In the global low-interest rate environment, PE funds are of great interest to investors as a part of alternative investments seeking higher absolute returns. For this reason, the Internal Rate of Return (IRR) and Total Value per $ invested (TVPI) with respect to absolute return have been widely used as performance indicators for PE funds (e.g., Gilligan and Wright [1] ). In recent years, PE funds have been the useful tool to enhance absolute return in the public pension investment. For example, the GPIF in Japan has set a target absolute return (annualized over the five-year accounting period from 2025 to 2029) of 1.9% over the wage growth rate. In addition to the target absolute return, the GPIF also tries to seek an excess return over the policy benchmark (e.g., Government Pension Investment Fund [2] ).

Because the cash flows from the investment in PE funds are complex, it is more difficult to evaluate the relative performance of PE funds than active funds, which invest in listed equities. Prior studies on relative performance measures are PME (Public Market Equivalent) such as PME (e.g., Long and Nickeles [3] ), PME+ (e.g., Rouvinez [4] ), mPME (e.g., Cambridge Associates [5] ), Direct Alpha (e.g., Gredi et al. [6] ), and Spread Based Direct Alpha (SBDA) (e.g., Miyazaki and Shimada [7] [8] ). Among these PMEs, SBDA has following three attractive features 1) The excess return (SBDA) is obtained by discounting the cash flows with the benchmark return from the time they occur to the time of commitment, like the credit spread of a corporate bond; 2) The distributed cash flows are not reinvested in the PE fund, but return to the benchmark, so that the cash flow after the time of distribution does not affect the excess return, and 3) Not only percentage display but also dollar value of the excess return can be easily obtained. For the derivation of the dollar value of the excess return, refer to Miyazaki and Shimada [7] .

While SBDA plays a role in measuring the ex-post excess return of PE funds, it is difficult to capture the mechanism by which factors have influence on SBDA, because cash flows of PE funds are complex and subject to many factors. Therefore, in this study, we newly introduce expected SBDA as a forward-looking measure of excess return of PE funds, and formulate factors such as drift and volatility of the investee companies, time for PE fund to find them, and target exit multiple to evaluate the expected SBDA, and discuss the mechanism to generate expected SBDA by examining the sensitivity of the key parameters of the model to the expected SBDA.

This paper is organized as follows. Section 2 reviews SBDA, the ex-post relative performance measure of PE funds. Section 3, as Valuation Model I, proposes the valuation model of the expected SBDA focusing on the case where the PE fund invests in only one investee company. Section 4 extends Valuation Model I to the case of multiple investee companies as Valuation Model II. Section 5 gives a numerical example focusing on Valuation Model I and derives the implications from the sensitivity of the key parameters of the model to the expected SBDA. In the final section, summary and future issues are added.

Since PE funds invest mainly in unlisted stocks, the benchmark of PE funds is an equity index. For example, for a PE fund that invests in unlisted stocks worldwide, the benchmark would be the MSCI All Country World Index (ACWI). When an investor commits to invest in a PE fund, he or she first sets the commitment amount, sourcing period (e.g., five years after commitment), and investment period, which is the total commitment period minus the sourcing period (e.g., up to 10 years after commitment). The PE fund finds investee companies during the sourcing period and then makes a capital call to raise funds from investors to invest in the investee company, and when the company’s business takes off, it exits (through IPO, M&A, etc.) and distributes the results back to the investors within the investment period. Thus, unlike investments in ordinary active equity funds, investments in PE funds have complex cash flows, and it is difficult to evaluate how well their performance is relative to the benchmark (relative performance).

When an investor who is evaluated against ACWI as a benchmark tries to invest in PE funds, the idea of SBDA is to “capture the excess return obtained on average over the commitment period”. Therefore, the commitment period of SBDA for a PE fund is defined as the period from the time point when the investor commits to invest in the PE fund (when the investor contracts to transfer funds from ACWI to the PE fund in response to a capital call) up to the time point when all investment results have been distributed by the last exit. Based on the above, the definition of SBDA is given by Equation (1).

(Definition 1) SBDA

SBDA is 𝑠 that satisfies the following equation.

${\displaystyle {\sum }_{j=1}^m\frac{Call\left(j\right)}{{\left(1+r_j\right)}^j}}={\displaystyle {\sum }_{i=1}^n\frac{Dist\left(i\right)}{{\left(1+r_i+s\right)}^i}}+\frac{NAV}{{\left(1+r_i+s\right)}^n}$ (1)

where $Call\left(j\right)$ and $r_j$ are the amount of funds invested in response to the capital call at $j$ and the annualized benchmark return such that the cumulative return from time 0 (the time of commitment to the PE fund) to time $j$ is ${\left(1+r_j\right)}^j$, respectively; $Dist\left(i\right)$ and $r_i$ are the amount of the distribution and the annualized benchmark rate such that the cumulative return on the benchmark from time 0 to time $i$ is ${\left(1+r_i\right)}^i$ respectively, and $NAV$ represents the valuation of the last distribution. The above formula shows that there are $m$ capital calls and $n$ distributions. In addition, the PE fund must meet the following conditions as stipulated in the original agreement: ${\displaystyle {\sum }_{j=1}^{m}Call\left(j\right)}$ is within the commitment amount, $m$ indicating the time of the last capital call is within the investment period, and 𝑛 expressing the time of the last distribution within the investment period. For more information on the mechanism of SBDA and other related issues, see Miyazaki and Shimada [7] .

The point in time when the PE fund invests the fund raised from investor through capital call in the investee companies is represented by the stochastic variable ${\tilde{t}}_C$, and the sourcing period $\left[0,T_1\right]$ ( $0\le {\tilde{t}}_C\le T_1$) is the period for the PE fund to find the investee companies. The stochastic variable ${\tilde{t}}_D$ represents the investment period from the time the PE fund invests at ${\tilde{t}}_C$ to the time it exits, and the commitment to the PE fund is completed at $T_2$. The total commitment period to the PE fund is $\left[0,T_2\right]$ ( $T_1<T_2$), so $0\le {\tilde{t}}_C+{\tilde{t}}_D\le T_2$ must be satisfied.

