GPIF Working Paper

Toward Integrated Active Management Based on SBDA

February 22, 2024
Koichi Miyazaki
Kazuhiro ShimadaRegarding the methodology for separating the investment performance of PE funds into abeta portion (investment performance of traditional assets), which is market performance,
and an alpha portion (excess return), we presented in our working paper dated April 24,
2023, "PE Fund Investment Performance Measurement Methodology as Spread-Based
Direct Alpha (SBDA)", dated April 24, 2023.

This study uses the SBDA as a starting point to estimate tracking error for alternative assets byintroducing various new concepts of excess returns. This will enable an integrated framework for
active management based on excess return and tracking error, including alternative assets inaddition to traditional assets.

（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
The "integrated active management" mentioned in the title is intended to incorporate alternative assets
into existing active management targeting traditional assets such as bonds and stocks, aiming for further
excess returns and improving performance from a risk-return perspective. First, we review existing active
management, mainly Ogishima and Yamamoto (2003) and Kikukawa et al. (2017) for domestic bond active
management, and Takehara (2012) for equity active management.
Ogishima and Yamamoto (2003) argue that the active management of domestic bonds required under low
interest rates is (1) to shift from conventional interest rate selection (changes in duration and remaining
maturity structure) to sector selection (changes in sector composition ratio) and issue selection (changes inissue composition ratio), and to improve investment efficiency by successfully combining sources of excess
return, (2) to take advantage of market inefficiencies, and (3) to incorporate new products (CDS, ABS, etc.).
Among these, Kikukawa et al. (2017) mainly incorporate (1) and (2) in the domestic bond active management.
Kikukawa et al. (2017) point out that domestic bond active funds for pension management in our country
are homogeneous and mainly use credit factors as a source of excess returns, and since credit has the
aspect of substituting for equity investments, there is a risk that returns will decline along with stocks whenthe economic environment deteriorates. On the other hand, carry rolldown factor has a high-risk premium andlow correlation with credit factor and equities, so if this factor were incorporated 36.9% into the existingbond active fund, the tracking error relative to the BPI（NOMURA Bond Performance Index） would remain at

p. 1

27.58BP, while the average excess return would increase from 17.25BP to 33.52BP.
In order to give a deeper insight into equity investment, Takehara (2012) asks the question, "Is it wrong
to use the empirically rejected CAPM as a benchmark model, and is equity management and risk management
using the Barra AEGIS inappropriate? The discussion is essentially an active management vs. passive
management debate. First, using portfolios on the efficient frontier, he derives the CAPM formula and showsthat mean-variance efficiency and the CAPM formula are in fact equivalent. Secondly, the mean-varianceefficiency of typical market value-weighted indices such as the S&P 500 in the U.S. and TOPIX in Japan is
not satisfied because most of them are located slightly to the right of the minimum variance portfolio on theefficient frontier (hyperbola) composed by stocks belonging to the universe. Due to the equivalence of theefficiency of mean-variance and CAPM, there are returns that cannot be explained by the CAPM (i.e., Jensen
alpha). Therefore, the article states that there is no clear advantage of passive management, nor does it
negate the potential of active management.
If the benchmark is not efficient, there will be significant variation in the fund's valuation ranking dependingon what benchmark is used. This leads to a discussion of how to define alpha, i.e., whether it is measured
relative to a single benchmark or multiple benchmarks. Takehara (2012) notes that for most practitioners,
alpha is the excess return relative to the benchmark, and for researchers it is the risk-adjusted return froma multi-beta model, and that such differences in definition also affect the results of equity valuation via theestimation of the cost of equity capital. In fact, from the researchers' perspective, Kubota and Takehara(2007) point out that the mean-variance efficiency of the Fama-French three-factor model or style index
is rejected as a long-term model and alpha is not zero even under the multi-beta model, whereas from thepractitioners' perspective Arai and Yamada (2002) find the rationale for investing in actively managed funds
to be to allow for appropriate factor risk exposure consistent with investment objectives, since alpha is small
under a multi-beta model. The differences of the opinions in the two articles support the discussion of how
to define alpha in Takehara (2012).
In addition to the articles introduced here, other studies on active management of domestic bonds include
Yamada (2000), Takatsu and Yamazaki (2000), Maeda and Koike (2002), Nakatani (2010), and Miyazaki,
Abe, and Shimada (2021), while those on active management of stocks include Oharazawa (1991), Ito et al.
(2009), Komai and Oka (2012), and Omori and Yano (2013), among others, and the accumulation of researchon these areas appears to be progressing steadily along with the progress of the asset management industry.
On the other hand, to the best of our knowledge, there are not many studies on active management ofalternative assets, much less integrated active management that incorporates alternative assets into existing
active management.
The introduction of alternative assets into Japanese pension fund management is thought to have started
in the early 2000s, and Miyai, Yamaguchi, Nakamura, Ishida, and Yamada (2005) summarized the efforts ofpension funds and pension consultants during this period in a vividly documented roundtable discussion memo.
Although this memo was written about 20 years ago, the awareness of the issues that were raised aboutalternative assets at that time still remain as issues that need to be addressed today. The main issues are:(1) alternative assets are difficult to evaluate at market value; (2) although it is said that diversification
effect can be obtained, how to measure the correlation coefficient; (3) whether diversification effect is reallyp. 2

