valuing a private equity carried interest …...valuing a private equity carried interest as a call...

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 57th Street, 28th Floor New York, NY 10019

Phone: (212) 845-4090 Email: [email protected]

© 2015 John D. Finnerty and Rachael W. Park. All rights reserved.

mailto:[email protected]

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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.

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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

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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.

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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).

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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

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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).

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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

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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:

• VP: 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 simple-

interest 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.

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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. 299-324).

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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 (S0), 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

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this premium in return as the fund’s alpha [α], where α is calculated as [μ - rf - β(rm-rf)].5 (μ =

expected rate of return of the portfolio, rf = risk free rate, and rm = 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 rf + 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. 1689-1690. 6 The sensitivity analysis in Section VI investigates the sensitivity of the value of the carried interest to this premium return.

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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 𝐼𝐼𝐼𝐶𝐶𝐶𝐼𝐼 = 𝑐𝐶−�𝑟𝑓+𝛼�𝑇 ∫ [𝑉𝑃(𝑇) − 𝐻1]∞𝐻1 𝑓(𝑉𝑃)𝐶𝑉𝑃 (1)

Equation (1) can be evaluated by applying the BSM call option pricing formula with the

following parameters: initial value of asset (S0) is Net Investment Capital at the inception of the

fund (VP(0)), volatility (σ) is the volatility of the value of the fund’s investment portfolio, rate of

return (r = rf + 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 VP(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

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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 𝐼𝐼𝐼𝐶𝐶𝐶𝐼𝐼 = 𝑐(𝑉𝑃(0)𝑁(𝐶1) − 𝐻1𝐶−(𝑟𝑓+𝛼)𝑇𝑁(𝐶2)) (2)

where 𝐶1 =ln�𝑉𝑃(0)

𝐻1 �+�𝑟𝑓+𝛼+𝜎2

2 �𝑇

σ√𝑇 and 𝐶2 =

ln�𝑉𝑃(0)𝐻1 �+�𝑟𝑓+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 𝐼𝐼𝐼𝐶𝐶𝐶𝐼𝐼 = 𝑐𝐶−�𝑟𝑓+𝛼�𝑇 ∫ [𝑉𝑃(𝑇) − 𝐻2]∞𝐻2 𝑓(𝑉𝑃)𝐶𝑉𝑃 (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 𝐼𝐼𝐼𝐶𝐶𝐶𝐼𝐼 = 𝑐(𝑉𝑃(0)𝑁(𝐶1) − 𝐻2𝐶−�𝑟𝑓+𝛼�𝑇𝑁(𝐶2)) (4)


𝐻2 �+�𝑟𝑓+𝛼+𝜎2

2 �𝑇

σ√𝑇 and 𝐶2 =


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)1−𝑐

� . When the portfolio value reaches this threshold, any further increase in value is

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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 𝐼𝐼𝐼𝐶𝐶𝐶𝐼𝐼 = 𝐶−�𝑟𝑓+𝛼�𝑇 ∫ [𝑉𝑃(𝑇) − 𝐻2]𝑓(𝑉𝑃)𝐶𝑉𝑝𝐻2+𝑐(𝐻2−𝐻1)

1−𝑐𝐻2

+𝑐𝐶−�𝑟𝑓+𝛼�𝑇 ∫ [𝑉𝑃(𝑇) − 𝐻1]∞𝐻2+𝑐(𝐻2−𝐻1)

1−𝑐𝑓(𝑉𝑃)𝐶𝑉𝑃 (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 𝑉𝑃(𝑇) = 𝐻2 and 𝑉𝑃(𝑇) = 𝐻2 + 𝑐(𝐻2−𝐻1)1−𝑐

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 𝑜𝑜𝐼𝐶𝑜𝐼 𝐼𝐼𝐶𝑠𝑐𝑠 𝐶𝐼 𝐻2 minus (b) 𝑐𝐶𝑐𝑐 𝑜𝑜𝐼𝐶𝑜𝐼 𝐼𝐼𝐶𝑠𝑐𝑠 𝐶𝐼 �𝐻2 + 𝑐(𝐻2−𝐻1)1−𝑐