(Modeling the stochastic variable ${\tilde{t}}_C$ which represents the beginning of the investment)

Following a standard approach for modeling random, independent events (see for example, Shreve [9] , Chan et al. [10] ), a lump-sum group of companies consisting of venture companies and buyout targets sourced by the PE fund will be assumed to emerge according to Poisson’s arrival with some arrival rate depending on the macroeconomic environment and the PE fund’s ability. $X\left(t\right)$ is the number of investee companies for the PE fund in the period [0, t] and this is modeled by the Poisson arrival in Equation (2) (Poisson distribution with parameter $\lambda t$).

$P\left(X\left(t\right)=k\right)=\frac{{\text{e}}^{-\lambda t}{\left(\lambda t\right)}^k}{k!}$ $k=0,1,2,\cdots$ (2)

Let $T$ denote the stochastic variable that represents the arrival interval of a group of firms that come one after another according to Poisson arrival, and consider its probability density function $f\left(t\right)$. Since the event $\left\{T>t\right\}$ represents that the arrival interval $T$ is greater than $t$, it is the same as the event $\left\{X\left(t\right)=0\right\}$ because it represents that the next group of firms has not appeared up to the time $t$, and we obtain Equation (3) by putting $k=0$ in Equation (2).

${\displaystyle {\int }_t^{\infty }f\left(x\right)\text{d}x}=P\left(T>t\right)=P\left(X\left(t\right)=0\right)={\text{e}}^{-\lambda t}$ (3)

By differentiating $F\left(t\right)={\displaystyle {\int }_0^{t}f\left(x\right)\text{d}x}=1-{\text{e}}^{-\lambda t}$ by $t$, the probability density function or probability of the stochastic variable ${\tilde{t}}_C$ representing the investment point in time is modeled by Equations (4) and (5).

Cases in which investment is made:

$f_{t_C}\left(t\right)=\lambda {\text{e}}^{-\lambda t}$ $\left(0\le t<T_1\right)$(4)

Probability that will not be invested:

$P\left(t_C>T_1\right)={\displaystyle {\int }_{T_1}^{\infty }\lambda {\text{e}}^{-\lambda t}\text{d}t}={\text{e}}^{-\lambda T_1}$(5)

(Modeling benchmark ( $i=0$) and investee company ( $i=1$) stock prices using stochastic process $S_i\left(t\right)$).

The drift and the volatility of the investee company return usually larger than those of the benchmark return, i.e. ${\mu }_1>{\mu }_0$ and ${\sigma }_1>{\sigma }_0$.

$\frac{\text{d}{S}_i\left(t\right)}{S_i\left(t\right)}={\mu }_i\text{d}t+{\sigma }_i\text{d}{B}_i\left(t\right)$ (6)

where ${\mu }_i$, ${\sigma }_i>0$, and $B_i\left(t\right)$ are the drift, volatility, and standard Brownian motion, respectively.

(Modeling the stochastic variable ${\tilde{t}}_D$ representing the investment period)

PE funds typically attempt to exit when the stock price of the investee company reaches several times the stock price $S_1\left({\tilde{t}}_C\right)$ at the starting time of investment ${\tilde{t}}_C$. Here, we assume that the PE funds exit at time ${\tilde{t}}_C+{\tilde{t}}_D$ when the stock price reaches $\gamma S_1\left({\tilde{t}}_C\right)$, which is $\gamma$ times $S_1\left({\tilde{t}}_C\right)$, for the first time (first arrival time). If the stock price never reaches $\gamma S_1\left({\tilde{t}}_C\right)$ during the commitment period $\left[0,T_2\right]$, we assume exit with the stock price $S_1\left(T_2-{\tilde{t}}_C\right)$ at $T_2$. If the stock price of the investee company follows the stochastic process in Equation (6), the stock price $S_1\left(t\right)$ ( $0<t<T_2-{\tilde{t}}_C$) at time $t$ from the starting time of investment ${\tilde{t}}_C$ is given by Equation (7).

$S_1\left(t\right)=S_1\left(t_C\right){\text{e}}^{\left({\mu }_1-\frac{1}{2}{\sigma }_1^2\right)t+{\sigma }_1{B}_1\left(t\right)}$ (7)

The probability density function or probability of the stochastic variable ${\tilde{t}}_D$ representing the investment period in the stock $S_1\left(t\right)$ is given by Lemma 1.

Lemma 1

The probability density function or not hitting probability of the stochastic variable ${\tilde{t}}_D$ representing the investment period from the starting point of investment ${\tilde{t}}_C$ is given by

Case 1 (the stock price reaches $\gamma S_1\left(t_C\right)$ within the investment period $\left[t_C,T_2\right]$)

The probability density function for the random variable ${\tilde{t}}_D$ is given by Equation (8).

$f_{t_D}\left(t_D\right)=\frac{\log \gamma }{\sqrt{2\pi t_D^3}{\sigma }_1}\exp \left[-\frac{{\left\{\log \gamma -\left({\mu }_1-\frac{1}{2}{\sigma }_1^2\right)t_D\right\}}^2}{2{\sigma }_1^2{t}_D}\right]$ $\left(0<t_D\le T_2-{\tilde{t}}_C\right)$(8)

Case 2 (the stock price does not reach $\gamma S_1\left(t_C\right)$ within the investment period $\left[t_C,T_2\right]$)

The probability $P\left({\tilde{t}}_D=T_2-{\tilde{t}}_C\right)$ is given by Equation (9).

$P\left({\tilde{t}}_D=T_2-{\tilde{t}}_C\right)\equiv P\left({\tilde{t}}_D>T_2-{\tilde{t}}_C\right)=1-{\displaystyle {\int }_0^{T_2-{\tilde{t}}_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}$(9)

(Proof)

From Equation (7), the stock price $S_1\left(t\right)$ reaching $\gamma S_1\left({\tilde{t}}_C\right)$ is the same as the Brownian motion with drift $\left({\mu }_1-\frac{1}{2}{\sigma }_1^2\right)t+{\sigma }_1{B}_1\left(t\right)$ reaching $\log \gamma$.

Regarding the standard Brownian motion $\left\{B\left(t\right),t\ge 0\right\}$, let $M^{*}\left(t\right)\stackrel{\text{def}}{=}\underset{0\le u\le t}{\max }B\left(u\right)$ be the maximum value up to the point in time $t$. Then, from the mirror image principle, we obtain Equation (10).