obtained under a large individuality impact of each fund; and (4) it is difficult to control the asset mix on an
actual value basis because the amount of commitment and the actual investment are different.
In Miyazaki and Shimada (2023-a), the authors reconfirmed (1) and (2) of their awareness of the problem:
With regard to J-REITs, both the value of the total investment amount (investment units), which is the realestate price given by a real estate appraiser minus interest-bearing debt, and the market value (investment
units), which has become a J-REIT after listing, are the values to the same real estate. Although both arevalues for the same real estate, the discrepancies between these values made us realize once again the
difficulty of evaluation of fair value. In addition, real estate appraisals are priced with reference to historicalvalues, so returns based on appraisals are subject to smoothing. Therefore, when determining correlation
coefficients between real estate returns and traditional assets, the correlation coefficients differ
correspondingly depending on whether J-REIT returns, returns based on appraisal prices, or returns based on
appraisal prices with de-smoothing are used as real estate returns. With these issues in mind, it is extremely
difficult to establish a return or risk matrix that is absolutely correct, or to accurately estimate thediversification effects of including alternative assets in traditional assets. Therefore, it seems better to thinkof alternative assets as tools that aim to achieve excess returns over the returns of traditional assets, while
fully acknowledging that their risk characteristics differ from those of traditional assets (under risk
management based on tracking error).
In the roundtable discussion memo, it was pointed out that in the case of alternative assets (hedge funds),"there are many cases where the factors of investment performance are not explained in a state where they
are properly broken down into beta and alpha. The fact that IRR, which expresses absolute returns, is currently
the mainstream method for evaluating the performance of alternative assets may also be subject to thispoint of view. Attempting to maximize excess returns over the returns of traditional assets under risk
constraints on tracking errors would help overcome the points raised here and the main issues (1) and (2).
SBDA (Spread Based Direct Alpha, hereafter "SBDA") proposed by Miyazaki and Shimada (2023-b) enables
us to divide the investment performance of a PE fund into a beta portion (investment performance oftraditional assets), which is market performance, and an alpha portion (alpha for practitioners as pointed out
by Takehara (2012)), which expresses the pure investment ability of a PE fund, and SBDA is a convenientmeasure to evaluate the excess returns. What remains is to estimate the tracking error in alternative assets,
and the objective of this study is to propose a method to overcome this problem.
The structure of this paper is as follows. In Section 2, we introduce various new concepts regarding excess
returns of alternative assets using SBDA as a starting point. In Section 3, we describe a framework for
integrating active management based on excess return and tracking error, including alternative assets, using
the new concepts introduced in Section 2. In Section 4, we confirm the mechanism of the new concepts withnumerical examples. In the final section, a summary and future issues are added.

