ＧＰＩＦ working Paper

Spread-based direct alpha (SBDA) as a performance measurefor PE funds

May 16 ,2023
Koichi Miyazaki
Kazuhiro Shimada

EXECUTIVE SUMMARY

Public Market Equivalent (“PME”) measures a Private Equity (“PE”) fund's performance relative tothe listed market. Various kinds of PME methods have been proposed, but the “direct alphamethod” has been pointed out to be the best method.

On the other hand, it is not always clear whether the direct alpha method is appropriate from theperspective of finance theory, given its derivation process, and there may be questions about its
concept as a method of measuring performance of PE funds relative to listed markets.

In this study, we propose a measurement method that can compare the performance of PE funds
and traditional assets more accurately than previous methods: Spread Based Direct Alpha(“SBDA”) and a method for deriving the alpha amount based on SBDA.

（Note）This working paper is a compilation of research results by GPIF staff, and the contents and opinions expressed in the text do not
represent the official views of the GPIF.

1．Introduction
Although the title of this study states “a performance measure for PE funds”, the approach taken here isa performance measure that is applicable not only to PE funds but also to other alternative assets such as
infrastructure and real estate. While the performance of traditional assets such as stocks and bonds is often
measured by time-weighted rates of return, the performance of alternative assets is generally measured by
the internal rate of return (IRR) since inception. The reason for this is explained in “(Column) Method of
Measuring the Rate of Return on Alternative Assets” on page 55 of the GPIF annual report (FY2021 version)[1] as follows; “In contrast to the fund manager for traditional assets, in the management of alternative
assets, the fund manager decides the timing of investing funds and selling out assets, asks investors tocontribute funds each time point of investing funds (capital call), and distributes investors the funds(distribution) each time point of selling out assets. Therefore, the internal rate of return (IRR) is used with
the notion that determining the timing and size of cash flows is part of the fund manager's capabilities”.
In measuring the performance of PE funds, according to BVCA's Limited Partner Committee and Investor
Relation Advisory Group [2], the abovementioned internal rate of return (“IRR”) and investment multiple, whichmeasure the absolute value of the investment, have been observed. While these are excellent for the purposep. 1

of understanding the absolute return of each PE fund, they are not suitable for comparing the performanceof PE funds with that of traditional assets. In contrast, the PME methodology assumes that, at the time pointof a capital call, the same amount in question was invested in the benchmark and the performance is comparedwith that of the real PE fund. The major existing PMEs include the PME of Long and Nickels [3], the PME+ of
Capital Dynamics and Rouvinez [4], the mPME of Cambridge Associate [5], and the Direct Alpha Method of
Gredil et. al [6]. Papers that explain these evaluation methods in an easy-to-understand manner using specificexamples include Gredil et. al [6] and Shiraki and Miyata [7]. In this study, we will refer to Shiraki and Miyata[7]. With regard to the existing major PME methods, in addition to giving an explanation of the valuation
method, the paper summarizes the advantages (that it reflects the essential valuation of cash flow) anddisadvantages (that it is not suitable for relative valuation) of the valuation method, and then states that “Forthe valuation of excess return against benchmarks, among the major PME methods, the direct alpha method,which has no mathematical defects and does not require any artificial corrections, is considered to be thebest method for measuring PE fund performance at present”.
Shiraki and Miyata [7] evaluated that the direct alpha method “seems to be the most reasonablemeasurement method currently considered”, but there may be some questions about the conception of thedirect alpha method. In this study, we propose the spread based direct alpha method (Spread Based Direct
Alpha, hereinafter referred to as “SBDA”.) as a measurement method that the performance of the PE fundcan be compared fairly accurately with those of traditional assets by splitting the performance of PE funds
into the beta part, which is the performance of the benchmark and the alpha part, which represents the pure
performance of PE funds. We will also clarify questions regarding the conception of the direct alpha method
using the tools used to introduce SBDA. Furthermore, based on numerical examples, the mechanisms andproperties of SBDA and alpha amounts will be understood.
The structure of this study is as follows; Section 2 clarifies the questions in the conception on the direct
alpha method, relying on specific examples by Shiraki and Miyata [7]. In Section 3, we discuss the conception
and definition of SBDA, the method for deriving the alpha amount (the amount of excess return due to thepure skill of the PE fund) from SBDA, and clarify the questions in the conception regarding the direct alphamethod. In Section 4, the mechanisms and properties of SBDA and alpha amount will be confirmed by numericalexamples. In the final section, a summary and future issues will be discussed.