$P\left[M^{*}\left(t\right)\ge x,B\left(t\right)\le x|B\left(0\right)=0\right]=P\left[B\left(t\right)\ge x|B\left(0\right)=0\right]$ $\left(x>0\right)$(10)

Using Equation (10),

$\begin{array}{l}P\left[M^{*}\left(t\right)\ge x|B\left(0\right)=0\right]\\ =P\left[M^{*}\left(t\right)\ge x,B\left(t\right)\le x|B\left(0\right)=0\right]+P\left[M^{*}\left(t\right)\ge x,B\left(t\right)\ge x|B\left(0\right)=0\right]\\ =P\left[M^{*}\left(t\right)\ge x,B\left(t\right)\le x|B\left(0\right)=0\right]+P\left[B\left(t\right)\ge x|B\left(0\right)=0\right]\\ =2P\left[B\left(t\right)\ge x|B\left(0\right)=0\right]\\ =2\left(1-\Phi \left(\frac{x}{\sqrt{t}}\right)\right)\end{array}$(11)

is obtained, where $\Phi (\cdot )$ represents the distribution function of the standard normal distribution. From this,

$P\left[M^{*}\left(t\right)\le x|B\left(0\right)=0\right]=1-2\left(1-\Phi \left(\frac{x}{\sqrt{t}}\right)\right)=2\Phi \left(\frac{x}{\sqrt{t}}\right)-1$(12)

Denoting the first arrival time of $\left\{B\left(t\right),t\ge 0\right\}$ to the state $x$ as $T_x$, we see $\left\{T_x\le t\right\}=\left\{M^{*}\left(t\right)\ge x\right\}$, so from Equation (11)

$P\left[T_x\le t|B\left(0\right)=0\right]=2\left(1-\Phi \left(\frac{x}{\sqrt{t}}\right)\right)$(13)

is obtained. Furthermore, by differentiating Equation (13) with time $t$, the probability density function $f_x\left(t\right)$ of $T_x$ is attained as,

$f_x\left(t\right)=\frac{x}{\sqrt{2\pi t^3}}\exp \left\{-\frac{x^2}{2t}\right\}$, $\left(t>0\right)$(14)

From this result, the transition probability density of the $\left(\mu ,{\sigma }_1\right)$-Brownian motion with a drift of $\mu$ and a volatility of ${\sigma }_1$ is given by Equation (15).

$f_x\left(t\right)=\frac{x}{\sqrt{2\pi t^3}{\sigma }_1}\exp \left\{-\frac{{\left(x-\mu t\right)}^2}{2{\sigma }_1^{2}t}\right\}$, $\left(t>0\right)$(15)

Substituting $\mu ={\mu }_1-\frac{1}{2}{\sigma }_1^2$ into Equation (15), we obtain the probability density function Equation (8) for the stochastic variable ${\tilde{t}}_D$.

The probability that the stock price does not reach $\gamma S_1\left(t_C\right)$ at $\left[0,T_2\right]$ is obtained by integrating Equation (8) over the interval ${\tilde{t}}_D>T_2-{\tilde{t}}_C$ as Equation (9).

(QED)

In Case 2, where the stock price $S_1\left(t\right)$ at time $t$ from the time ${\tilde{t}}_C$ does not reach $\gamma S_1\left({\tilde{t}}_C\right)$ within the investment period $\left[t_C,T_2\right]$, the expected exit price is assumed to be the expected price of $S_1\left(T_2-{\tilde{t}}_C\right)$ at the end of the investment period $T_2$ and given by Lemma 2.

Lemma 2

When the stock price $S_1\left(u\right)=S_1\left({\tilde{t}}_c\right)\exp \left\{\left({\mu }_1-\frac{1}{2}{\sigma }_1^2\right)u+{\sigma }_1{B}_1\left(u\right)\right\}$ does not reach $\gamma S_1\left({\tilde{t}}_C\right)$ in the investment period $\left[t_C,T_2\right]$, with the definition $R_1\left(u\right)\stackrel{\text{def}}{=}\left({\mu }_1-\frac{1}{2}{\sigma }_1^2\right)u+{\sigma }_1{B}_1\left(u\right)=\mu u+{\sigma }_1{B}_1\left(u\right)$, the expected exit price of $S_1\left(T_2-{\tilde{t}}_C\right)$ at time $T_2$ is given as Equation (16) by calculating $E\left[S_1\left({\tilde{t}}_c\right){\text{e}}^{R_1\left(T_2-{\tilde{t}}_C\right)}{1}_{\left\{R_1\left(T_2-{\tilde{t}}_C\right)\le \log \gamma \right\}}\cdot 1_{\left\{\underset{0\le u\le T_2-{\tilde{t}}_C}{\max }{R}_1\left(u\right)\le \log \gamma \right\}}|{ℱ}_{{\tilde{t}}_c}\right]$.

$S_1\left({\tilde{t}}_c\right){\text{e}}^{{\mu }_1\left(T_2-{\tilde{t}}_C\right)}\left\{\Phi \left({K^{\prime }}_1\right)-{\gamma }^{\left(1+\frac{2{\mu }_1}{{\sigma }^2}\right)}\left(1-\Phi \left({K^{\prime }}_2\right)\right)\right\}$(16)

where ${K^{\prime }}_1=\frac{\log \gamma -\left({\mu }_1+\frac{1}{2}{\sigma }_1^2\right)\left(T_2-{\tilde{t}}_C\right)}{{\sigma }_1\sqrt{T_2-{\tilde{t}}_C}}$ and ${K^{\prime }}_2=\frac{\log \gamma +\left({\mu }_1+\frac{1}{2}{\sigma }_1^2\right)\left(T_2-{\tilde{t}}_C\right)}{{\sigma }_1\sqrt{T_2-{\tilde{t}}_C}}$ and $\Phi (\cdot )$ represents the distribution function of the standard normal distribution.

