VALUING A PRIVATE EQUITY CARRIED INTEREST AS A CALL OPTION ON THE FUND S PERFORMANCE

Transcription

1 VALUING A PRIVATE EQUITY CARRIED INTEREST AS A CALL OPTION ON THE FUND S PERFORMANCE John D. Finnerty Managing Director, AlixPartners LLP Professor of Finance, Fordham University Rachael W. Park Vice President, AlixPartners LLP January 2015 John D. Finnerty Managing Director AlixPartners LLP 40 West 57 th Street, 28 th Floor New York, NY Phone: (212) jfinnerty@alixpartners.com 2015 John D. Finnerty and Rachael W. Park. All rights reserved.

2 VALUING A PRIVATE EQUITY CARRIED INTEREST AS A CALL OPTION ON THE FUND S PERFORMANCE Abstract A PE fund manager receives a valuable call option in the form of a carried interest in the fund. This paper makes three contributions to the alternative investments literature. We model the carried interest within a call option pricing framework that is tailored to fit the characteristics of the PE fund s carried interest. The value of the carried interest is the difference between the values of two BSM call option positions. Second, we quantify the sensitivity of the carried interest s value to the investors preferred rate of return and to the fund s expected rate of return and return volatility. Third, we find that under reasonable assumptions, the carried interest accounts for less than one-third of the fund manager s expected total compensation. 1

3 I. Introduction Contingent claims analysis has been applied to value securities, derivative instruments, and a variety of other contingent economic interests, including loan guarantees (Margrabe, 1978; Hull, 2012), insurance contracts (Briys and Varenne, 1997), employee stock options (Johnson and Tian, 2000a, 2000b; Brisley and Anderson, 2008), and earn-outs and other profit-sharing interests (Howell, 2014). The last category includes a fund manager s carried interest in a hedge fund (Goetzmann, Ingersoll, and Ross, 2003) or a PE (PE) fund (Rouvinez, 2005; Metrick and Yasuda, 2010). A PE fund manager receives a valuable carried interest in the fund, which represents a call option on the fund s performance. This paper makes three contributions to the alternative investments literature. We model the carried interest within a call option pricing framework, which takes into account the hurdle rate of return and catch-up features typically found in PE contracts and the other characteristics of the PE fund s carried interest. The value of the carried interest can be expressed as the difference between the values of two BSM call option positions. Second, we perform comparative statics to assess the sensitivity of the value of the carried interest to the limited partners hurdle rate of return and to the PE fund s expected rate of return and return volatility. Third, we investigate how important the carried interest is to a PE fund manager s overall compensation. We find that under reasonable assumptions, the carried interest accounts for less than one-third of the fund manager s expected total compensation. The paper is organized as follows. Section II summarizes the distinguishing economic characteristics of PE funds and the compensation provisions of the fund manager s contract, which we rely on in modeling the carried interest. Section III describes the carried interest provision, and Section IV models it as a call option on the fund s performance. Section V 2

4 presents an example illustrating how to apply the model to value the fund manager s carried interest and the fund management contract of which it is an important component. Section VI quantifies the sensitivity of the value of the carried interest to the investors hurdle rate of return and the fund investment portfolio s expected return and volatility. Section VII concludes. II. Economic Characteristics of PE Finds PE funds are professionally managed pools of capital, which invest in companies either by buying private companies from their owners, including other PE funds, or by buying public companies and taking them private. Many PE funds also invest in financial instruments, including equity, debt, derivatives, currencies, commodities, and other financial securities, but the focus of their investment activity is buying, managing, and reselling companies for profit. Metrick and Yasuda (2010) provide a comprehensive economic analysis of PE funds. This section of the paper summarizes those features that are important in modeling the carried interest. PE funds are typically organized as limited partnerships with limited partners, who are outside investors who provide the bulk of the fund s investment capital, and a general partner, who is the organizer of the fund and who serves as the general partner of the PE fund and as the general manager of the fund s investments. 1 The general manager selects, purchases, manages, and sells the investments on behalf of the fund. The PE fund s investments are usually phased over a period of up to five years from the inception date of the fund. The average length of the investing period is about three years, and the fund s investments are harvested over roughly six to ten years expressed with respect to the inception date of the fund (Metrick and Yasuda, 2010). The limited partners contribute capital and usually receive a hurdle rate of return on their contributed capital (also referred to as the preferred rate of return) and an additional contingent return on their capital, which is typically 80% of the residual profit of the fund (Metrick and 1 We use the terms general partner and fund manager interchangeably. 3

5 Yasuda, 2010). The hurdle rate of return is generally between 7% and 10% per annum with 8% being the most common figure (Rouvinez, 2005). The general partner contributes capital, manages the fund s investments, and receives annual management fees, which are typically 2% of assets under management, and a carried interest, which provides an additional contingent return. 2 Rouvinez (2005) describes the carried interest, which he states typically accounts for most of the manager s compensation. We model the carried interest and find that under reasonable assumptions, it accounts for less than half the manager s total compensation. The carried interest provides an economic benefit to the general partner independent of the general partner s contribution of capital to the fund. This contingent claim is designed to align the economic interests of the fund manager and the fund s investors. However, it will do this only if the carried interest is expected to provide a significant portion of the manager s compensation. The percentage of the profit of the fund assigned to the general partner is referred to as the carry level. It is typically 20% (with a range of between 10% and 40%) (Rouvinez, 2005; Metrick and Yasuda, 2010). The general partner s carried interest represents a residual claim on a PE fund s value after the fund pays management fees and other expenses and returns the limited partners contributed capital together with any preferred rate of return. Two types of models are commonly utilized to value PE carried interests: the discounted cash flow (DCF) model (Zhou and Kam, 2013) and the conventional Black-Scholes-Merton (BSM) call option pricing model (Fleischer, 2005; Anson, 2012). A DCF model projects future cash flows associated with the carried interest and discounts them at an appropriate rate of return, which reflects the riskiness of these cash flows. The cash flow s riskiness is generally greater than the riskiness of the fund itself, because the carried interest is subordinated in right of 2 Larger funds often agree to smaller management fees, sometimes as low as 1.5% of assets under management (Metrick and Yasuda, 2010). 4