� plus

(c) 𝑐 × 𝑐𝐶𝑐𝑐 𝑜𝑜𝐼𝐶𝑜𝐼 𝐼𝐼𝐶𝑠𝑐𝑠 𝐶𝐼 �𝐻2 + 𝑐(𝐻2−𝐻1)1−𝑐

�. 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)1−𝑐

� 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 𝐼𝐼𝐼𝐶𝐶𝐶𝐼𝐼

= 𝐵𝐵𝐵 𝑣𝐶𝑐𝑠𝐶 𝑜𝑓 𝑐𝐶𝑐𝑐 𝑜𝑜𝐼𝐶𝑜𝐼 𝐼𝐼𝐶𝑠𝑐𝑠 𝐶𝐼 𝐻2

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−(1 − 𝑐) × 𝐵𝐵𝐵 𝑣𝐶𝑐𝑠𝐶 𝑜𝑓 𝑐𝐶𝑐𝑐 𝑜𝑜𝐼𝐶𝑜𝐼 𝐼𝐼𝐶𝑠𝑐𝑠 𝐶𝐼 �𝐻2 + 𝑐(𝐻2−𝐻1)1−𝑐

�

= �𝑉𝑃(0)𝑁(𝐶1) − 𝐻2𝐶−�𝑟𝑓+𝛼�𝑇𝑁(𝐶2)�

−(1 − 𝑐) �𝑉𝑃(0)𝑁(𝐶1′ ) − (𝐻2 + 𝑐(𝐻2−𝐻1)1−𝑐

)𝐶−�𝑟𝑓+𝛼�𝑇𝑁(𝐶2′ )� (6)


𝐻2 �+�𝑟𝑓+𝛼+𝜎2

2 �𝑇

σ√𝑇, 𝐶2 =


𝜎2

2 �𝑇

σ√𝑇,

𝐶1′ =ln� 𝑉𝑃(0)

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐

�+�𝑟𝑓+𝛼+𝜎2

2 �𝑇

σ√𝑇, and 𝐶2′ =

ln� 𝑉𝑃(0)

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐

�+�𝑟𝑓+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

16

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).

17

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 $13.55.

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 $85.45.

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).

18

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 $83.81.

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 $63.90. Therefore, the strike price for BSM Option 1 for the carried interest

valuation is the sum of $100 and $63.90, or $163.90.

19

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)1−𝑐

�.

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 = $163.90 and �𝐻2 + 𝑐(𝐻2−𝐻1)1−𝑐

� = $179.87.

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 2014. We only consider companies for which at least one year’s historical common stock prices are available on Bloomberg.

20

Carried Interest, the fund manager’s expected present value total compensation as of the fund’s

inception is $17.39. 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,

21

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


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 one-

third 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


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

22

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 catch-

up 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.

23

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.

24

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 𝑉𝑃(𝑇) = H2 and 𝑉𝑃(𝑇) = H2 + c(H2−H1)1−c

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

= 𝐶−�𝑟𝑓+𝛼�𝑇 �� [𝑉𝑃(𝑇) − 𝐻2]𝑓(𝑉𝑃)𝐶𝑉𝑝𝐻2+𝑐(𝐻2−𝐻1)

1−𝑐

𝐻2�

+ 𝐶−�𝑟𝑓+𝛼�𝑇 �𝑐 � [𝑉𝑃(𝑇)− 𝐻1]∞

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐

𝑓(𝑉𝑃)𝐶𝑉𝑃�

= 𝐶−�𝑟𝑓+𝛼�𝑇 �� [𝑉𝑃(𝑇) − 𝐻2]𝑓(𝑉𝑃)𝐶𝑉𝑝∞

𝐻2− � [𝑉𝑃(𝑇) − 𝐻2]𝑓(𝑉𝑃)𝐶𝑉𝑝

∞

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐

�

+ 𝐶−�𝑟𝑓+𝛼�𝑇 �𝑐 � [𝑉𝑃(𝑇)− 𝐻1]∞

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐


25


𝐻2

− � �𝑉𝑃(𝑇) − �𝐻2 +𝑐(𝐻2 − 𝐻1)