2．Various New Concepts of Excess Returns on Alternative Assets
2.1 Excess return on alternative assets（SBDA）
The excess return of alternative assets over the performance evaluation period (N years) can be captured

p. 3

by the SBDA; see Miyazaki and Shimada (2023-b) for the detailed mechanism of the SBDA. In this section,
we only describe the definition of SBDA. The time point 𝑖 used in the definition is in months, where 𝑖 = 0
represents the month in which the alternative investment was committed, and thereafter 𝑖 = 1, 2,…,
represents one month later, two months later,…. The point in time at which the SBDA is evaluated is 𝑛months after the month of commitment. Hereafter, symbols representing rates and returns are treated asmonthly rates.
（Definition 1）SBDAThe SBDA at the valuation time 𝑛 months from time 0, the point of commitment to the alternative asset,
is that 𝑠 which satisfies the following equality.
∑∑

where 𝐶𝑎𝑙𝑙 (𝑖 ), 𝐷𝑖𝑠𝑡(𝑖 ), and 𝑟（1）
denote the amount of capital call, the amount of distribution, and thebenchmark rate (the monthly rate of the benchmark return from month 0 to month 𝑖) at month 𝑖, respectively,
and 𝑁𝐴𝑉 denotes the value of alternative assets under management at the valuation point 𝑛 (Here, we
assume that capital calls and distributions occur at the end of the month). In equation (1), 𝐶𝑎𝑙𝑙(𝑖) = 𝐷𝑖𝑠𝑡(𝑖)=0
if there is no capital call or distribution at time 𝑖 months later.
Using the topmost schematic in Figure 1, we will review the information used to determine the SBDA 𝑠
with the valuation point being 60 months later. Over the period from month 0 to month 60, we use monthly
information for capital calls and distributions, and quarterly information for NAV (3 months later, 6 months
later,….). From month 0 to month 6, we can see that capital calls occur after 1 month and 4 months, anddistributions occur after 2 months and 5 months, etc. From month 59 to month 60, there are no capital calls
and distributions occur after 59 months, etc. Since the time of valuation is after 60 months, 𝑁𝐴𝑉 adopts
𝑁𝐴𝑉 , which is shown after 60 months. The benchmark rate 𝑟

is the one at the maturity of each monthafter the point 𝑖 =1…60.

【Figure 1：Schematic for Constant Maturity 5-year SBDA calculation】（Source: Prepared by the authors.）

p. 4

2.2 Constant Maturity 5-year SBDA
The constant maturity 5-year SBDA that we are newly introducing here is an SBDA that is based on
information from 5 years back from the time of valuation at any future valuation point (which appears every
3 months after the 5-year period). We consider such SBDAs because we view SBDAs as analogous to credit
spreads. For example, if a corporate bond with a maturity of 6 years is issued, its credit spread is the credit
spread of the 6-year maturity, but after one year, this bond's credit spread becomes the credit spread of
the 5-year maturity, after two years, this bond's credit spread becomes the credit spread of the 4-year
maturity. In this way, when we observe changes in the credit spread of a corporate bond, we also include theeffect of the term structure of the credit spread. To remove this effect, we often observe variation in theconstant maturity 5-year credit spread, which is the spread between the generic constant maturity 5-year
corporate bond yield and the generic constant maturity 5-year government bond yield. As a corresponding
concept for alternative assets, we introduce the Constant Maturity 5-year SBDA.
The SBDA in (Definition 1) also has the same inherent observational problems as corporate bond credit
spreads. If we follow SBDAs with valuation points of 4, 5, and 6 years (𝑠 , 𝑠 , 𝑠 , in that order), etc.,
when we observe changes in SBDAs, we also include the term structure of SBDAs in our observations. To
remove this effect, we propose a Constant Maturity 5-year SBDA.
First, we obtain the constant maturity 5-year SBDA at the time 60 months after the 5-year elapsed fromthe time zero month. This is nothing other than obtaining the SBDA 𝑠
we introduce 𝑠 ,shown in Section 2.1. In the following,

to denote the Constant Maturity 5-year SBDA at the point 𝑗 months further after the 5-

year lapse, and denote this SBDA 𝑠

newly as SBDA 𝑠 , .