6 payment to the distribution of the limited partners preferred returns. However, a DCF model cannot accurately capture the contingent nature of the carried interest, especially the catch-up feature. An option pricing model recognizes that a PE carried interest has the character of a call option, because the carried interest gives the general partner the right to share in the profits of the fund that exceed a predetermined threshold (the investors capital and preferred return) when the fund s investments are harvested. The conventional BSM call option pricing model by itself can be used when the fund has a very simple structure that excludes a preferred return. But it cannot accommodate the preferred return and catch-up features typically found in PE funds, which we describe in the next section of the paper. We show that a workable carried interest valuation model that respects the critical features of the PE carried interest can be developed. Our model contains a combination of BSM call option values as components. III. Characteristics of the Carried Interest In order to value a carried interest, one needs to consider four basic elements return of investors capital requirement, hurdle rate of return, catch-up provision, and clawback feature. Features of a PE Carried Interest The return of investors capital requirement means that the limited partners must recover their contributed capital in full before the fund can make any distributions to the general partner. This return of capital includes repayment of the portion of their capital the fund spent on establishing the fund or paying management fees, monitoring fees, and other fund expenses. Preferred return (or hurdle rate of return) represents the rate of return that the fund must provide to the limited partners (including the return of their capital) before it can make any carried interest distributions to the general partner. The typical hurdle rate of return is 8% of 5

7 committed capital. Therefore, a carried interest has no value until the fund generates a return in excess of the investors preferred return. Our model incorporates this feature. The catch-up provision is a feature that allows the fund manager to receive a disproportionate share of the fund s profits once the fund generates sufficient profits to cover the investors hurdle rate of return. The catch-up provision allows the fund manager to receive all (or at least a disproportionate share) of profits until the manager s profits distribution catches up with the investors profits interest. 3 The fund manager catches up to the investors when its profit distribution reaches the agreed percentage (typically 20%) of the total profits distributed (including the catch-up). Our model allows for a catch-up provision to accompany the hurdle rate of return feature. The clawback provision is a device that protects the limited partners if the fund loses money after the managers have started receiving profits distributions. If the fund manager can be paid some of her carried interest during the life of the fund due to the catch-up provision, a clawback provision protects investors from the possibility that subsequent losses drive the fund s rate of return under the hurdle rate of return. The clawback provision claws back distributions from the general partner in that case so as to ensure that first, the limited partners receive their promised preferred return and second, the general partner does not receive more than the agreed carried interest percentage of the PE fund s total profits over the life of the fund. The amount of the clawback, if any, is usually calculated at the end of the investment holding period when all the fund s investments have been harvested. We assume that the profits distributions to the fund manager occur at the end of the holding period. At this point, the fund has realized its total profit, and the carried interest can be calculated taking the profits and losses on all the fund s 3 During the catch-up period, the general partner usually receives 100% of the incremental profits, although an 80/20 (general partner/limited partners) split or something similar is sometimes found in the partnership agreement (Fleischer, 2005). 6

8 investments into account. This procedure obviates the need to perform a separate clawback calculation. This simplifying assumption improves our model s tractability. Option-Like Character of the Carried Interest A carried interest has a call option-like quality. The contingent payoff profile of a carried interest is similar to that of a call option. A call option conveys the right, without an obligation, to buy an asset at a stated strike price before the option expires. If the asset price is below the strike price, the option holder will let the option expire worthless. If the asset price exceeds the strike price, the option holder will exercise it and receive the difference between the asset price and the strike price. The value of the call option derives from the appreciation in the value of the underlying asset above the stated strike price before the call option expires. Similarly, the value of a PE carried interest is contingent on what portion of the fund s profits remains after the fund has returned all the limited partners capital and paid their preferred return. In our model, the limited partners committed capital compounded forward to the terminal date to reflect the preferred rate of return is the carried interest option s strike price. The payoff of the carried interest option will be zero if the terminal value of the portfolio is below the strike price. If the terminal value is above the strike price, the option payoff is equal to the general partner s carried interest percentage of the difference between the terminal value of the portfolio and the strike price. Illustration of How the Carried Interest Feature Works First, I define five terms that are used to describe different measures of the investors investment in a PE fund. Committed Capital ( CP ) is the total amount of money that the limited partners contribute to a PE fund. A portion of the Committed Capital is used to pay certain fees. The fund s Investment Capital is the amount in a fund net of both the cost of 7