1 − 𝑐�� 𝑓(𝑉𝑃)𝐶𝑉𝑝

∞

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐

− � �𝑐(𝐻2 − 𝐻1)

1 − 𝑐� 𝑓(𝑉𝑃)𝐶𝑉𝑝

∞

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐

�

+ 𝐶−�𝑟𝑓+𝛼�𝑇 �𝑐 � �𝑉𝑃(𝑇) − �𝐻2 +𝑐(𝐻2 − 𝐻1)

1 − 𝑐��

∞

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐

𝑓(𝑉𝑃)𝐶𝑉𝑃

+ 𝑐 � �(𝐻2 −𝐻1)

1 − 𝑐� 𝑓(𝑉𝑃)𝐶𝑉𝑝

∞

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐

�


𝐻2

− � �𝑉𝑃(𝑇) − �𝐻2 +𝑐(𝐻2 − 𝐻1)

1 − 𝑐�� 𝑓(𝑉𝑃)𝐶𝑉𝑝

∞

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐

�

+ 𝐶−�𝑟𝑓+𝛼�𝑇 �𝑐 � �𝑉𝑃(𝑇) − �𝐻2 +𝑐(𝐻2 − 𝐻1)

1 − 𝑐��

∞

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐


= �𝑉𝑃(0)𝑁(𝐶1) − 𝐻2𝐶−(𝑟𝑓+𝛼)𝑇𝑁(𝐶2)� − �𝑉𝑃(0)𝑁(𝐶1′ ) − (𝐻2 + 𝑐(𝐻2−𝐻1)1−𝑐

)𝐶−(𝑟𝑓+𝛼)𝑇𝑁(𝐶2′ )�

+ c �𝑉𝑃(0)𝑁(𝐶1′ ) − (𝐻2 + 𝑐(𝐻2−𝐻1)1−𝑐

)𝐶−(𝑟𝑓+𝛼)𝑇𝑁(𝐶2′ )�

=�𝑉𝑃(0)𝑁(𝐶1) − 𝐻2𝐶−(𝑟𝑓+𝛼)𝑇𝑁(𝐶2)�

− (1 − 𝑐)�𝑉𝑃(0)𝑁(𝐶1′ ) − (𝐻2 +𝑐(𝐻2 − 𝐻1)

1 − 𝑐)𝐶−(𝑟𝑓+𝛼)𝑇𝑁(𝐶2′ )�


𝐻2 �+�𝑟𝑓+𝛼+𝜎2

2 �𝑇

σ√𝑇, 𝐶2 =


𝜎2

2 �𝑇

σ√𝑇,

𝐶1′ =ln� 𝑉𝑃(0)

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐

�+�𝑟𝑓+𝛼+𝜎2

2 �𝑇

σ√𝑇, and 𝐶2′ =

ln� 𝑉𝑃(0)

𝐻2+𝑐(𝐻2−𝐻1)1−𝑐

�+�𝑟𝑓+𝛼−𝜎2

2 �𝑇

σ√𝑇

26

REFERENCES

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Information in Alternative Assets,” Journal of Portfolio Management 38 (3) (Spring), 89-

103.

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Briys, Eric and Francois de Varenne, 1997, “On the Risk of Insurance Liabilities: Debunking

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Fleischer, Victor, 2005, “The Missing Preferred Return,” Journal of Corporation Law 31 (Fall),

77-117.

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), 1685-

1717.

Howell, David W., 2014, “Valuing Profits Interests Used as Equity Compensation in LLC,”

Valuation Strategies (May/June), 4-46.

Hull, John C., 2012, Options, Futures, and Other Derivatives, 8th ed., Pearson, Boston,.

Johnson, Shane A., and Yisong S. Tian, 2000a, “Indexed Executive Stock Options,” Journal of

Financial Economics 57 (July), 35-64.

Johnson, Shane A., and Yisong S. Tian, 2000b, “The Value and Incentive Effects of

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3-34.

Margrabe, William, 1978, “The Value of an Option to Exchange One Asset for Another,”

Journal of Finance 33 (March), 177-186.