Next, we obtain the Constant Maturity 5-year SBDA 𝑠 ,at the 63-month point in time, which is three

more months after the 5-year lapse. This is the SBDA evaluated based on information from 3 months after
the time point, which is 5 years back from 63 months after the time point. Therefore, in the cash flow indetermining SBDA 𝑠 , , there are no capital calls or distributions from the middle of Figure 1 from the time
point 0 month to 3 months later, and the amount equivalent to 𝑁𝐴𝑉 , the value of the NAV after 3 months,
is considered to be newly capitalized. On the other hand, the capital call after 61 months and the distribution
after 62 months will be added. In addition, 𝑁𝐴𝑉 , which was at the time 60 months later, will be replaced by
𝑁𝐴𝑉 , which is the NAV at that time 63 months later. Taking these factors into account, the cash flow indetermining SBDA 𝑠 ,would be as shown in the lower part of Figure 1. To obtain the Constant Maturity

5-year SBDA 𝑠 , , we can use this cash flow to obtain 𝑠 ,defined by Equation (2) with 𝑗=3.

（Definition 2）Constant Maturity 5-year SBDA
The Constant Maturity 5-year SBDA, which is measured at the point in time when 5 years plus 𝑗 months
have passed from the point in time 0, the point in time of commitment to the alternative asset, is 𝑠 ,satisfying the following equation. However, the valuation time points are the quarterly time points 𝑗 =3,6,9,・・・.
∑

∑
,

（2）
,

,

,

,

where 𝐶𝑎𝑙𝑙(𝑖) and 𝐷𝑖𝑠𝑡(𝑖) represent the amount of capital call (but 𝐶𝑎𝑙𝑙(𝑗)=𝑁𝐴𝑉 ) and the amount of
distribution at the time 𝑖 months later, respectively. 𝑓 , represents the benchmark forward rate (the monthly
rate of benchmark forward return from the time 𝑗 months later to the time 𝑖 months later). 𝑁𝐴𝑉

p. 5

represents the value of the alternative assets under management at the valuation point 𝑗+60 months later.
Equation (2) also assumes that 𝐶𝑎𝑙𝑙(𝑖) = 𝐷𝑖𝑠𝑡(𝑖) = 0 if there are no capital calls or distributions at time point𝑖 months later.
Over time, information on the amount of capital calls, the amount of distributions, NAV, benchmark rates,and benchmark forward rates is accumulated, generating Constant Maturity 5-year SBDA 𝑠 , , 𝑗 =3,6,9,…
every 3 months starting after 5 years.
2.3 Two Constant Maturity 5-Year Performance Evaluation Bond Prices and Rate of Price Changes
The credit risk required to obtain the credit spread, which is the excess return on corporate bonds (herewe assume high grade bonds that do not need to factor in bankruptcy risk), is considered to be the risk ofprice fluctuations of corporate bonds due to changes in the credit spread. In the case of corporate bonds,
since the coupon and face value are fixed, the credit risk of corporate bonds can be determined by tracking
changes in the credit spread. For alternative assets, there is no equivalent to these asset prices (because
the investment amount varies from point in time even when looking at the NAV at each point in time), so it isnecessary to calculate the price volatility risk of alternative assets by assuming an appropriate one. In thissection, we introduce two types of constant maturity 5-year performance evaluation bond prices to derive
the tracking error of alternative assets: one is the realized constant maturity 5-year performance evaluationbond price (Definition 3), and the other is the pre-Constant Maturity 5-year performance evaluation bondprice (Definition 4).
（Definition 3）Realized Constant Maturity 5-Year Performance Evaluation Bond Price (𝑃 )The realized constant maturity 5-year performance evaluation bond price (𝑃 ) at the point in time 𝑗 months
back from the point in time when 5 years plus 𝑗 months (𝑗 =3,6,9,… have passed is defined as in Equation(3) as the right-hand side of Equation (2) with the constant maturity 5-year SBDA 𝑠 ,that satisfies