9 establishing the fund and the management fees paid during the life of the fund. Net Investment Capital is the amount invested net of any fees, including monitoring fees and transaction fees, and any other costs incurred. Invested Capital represents, at any point in time during the life of the fund, the portion of investment capital that has been invested in portfolio companies. Net Invested Capital represents, at any point in time, the Invested Capital net of the cost basis of the investments the fund has exited. Table 1 illustrates a distribution of PE fund profits between limited partners and the general partner when there is hurdle rate of return provision and a catch-up feature. Table 1 includes the following variables, which we will use in developing the carried interest valuation model: V P : value of the PE fund s equity investment portfolio (exclusive of the leveraging of each of its investments and after netting out any fees incurred). c: General Partner s percentage carried interest (carry level). h: hurdle rate of return on the portfolio, which is usually expressed on a simpleinterest basis. H1: the threshold amount of the terminal portfolio value for the carried interest when there is no hurdle rate of return. To simplify this example, we have assumed zero costs to establish and operate the fund. Our model in the next section of the paper does take these costs into account. 4 Therefore, the initial Committed Capital equals the Net Investment Capital. We have also assumed that the initial value of the fund s portfolio is $100, the holding period is eight years, the hurdle rate of return is 8% per annum, the carry level is 20%, and the expected investment return is 15% per 4 Section V provides a more realistic calculation of a PE carried interest. 8

10 annum. After the eight-year holding period, the terminal value of the portfolio is expected to be $306 (= $100 (1 + 15%) 8 ). The fund s total profit is $206 (= $306 $100). With an 8% hurdle rate, the limited partners are expected to receive $164 (= $100 (1+ 8 8%)) during the holding period. The residual profits in the fund would be $142 (= $306 $164). These residual profits are allocated between the limited partners and the general partner in accordance with the carry level. Because we assume there is a catch-up feature in this example, the fund manager s profit distributions will catch up to the investors distributions when its profit distribution reaches 20% of the total profits distributed. The catch-up proceeds are calculated as $16 (= 20% ($164 $100) 1 20% ). Consequently, the residual profit less the catch-up proceeds would be $126 (= $142 $16), of which 80% would be allocated to the limited partners, or $101, and 20% would be allocated to the general partner, or $25. Therefore, the value of the carried interest is equal to $41, which is the sum of the catch-up distribution ($16) and the allocation of the residual profits to the general partner ($25). Note that as a result of the catch-up feature, the fund manager receives 20% of the total profits ($206 20% = $41) and the investors get 80% ($206 80% = $165). IV. Carried Interest Option Pricing Model We develop our carried interest option pricing model in this section. We show that a PE carried interest can be viewed as a single simple call option or as a combination of two simple call options, depending on the structure of the carried interest feature in the fund manager s contract. In particular, the option structure depends on whether the carried interest includes a preferred rate of return with a catch-up feature. Each of these call options can be valued using the conventional BSM call option pricing model (Hull, 2012, pp ). 9

11 The price of a call option on a stock depends on six parameters: the market price of the stock at the time the option is valued (S 0 ), the time until the option is scheduled to expire (T), the interest rate on riskless (i.e., government) debt with a maturity equal to the option s time to expiration (r), the option strike price (K), the stock s dividend yield (D), and the volatility of the stock price (σ). Next, we formulate the carried interest valuation model and specify the six parameters to reflect the characteristics of the PE carried interest. Three Possible Option Structures for a PE Carried Interest We model the value of a PE carried interest within the risk-neutral framework. Asset value grows at the riskless rate in this framework because the investor s interest in the contingent claim is assumed to be perfectly hedged. However, PE funds offer investors the opportunity to realize a premium rate of return due to the fund manager s skill in choosing, managing, and harvesting the fund s investments (Metrick and Yasuda, 2010). PE fund managers claim to be able to add value due to their special industry knowledge, which provides an economic rent on their intellectual capital. Second, PE funds are not publicly traded, which requires a higher rate of return to compensate investors for this lack of marketability. Third, PE funds invest in various companies and incur related transaction costs in this process, which must be covered by the rate of return if the fund is to compete effectively for investors capital. Last, there are significant investment management costs, which must also be covered by the fund s rate of return. These features of private equity all act so as to raise the investors required rate of return, including in the risk-neutral framework. Considering these characteristics of PE funds, we assume that PE fund managers have the ability to provide a premium rate of return (expressed as an average annualized premium) over the life of the fund. We model this premium rate of return as a constant α > 0. Investors refer to 10

12 this premium in return as the fund s alpha [α], where α is calculated as [μ - r f - β(r m -r f )]. 5 (μ = expected rate of return of the portfolio, r f = risk free rate, and r m = market rate of return.) We assume that prospective investors can estimate α based on the information in the PE fund s offering materials when the fund managers market the limited partnership interests to investors. Therefore, we assume that the value of the PE fund s investment portfolio grows from the inception date at an average annual rate r f + α in the risk-neutral framework. Because of this premium, the rate of return is more appropriately interpreted as a risk-neutralized expected rate, rather than a risk-free rate. In effect, the risk-free rate should be adjusted upward for this premium rate of return when applying an option model to value the carried interest assuming the investors believe that the fund manager is capable of generating a premium rate of return. 6 Since the PE fund is not publicly traded, the value of the carried interest cannot be perfectly hedged (as the arbitrage-free framework assumes). Thus, our valuation model provides only an approximate value of the carried interest. The accuracy depends on how much of a premium in rate of return the general partner requires on account of any unhedged risk. We make the following additional simplifying assumptions in developing our carried interest valuation model. We calculate the present value of the fund s management fees and monitoring fees under the assumption that all the fund s investments are made at the midpoint of the investing period and harvested at the midpoint of the holding period. This assumption improves the model s tractability. We also assume that the limited partners must be paid their hurdle rate of return for the entire investing period and the entire holding period. This assumption is reasonable because the limited partners want the fund manager to have an 5 This method of modeling the PE fund s rate of return on investment as the riskless rate plus a premium return follows Goetzmann, Ingersoll, and Ross, 2003, pp The sensitivity analysis in Section VI investigates the sensitivity of the value of the carried interest to this premium return. 11