27

Metrick, Andrew, and AyakoYasuda, 2010, “The Economics of PE Funds,” Review of Financial

Studies 23 (6), 2303-2341.

Rouvinez, Christophe, 2005, “The Value of the Carry,” PE International, (July/August), 55-57.

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

http://www.cogentvaluation.com/pdf/CarriedInterestTechniques2pp.pdf

H1 H2 Terminal Fund Asset Value [VP(T)]

Figure 1 Payoff Diagram of the Value 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 without Catch Up

Hurdle Rate of Return with Catch Up

H2 + 𝑐(𝐻𝐻−𝐻𝐻)𝐻−𝑐

Am

ount

of a

Car

ried

Inte

rest

-30

-20

-10

0

10

20

30

40

90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210

Payo

ff A

mou

nt

Terminal Portfolio Value

Call Option Positon 1 Call Option Posiotn 2 Call Option Position 3

Figure 2 Payoff Diagram for Three Call Option Positions

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)

0

10

20

30

40

90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210

Payo

ff A

mou

nt

Terminal Portfolio Value

Call Option Position 1 + Call Option Position 2 + Call Option Position 3

Panel B. The Payoff Diagram for the Combination of the Call Option Positions 1, 2, and 3.

Assumptions[1] Parameter ValueInitial Value of the Portfolio V P (0) = H1 $100Holding Period (years) T 8Preferred (Hurdle) Rate (per annum) h 8%Carry Level c 20% [2]

Investment Return (per annum) µ 15%

Distribution at the End of the TermTerminal 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]

[2]

[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]

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.

Table 1Example of Distribution of Private Equity Fund Profits

When There Is a Hurdle Rate of Return and a Catch-Up Feature

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.

Year 1 2 3 4 5Average Year 0.5 1.5 2.5 3.5 4.5Fraction Invested[1] 26% 23% 25% 18% 8%(as a Percentage of Investment Capital)

Weighted Average Time to Invest 2.09 Years

Year 5 6 7 8 9 10Exit Rate (λ)[2] Average Year 4.5 5.5 6.5 7.5 8.5 9.5

10%Percentage Remaining 100% 90.48% 81.87% 74.08% 67.03% 60.65%Percentage Exit 0.00% 9.52% 8.61% 7.79% 7.05% 67.03% 8.63 Years



Notes:[1] The fractions are based on empirical research for buy-out funds. See Metrick and Yasuda (2010).[2] Assuming the exits follow an exponential distribution with a continuously compounded rate of decay. The percentage remaining is calculated

as e-λ(t-5).

Table 2Average Time to Invest and Average Date of Exit

Weighted Average Date of Exit

Panel B. Baseline Weighted Average Date of Exit Assuming the Final Exit Year is Year 10 Calculated from the Inception Date for the Fund

Panel A. Baseline Weighted Average Time to Invest from the Inception Date for the Fund

Panel A. Calculation of Net Investment Capital at the Inception of the FundBasis Value

[1] Committed Capital 100$ 100.00$ − [2] Establishment Cost 1.00% 1.00 − [3] Present Value of Management Fees 2.00% 13.55 = [4] Investment Capital 85.45$ − [5] Present Value of Monitoring Fees 0.40% 0.47 − [6] Transaction Fees (as Percentage of Transaction Value) 1.37% 1.17 = [7] Net Investment Capital at the Inception of the Fund 83.81$

Panel B. Calculation of Strike Price for Carried Interest Option Pricing Model

[1] Committed Capital 100.00$ + [8] Hurdle Return 8.00% 63.90 = [9] Carried Interest Option Pricing Model Strike Price 163.90$

Notes:[1] Assumed to be $100 at the inception of the fund for the purposes of illustration. [2] Establishment Costs equal 1% of Committed Capital, which is paid at the inception of the fund (t=0).[3] Management fee and risk-free rate are assumed to be 2% per annum. See Table 4.[4] Represents Investment Capital before Transaction Fees and Monitoring Fees.[5] Monitoring Fee is assumed to be 0.4% of the Investment Capital per annum. See Table 5. [6][7] Equals Investment Capital minus Present Value of Monitoring Fees minus Transaction Fees.[8]

[9] Strike price for the Carried Interest option pricing model.