2．Questions regarding the Direct Alpha method
Questions in the conception of the direct alpha method are explained using Figure 4 (reproduced here as
Table 1), which was used by Shiraki and Miyata [7] in their explanation of the direct alpha method. In the cash
flow column of the PE fund in Table 1, capital call is the amount of the contribution of investors, the distribution
is the amount of fund that the PE fund distributes to investors each time point of selling out assets, and NAV
is the residual market value. The sum of these three amounts at each time point (from Year 0 to Year 4,
year by year) is shown in the Net column. The IRR is nothing other than the rate of return obtained by relying
on the cash flows listed in the net column.
Next, in the Benchmark column, the level of the benchmark against which the PE returns are compared isp. 2

noted for each year. Depending on the level of these benchmarks, each of the contributions, distributions,remaining market value, and net amounts in the PE fund's cash flow column are all converted to present value(value in Year 0) and listed in the Investment in Benchmark column. As an example, the amount “-60” in the
cash flow column of the PE fund that will be the fund contribution in 2 years is converted to the present
value “-50” as in “-60 x 100 ÷ 120 = -50” because the level of the benchmark in 2 years is “120”. The
amount “-60” is converted to the present value “-50” and is listed in the fund contribution after 2 years(present value) in the “Investment in Benchmark” column.
The question in the direct alpha method is whether it is appropriate, from the perspective of finance theory,
to obtain the IRR by converting the cash flows generated at each year to present value at the benchmark
return and then considering these as having occurred at the year in question. This question is clarified by themathematical expression of the direct alpha method using the tools we prepared to define our SBDA in Section
3 and the implications of the direct alpha method in terms of finance theory are not always clear.
【Table 1：Direct Alpha Method】（Source: Prepared by the authors from Shiraki and Miyata [7]）3．SBDA
3.1 Conception and Definition of SBDA
The inspiration for the SBDA comes from the concept of credit spreads in the bond market, more specifically,
the credit spread over the spot rate of bonds with the relevant maturity. In the spot rate curve, the current
time point is 0, and the future time point is taken on the horizontal axis. In the SBDA, a “benchmark spot rate
curve” is defined as the equivalent of the spot rate curve. Since the SBDA is a measure of the performanceof alternative assets, it is presented as a value obtained at the time point of the valuation. The benchmark
spot rate curve is the performance of the benchmark from the time point of investment commitment to thetime point of valuation (see Figure 1 with numerical examples in Table 1 for the benchmark dynamics). Thus,in the benchmark spot rate curve, Year 0 on the horizontal axis is the investment commitment time point andp. 3

the final time point on the horizontal axis is the valuation time point. The “benchmark spot rate” in thebenchmark spot rate curve is the annualized cumulative rate of return of the benchmark from the time pointof investment commitment to each time point up to the valuation time point (see Figure 2 with numericalexamples from Table 1).
Normally, in bond investments, the investor invests in the bond at present and receives the coupon payment
and face value at a pre-determined future time point. In contrast, with alternative assets, the investment is
not made immediately at the time point of commitment, but rather, the commitment begins, sometime later
when contributions of funds are made at first time in response to a capital call, which continues until the
cumulative amount of contributions reaches the commitment limit (In Figure 3, which uses the numericalexample in Table 1, there are capital calls at Year 0 and Year 2). The amount of the capital call at each time
point is invested in alternative assets and the investor receives a certain amount as distribution at a certain
time point (in Figure 3 using the numerical example in Table 1, there are distributions at Year 1, Year 3, andYear 4). At the time point of valuation (Year 4 in Figure 3, which uses the numerical example in Table 1), not
all the funds invested in response to the capital call are collected as a distribution, in which case the value ofthe alternative assets under management is recognized as NAV. The SBDA represents the degree to which,on average, the PE fund has exceeded the benchmark return over the investment period in the process ofobtaining distribution.
The valuation method for SBDA relies on the valuation method for interest rate swaps with credit risk.
First, we consider the funding side. As an alternative asset, we will use a foreign PE as an example. This isan alternative asset that falls under the category of foreign (Non-Japanese) equities in the GPIF's policy
benchmark portfolio, and the GPIF's policy benchmark for foreign equities is the MSCI ACWI ex Japan.
Therefore, the benchmark spot rate at a given time point is the annualized cumulative rate of return of the
MSCI ACWI ex Japan from the time point of commitment to that time point. For the funding side, the total
investment is considered to be the sum of the funds invested in each capital call, discounted by the benchmark
spot rate to the time point of commitment. The total investment is then considered to be the sum of the
distributions (including NAV) available at each time point on the investment side, discounted to the time pointof commitment at the benchmark rate plus the SBDA at that time point. Relying on the swap valuation method,
the SBDA is determined so that the present value of the funding side equals the present value of theinvestment side at the time point of commitment.
We now define SBDA by expressing the above ideas in a mathematical formula.
（Definition）SBDA
SBDA is defined as s satisfying the following equation;
∑