(Proof)

To calculate the expected value $\left[S_1\left({\tilde{t}}_c\right){\text{e}}^{R_1\left(T_2-{\tilde{t}}_C\right)}{1}_{\left\{R_1\left(T_2-{\tilde{t}}_C\right)\le \log \gamma \right\}}\cdot 1_{\left\{\underset{0\le u\le T_2-{\tilde{t}}_C}{\max }{R}_1\left(u\right)\le \log \gamma \right\}}|{ℱ}_{{\tilde{t}}_c}\right]$, we need the transition probability density of the $\left(\mu ,{\sigma }_1\right)$-Brownian motion with an absorbing boundary at $\log \gamma$. Referring to Kijima [11] , the transition probability density $a\left(x,y,t\right)$ of the $\left(\mu ,{\sigma }_1\right)$-Brownian motion with an absorbing boundary at $\log \gamma$ from state $x$ at time 0 to state $y$ at time $t$ is given by Equation (17). $x,y>0$; at $t>0$ time,

$\begin{array}{c}a\left(x,y,t\right)=\frac{1}{\sqrt{2\pi t}\cdot {\sigma }_1}\exp \left\{-\frac{{\left(y-x-\mu t\right)}^2}{2{\sigma }_1^{2}t}\right\}\\ -{\text{e}}^{-\frac{2\mu \left(x-\log \gamma \right)}{{\sigma }_1^2}}\frac{1}{\sqrt{2\pi t}\cdot {\sigma }_1}\exp \left\{-\frac{{\left(y+x-2\log \gamma -\mu t\right)}^2}{2{\sigma }_1^{2}t}\right\}\end{array}$ (17)

The derivation of Equation (17) is provided in Appendix.

Since we are interested in the transition probability density $\tilde{a}\left(0,y,t\right)$ for the $\left(\mu ,{\sigma }_1\right)$-Brownian motion with an absorbing boundary at state $\log \gamma$ starting from the state $R_1\left(0\right)=\mu \cdot 0+{\sigma }_1{B}_1\left(0\right)=0$ at time 0, to the state $y$ at time $t$, it is given by Equation (18).

$\begin{array}{c}\tilde{a}\left(0,y,t\right)=\frac{1}{\sqrt{2\pi t}\cdot {\sigma }_1}\exp \left\{-\frac{{\left(y-\mu t\right)}^2}{2{\sigma }_1^{2}t}\right\}\\ -{\text{e}}^{\frac{2\mu \log \gamma }{{\sigma }_1^2}}\frac{1}{\sqrt{2\pi t}\cdot {\sigma }_1}\exp \left\{-\frac{{\left(y-2\log \gamma -\mu t\right)}^2}{2{\sigma }_1^{2}t}\right\}\end{array}$ (18)

To obtain the expected value, we utilize Equation (18) to calculate Equation (19).

$\begin{array}{l}{\displaystyle {\int }_{-\infty }^{\log \gamma }{\text{e}}^y\tilde{a}\left(0,y,t\right)\text{d}y}\\ ={\displaystyle {\int }_{-\infty }^{\log \gamma }{\text{e}}^y\frac{1}{\sqrt{2\pi t}\cdot {\sigma }_1}\exp \left\{-\frac{{\left(y-\mu t\right)}^2}{2{\sigma }_1^{2}t}\right\}\text{d}y}\\ -{\text{e}}^{\frac{2\mu \log \gamma }{{\sigma }_1^2}}{\displaystyle {\int }_{-\infty }^{\log \gamma }{\text{e}}^y\frac{1}{\sqrt{2\pi t}\cdot {\sigma }_1}\exp \left\{-\frac{{\left(y-2\log \gamma -\mu t\right)}^2}{2{\sigma }_1^{2}t}\right\}\text{d}y}\end{array}$(19)

Compute the first term on the right-hand side of Equation (19),

${\displaystyle {\int }_{-\infty }^{\log \gamma }{\text{e}}^y\frac{1}{\sqrt{2\pi t}\cdot {\sigma }_1}\exp \left\{-\frac{{\left(y-\mu t\right)}^2}{2{\sigma }_1^{2}t}\right\}\text{d}y}$.

Convert the variable to $\frac{y-\mu t}{{\sigma }_1\sqrt{t}}=z_1$ and proceed with the calculation, paying attention to $\text{d}y={\sigma }_1\sqrt{t}\text{d}{z}_1$ and $z_1:-\infty \to \frac{\log \gamma -\mu t}{{\sigma }_1\sqrt{t}}$, to obtain ${\displaystyle {\int }_{-\infty }^{\frac{\log \gamma -\mu t}{{\sigma }_1\sqrt{t}}}{\text{e}}^{\mu t+{\sigma }_1\sqrt{t}{z}_1}\frac{1}{\sqrt{2\pi }}\exp \left\{-\frac{z_1^2}{2}\right\}\text{d}{z}_1}$.

$\begin{array}{l}{\displaystyle {\int }_{-\infty }^{\frac{\log \gamma -\mu t}{{\sigma }_1\sqrt{t}}}{\text{e}}^{\mu t+{\sigma }_1\sqrt{t}{z}_1}\frac{1}{\sqrt{2\pi }}\exp \left\{-\frac{z_1^2}{2}\right\}\text{d}{z}_1}\\ ={\text{e}}^{\left(\mu +\frac{1}{2}{\sigma }_1^2\right)t}{\displaystyle {\int }_{-\infty }^{\frac{\log \gamma -\mu t}{{\sigma }_1\sqrt{t}}}\frac{1}{\sqrt{2\pi }}\exp \left\{-\frac{{\left(z_1-{\sigma }_1\sqrt{t}\right)}^2}{2}\right\}\text{d}{z}_1}\end{array}$

Now, convert the variable to $z_1-{\sigma }_1\sqrt{t}=v_1$ and proceed with the calculation, paying attention to $\text{d}{z}_1=\text{d}{v}_1$, and $v_1:-\infty \to \frac{\log \gamma -\left(\mu +{\sigma }_1^2\right)t}{{\sigma }_1\sqrt{t}}$.

$\begin{array}{l}{\text{e}}^{\left(\mu +\frac{1}{2}{\sigma }_1^2\right)t}{\displaystyle {\int }_{-\infty }^{\frac{\log \gamma -\mu t}{{\sigma }_1\sqrt{t}}}\frac{1}{\sqrt{2\pi }}\exp \left\{-\frac{{\left(z_1-{\sigma }_1\sqrt{t}\right)}^2}{2}\right\}\text{d}{z}_1}\\ ={\text{e}}^{\left(\mu +\frac{1}{2}{\sigma }_1^2\right)t}{\displaystyle {\int }_{-\infty }^{\frac{\log \gamma -\left(\mu +{\sigma }_1^2\right)t}{{\sigma }_1\sqrt{t}}}\frac{1}{\sqrt{2\pi }}{\text{e}}^{-\frac{v_1^2}{2}}\text{d}{v}_1}\end{array}$

Finally, substituting $\mu ={\mu }_1-\frac{1}{2}{\sigma }_1^2$, the first term on the right side of Equation (19) becomes Equation (20).