13 economic incentive to invest the fund reasonably quickly as well as an economic incentive to harvest the investments efficiently -- and not delay the investing or the harvesting simply to collect more management fees. We describe three possible structures for the carried interest, which differ according to whether there is a hurdle rate of return feature and a catch-up provision for the carried interest. The payoff diagrams for the three structures are illustrated in Figure 1. a) Structure 1: No Hurdle Rate of Return When there is no hurdle rate of return (and therefore no catch-up provision either), the threshold amount at the date of exit is defined as H1, which is equal to the amount of Committed Capital, CP. Therefore, the value of a carried interest is expressed as the expected present value of the contingent profits interest according to the following formula: CCCCCCC IIIIIIII = ce rf+α T [V P (T) H1] f(v H1 P )dv P (1) Equation (1) can be evaluated by applying the BSM call option pricing formula with the following parameters: initial value of asset (S 0 ) is Net Investment Capital at the inception of the fund (V P (0)), volatility (σ) is the volatility of the value of the fund s investment portfolio, rate of return (r = r f + α) is the risk-neutralized rate of return of the fund s investment portfolio, dividend rate (D) is assumed to be zero, strike price (K) is Committed Capital (H1), and time to expiration (T) is the weighted average date of exit for the investments in the fund s portfolio where T is expressed with respect to the inception date of the fund. In addition, the terminal value of the fund s investment portfolio V P (T) is determined as the Net Investment Capital at the inception of the fund compounded forward to the expiration date at the expected rate of return of the fund s investment portfolio. Therefore, the value of the carried interest is calculated as the 12

14 carry level c multiplied by the conventional BSM model call option value when the option is struck at H1 according to the following formula: CCCCCCC IIIIIIII = c(v P (0)N(d 1 ) H1e (r f+α)t N(d 2 )) (2) where d 1 = ln VP(0) H1 + r f+α+ σ2 2 T σ T and d 2 = ln VP(0) H1 + r f+α σ2 2 T σ T b) Structure 2: Hurdle Rate of Return without Catch-Up When there is hurdle rate of return but no catch-up provision, the threshold amount at the date of exit is defined as H2, which is equal to CP (1 + h T). Therefore, the value of a carried interest can be expressed by the following formula: CCCCCCC IIIIIIII = ce rf+α T [V P (T) H2] f(v H2 P )dv P (3) Equation (3) can be evaluated by applying the BSM call option pricing formula with the same parameters as Structure 1 except that the strike price (K) is defined as Committed Capital (1 + hurdle rate of return weighted average date of exit). Therefore, the value of the carried interest is calculated as the carry level c multiplied by the conventional BSM model call option value when the option is struck at H2 according to the following formula: CCCCCCC IIIIIIII = c(v P (0)N(d 1 ) H2e r f+α T N(d 2 )) (4) where d 1 = ln VP(0) H2 + r f+α+ σ2 2 T σ T and d 2 = ln VP(0) H2 + r f+α σ2 2 T σ T c) Structure 3: Hurdle Rate of Return with Catch-Up When there is hurdle rate of return and a catch-up provision, the threshold amount at the date of exit is defined as H2, which is equal to CP (1 + h T). In this structure, the general partner/fund manager gets all the added return until the terminal portfolio value reaches H2 + c(h2 H1). When the portfolio value reaches this threshold, any further increase in value is 13

15 shared between the limited partners and the general partner/fund manager in accordance with the agreed carried interest formula. Therefore, the value of a carried interest can be expressed by the following formula: CCCCCCC IIIIIIII = e r f+α T +ce r f+α T H2+ c(h2 H1) H2 H2+ c(h2 H1) [V P (T) H2]f(V P )dv p [V P (T) H1] f(v P )dv P (5) Equation (5) can be interpreted with reference to Figure 1. The first term represents the value attributable to the catch-up period. The fund manager earns, as a carried interest, the full increase in portfolio value between V P (T) = H2 and V P (T) = H2 + c(h2 H1) to catch up to the limited partners after these investors have received their promised hurdle rate of return. The second term represents the carried interest on any subsequent increase in portfolio value in the fund, which is equal to the fraction c of this increase. Equation (5) can be expressed as the composite of three call option positions: (a) cccc oooooo ssssss aa H2 minus (b) cccc oooooo ssssss aa H2 + c(h2 H1) plus (c) c cccc oooooo ssssss aa H2 + c(h2 H1). A graphical illustration is provided in Figure 2, and the mathematical derivation of the formula is presented in the Appendix. The BSM formula applies to each call option with the same parameters as Structure 1 except the strike price (K) is defined as either H2 or H2 + c(h2 H1) depending on which option is being valued. Combining call option positions (b) and (c), the value of the carried interest is calculated as the difference between the values of two BSM call option positions: CCCCCCC IIIIIIII = BBB vvvvv oo cccc oooooo ssssss aa H2 14