∑

(1)

Here, 𝐶𝑎𝑙𝑙(𝑗) and 𝑟j are, respectively, the amount of funds invested in response to the capital call and thebenchmark rate at the time point 𝑗, 𝐷𝑖𝑠𝑡(𝑖) and 𝑟𝑖 are the amount of distribution and the benchmark rate at
the time point 𝑖, respectively, 𝑁𝐴𝑉 and 𝑟n represent the amount of alternative assets under management andthe benchmark rate at the time point 𝑛 of valuation. The above equation shows that there were 𝑚 capitalcalls and 𝑛 distributions.

p. 4

（Remark）
The usual derivation of IRR does not separate the funding side cash flows from the investment side cash
flows, and also assumes that distributions made during the period are reinvested in IRR. Therefore, the IRR
can be derived either based on the present value of the cash flows (value at the time point of commitment)
or based on the future value of the cash flows (value at the time point of valuation), and the IRRs from bothderivation methods are consistent. In this study, the objective is to compare the performance of the PE fundwith that of traditional assets as accurately as possible after separating the alpha portion of the PE fund,which is the 𝑠 in equation (1) from the beta portion of it, which is the 𝑟𝑖 and 𝑟𝑗 in equation (1). After the
distribution 𝐷𝑖𝑠𝑡(𝑖) at the time point 𝑖 is distributed to investors, the corresponding present value of capitalcall

is no longer invested in the PE fund, and therefore the concept of reinvestment after the time

point 𝑖 does not exist. In fact, once the distribution is made, GPIF will invest it again in the MSCI ACWI ex
Japan, so there is no room for an alpha portion that expresses the pure management skill of the PE fund,since the PE fund has not managed it. Therefore, unlike the IRR derivation method, SBDA can only be derived
based on the present value of cash flows (value at the time point of commitment).
【Figure 1：Benchmark dynamics】【Figure 2：Benchmark spot rate curve (annualized)】p. 5

【Figure 3：Diagram of cashflows in Table 1】

3.2 Conversion from SBDA to Alpha AmountThe alpha, which is the excess return from the benchmark in the management of traditional assets,corresponds to the SBDA obtained in a way that satisfies equation (1) in the case of PE funds. In this section,
we describe a method for deriving the alpha amount, which is the amount of excess return from the benchmark,using the SBDA. There are three possible ways to capture the alpha amount assumed to have been generated
by the PE fund. Before presenting the formula for deriving the alpha amount, we will explain the backgroundconception using Figure 4.
The amount of distribution at time point 2 is shown as the area of the largest square in Figure 4. We
decompose this and consider three different alpha amounts in terms of which part can be regarded as thealpha amount generated by the PE fund. The area of the largest square in Figure 4 is 1𝑟𝑠

and thisis the amount of distribution at time point 2. This can be decomposed into nine parts (①, two ②, ③, two A,
two C, and B in Figure 4) since it is the square of the trinomial. The area of the square in ① represents thepresent value at the time point of the commitment (the investment amount at time point 0 that generates the
distribution at time point 2). The area of 2 × ② + ③ should be called beta because it is the amount of returnobtained by investing the area of ① in the MSCI ACWI ex Japan at the time point of commitment and managing
it until the time point 2. The area of 2×A+B can be obviously regarded as the amount of PE's alpha in relation
to the investment amount of area ①, and the alpha amount based on this conception is called the alphaamount (1). The area of 2×C is the alpha amount obtained with respect to the part where the capital call isgrown with beta, and can be viewed as the alpha amount mixed with beta. If we consider this portion is thealpha amount that have been generated entirely by the PE, and we call 2×A+B+2×C alpha amount (2). The
alpha amount 2×A+B+C that we consider half of the portion to have been generated by the PE is called thealpha amount (3). Note that the amount of distribution at time point 3 is the volume of a cube with one side1