${\text{e}}^{\mu t+\frac{{\sigma }_1^{2}t}{2}}\Phi \left(\frac{\log \gamma -\left(\mu +{\sigma }_1^2\right)t}{{\sigma }_1\sqrt{t}}\right)={\text{e}}^{{\mu }_{1}t}\Phi \left(\frac{\log \gamma -\left({\mu }_1+\frac{{\sigma }_1^2}{2}\right)t}{{\sigma }_1\sqrt{t}}\right)$(20)

In a similar manner, the second term on the right-hand side of Equation (19) is calculated.

${\text{e}}^{\frac{2\mu \log \gamma }{{\sigma }_1^2}}{\displaystyle {\int }_{-\infty }^{\log \gamma }{\text{e}}^y\frac{1}{\sqrt{2\pi t}\cdot {\sigma }_1}\exp \left\{-\frac{{\left(y-2\log \gamma -\mu t\right)}^2}{2{\sigma }_1^{2}t}\right\}\text{d}y}$

is calculated as follows.

Convert the variable as $\frac{y-2\log \gamma -\mu t}{{\sigma }_1\sqrt{t}}=v_2$ and proceed with the calculation, paying attention to $\text{d}y={\sigma }_1\sqrt{t}\text{d}{v}_2$ and $v_2:-\infty \to \frac{-\log \gamma -\mu t}{{\sigma }_1\sqrt{t}}$.

$\begin{array}{l}{\text{e}}^{\frac{2\mu \log \gamma }{{\sigma }_1^2}}{\displaystyle {\int }_{-\infty }^{\frac{-\log \gamma -\mu t}{{\sigma }_1\sqrt{t}}}{\text{e}}^{2\log \gamma +\mu t+{\sigma }_1\sqrt{t}{v}_2}\frac{1}{\sqrt{2\pi }}{\text{e}}^{-\frac{v_2^2}{2}}\text{d}{v}_2}\\ ={\text{e}}^{\frac{2\mu \log \gamma }{{\sigma }_1^2}+2\log \gamma +\mu t+\frac{1}{2}{\sigma }_1^{2}t}{\displaystyle {\int }_{-\infty }^{\frac{-\log \gamma -\mu t}{{\sigma }_1\sqrt{t}}}\frac{1}{\sqrt{2\pi }}{\text{e}}^{-\frac{{\left(v_2-{\sigma }_1\sqrt{t}\right)}^2}{2}}\text{d}{v}_2}\end{array}$

Finally, after variable conversion and setting $v_2-{\sigma }_1\sqrt{t}=w_2$, noting $\text{d}{v}_2=\text{d}{w}_2$ and $w_2:-\infty \to \frac{-\log \gamma -\left(\mu +{\sigma }_1^2\right)t}{{\sigma }_1\sqrt{t}}$, and further substituting $\mu ={\mu }_1-\frac{1}{2}{\sigma }_1^2$, the first term on the right side of Equation (19) becomes Equation (21).

$\begin{array}{l}{\text{e}}^{\frac{2\mu \log \gamma }{{\sigma }_1^2}+2\log \gamma +\mu t+\frac{1}{2}{\sigma }_1^{2}t}{\displaystyle {\int }_{-\infty }^{\frac{-\log \gamma -\mu t}{{\sigma }_1\sqrt{t}}}\frac{1}{\sqrt{2\pi }}{\text{e}}^{-\frac{{\left(v_2-{\sigma }_1\sqrt{t}\right)}^2}{2}}\text{d}{v}_2}\\ ={\text{e}}^{\frac{2\mu \log \gamma }{{\sigma }_1^2}+2\log \gamma +\mu t+\frac{{\sigma }_1^{2}t}{2}}{\displaystyle {\int }_{-\infty }^{\frac{-\log \gamma -\left(\mu +{\sigma }_1^2\right)t}{{\sigma }_1\sqrt{t}}}\frac{1}{\sqrt{2\pi }}{\text{e}}^{-\frac{w_2^2}{2}}\text{d}{w}_2}\\ ={\text{e}}^{\frac{2\mu \log \gamma }{{\sigma }_1^2}+2\log \gamma +\left(\mu +\frac{{\sigma }_1^2}{2}\right)t}\left\{1-\Phi \left(\frac{\log \gamma +\left(\mu +{\sigma }_1^2\right)t}{{\sigma }_1\sqrt{t}}\right)\right\}\\ ={\text{e}}^{\frac{2\left({\mu }_1-\frac{1}{2}{\sigma }_1^2\right)\log \gamma }{{\sigma }_1^2}+2\log \gamma }{\text{e}}^{\left({\mu }_1-\frac{1}{2}{\sigma }_1^2+\frac{{\sigma }_1^2}{2}\right)t}\left\{1-\Phi \left(\frac{\log \gamma +\left({\mu }_1-\frac{1}{2}{\sigma }_1^2+{\sigma }_1^2\right)t}{{\sigma }_1\sqrt{t}}\right)\right\}\\ ={\gamma }^{1+\frac{2\mu }{{\sigma }_1^2}}\cdot {\text{e}}^{{\mu }_{1}t}\left\{1-\Phi \left(\frac{\log \gamma +\left({\mu }_1+\frac{1}{2}{\sigma }_1^2\right)t}{{\sigma }_1\sqrt{t}}\right)\right\}\end{array}$(21)

In Equations (20) and (21), substituting $t=T_2-{\tilde{t}}_C$ and subtracting Equation (21) from Equation (20), we obtain Equation (16). (QED)

The expected return from committing to a PE fund is shown in Proposition 1, and the expected time from the beginning of commitment to exit is shown in Proposition 2.

Proposition 1

The expected absolute return $E\left({\tilde{R}}_{PE}\right)$ obtained from committing to the PE fund at time 0 until the PE fund exits at time ${\tilde{t}}_C+{\tilde{t}}_D$ is given by Equation (22).