16 (1 c) BBB vvvvv oo cccc oooooo ssssss aa H2 + c(h2 H1) = V P (0)N(d 1 ) H2e r f+α T N(d 2 ) (1 c) V P (0)N(d 1 ) (H2 + c(h2 H1) )e rf+α T N(d 2 ) (6) where d 1 = ln VP(0) H2 + r f+α+ σ2 d 1 = σ T 2 T, d 2 = ln VP(0) H2 V ln P (0) H2+ c(h2 H1) + r f +α+ σ2 2 T σ T + r f+α σ2 2 T σ T, and d 2 =, V ln P (0) H2+ c(h2 H1) + r f +α σ2 2 T σ T V. Example of a PE Carried Interest Valuation Tables 2 through 7 illustrate the valuation process for a carried interest assuming the carried interest has a hurdle rate of return and a catch-up provision (Structure 3). The conventional BSM call option pricing model is used to value the option components. Table 2 calculates the weighted average time to invest and the weighted average exit date of the fund s investments. Table 3 illustrates the calculation of the Net Investment Capital at the inception of the fund and the strike price for the conventional BSM call option pricing model. Tables 4 and 5 calculate the present value of the Management Fees and the Monitoring Fees, respectively. Table 6 provides the calculation of the expected return and volatility for a portfolio consisting of ten investments. Table 7 calculates the value of the carried interest using the conventional BSM call option pricing model based on the calculations performed in Tables 2 through 6. We apply the following four-step process to value the carried interest: STEP1: We calculate the Weighted Average Time to Invest and the Weighted Average Date of Exit as shown in Table 2. While a PE fund carried interest can be modeled as a call option on the performance of the fund, the exit dates of the underlying investments are unknown 15

17 until the investments have been harvested. We first assume that the life of the PE fund is ten years, with the first five years being an Investing Period and the second five years being a Holding Period. 7 As illustrated in Panel A of Table 2, for the Investing Period, we assume that the pace of investment across the five years is 26%, 23%, 25%, 18%, 8% of the fund s Investment Capital in year 1 through year 5. 8 Therefore, the weighted average time to invest is approximately 2.09 years. Next, as shown in Panel B of Table 2, for the Holding Period, we calculate the weighted average date of exit based on the assumption that the time to exit follows an exponential distribution. 9 We consider continuously compounded exit rates of 10%, 20%, and 25% with the 20% exit rate being the base case scenario. The weighted average date of exit is approximately 7.99 years when the exit rate is assumed to be 20%. STEP 2: We calculate the amount of Net Investment Capital at the inception of the fund, which is illustrated in Panel A of Table 3. In order to calculate a carried interest using the BSM call option pricing model, the initial value of the portfolio, which serves as the spot price for the BSM model, has to be estimated. This initial value is the Net Investment Capital at the inception of the fund, which is the net amount that would be used to make investments. We assume the initial Committed Capital is $100 at the inception of the fund. This is the amount of money that investors initially contributed to the fund. Second, we estimate the establishment cost, which is a one-time fee, which is charged to the limited partners to cover the cost of establishing the fund. We assume the initial establishment cost is equal to 1% of the Committed Capital, or $1. Third, we estimate the Management Fees, which are paid to the general partner/manager for investment and portfolio management services. Typically, funds charge a certain percentage 7 These assumptions are consistent with data furnished in Metrick and Yasuda (2010, p. 2309). 8 The fractions are based on empirical research for buy-out funds. See Metrick and Yasuda (2010, p.2315). 9 These assumptions are consistent with data furnished in Metrick and Yasuda (2010, p. 2309). 16

18 of Committed Capital each year or have a decreasing fee schedule during the Holding Period. We assume the Management Fees have a decreasing fee schedule, i.e., the fee rate of 2% is constant but the basis for this rate changes from the amount of Committed Capital during the Investing Period to the amount of Net Invested Capital during the Holding Period. The calculation of the Management Fees is illustrated in Table 4. Because the Committed Capital is fixed but the Investment Capital is a function of the Management Fees, we solve for the Investment Capital and Management Fees such that Committed Capital is equal to the sum of Investment Capital, Establishment Cost, and Management Fees. Panels A and B of Table 4 show the scenario analysis varying the assumptions; Management Fees range from 1% to 2% and the risk-free rate ranges from 1% to 5%. The present value of the Management Fees under the base case scenario - Management Fees of 2% and a risk-free rate of 5% - is $ Fourth, the amount of Investment Capital is calculated as the Committed Capital minus the Establishment Costs minus the present value of the Management Fees, or $ Fifth, we calculate the amount of the Monitoring Fees, which the fund charges to their portfolio companies. In most cases, the fees are shared between the limited partners and the general partner. 10 We assume that the limited partners receive 80% and the general partner receives 20% of the fees. In practice, Monitoring Fees are typically calculated at a rate between 1% and 5% of the total annual earnings before interest, income taxes, depreciation, and amortization (EBITDA) for the portfolio companies, and this fee is calculated each year. 11 We 10 Current practice reflects a change from prior practice, in which the general partners typically kept all the Monitoring Fees. Pension funds and other institutional investors have increasingly insisted as a condition for their agreeing to invest in a PE fund that the limited partners, and not the general partner, get the Monitoring Fees (Spector and Maremont, 2014). 11 This assumption is consistent with data furnished in Metrick and Yasuda (2010, p. 2014). 17