𝑟3

𝑠. The volume of the cube 1

𝑟3

𝑠 3 can be decomposed into 27 parts by the third power of 3

and considered in the same way as the amount of distribution at time point 2. In the same manner, the amountof distribution at time point 𝑛 can be assumed to be the volume of a n-dimensional cube.

p. 6

【Figure 4：Illustration of the alpha amount (case at time point 2)】

In line with the above approach, it can be easily confirmed that alpha amounts (1) through (3) can be
expressed as in equations (2) through (4) using the capital call amount, distribution amount, NAV, benchmark
spot rate, and SBDA that appear in equation (1).
𝐴𝑙𝑝ℎ𝑎 𝐴𝑚𝑜𝑢𝑛𝑡 (1)∑𝑛𝑖 1𝐷𝑖𝑠𝑡 𝑖
1 𝑟𝑖 𝑠 𝑖

1

𝑠 𝑖

1

1

𝑟𝑖

𝑠 𝑖

𝑁𝐴𝑉

𝑠 𝑛

1

1 𝑟𝑛 𝑠 𝑛

1(2)

𝐴𝑙𝑝ℎ𝑎 𝐴𝑚𝑜𝑢𝑛𝑡 (2)∑𝑛𝑖 1𝐷𝑖𝑠𝑡 𝑖
1 𝑟𝑖

𝑠 𝑖

𝑁𝐴𝑉

𝑟𝑖 𝑖

1

1

1 𝑟𝑛 𝑠 𝑛

𝑟𝑛

𝑠 𝑛

1

𝑟𝑛 𝑛

(3)

𝐴𝑙𝑝ℎ𝑎 𝐴𝑚𝑜𝑢𝑛𝑡 (3)𝐴𝑙𝑝ℎ𝑎 𝐴𝑚𝑜𝑢𝑛𝑡 1

𝐴𝑙𝑝ℎ𝑎 𝐴𝑚𝑜𝑢𝑛𝑡 2

𝐴𝑙𝑝ℎ𝑎 𝐴𝑚𝑜𝑢𝑛𝑡 1(4)

3.3 Clarification of questions in the Direct Alpha method
We define the direct alpha using the tools prepared in Section 3.1 in line with the method of deriving direct
alpha presented with concrete examples in Section 2. The present value of the invested funds in response to
the capital call at time point 𝑗 is𝐶𝑎𝑙𝑙 𝑗1 𝑟𝑗

𝑗and the present value of the distribution at time point 𝑖 and thepresent value of NAV at time point 𝑛 are

𝐷𝑖𝑠𝑡 𝑖
1 𝑟𝑖

𝑖

and

𝑁𝐴𝑉
1 𝑟𝑛 𝑛, respectively. Since direct alpha is the internalrate of return obtained by considering each of these amounts converted to present value as if they occurredat the same time point, it can be defined as follows;（Definition）Direct Alpha
Direct alpha is α that satisfies the following equation;∑𝑚𝑗 1

𝐶𝑎𝑙𝑙 𝑗
1 𝑟𝑗

𝑗

∙

1
1 𝛼 𝑗

∑𝑛𝑖 1

𝐷𝑖𝑠𝑡 𝑖
1 𝑟𝑖 𝑖

∙

1

𝑁𝐴𝑉

1 𝛼 𝑖

1 𝑟𝑛 𝑛

∙

1
1 𝛼 𝑛

(5)

Here, all notations except for α are the same as those used in defining SBDA. Although the original
equation to define direct alpha is the one that the left-hand side of equation (5) is transposed to the righthandside, it is defined in the form of equation (5) for ease of comparison with the equation (1) to define SBDA.

p. 7

The denominators of both sides of equation (5) are organized as follows;
∑

∙

∑

∙

(5ʼ)

∙

It follows that the financial implications are not certain for 𝑟𝑖 ∙ 𝛼 and 𝑟𝑛 ∙ 𝛼 in the denominator of equation (5').
Also, since 𝑟𝑖 ∙ 𝛼 and 𝑟𝑛 ∙ 𝛼 are considered to be second-order of small terms and ignorable, an approximate
equation 1