Transaction Fees are equal to 1.37% of Investment Capital, which are paid at the inception of the fund.

Equals Committed Capital × (1 + Hurdle Rate × Weighted Average Date of Exit). Calculation assumes Hurdle Rate of 8% on a simple interest basis and Weighted Average Date of Exit of 7.99 years. See Table 2.

Table 3Amount of Capital Invested and Option Strike Price

Assumptions:• Present value as of the inception of the fund.• Calculation base:[1]

(a) Committed Capital for the first five years (Investing Period)(b) Net Invested Capital for the last five years (Holding Period)

• Solve for the Investment Capital and fees such thatCommitted Capital = Investment Capital + Establishment Cost + Management Fees.

Panel A. The First Five Years (Investing Period)

Risk-Free Rate 1.00% 1.25% 1.50% 1.75% 2.00%1% 4.85$ 6.07$ 7.28$ 8.49$ 9.71$ 2% 4.71$ 5.89$ 7.07$ 8.24$ 9.42$ 3% 4.57$ 5.72$ 6.86$ 8.00$ 9.15$ 4% 4.44$ 5.55$ 6.66$ 7.77$ 8.88$ 5% 4.31$ 5.39$ 6.47$ 7.55$ 8.63$

Panel B. The Last Five Years (Holding Period)

Risk-Free Rate 1.00% 1.25% 1.50% 1.75% 2.00%1% 2.42$ 2.96$ 3.48$ 3.98$ 4.46$ 2% 2.24$ 2.75$ 3.23$ 3.69$ 4.13$ 3% 2.08$ 2.55$ 2.99$ 3.42$ 3.83$ 4% 1.93$ 2.36$ 2.78$ 3.17$ 3.55$ 5% 1.79$ 2.19$ 2.57$ 2.94$ 3.29$

Notes:[1] Per $100 of Committed Capital at the inception of the fund.[2]

Present Value of Investing Period Management FeesAssuming Annual Management Fee of[3]

Present Value of Holding Period Management FeesAssuming Annual Management Fee of[3]

Table 4Calculation of the Present Value of Management Fees

Management fee may be different in the Investing Period and the Holding Period. Assummed 2% management fees on Committed Capital for the first five years (Investing Period) and 2% fees on Invested Capital in each year for the last five years (Holding Period).

Assumptions:Present value as of the inception of the fundCalculation base:

Investment Capital 85.45$ Monitoring Fees (per annum)[1] 0.40%

Panel A. General Partner Keeps Entire Monitoring Fees

Discount Rate 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.51% 1.93$ 2.08$ 2.23$ 2.38$ 2.52$ 2.67$ 2.81$ 2.95$ 2% 1.82$ 1.95$ 2.08$ 2.21$ 2.33$ 2.45$ 2.57$ 2.69$ 3% 1.71$ 1.83$ 1.94$ 2.05$ 2.15$ 2.25$ 2.35$ 2.44$ 4% 1.61$ 1.71$ 1.81$ 1.90$ 1.99$ 2.07$ 2.15$ 2.22$ 5% 1.52$ 1.61$ 1.69$ 1.76$ 1.83$ 1.90$ 1.96$ 2.02$

Panel B. General Partner Keeps 20% of Monitoring Fees

Discount Rate 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.51% 0.39$ 0.42$ 0.45$ 0.48$ 0.50$ 0.53$ 0.56$ 0.59$ 2% 0.36$ 0.39$ 0.42$ 0.44$ 0.47$ 0.49$ 0.51$ 0.54$ 3% 0.34$ 0.37$ 0.39$ 0.41$ 0.43$ 0.45$ 0.47$ 0.49$ 4% 0.32$ 0.34$ 0.36$ 0.38$ 0.40$ 0.41$ 0.43$ 0.44$ 5% 0.30$ 0.32$ 0.34$ 0.35$ 0.37$ 0.38$ 0.39$ 0.40$

Notes:[1]

[2]

Table 5Calculation of the Present Value of Monitoring Fees

The Investment Capital ending on the date that corresponds to the Weighted Average Date of Exit (7.99 years). See Table 3.Based on 2% of EBITDA and an average EBITDA multiple of 5x for the fund's investments.