$\begin{array}{c}E\left({\tilde{R}}_{PE}\right)={\mu }_0{T}_1{\text{e}}^{-\lambda T_1}+{\displaystyle {\int }_0^{T_1}\{{\mu }_0{t}_C\cdot \left(1+{\displaystyle {\int }_0^{T_2-t_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}\right)}\\ +\left(\log \gamma \right){\displaystyle {\int }_0^{T_2-t_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}+\log \Psi \left(t_C\right)\}\lambda {\text{e}}^{-\lambda t_C}\text{d}{t}_C\end{array}$(22)

where $\Psi \left(t_C\right)={\text{e}}^{{\mu }_1\left(T_2-t_C\right)}\left\{\Phi \left({K^{\prime }}_1\right)-{\gamma }^{\left(1+\frac{2{\mu }_1}{{\sigma }^2}\right)}\left(1-\Phi \left({K^{\prime }}_2\right)\right)\right\}$.

(Proof)

Derive the expected returns from committing to a PE fund in three cases and sum the three of them with appropriate probability weight.

1) PE fund cannot find an investee company within the sourcing period $\left[0,T_1\right]$ ( $T_1\le {\tilde{t}}_C$)

In this case, the commitment to the PE fund ends at the time $T_1$, so the return for this period is the return on the benchmark only, and given by Equation (23).

$\left\{\log \frac{E\left[S_0\left(0\right){\text{e}}^{\left({\mu }_0-\frac{1}{2}{\sigma }_0^2\right)T_1+{\sigma }_{0}B\left(T_1\right)}\right]}{S_0\left(0\right)}\right\}\cdot {\displaystyle {\int }_{T_1}^{\infty }\lambda {\text{e}}^{-\lambda t_c}\text{d}{t}_c}={\mu }_0{T}_1{\text{e}}^{-\lambda T_1}$(23)

2) PE fund invests an amount $S_1\left({\tilde{t}}_C\right)=S_0\left(0\right){\text{e}}^{\left({\mu }_0-\frac{1}{2}{\sigma }_0^2\right){\tilde{t}}_C+{\sigma }_{0}B\left({\tilde{t}}_C\right)}$ in an investee company at ${\tilde{t}}_C$ in the sourcing period $\left[0,T_1\right]$ and is able to exit at ${\tilde{t}}_C+{\tilde{t}}_D$ during the investment period $\left[{\tilde{t}}_C,T_2\right]$ at $\gamma S_1\left({\tilde{t}}_C\right)$.

In this case, the return at a given investment point in time ${\tilde{t}}_C=t_c$ is $\left\{\log \frac{E\left[\gamma \cdot S_0\left(0\right){\text{e}}^{\left({\mu }_0-\frac{1}{2}{\sigma }_0^2\right){\tilde{t}}_C+{\sigma }_{0}B\left({\tilde{t}}_C\right)}\right]}{S_0\left(0\right)}\right\}\cdot {\displaystyle {\int }_0^{T_2-{\tilde{t}}_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}$ using the exit probability ${\displaystyle {\int }_0^{T_2-{\tilde{t}}_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}$ from Lemma 1, so the expected return is given by Equation (24).

$\begin{array}{l}{\displaystyle {\int }_0^{T_1}\left\{\log \frac{E\left[\gamma \cdot S_0\left(0\right){\text{e}}^{\left({\mu }_0-\frac{1}{2}{\sigma }_0^2\right){\tilde{t}}_C+{\sigma }_{0}B\left({\tilde{t}}_C\right)}\right]}{S_0\left(0\right)}\right\}\cdot {\displaystyle {\int }_0^{T_2-{\tilde{t}}_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D\lambda {\text{e}}^{-\lambda t_c}\text{d}{t}_c}}\\ ={\displaystyle {\int }_0^{T_1}\left({\mu }_0{t}_c+\log \gamma \right)}\cdot {\displaystyle {\int }_0^{T_2-{\tilde{t}}_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D\lambda {\text{e}}^{-\lambda t_c}\text{d}{t}_c}\end{array}$(24)

3) PE fund invests an amount $S_1\left({\tilde{t}}_C\right)=S_0\left(0\right){\text{e}}^{\left({\mu }_0-\frac{1}{2}{\sigma }_0^2\right)T_1+{\sigma }_{0}B\left(T_1\right)}$ in an investee company at ${\tilde{t}}_C$ in the sourcing period $\left[0,T_1\right]$ but fails to exit in the investment period $\left[{\tilde{t}}_C,T_2\right]$ and sells it with the price $S_0\left(0\right){\text{e}}^{\left({\mu }_0-\frac{1}{2}{\sigma }_0^2\right){\tilde{t}}_C+{\sigma }_{0}B\left({\tilde{t}}_C\right)+\left({\mu }_1-\frac{1}{2}{\sigma }_1^2\right)\left(T_2-{\tilde{t}}_C\right)+{\sigma }_{1}B\left(T_2-{\tilde{t}}_C\right)}=S_1\left({\tilde{t}}_C\right){\text{e}}^{R_1\left(T_2-{\tilde{t}}_C\right)}$ at the point of time $T_2$.

The expected return is given by Equation (25), using Equation (16) in Lemma 2.

${\displaystyle {\int }_0^{T_1}\log \frac{E\left[S_1\left({\tilde{t}}_c\right){\text{e}}^{R_1\left(T_2-{\tilde{t}}_C\right)}{1}_{\left\{R_1\left(T_2-{\tilde{t}}_C\right)\le \log \gamma \right\}}\cdot 1_{\left\{\underset{0\le u\le T_2-{\tilde{t}}_C}{\max }{R}_1\left(u\right)\le \log \gamma \right\}}|{ℱ}_{{\tilde{t}}_c}\right]}{S_0\left(0\right)}\text{d}{\tilde{t}}_c}$(25)

Summing Equations (23)-(25), we obtain Equation (22). (QED)

Proposition 2

The expected time $E\left({\tilde{T}}_{PE}\right)$ from the beginning of the commitment to the PE fund at time 0 to the final exiting at time ${\tilde{t}}_C+{\tilde{t}}_D$ is given by Equation (26).