𝑟𝑗
∑

1

𝛼 ≅ 1 𝑟𝑗

𝛼 is attained and rewriting both sides of equation (5) using it;

≅∑

(6)

is obtained. Comparing the approximate equation (6) with equation (1), which is the defining equation of theSBDA, α in the investment side of the PE fund implied by the right-hand side of approximate equation (6) is
identical to 𝑠 in equation (1). However, in the funding side of the PE fund implied by the left-hand side of theapproximate equation (6), capital call 𝐶𝑎𝑙𝑙 𝑗 at time point 𝑗 is discounted by 1

𝑟𝑗

𝛼 instead of 1

𝑟𝑗 in

equation (1). In the funding side of the PE fund, the fund raised from the funds invested in MSCI ACWI ex
Japan, which is the benchmark, so when discounting capital call to present value, the return of MSCI ACWI ex
Japan should be used and capital call should be discounted by 1

𝑟𝑗. But instead, the return attained byadding direct alpha to this return is used and capital call is discounted by 1

𝑟𝑗

𝛼. Therefore, in the usualcase where capital call occurs after the time point of commitment and α is positive, the left side of equation(1) defining SBDA will be larger than the left side of equation (5) defining direct alpha, and SBDA will besmaller than direct alpha due to this effect. In other words, using direct alpha in evaluating the performanceof a PE fund would result in an overestimation of the fund's performance. Although it is a special case, weshould also check the case where a capital call occurs only at the time point of commitment. In this case, theleft-hand side of equation (1) defining SBDA and the left-hand side of equation (5') defining direct alphahave the same value, whereas the denominator of the right-hand side of equation (5') has 𝑟𝑖 ∙ 𝛼, which does
not exist in the denominator of the right-hand side of equation (1). Under the influence of such factors, as
𝑟𝑖 and α are positive, direct alpha is slightly smaller than SBDA in the usual case.

4．Numerical examples
In this section, based on numerical examples, we first compare SBDA and direct alpha (“DA” in Section 4
below.) and see to what extent using DA in evaluating the performance of a PE fund will lead to an
overestimation of the fund's performance in many cases. In addition, we will reconfirm the pricing mechanismof SBDA and DA through numerical examples. Furthermore, we will obtain three different alpha amounts andreview the magnitude of the alpha amounts for SBDA and capital calls, as well as the differences in magnitude
among the three alpha amounts.
As numerical examples, we consider five different PE funds from Case 1 to Case 5, i.e., the 5 kinds ofsets of capital calls and distributions (hereinafter referred to as the “set”). Case 1 is the set shown in Table
1; Case 2 and Case 3 are the sets of Case 1 in which 500, half of the invested funds in response to thecapital call at Year 0, is moved to Year 1 and Year 2, respectively. Case 4 and Case 5 are the sets of Case
1 in which the distribution of 280 at Year 3 is moved to Year 4 and Year 2, respectively. The benchmark

p. 8

spot rates to each year were obtained from and the numerical examples in Table 1 and shown in Figure 2.
Table 2 shows the DA, SBDA, and 3 kinds of alpha amounts for these 5 PE funds.
4.1