19 assume that the Monitoring Fees are 2% of EBITDA, or 40 bps of the portfolio investments value, based on an assumed EBITDA multiple of 5. Table 5 has the calculation of the present value of the Monitoring Fees. We use the amount of Investment Capital as the basis for calculating the Monitoring Fees. The scenario analysis is performed with the weighted average date of exit ranging from 6 years to 9.5 years and the discount rate ranging from 1% to 5%. Panel A of Table 5 provides a scenario analysis, which assumes that the general partner keeps the entire Monitoring Fees. Panel B of Table 5 furnishes a scenario analysis, which assumes that the general partner keeps 20% of the Monitoring Fees (limited partners keep 80%). For this example, we assume that the general partner keeps 20% of the Monitoring Fees. The present value of the Monitoring Fees is $0.47 in the base case scenario, in which the average date of exit is 8 years and the discount rate is 2%. Sixth, the Transaction Fees are assumed to be 1.37% of Investment Capital, or $1.17, at the inception of the fund. Finally, the Net Investment Capital at the inception of the fund is calculated as the Investment Capital minus the present value of the Monitoring Fees minus the Transaction Fees, or $ STEP 3: We calculate the strike price for the carried interest option pricing model in Panel B of Table 3. The strike price is the amount of Committed Capital compounded at the hurdle rate, which we assume is an 8% simple interest rate of return on Committed Capital. As illustrated in Table 2, the weighted average date of exit is assumed to be 7.99 years. The hurdle return, therefore, is calculated as (Committed Capital Hurdle Rate of Return Weighted Average Date of Exit), or $ Therefore, the strike price for BSM Option 1 for the carried interest valuation is the sum of $100 and $63.90, or $

20 STEP 4: We calculate the value of the carried interest utilizing the conventional BSM call option pricing model in Table 7. As noted, we assume that the carry level, the percentage of the profits claimed by the general partner, is 20% of the Committed Capital. Therefore, the BSM model applies with the following parameter values: the spot price is the Net Investment Capital (the initial value of the portfolio), which is $83.81; the risk-neutralized expected rate of return and the volatility are estimated in Table 6 based on the return and volatility characteristics of the ten selected portfolio companies, 12 where the expected rates of return, volatilities, and correlations for these ten companies equity returns are shown in Table 6. Based on the historical data, the risk-neutralized expected rate of return of the portfolio is 7% and the portfolio volatility is 19%; the time to expiration is equal to 7.99 years, which is the weighted average date of exit; and the strike price for Option 1 is equal to $163.90, as illustrated in Table 3. The strike price for Option 2 is equal to $179.87, which is calculated as H2 + c(h2 H1). Therefore, the value of the Carried Interest can be expressed as the value of a combination of conventional BSM call option positions by applying equation (6). The carry level is 20%, the preferred rate of return is 8%, the initial value of the asset is $83.81, and the strike prices of the component options are H2 = $ and H2 + c(h2 H1) = $ As shown in Table 7, the value of the Carried Interest is equal to [the value of Option 1] minus [(1-20%) the value of Option 2], or $3.37. Next, we value the fund manager s total compensation. Adding the present value of the Management Fees from Table 4 in the amount of $13.55 ($9.42 for the investing period and $4.13 for the holding period) and the $0.47 of Monitoring Fees from Table 5 to the value of the 12 In order to estimate the expected rate of return and the volatility of the fund s investment portfolio, we randomly selected ten companies that were merged or acquired in We only consider companies for which at least one year s historical common stock prices are available on Bloomberg. 19

21 Carried Interest, the fund manager s expected present value total compensation as of the fund s inception is $ The Carried Interest represents only 19% of the fund manager s expected total compensation on a present-value basis. VI. Sensitivity Analysis for the Value of the Carried Interest The value of a carried interest is sensitive to the preferred rate of return that is promised the fund s investors, the expected rate of return on the fund s investment portfolio, and the volatility of those portfolio returns. Similarly, the fraction of the fund manager s expected total compensation that is in the form of the carried interest depends on these key parameters. This section extends the previous example by quantifying these sensitivities. Sensitivity to the Hurdle Rate of Return Table 8 illustrates the sensitivity of the value of the carried interest to variation in the hurdle rate of return for Structures 1, 2, and 3. We make the exact same assumptions in applying the BSM model as in the previous example, including the Management Fees and the Monitoring Fees. Panel A shows the value of the carried interest when the hurdle rate of return ranges from 0% to 10%, and Panel B calculates the value of the carried interest as a percentage of the fund manager s expected total compensation, which consists of the carried interest, the Management Fees, and 20% of the Monitoring Fees. The value of the carried interest decreases as the hurdle rate of return increases. Accordingly, the percentage of the total compensation the fund manager receives in the form of a carried interest also decreases, for example, from 27% to 17% for Structure 3, as the hurdle rate of return increases. The value of the carried interest never accounts for more than one-third of the fund manager s expected total compensation. At the customary 8% preferred rate of return, 20