$\begin{array}{l}E\left({\tilde{T}}_{PE}\right)=E\left({\tilde{t}}_C+{\tilde{t}}_D\right)=T_1{\text{e}}^{-\lambda T_1}+T_2\left(1-{\text{e}}^{-\lambda T_1}\right)\\ +{\displaystyle {\int }_0^{T_1}\left(\left(\left(t_C+{\displaystyle {\int }_0^{T_2-t_C}{t}_D{f}_{t_D}\left(t_D\right)\text{d}{t}_D}\right)-T_2\right)\left({\displaystyle {\int }_0^{T_2-t_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}\right)\right)\lambda {\text{e}}^{-\lambda t_c}\text{d}{t}_c}\end{array}$ (26)

(Proof)

Derive the expected times committing to a PE fund in three cases and sum the three of them with appropriate probability weights.

1) PE fund cannot find an investee company in the sourcing period $\left[0,T_1\right]$ ( $T_1\le {\tilde{t}}_C$).

In this case, the expected time is given by Equation (27) because the commitment to the PE fund ends at the time $T_1$.

$T_1\cdot {\displaystyle {\int }_{T_1}^{\infty }\lambda {\text{e}}^{-\lambda t_c}\text{d}{t}_c}=T_1{\text{e}}^{-\lambda T_1}$(27)

2) PE fund invests in an investee company at ${\tilde{t}}_C$ in the sourcing period $\left[0,T_1\right]$ and exits at ${\tilde{t}}_C+{\tilde{t}}_D$ in the investment period $\left[{\tilde{t}}_C,T_2\right]$.

In this case, the expected time is given by Equation (28), since the probability density function of the time to exit given the point in time of investment ${\tilde{t}}_C=t_c$ is Equation (8) in Lemma 1.

${\displaystyle {\int }_0^{T_1}\left(t_C+{\displaystyle {\int }_0^{T_2-t_C}{t}_D{f}_{t_D}\left(t_D\right)\text{d}{t}_D}\right)\left({\displaystyle {\int }_0^{T_2-t_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}\right)\lambda {\text{e}}^{-\lambda t_c}\text{d}{t}_c}$(28)

3) PE fund invests in an investee company at ${\tilde{t}}_C$ in the sourcing period $\left[0,T_1\right]$ but fails to exit in the investment period $\left[{\tilde{t}}_C,T_2\right]$ and sells it at time $T_2$.

The expected time is given by Equation (29), using Equation (9) in Lemma 1.

$\begin{array}{l}{\displaystyle {\int }_0^{T_1}\left(t_C+\left(T_2-t_C\right)\right)\left(1-{\displaystyle {\int }_0^{T_2-t_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}\right)\lambda {\text{e}}^{-\lambda t_c}\text{d}{t}_c}\\ =T_2{\displaystyle {\int }_0^{T_1}\left(1-{\displaystyle {\int }_0^{T_2-t_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}\right)\lambda {\text{e}}^{-\lambda t_c}\text{d}{t}_c}\\ =T_2\left(1-{\text{e}}^{-\lambda T_1}\right)-{\displaystyle {\int }_0^{T_1}{T}_2\left({\displaystyle {\int }_0^{T_2-t_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}\right)\lambda {\text{e}}^{-\lambda t_c}\text{d}{t}_c}\end{array}$(29)

Summing Equations (27)-(29), we obtain Equation (26). (QED)

Here, corresponding to SBDA in (Definition 1) that evaluates the ex-post return of the PE fund, we propose the expected SBDA given by Equation (30) in (Definition 2) that allows us to compare the expected excess return of the PE funds with that of the regular active equity funds.

(Definition 2) Expected SBDA

The expected SBDA is $\alpha$ that satisfies the following equation

$\left({\mu }_0+\alpha \right)E\left({\tilde{T}}_{PE}\right)=E\left({\tilde{R}}_{PE}\right)$(30)

where ${\mu }_0$, $E\left({\tilde{R}}_{PE}\right)$, and $E\left({\tilde{T}}_{PE}\right)$ are the benchmark drift, Equation (22) in Proposition 1, and Equation (26) in Proposition 2, respectively.

The expected SBDA is given by Equation (31).

$-{\mu }_0+\frac{{\mu }_0{T}_1{\text{e}}^{-\lambda T_1}+{\displaystyle {\int }_0^{T_1}\left\{{\mu }_0{t}_C\cdot \left(1+{\displaystyle {\int }_0^{T_2-{\tilde{t}}_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}\right)+\left(\log \gamma \right){\displaystyle {\int }_0^{T_2-{\tilde{t}}_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}+\log \Psi \left(t_C\right)\right\}\lambda {\text{e}}^{-\lambda t_C}\text{d}{t}_C}}{T_1{\text{e}}^{-\lambda T_1}+T_2\left(1-{\text{e}}^{-\lambda T_1}\right)+{\displaystyle {\int }_0^{T_1}\left(\left(\left(t_C+{\displaystyle {\int }_0^{T_2-t_C}{t}_D{f}_{t_D}\left(t_D\right)\text{d}{t}_D}\right)-T_2\right)\left({\displaystyle {\int }_0^{T_2-t_C}{f}_{t_D}\left(t_D\right)\text{d}{t}_D}\right)\right)\lambda {\text{e}}^{-\lambda t_c}\text{d}{t}_c}}$ (31)

(Proof)

We have only to substitute Equation (22) and Equation (26) into $\alpha =-{\mu }_0+\frac{E\left({\tilde{R}}_{PE}\right)}{E\left({\tilde{T}}_{PE}\right)}$. (QED)

First, using Valuation Model I, we consider the case in which a PE fund invests in two investee companies whose returns are assumed to be independent at two points in time. The assumptions introduced in Valuation Model I are retained. However, for the purpose of distinction between the first and second investee companies, they are represented by adding suffixes 1 and 2, respectively. More specifically, the stochastic variable ${\tilde{t}}_C$ representing the point in time of investment becomes ${\tilde{t}}_{C1}$ for the first time and ${\tilde{t}}_{C2}={\tilde{t}}_{C1}+{\tilde{t}}_{C1\to 2}$ for the second time, where ${\tilde{t}}_{C1\to 2}$ is a stochastic variable representing the time interval from the first investment point in time to the second one and follows the same, independent exponential distribution as ${\tilde{t}}_{C1}$. ${\tilde{t}}_{D1}$ and ${\tilde{t}}_{D2}$ are stochastic variables representing the investment period from the time the PE fund invests at ${\tilde{t}}_{C1}$ and ${\tilde{t}}_{C2}$ until it exits, respectively. ${\gamma }_1{S}_1\left({\tilde{t}}_{C1}\right)$ and ${\gamma }_2{S}_1\left({\tilde{t}}_{C2}\right)$ are stochastic variables representing the exit price of the first investee company and the second one, respectively. The weights for the first and second investee companies are $w_1$ and $w_2$ ( $w_1\ge 0$, $w_2\ge 0$, and $w_1+w_2=1$), respectively.