SBDA と DA

First, we compare Case 1 with Case 2 and Case 3. The present value (hereinafter referred to as PV) is
evaluated on the time point of commitment to the PE (at Year 0). The PV of the invested funds in responseto the capital call becomes smaller from Case1 to Case2 and Case3. On the other hand, since the distribution
Case1 to Case3 are all the same, the excess rate of return is expected to increase from Case1 to Case2and Case3. In fact, Table 2 shows that SBDA and DA both increase from Case1 to Case3. It is important tonote that while there is only a slight difference between SBDA and DA in Case 1, DA increases significantly
more than SBDA in Case 2 and Case 3. This is because, as pointed out in Section 3.3, the funds invested in
response to the capital call are raised from funds invested in the benchmark MSCI ACWI ex Japan, so when
discounting to present value, the return of the MSCI ACWI ex Japan should be used and discounted with 1
𝑟𝑗. In the DA method, however, they are discounted by 1 + 𝑟𝑗 + 𝛼 + 𝑟𝑖 ∙ 𝛼. As expected, it is confirmed thatthe DA method considerably overestimates excess returns.
Next, compare Case 1 with Case 4 and Case 5. This time, the PV of the invested funds in response to thecapital call is the same in all cases, but a part of the distribution is moved to a later time point (Case 4) anda part of the distribution is moved to an earlier time point (Case 5), respectively. The point of interest is thecomparison between Case 1 and Case 4, since the distribution (280) that was at Year 3 in Case 1 is moved
to a later time point (Year 4) in Case 4, it is usually assumed that the PV of the distribution in Case 4 issmaller than that in Case 1. However, Figure 2 shows that the benchmark spot rates up to Year 3 and Year
4 are 11.9% and 4.7%, respectively, and the values (discount function values) to be multiplied when discounting
the PV from Year 3 and Year 4 are 0.714 and 0.832, respectively, indicating that the PV discounted fromYear 4 is larger than the PV discounted from Year 3. From this effect, both SBDA and DA are larger in Case
4 than in Case 1. In a similar fashion, both SBDA and DA are larger in Case 5 than in Case 4. A closer lookat Case 1, Case 4, and Case 5 shows that DA is slightly smaller than SBDA in all cases. When considering
the reason for this, it is important to first confirm that in these cases, the PV of invested funds according
to the capital call used in the definition of SBDA and that used in the definition of DA are almost the same.
This is because the majority (1000) of the funds invested in response to the capital call (1060 in total)
occurred at Year 0, so the difference in discount rates (i.e., the difference between 1

𝑟𝑗 and 1

𝑟𝑗𝛼

𝑟𝑗 ∙ 𝛼) hardly appears. Thus, the reason is related to distribution; in the definition of DA, the discount rate in
the denominator of the right-hand side of equation (5') has terms such as 𝑟𝑖 ∙ 𝛼 and 𝑟𝑛 ∙ 𝛼 that are not present
in SBDA and have no concrete financial implications. Because these terms were positive, DA was smaller than
SBDA.
4.2

Three kinds of alpha amounts

The size of the SBDA is the most important factor because alpha amount (1) through alpha amount (3)are, in turn, derived based on equations (2) through (4) using the SBDA. In Section 4.1, we observed that the
SBDA increases in order from Case 1 to Case 2 and Case 3. Table 2 shows that the alpha amounts (1)through (3) corresponding to these cases also increase from Case 1 to Case 2 and Case 3. In other words,

p. 9

the order of magnitude of SBDA, which was introduced as an indicator of (excess return) rate, is maintained
when converted to an indicator of (excess return) amount, i.e., the alpha amount, if the SBDA is comparedamong PE funds with the same distribution.
Next, we consider the use of SBDA to calculate the alpha amount in a comparison between PE funds withthe same invested fund in response to a capital call and different distributions. In this section, we consider
the alpha amount based on the alpha amount (1), which is representative of the alpha amount. It is interesting
to note that Case 4 has an increase of 33.8 in alpha amount (1) even though SBDA is only 0.82% larger thanCase 1, while Case 5 has an increase of only 1.4 in alpha amount (1) even though SBDA is 0.91% larger thanCase 4. The reason for this can be largely explained by the difference in the time point at which distributions
occur; distribution 280 at Year 3 in Case 1 is moved to Year 4 in Case 4 and to Year 2 in Case 5. In Case4, the portion of the alpha amount in distribution 280 that is the increased amount in PV of the capital callcompounded over four years (20.72%) at the rate of SBDA (4.82%), while in Case 5, the portion of the alphaamount in distribution 280 that is the increased amount in PV of the capital call compounded over only two
years (11.79%) at the rate of SBDA (5.73%). This effect is so strong that even though SBDA is about 0.91%
larger in Case 5 than in Case 4, the alpha amount (1) remains at the same level. Thus, if SBDA is positive andthe same among PE funds, the alpha amount (1) tends to be larger for PE fund with distribution at a latertime point due to the compounding effect of SBDA over a longer period of time.
Finally, we examine the relative comparison of alpha amounts (1) through (3) in each case. Table 2 showsthat in all cases, alpha amount (1) is the smallest, alpha amount (2) is the largest, and alpha amount (3) is
between the two. The reason for this is the benchmark dynamics (see Figure 1), which is another factor that
determines the SBDA. Figure 1 shows that in this numerical example, the benchmark dynamics are generally
upward trending, i.e., as shown in Figure 2, the benchmark spot rate is positive at all time points, so the PVof invested funds in response to a capital call at Year 0 (① in Figure 4) will be inflated at the benchmark
spot rate (two ② and ③ in Figure 4 will be added). This generates an alpha amount (C in Figure 4) for the
portion of the PV inflated by the benchmark spot rate. Thus, in this numerical example, the alpha amount (1)was the smallest and the alpha amount (2) was the largest in all cases. Contrary to the benchmark dynamics
in this example, if the benchmark dynamics are generally downward trending, i.e., the benchmark spot rate is
negative at all time points, the PV of the capital call at Year 0 will tend to shrink with the benchmark spot
rate, and the alpha amount associated with this portion (the two C) will be recognized as a negative value.
Therefore, in this case, alpha amount (1) is the maximum and alpha amount (2) is the minimum. If the benchmark
dynamics are horizontal, i.e., the benchmark spot rate is zero at all times, the PV of the invested funds in
response to the capital call at Year 0 is not affected by the benchmark spot rate, and therefore the alphaamount related to this portion (the two C) is not generated. The PV of funds invested in response to a capitalcall at Year 0 at all times is not affected by the benchmark spot rate. Thus, in this case, the three kinds ofalpha amounts will have the same value.