Present Value of Monitoring FeesAssuming Weighted Average Date of Exit (in Years)[2]

Present Value of Monitoring FeesAssuming Weighted Average Date of Exit (in Years)[2]

Panel A. Investment Expected Return and VolatilityInvestment (i ) 1 2 3 4 5 6 7 8 9 10Expected Return 10% 8% 9% 12% 15% 20% 25% 30% 15% 10%Volatility (σ i ) 31% 49% 62% 49% 34% 27% 44% 24% 47% 46%Weight (w i ) 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%

Panel B. Correlation Matrix (ρ)Investment 1 2 3 4 5 6 7 8 9 10

1 1.0000 0.1545 0.1846 0.0903 0.1114 0.0396 0.1693 -0.0192 0.1229 0.18192 0.1545 1.0000 0.1772 0.0184 0.3222 0.0910 0.0821 0.2177 0.0235 0.41633 0.1846 0.1772 1.0000 -0.0913 0.2050 0.1204 0.0758 0.0593 0.0515 0.07634 0.0903 0.0184 -0.0913 1.0000 0.1316 0.1255 0.0910 0.1691 0.0659 0.14195 0.1114 0.3222 0.2050 0.1316 1.0000 0.1550 0.0615 0.2635 0.1293 0.17726 0.0396 0.0910 0.1204 0.1255 0.1550 1.0000 0.0303 -0.0094 0.0284 0.09337 0.1693 0.0821 0.0758 0.0910 0.0615 0.0303 1.0000 0.1115 0.0026 0.05798 -0.0192 0.2177 0.0593 0.1691 0.2635 -0.0094 0.1115 1.0000 0.0396 0.16879 0.1229 0.0235 0.0515 0.0659 0.1293 0.0284 0.0026 0.0396 1.0000 -0.0122

10 0.1819 0.4163 0.0763 0.1419 0.1772 0.0933 0.0579 0.1687 -0.0122 1.0000

Panel C. Investment Portfolio Expected Return and VolatilityPortfolio Expected Return (μ )[1] 15.40%Portfolio Volatility[2] 18.51%

Panel D. Risk-Neutralized Expected Return of the Investment Portfolio[3]

Risk-Free Rate (r f ) 2.00%Market Rate (r m ) 9.00%Beta of the Portfolio (β ) 1.16Premium Return (α ) 5.28%

Risk-Neutralized Expected Return (r f +α ) 7.28%Notes:

[1]

[2] The expected volatility of the investment is calculated as the square root of the portfolio variance

[3]

Table 6Investment Portfolio Expected Return and Volatility

Expected average annual rate of return on fund equity net of monitoring fees. This calculation takes into account the average amount the fund invests in each portfolio company.

The risk-neutralized expected return is calculated based on the following formula: α = μ - r f - β(r m -r f ) . See Goetzmann, Ingersoll and Ross (2013), p. 1690.

�.𝑛

𝑖=𝐻

�ρ𝑖𝑖𝑤𝑖𝑤𝑖σ𝑖σ𝑖

𝑛

𝑗=𝐻.

Assumptions Parameter ValueCommitted Capital CP or H1 100.00$ Net Investment Capital at Inception of the Fund V P (0) 83.81$ Holding Period (years) T 7.99 Preferred (Hurdle) Rate (per annum) h 8%Carried Interest Percentage c 20%

Valuation of a Carried Interest

Input Parameters for the BSM ModelValue for Option 1

Value for Option 2

Initial Value of Asset (Net Investment Capital) S 0 83.81$ 83.81$ Risk-Neutralized Expected Return of the Portfolio r 7.28% 7.28%Volatility of the Portfolio σ 18.51% 18.51%Time to Expiration T 7.99 7.99 Dividend d 0% 0%Strike Price K 163.90$ [1] 179.87$ [2]