22 the value of the carried interest accounts for only one-fifth of the fund manager s expected total compensation in Structure 3. Sensitivity to the Expected Rate of Return of the Portfolio Table 9 illustrates the sensitivity of the value of the carried interest to variation in the risk-neutralized expected rate of return of the portfolio for Structures 1, 2, and 3. Panel A shows the value of the carried interest when the risk-neutralized expected rate of return of the portfolio ranges from 4% to 15%, and Panel B calculates the value of the carried interest as a percentage of the fund manager s expected total compensation. The value of the carried interest increases as the risk-neutralized expected rate of return of the portfolio increases. Accordingly, the percentage of the total compensation the fund manager receives in the form of a carried interest also increases, for example, from 15% to 28% for Structure 3, as the risk-neutralized expected rate of return increases. With an 8% preferred return, and even with a catch-up feature, the carried interest never accounts for more than onethird of the fund manager s expected total compensation, even if the risk-neutralized expected portfolio rate of return reaches 15%. Sensitivity to the Volatility of the Portfolio Returns Table 10 illustrates the sensitivity of the value of the carried interest to variation in the volatility of the portfolio returns for Structures 1, 2, and 3. Panel A shows the value of the carried interest when the volatility of the portfolio returns ranges from 5% to 60%, and Panel B calculates the value of the carried interest as a percentage of the fund manager s expected total compensation. The value of the carried interest increases as the volatility of the portfolio returns increases. Accordingly, the percentage of the total compensation the fund manager receives in 21

23 the form of a carried interest also increases, for example, from 0% to 43% for Structure 3, as the volatility of the portfolio returns increases. With an 8% preferred return, and even with a catchup feature, the carried interest never accounts for more than one-half of the fund manager s expected total compensation, even if the volatility of the portfolio returns reaches 60%. In practice, portfolio volatility greater than 30% would be unusual. 13 With 30% volatility, the carried interest would not account for more than one-third of the fund manager s expected total compensation when the preferred return is 8%, even with a catch-up. Conclusions Drawn from the Sensitivity Analyses As illustrated in Table 8 through Table 10, the value of a carried interest is sensitive to the hurdle rate of return, the risk-neutralized expected portfolio rate of return, and the volatility of the portfolio returns. Throughout all the scenarios tested, we find that under reasonable assumptions, the carried interest accounts for no more than one-third of the fund manager s expected total compensation. In many cases, it would be substantially less than one-third. VII. Conclusions The manager of a PE fund receives a valuable contingent claim in the form of a carried interest in the fund. The contingent claim represents a call option on the fund s future performance. Business appraisers typically employ either a discounted cash flow model or the conventional BSM call option pricing model to value a PE carried interest. The BSM call option pricing model cannot accommodate the preferred return and catch-up features of the carried interest when those features are present. We show in this paper that these features can be modeled within a call option pricing framework that is tailored to fit the characteristics of the PE 13 The volatility of the S&P 500 Index has averaged approximately 20% per year since 2010, so a portfolio volatility of 30% would represent about 1.5 times the volatility of the S&P 500 Index. 22

24 fund s carried interest. The value of the carried interest can be expressed as the difference between the values of two BSM call option positions. We also quantified the sensitivity of the carried interest s value to the investors preferred rate of return and to the investment portfolio s expected rate of return and return volatility. We found that under reasonable assumptions, the carried interest accounts for less than one-third and in some cases substantially less than a third of the fund manager s expected total compensation. 23

25 APPENDIX Derivation of the Formula for the Value of a PE Carried Interest Assuming a Hurdle Rate Requirement with a Catch-Up Feature Figures 1 and 2 provide the intuition underlying the derivation of the PE carried interest option pricing model. In Figure 1, the fund manager has a carried interest consisting of (a) the full increase in asset value between V P (T) = H2 and V P (T) = H2 + c(h2 H1) to catch up to the fund investors after these investors have received their promised hurdle rate of return and (b) the full carried interest on any subsequent increase in fund asset value, which is equal to the fraction c of this increase in value. Figure 2 provides the intuition behind simplifying the option expression to a difference between two BSM call option values in equation (4). Expressing the value of the carried interest in integral form, we have Value of a Carried Interest H2+ c(h2 H1) = e rf+α T [V P (T) H2]f(V P )dv p H2 + e rf+α T c [V P (T) H1] f(v H2+ c(h2 H1) P )dv P = e rf+α T [V P (T) H2]f(V P )dv p H2 H2+ c(h2 H1) [V P (T) H2]f(V P )dv p + e rf+α T c [V P (T) H1] f(v P )dv P H2+ c(h2 H1) 24

26 = e rf+α T [V P (T) H2]f(V P )dv p H2 H2+ c(h2 H1) H2+ c(h2 H1) V P (T) H2 + c(h2 H1) f(v 1 c P )dv p + e r f+α T c V P (T) H2 + H2+ c(h2 H1) (H2 H1) + c f(v 1 c P )dv p H2+ c(h2 H1) c(h2 H1) f(v 1 c P )dv p c(h2 H1) 1 c f(v P )dv P = e rf+α T [V P (T) H2]f(V P )dv p H2 H2+ c(h2 H1) V P (T) H2 + c(h2 H1) 1 c + e r f+α T c V P (T) H2 + H2+ c(h2 H1) f(v P )dv p c(h2 H1) 1 c f(v P )dv P = V P (0)N(d 1 ) H2e (rf+α)t N(d 2 ) V P (0)N(d 1 ) (H2 + c(h2 H1) )e (rf+α)t N(d 2 ) + c V P (0)N(d 1 ) (H2 + c(h2 H1) )e (rf+α)t N(d 2 ) = V P (0)N(d 1 ) H2e (r f+α)t N(d 2 ) (1 c) V P (0)N(d 1 ) (H2 + c(h2 H1) )e (rf+α)t N(d 1 c 2 ) where d 1 = ln VP(0) H2 + r f+α+ σ2 d 1 = σ T 2 T, d 2 = ln VP(0) H2 V ln P (0) H2+ c(h2 H1) + r f +α+ σ2 2 T σ T + r f+α σ2 2 T σ T, and d 2 =, V ln P (0) H2+ c(h2 H1) + r f +α σ2 2 T σ T 25