p. 10

【Table 2：Numerical example】
Year 0 Year 1 Year 2 Year 3 Year 4

Case1

Case2

Case3

Case4

Case5

Capital Call

1000

0

60

0

0

Distribution

0

560

0

280

240

Capital Call

500

500

60

0

0

Distribution

0

560

0

280

240

Capital Call

500

0

560

0

0

Distribution

0

560

0

280

240

Capital Call

1000

0

60

0

0

Distribution

0

560

0

0

520

Capital Call

1000

0

60

0

0

Distribution

0

560

280

0

240

Alpha amount

NAV

DA

SBDA

300

3.87%

4.00% 107.64 122.62 115.13

300

7.77%

6.43% 165.61 188.29 176.95

300

12.24% 7.89% 197.92 224.76 211.34

300

4.72%

4.82% 141.44 158.35 149.90

300

5.59%

5.73% 142.84 158.05 150.45（１） （２） （３）5．Summary and future issues
In this study, we proposed a measurement method that can compare the performance of PE funds quite
accurately with that of traditional assets, splitting the performance of PE funds into a beta portion, which is
the market performance, and an alpha portion, which expresses the pure investment skill of PE funds, by wayof the spread based direct Alpha (SBDA) and the alpha amount based on SBDA. The tools prepared in
introducing these concepts also clarified the question of the conception of the direct alpha method, which isregarded as the best among the existing PMEs. Furthermore, based on numerical examples, the mechanismand nature of the SBDA and alpha amounts were ascertained in terms of the time point of occurrence ofcapital calls and distributions and the benchmark spot rate.
The SBDA and the alpha amount based on the SBDA were devised to measure the performance of PE funds
in order to satisfy the double mandate; (1) the alpha portion, which expresses the pure skill of the PE fund,should be extractable, and (2) the performance relative to the MSCI ACWI ex Japan, the GPIF's policy
benchmark for foreign equities, should be measurable. Whether or not the double mandate is actually met inpractice will need to be examined from various perspectives in the future. In the process, it will also beessential to improve the SBDA and the corresponding alpha amounts.
We hope this paper will be of some help when considering “better PE fund performance measurement
methods”.

References[1] Government Pension Investment Fund（2022） “Annual Report Fiscal Year 2021”.
[2] BVCA’s Limited Partner Committee and Investor Relation Advisory Group (2015) “Private Equity
Performance Measurement (BVCA Perspectives Series).”[3] Long, A. M., and Nickeles, C. J. (1996) “A private investment benchmark.” Working paper.
[4] Rouvinez, C., (2003) “Private Equity Benchmarking with PME+,” Venture Capital Journal, August, pp. 3438.

p. 11

[5] Cambridge Associates (2013) “Private Equity and Venture Capital Benchmark - An Introduction for
Readers of Quarterly Commentaries.”[6] Gredil, Oleg and Griffiths, Barry E and Stucke, Rüdiger, Benchmarking Private Equity: The Direct Alpha
Method(February28,2014).

Available

at

SSRN:

https://ssrn.com/abstract=2403521

or

http://dx.doi.org/10.2139/ssrn.2403521[7] Shiraki, S. and Miyata, S. (2018), “On the investment performance measurement method of PE funds,”
Securities Analyst Journal, 2018.10 pp. 42-52.

p. 12