The probability that no single investee company arrives in the sourcing period, which is calculated $P\left({\tilde{t}}_{C1}>T_1\right)$ as in Equation (5), resulting in Equation (32).

$P\left({\tilde{t}}_{C1}>T_1\right)={\displaystyle {\int }_{T_1}^{\infty }\lambda {\text{e}}^{-\lambda t}\text{d}t}={\text{e}}^{-\lambda T_1}$(32)

The probability that both the first and the second investee companies are invested is given by the probability $P\left({\tilde{t}}_{C2}\le T_1\right)$ representing that the second investment time point ${\tilde{t}}_{C2}$ falls on the sourcing period $\left[0,T_1\right]$. The probability density function of ${\tilde{t}}_{C2}={\tilde{t}}_{C1}+{\tilde{t}}_{C1\to 2}$ composed by ${\tilde{t}}_{C1\to 2}$ and ${\tilde{t}}_{C1}$, which follow the same independent exponential distribution with parameter $\lambda$, is the gamma distribution with parameters 2, $\lambda$ and is given by Equation (33).

$f_{tC2}\left(t\right)=\{\begin{array}{l}\frac{{\lambda }^2}{\Gamma \left(2\right)}{t}^{2-1}{\text{e}}^{-\lambda t}={\lambda }^{2}t{\text{e}}^{-\lambda t}t>0\\ 0\text{otherwise}\end{array}$(33)

Integrating the probability density function in Equation (33) gives $P\left({\tilde{t}}_{C2}\le T_1\right)$ as Equation (34).

$P\left({\tilde{t}}_{C2}\le T_1\right)={\displaystyle {\int }_0^{T_1}{\lambda }^{2}t{\text{e}}^{-\lambda t}\text{d}t}=1-{\text{e}}^{-\lambda T_1}-\lambda T_1{\text{e}}^{-\lambda T_1}$(34)

Find the probability that the first investment is made in the investment period $\left[0,T_1\right]$ but not the second one. This probability is obtained by subtracting the probability of not making a single investment ${\text{e}}^{-\lambda T_1}$ and the probability of making both the first and second investments $1-{\text{e}}^{-\lambda T_1}-\lambda T_1{\text{e}}^{-\lambda T_1}$ from the total probability 1, resulting in Equation (35).

$1-{\text{e}}^{-\lambda T_1}-\left(1-{\text{e}}^{-\lambda T_1}-\lambda T_1{\text{e}}^{-\lambda T_1}\right)=\lambda T_1{\text{e}}^{-\lambda T_1}$(35)

This probability is the same as in Equation (36), which is obtained by integrating the probability density function $g_{{\tilde{t}}_{C1\times }}\left(t_{C1\times }\right)=\lambda {\text{e}}^{-\lambda t_{C1\times }}-{\lambda }^2{t}_{C1\times }{\text{e}}^{-\lambda t_{C1\times }}$ of the random variable ${\tilde{t}}_{C1\times }$ representing the point in time of the first investment when the second investment is not made, over the investment period $\left[0,T_1\right]$.

$P\left({\tilde{t}}_{C1\times }\le T_1\right)={\displaystyle {\int }_0^{T_1}\lambda {\text{e}}^{-\lambda t_{C1\times }}-{\lambda }^2{\tilde{t}}_{C1\times }{\text{e}}^{-\lambda {\tilde{t}}_{C1\times }}\text{d}{\tilde{t}}_{C1\times }}=\lambda T_1{\text{e}}^{-\lambda T_1}$(36)

Based on the above, Proposition 3 shows the expected absolute return from committing to the PE fund in the case where the PE fund invests in two investee companies at two points in time, and Proposition 4 shows the expected commitment time to the PE fund.

Proposition 3

The expected absolute return $E\left({\tilde{R}}_{PE2}\right)$ obtained from committing to the PE fund at time 0 until the PE fund exits the first and the second investee companies at time ${\tilde{t}}_{C1}+{\tilde{t}}_{D1}$ and ${\tilde{t}}_{C2}+{\tilde{t}}_{D2}$, respectively is given by Equation (37).

$\begin{array}{c}E\left({\tilde{R}}_{PE}\right)={\mu }_0{T}_1{\text{e}}^{-\lambda T_1}+w_1{\displaystyle {\int }_0^{T_1}\{{\mu }_0{t}_{C1}\cdot \left(1+{\displaystyle {\int }_0^{T_2-t_{C1}}{f}_{t_{D1}}\left(t_{D1}\right)\text{d}{t}_{D1}}\right)}\\ +\left(\log {\gamma }_1\right){\displaystyle {\int }_0^{T_2-t_{C1}}{f}_{t_{D}1}\left(t_{D1}\right)\text{d}{t}_{D1}}+\log \Psi \left(t_{C1}\right)\}\lambda {\text{e}}^{-\lambda t_{C1}}\text{d}{t}_{C1}\\ +w_2{\displaystyle {\int }_0^{T_1}\{{\mu }_0{t}_{C2}\cdot \left(1+{\displaystyle {\int }_0^{T_2-t_{C2}}{f}_{t_{D2}}\left(t_{D2}\right)\text{d}{t}_{D2}}\right)}\\ +\left(\log {\gamma }_2\right){\displaystyle {\int }_0^{T_2-t_{C2}}{f}_{t_{D2}}\left(t_{D2}\right)\text{d}{t}_{D2}}+\log \Psi \left(t_{C2}\right)\}{\lambda }^2{t}_{C2}{\text{e}}^{-\lambda t_{C2}}\text{d}{t}_{C2}\end{array}$(37)