27 REFERENCES Anson, Mark, 2012, Asset Owners versus Asset Managers: Agency Costs and Asymmetries of Information in Alternative Assets, Journal of Portfolio Management 38 (3) (Spring), Brisley, Neil and Chris K. Anderson, 2008, Employee Stock Option Valuation with an Early Exercise Boundary, Financial Analysts Journal 64 (September/October), Briys, Eric and Francois de Varenne, 1997, On the Risk of Insurance Liabilities: Debunking Some Common Pitfalls, Journal of Risk and Insurance 64 (4), Fleischer, Victor, 2005, The Missing Preferred Return, Journal of Corporation Law 31 (Fall), Goetzmann, William N., Jonathan E. Ingersoll, Jr., and Stephen A. Ross, 2003, High-Water Marks and Hedge Fund Management Contracts, Journal of Finance 58 (August), Howell, David W., 2014, Valuing Profits Interests Used as Equity Compensation in LLC, Valuation Strategies (May/June), Hull, John C., 2012, Options, Futures, and Other Derivatives, 8 th ed., Pearson, Boston,. Johnson, Shane A., and Yisong S. Tian, 2000a, Indexed Executive Stock Options, Journal of Financial Economics 57 (July), Johnson, Shane A., and Yisong S. Tian, 2000b, The Value and Incentive Effects of Nontraditional Executive Stock Option Plans, Journal of Financial Economics 57 (July), Margrabe, William, 1978, The Value of an Option to Exchange One Asset for Another, Journal of Finance 33 (March),

28 Metrick, Andrew, and AyakoYasuda, 2010, The Economics of PE Funds, Review of Financial Studies 23 (6), Rouvinez, Christophe, 2005, The Value of the Carry, PE International, (July/August), Spector, Mike, and Mark Maremont, Fees Get Leaner on PE, Wall Street Journal (December 29, 2014), A1, A2. Zhou, Phillip and Steven Kam, 2013, Carried Interest Valuation Techniques: The First in a Two Part Series, Cogent Valuation at 27

29 Figure 1 Payoff Diagram of the Value of a Carried Interest Amount of a Carried Interest Payoff diagram for a carried interest expressed as a function of the terminal fund asset value under Three Alternative Structures: (i) no hurdle rate of return, (ii) hurdle rate of return without catch up, and (iii) hurdle rate of return with catch up. c is defined as a percentage carried interest. H1 and H2 represent the value of net investment capital and the value of the hurdle (the minimum asset value to meet the preferred return at the date of exit), respectively. No Hurdle Rate of Return Hurdle Rate of Return with Catch Up Hurdle Rate of Return without Catch Up H1 H2 H2 + c(hh HH) Terminal Fund Asset Value [V P (T)]

30 Payoff Amount Panel A. Payoff Diagram for three call option positions under the following assumptions: the carry level is 20%, the term of the fund is 8 years, the hurdle rate of return is 8%, and the value of Committed Capital is $100. (H1=$100 and H2=$164) Call Option Position 1: Long call option with strike price $164 Call Option Position 2: Long call option with strike price $180 multiplied by 20% Call Option Position 3: Short call option with strike price $180 (negative payoffs above the strike price) Figure 2 Payoff Diagram for Three Call Option Positions Terminal Portfolio Value Call Option Positon 1 Call Option Posiotn 2 Call Option Position 3 Payoff Amount Panel B. The Payoff Diagram for the Combination of the Call Option Positions 1, 2, and Terminal Portfolio Value Call Option Position 1 + Call Option Position 2 + Call Option Position 3

31 Table 1 Example of Distribution of Private Equity Fund Profits When There Is a Hurdle Rate of Return and a Catch-Up Feature Assumptions [1] Parameter Value Initial Value of the Portfolio V P (0) = H1 $100 Holding Period (years) T 8 Preferred (Hurdle) Rate (per annum) h 8% Carry Level c 20% [2] Investment Return (per annum) µ 15% Distribution at the End of the Term Terminal Value of the Portfolio V P (T) $306 [3] Preferred Return to Limited Partners H2 $164 [4] Residual Profit $142 [5] Carried Interest Catch-Up Proceeds $16 [6] Allocation of the Residual Profit to Limited Partners $101 [7] Allocation of the Residual Profit to General Partner $25 [8] Value of the Carried Interest $41 [9] Notes: [1] This example assumes zero cost to establish and to operate the fund only for illustration purposes. Next tables provide a more realistic calculation fo a PE carried interest. [2] Represents the percentage of residual profit of the fund distributed to the general partner. Consequently, 80% of the residual profit is distributed to the limited partners. [3] = V P (0) (1+µ ) T [4] = V P (0) (1+h T ) [5] = V P (T) -H2 [6] = c (H2 -H1 )/(1-c ) [7] = (1-c ) ([4]-[5]) [8] = c ([4]-[5]) [9] = [5]+[7]