Value of the Carried Interest 8.51$ 6.43$ 3.37$

Notes:[1] = H2 = CP × (1+h ×T)[2] = H2 + c×(H2-H1)/(1-c)

Table 7Valuing the Carried Interest Based on the Black-Scholes-Merton Option Pricing Model

Option 1- (1 - 20%) × Option 2

Calculation Base:Management Fees 2.00% per annum (PV $13.55)Monitoring Fees 0.40% per annum (PV $0.47)Net Investment Capital 83.81$

Panel A. Value of a Carried Interest

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%Structure 1 5.19$ - - - - - - - - - -Structure 2 5.19$ 4.54$ 3.96$ 3.44$ 2.99$ 2.60$ 2.26$ 1.96$ 1.70$ 1.48$ 1.29$ Structure 3 5.19$ 5.15$ 5.02$ 4.82$ 4.57$ 4.28$ 3.98$ 3.68$ 3.37$ 3.08$ 2.80$

Panel B. Carried Interest as a Percentage of Total Fees

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%Structure 1 27% - - - - - - - - - -Structure 2 27% 24% 22% 20% 18% 16% 14% 12% 11% 10% 8%Structure 3 27% 27% 26% 26% 25% 23% 22% 21% 19% 18% 17%

Value of a Carried Interest as a percentage of Fund Manager's Expected Total Compenssation (Carried Interest, Management Fees, and Monitoring Fees)

Assuming Hurdle Rate of Return of

Table 8Sensitivity Analysis for the Value of the Carried Interest

- Sensitivity to Hurdle Rate of Return

Value of a Carried Interest Assuming Hurdle Rate of Return of



4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%Structure 1 4.02$ 4.39$ 4.75$ 5.10$ 5.44$ 5.76$ 6.08$ 6.38$ 6.67$ 6.96$ 7.24$ 7.50$ Structure 2 1.20$ 1.35$ 1.51$ 1.66$ 1.81$ 1.96$ 2.11$ 2.26$ 2.41$ 2.56$ 2.70$ 2.85$ Structure 3 2.44$ 2.73$ 3.01$ 3.29$ 3.57$ 3.84$ 4.11$ 4.38$ 4.64$ 4.89$ 5.15$ 5.39$

Panel B. Carried Interest as a Percentage of Total Fees

4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%Structure 1 22% 24% 25% 27% 28% 29% 30% 31% 32% 33% 34% 35%Structure 2 8% 9% 10% 11% 11% 12% 13% 14% 15% 15% 16% 17%Structure 3 15% 16% 18% 19% 20% 22% 23% 24% 25% 26% 27% 28%

Value of a Carried Interest Assuming Risk-Neutralized Expected Return of the Portfolio of


Assuming Risk-Neutralized Expected Return of the Portfolio of


- Sensitivity to Risk-Neutralized Expected Return of the Portfolio



5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60%Structure 1 1.90$ 3.47$ 4.54$ 5.46$ 6.30$ 7.10$ 7.85$ 8.57$ 9.26$ 9.91$ 10.53$ 11.11$ Structure 2 0.00$ 0.19$ 0.97$ 2.02$ 3.10$ 4.15$ 5.16$ 6.11$ 7.02$ 7.88$ 8.69$ 9.46$ Structure 3 0.00$ 0.60$ 2.27$ 3.79$ 5.04$ 6.10$ 7.05$ 7.91$ 8.70$ 9.44$ 10.12$ 10.76$

Pa Panel B. Carried Interest as a Percentage of Total Fees

5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60%Structure 1 12% 20% 24% 28% 31% 34% 36% 38% 40% 41% 43% 44%Structure 2 0% 1% 6% 13% 18% 23% 27% 30% 33% 36% 38% 40%Structure 3 0% 4% 14% 21% 26% 30% 33% 36% 38% 40% 42% 43%

Value of a Carried Interest Assuming Volatility of the Portfolio Returns of


Assuming Volatility of the Portfolio Returns of


- Sensitivity to the Volatility of the Portfolio Returns

valuing a private equity carried interest …...valuing a private equity carried interest as a call...

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