Equation (2).
𝑃

∑

（3）
,

,

,

,

（Definition 4）Pre-Constant Maturity 5-Year Performance Evaluation Bond Price (𝑃 ,The pre-constant maturity 5-year performance evaluation bond price (𝑃 ,)) at the point in time 𝑗 months

back from the point in time when 5 years plus 𝑗 months (𝑗=3,6,9,…) have passed is defined as in Equation(4), replacing the amount of real distributions 𝐷𝑖𝑠𝑡 ( 𝑖 ) or 𝑁𝐴𝑉
,

,
,

,

,

or

,

𝑃,

,

computed assuming that the fund managerʼs skill had,,

remained at the SBDA 𝑠

used in (Definition 3) withmeasured three months earlier.
,

,

∑
,

,

,

,

,

（4）

,

Using the two constant maturity 5-year performance evaluation bond prices, we define the rate of pricechange (𝑅, ) in the constant-maturity 5-year performance evaluation bond price due to the change onlyin the constant maturity 5-year SBDA (from 𝑠,

to 𝑠 , ) alone (eliminating changes in the benchmark

return), as in equation (5)

p. 6

𝑅

,

,

（5）

,

2.4 The case of alternative assets whose prices are given as index values
Let the index value of the alternative asset at the point in time 𝑗 months later, which is 5 years back fromthe point in time when 5 years plus 𝑗 months (𝑗 =3,6,9,…) have passed, be (𝑃 ) and the index value of thealternative asset at 5 years plus 𝑗 months later be (𝑃𝑃equation (3), where 𝐷𝑖𝑠𝑡(𝑖) = 0, 𝑁𝐴𝑉), the relationship between the two is shown inin equation (3). From equation (6), SBDA 𝑠 ,𝑃

is obtained.

（6）
,

,

To get an intuitive understanding, we express equation (6) as equation (6)' on a continuously compoundedbasis.
𝑃

𝑃

∙ 𝑒𝑥𝑝

𝑓,

𝑠,

∙ 60

(6)’

In exactly the same way, if the index value of the alternative asset at the point 𝑗-3 months after 5 years
plus 𝑗-3 months (𝑗 =3,6,9,…) and the index value of the alternative asset at 5 years plus 𝑗-3 months after
𝑗-3 months is (𝑃SBDA 𝑠,), the relationship between the two satisfies Equation (7). From equation (7), we obtain

.
𝑃

𝑃

∙ 𝑒𝑥𝑝

𝑓

𝑠

,

∙ 60

,

（7）

The pre-constant maturity 5-year performance evaluation bond (𝑃 ,) is defined as the price that would

have been assumed if the fund managerʼs skill had remained at the SBDA 𝑠,

measured three months

earlier in equation (7), as in equation (8).
𝑃,

𝑒𝑥𝑝

𝑓,

𝑠,

∙ 120 𝑃∙ 𝑒𝑥𝑝 𝑓 ,

𝑠

∙ 60

,

（8）

The log change rate in the constant maturity 5-year performance evaluation bond price, 𝑅, , from 𝑗-3

months to 𝑗 months due to the change in the constant maturity 5-year SBDA alone (removing changes inthe benchmark return) is expressed by Equation (9). This is the change in SBDA multiplied by 60 (months) ofduration.
𝑅

,

𝑙𝑛
𝑠,

∙

𝑙𝑛

,

,

𝑠

,

,

∙

∙

,

,

∙

,

,

∙ 60

∙

(9)

3．Integrated Active Management
3.1 Expected excess return, tracking error, and expected information ratio for a single alternative asset
We define the expected excess return (annualized), tracking error (annualized), and expected information
ratio of the alternative assets using the tools prepared in Section 2.1 through Section 2.4.
（Expected excess return (annualized)）
Expected value of the constant maturity 5-year SBDA 𝑠 ,𝐸 𝑠,(𝑗 =3,6,9,…)

∙ 12

p. 7

（Tracking error (annualized)）
Annualized standard deviation of the rate of quarterly change in constant maturity 5-year performance
evaluation bond price (𝑅
𝜎 𝑅

,

, )

∙2

（expected information ratio）
Expected excess return (annualized) divided by tracking error (annualized)
𝐼. 𝑅.

