sell the The ASP Model. Another DLOM security with equal probability at any model that captures opportunity cost given time within the option term. is the ASP model. The ASP model Like the BSM model, the ASP Mod- derives a DLOM based on the value of el can be applied to securities that an average strike put option, a specif- have partial marketability. Exhibit 8 ic type of whose strike presents the application of the ASP price depends on the arithmetic aver- model to securities that have partial age of the security’s price during the marketability.6 life of the option. Unlike the Longstaff model, under the ASP model the investor is not assumed to have any Conclusion special market timing ability. Rather, Valuation of a protective put option the investor is assumed to be equally using the BSM model is a method used likely to sell his security at any time to determine a lack of marketability during the assumed holding period. discount for a nonmarketable securi- Consequently, the strike price of the ty (e.g., a closely held common stock, put option in the ASP model is the minority LP interest, or minority LLC arithmetic average of the forward stock interest). Although quantitative mod- prices during the life of the option. els, such as the BSM model, yield dis- Comparatively, the strike price of counts for fully marketable securities, the lookback put option in the the application of protective put the- Longstaff model is the maximum for- ory can be extended to measure the ward price during the life of the option. partial discounts for lack of mar- In practice, therefore, the ASP model is ketability on RELP interests in com- a promising alternative to the BSM parison to closely held minority model, as it incorporates the oppor- securities. tunity cost of value foregone by hold- The Longstaff and the ASP models ing an investment until option serve as alternatives to the BSM mod- maturity rather than selling at some el. The primary advantage of the point within the option term, but it Longstaff and ASP models is their does not assume that the investor has ability to capture the opportunity cost any ability to time the sale of the secu- of the value foregone by holding an rity at the highest price. The ASP mod- investment until option maturity, el is shown in Exhibit 6. rather than selling at some point The BSM and the ASP Models Com- within the option term. One draw- compared to DLOM indications pared. As in the example presented ear- back of the Longstaff model, howev- derived from other models or studies. lier, consider a hypothetical investor er, is that it assumes the investor has Arguably, all DLOM indications who owns a security with a marketable, perfect market timing ability, and that derived from other models should be minority value of $100 at the valuation the sale of the security will be at the lower than the indication derived from date with a holding period of five years. highest possible price during the the Longstaff model. But since Also, assume an annualized volatility option term. The ASP model resolves investors do not have the ability to for the security of 0.45, a risk-free rate this issue, as it incorporates the time the market perfectly, the of 4.0%, and a dividend yield of 0.0%. opportunity cost of value foregone Longstaff model does not produce Exhibit 7 provides a comparison of by holding an investment until option meaningful point estimates of DLOM DLOM indications using the BSM maturity rather than selling at some that can be applied to nonmarketable model versus the ASP model for a secu- point within the option term, but securities. Therefore, while it is impor- rity that is completely nonmarketable. does not assume that the investor has In Exhibit 7, the ASP model yields any type of special market timing a discount (35.93%) that is 933 basis ability. This feature of the ASP mod- 6 See Exhibit 5 and the accompanying text for a full discussion regarding the determination of points higher than the discount from el makes it a promising alternative to DLOM for partially marketable securities. the BSM (26.60%). The 933-basis- the BSM model. G

INCREMENTAL DLOM November/December 2010 VALUATION STRATEGIES 13 601 California Street, Suite 800 San Francisco, CA 94108 415.392.0888 voice 415.392.7070 fax

REPRINTED WITH PERMISSION WARREN, GORHAM & LAMONT OF TTA