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 option the put option, 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 volatility 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. Strike price. 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=exercise price or strike price σ = annualized volatility of the underlying stock T=time to expiration (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.