∙ 12

𝐸 𝑠,

𝜎 𝑅

,

∙2

3.2 Integrated Active Portfolio Optimization
Integrated active management refers to the management of an optimal portfolio from a risk-return
perspective, covering not only various active strategies in bonds and stocks, but also investments inalternative assets as one active strategy, and covering all these active strategies. In order to use the notation
introduced so far for multiple assets, the asset 𝑘 is denoted by (𝑘) in the upper right corner of the symbol.
For example, the log price change in the constant maturity 5-year performance evaluation bond price from𝑗-3 months to 𝑗 months (𝑅, ) due only to changes in the constant maturity 5-year SBDA for alternative(removing changes in the benchmark return) is written as 𝑅 , , , etc. In addition, since the excess

asset 𝑘returns of various active strategies in bonds and stocks are the same as in the case of alternative assets,
where the prices in Section 2.4 are given as index values, all notation here is unified with that of alternative
assets. The optimization of the integrated portfolio requires the covariance between alternative asset 𝑘 andalternative asset 𝑙 in addition to the indices introduced in Section 3.1. The covariance (annualized) of therate of quarterly price change of the constant maturity 5-year performance evaluation bond 𝑅 ,∗ , (𝑗=3,6,9,…) with alternative asset 𝑘 and alternative asset 𝑙 is given by 𝐶𝑜𝑣 𝑅 , , , 𝑅 , ,∙ 4.

Optimization of the integrated active portfolio, in which investment in alternative assets is one of the active
strategies, is an attempt to maximize the expected excess return (Equation (10)) while keeping the tracking
error of the integrated active portfolio within a given constraint (Equation (11)) and the amount allocated toeach active strategy within a given constraint (Equation (12)).
[Integrated Active Portfolio Optimization Model]
max ∑
,⋯,

𝑤 ∙𝐸 𝑠 ,Constraints; 𝒘𝑻 𝑪𝑶𝑽𝒘
0
where 𝒘𝑻
𝐶𝑜𝑣 𝑅

,

, ,𝑅

𝑤 ,⋯,𝑤
,

,

𝑤

∙ 12

(10)

𝑟𝑖𝑠𝑘 𝑙𝑖𝑚𝑖𝑡(11)

𝑤𝑒𝑖𝑔h𝑡 𝑘 𝑙𝑖𝑚𝑖𝑡𝑘1, ⋯ , 𝑛(12)

and 𝑪𝑶𝑽 is the variance-covariance matrix whose ( 𝑘 , 𝑙 ) component is∙4 .

（Remark）
The expected information ratio of an integrated active portfolio is the expected excess return (annualized)
divided by the tracking error (annualized) and is given by equation (13).
𝐼. 𝑅.

∑

𝑤 ∙𝐸 𝑠 ,∙ 12√𝒘𝑻 𝑪𝑶𝑽𝒘

(13)

p. 8

To find the optimal weights for maximizing the expected information ratio of the integrated active
portfolio, equation (13) is used as the objective function in (10) and equation (11) can be removed fromthe constraints. Therefore, the concept of optimal weights here is the same as the concept of fundallocation according to the information ratio in credit bonds active management as argued by Kasuga(2009).

4．Numerical examples for various new concepts
In this section, through numerical examples, we will examine the time series of the constant maturity (3and 5-year) SBDA, the adjusted duration of the realized constant maturity (3- and 5-year) performance
evaluation bonds, and the rate of the change 𝑅,

in constant maturity (3- and 5-year) performance

evaluation bond price due only to the change in SBDA (removing changes in the benchmark return) from 𝑗 3 months to 𝑗 months later. Figure 2 shows the hypothetical cash flow (corresponding to the top row in theschematic diagram in Figure 1) as the fund accumulates, which is used to derive the numerical example
corresponding to these concepts. In Figure 2, the downward-facing blue bar represents the amount of capitalcalls, the upward-facing red bar represents the amount of distributions, and the blue circle represents theamount of NAV, respectively. From Figure 2, it can be seen that the NAV rises immediately after a capitalcall and falls immediately after a distribution, and that the NAV gradually builds up as the fund accumulates.
The trends of the constant maturity (3-year and 5-year) SBDA are shown in Figure 3. （In order to obtainthe (3- and 5-year) SBDA, data for the past 3 and 5 years, respectively, are required, so the starting points
are March 2008 for the 3-year SBDA and March 2010, two years later, for the 5-year SBDA. The 3-year
SBDA fluctuates between -5% and 18%, while the 5-year SBDA fluctuates between 0% and 15%. However,all SBDAs are confirmed to be stable as the measurement period becomes longer. In addition, prior to 2017,
there were periods when the 3-year SBDA and 5-year SBDA diverged significantly, but after 2018, thedivergence between the two is negligible.
The modified duration of the realized constant maturity (3-year and 5-year) performance evaluation
bonds is shown in Figure 4. The modified duration is the percentage change in the price (𝑃 ) of the realized
constant maturity 5-year performance evaluation bond calculated by decreasing the constant maturity 5year SBDA 𝑠 ,by 1BP in equation (3) from the original price. Looking at the entire period from Figure 4,

the 3-year modified duration ranges from 1.3 to 2.9, while the 5-year modified duration ranges from 2.1
to 4.3, confirming that the modified duration fluctuates significantly as the measurement period becomes
longer. This is because the modified duration tends to be smaller when there are more capital calls in thefirst half of the measurement period and more distributions in the second half, and larger when the opposite
is true, as the measurement period becomes longer.
The rate of the change 𝑅,

in constant maturity (3- and 5-year) performance evaluation bond pricedue only to the change in SBDA is shown in Figure 5. In the first half of the period, the variation in the rateof the change 𝑅,

is greater for the 3-year than for the 5-year, reflecting the fact that the variation in

the 3-year SBDA is greater than that in the 5-year SBDA, but in the second half of the period, thevariation in the rates for 3- and 5-year are both stable at low levels.

p. 9

【Figure 2: Cash Flow and NAV】

【Figure 3: Changes in SBDA】【Figure 4: Duration】【Figure 5: Returns】（Source: Authors' calculations for Figures 2 through 5.）p. 10

5．Summary and Future Issues
Based on our knowledge of the issues and analysis of alternative assets to date, we believe that whenincorporating alternative assets into traditional assets, it is more cautious to try to get excess returns overtraditional asset returns under some tracking error risk constraints, rather than to expect diversification
effects that are difficult to measure. We have previously devised the SBDA as a measure to estimate
excess returns on alternative assets, and in this paper, we propose a method to quantify tracking error
required for integrated active management by applying the SBDA approach. There are three main issues tobe addressed in the future.
(1) The challenge from finance theory is the decomposition of SBDA. In light of Takehara (2012), SBDA is"alpha (return in excess of benchmark) for practitioners," but it is necessary to verify by some method
how much alpha (risk-adjusted return by multi-beta model) for researchers remains in SBDA. The
multi-beta model is a method to examine how much alpha (risk-adjusted return based on the multi-betamodel) remains in the SBDA. Multi-beta here mainly includes liquidity risk and credit risk derived fromcredit facilities. Once this is possible, integrated active management including active strategies forother traditional assets as described in "1. Introduction" will be possible.
(2) The challenges from finance practice are, first, if the benchmark adopted for the PE funds is different
from the policy benchmark, this part needs to be adjusted in the derivation of the integrated excess
return and tracking error. Second, if the currency risk is to be neutral, the currency composition of thepolicy benchmark and the currency composition of the alternative asset should be adjusted by currency
hedging or other means. It is also necessary to examine what value should be selected as N for theconstant maturity N-year performance evaluation bond price to ensure that the estimated tracking error
is consistent with the realized tracking error.
(3) An issue that bridges finance theory and practice is the comparison of tracking error in this study withtracking error in risk models employed by asset management companies. Since risk models usually employ
factor models, a detailed examination of tracking errors of risk models in a factor-dependent manner
may provide a clue to solving the issue described in (1). This seems to be an interesting subject foranalysis where theory and practice entangle.

References[1] Tomio Arai and Hiroyuki Yamada (2002), "Is it rational to invest in actively managed funds?", Securities
Analyst Journal, 40 (11), pp. 74-95. (in Japanese)[2] Keisuke Ito, Satoru Ikeda, and Toyokazu Kido (2009), "Investment Style, Liquidity and Active
Management," Securities Analyst Journal, 47 (6), pp. 48-60. (in Japanese)[3] Kozo Omori and Manabu Yano (2013), "Risk-Based Portfolio and Active Management," Securities
Analyst Journal, 51 (11), pp. 17-26. (in Japanese)[4] Nobukazu Ogishima and Kei Yamamoto (2003), "Domestic Bond Active Management under Low Interest
Rates," Securities Analyst Journal, 41 (9), pp. 48-63. (in Japanese)p. 11

[5] Noriyuki Oharazawa (1991), "The Factor Model and Its Application to Active Strategies," Securities
Analyst Journal, 29 (9), pp. 36-55. (in Japanese)[6] Kenichi Kasuga (2009), "Issues in Credit Management - Necessity of Risk Allocation commensuratewith Information Ratio", Securities Analyst Journal, 47 (10), pp. 71-80. (in Japanese)[7] Tadashi Kikukawa, Tomonori Uchiyama, Mamoru Motohiro, and Sho Nishiuchi (2017), "Performance ofDomestic Bond Active Management and Smart Beta Strategies," Securities Analyst Journal. 55 (2),pp. 69-80. (in Japanese)[8] Keiichi Kubota and Hitoshi Takehara (2007), "Reexamination of the Validity of the Fama-French Factor
Model," Contemporary Finance,22、83-23. (in Japanese)[9] Hayato Komai and Shuhei Oka (2012), "Japanese Equity Active Fund Performance: Diversification andEconomies of Scale," Securities Analyst Journal, 50 (8), pp. 38-46. (in Japanese)[10] Takehiko Shikata (2010), "Assessing the Management Performance of Equity Active Funds,"
Securities Analyst Journal, 48 (9), pp. 76-87. (in Japanese)[11] Manabu Takatsu and Makoto Yamazaki (2000), "A Study of Bond Portfolio Management by Yield
Curve Analysis," Securities Analyst Journal 、38 (12), pp. 16-31. (in Japanese)[12] Hitoshi Takehara (2012), "Equity Investment: A Reexamination from an Asset Pricing Perspective,"
Securities Analyst Journal, 50 (11), pp. 22-36. (in Japanese)[13] Yoshihiro Nakatani (2010), "The Relationship between the Curvature of the Yield Curve and Interest
Rate Volatility: Verification by Measuring Butterfly Trade Returns," Securities Analyst Journal, 48 (8),pp. 26-35. (in Japanese)[14] Government Pension Investment Fund (2022), "Annual Report."[15] Katsuji Maeda and Takuji Koike (2002), "On Yield Curve Tactics Using Market Overreaction,"
Securities Analyst Journal, 40 (11), pp. 125-140. (in Japanese)[16] Miyai, H., Nakamura, A., Yamaguchi, N., Ishida, H., and Yamada, M. (2005), "Roundtable Discussion:Pension Management and Alternative Investments," Securities Analysts Journal, 43 (6), pp. 77-102.
(in Japanese)[17] Koichi Miyazaki, Yuki Abe, and Kazuhiro Shimada (2021), "Bond Investment Strategy Based on Factor
Analysis: Revisited Application to Carry and Roll-down Strategy in the U.S. Treasury Market," GPIF
Working Paper[18] Koichi Miyazaki and Kazuhiro Shimada (2023-a), "Japanese Private Real Estate Models and Portfolio
Selection," GPIF Working Paper[19] Koichi Miyazaki and Kazuhiro Shimada (2023-b), "Spread-Based Direct Alpha (SBDA) as a
Performance Measure for PE Funds," GPIF Working Paper[20] Satoshi Yamada (2000), "Risk Premium of Japanese Government Bonds and Its Application to
Investment Strategies", Securities Analyst Journal, 38 (12), pp. 32-62. (in Japanese)

p. 12