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The Benchmark Inclusion Subsidy
Anil K Kashyap∗ , Natalia Kovrijnykh† , Jian Li‡ , and Anna Pavlova§
February 2019
Abstract We study the impact of evaluating the performance of asset managers relative to a benchmark portfolio on firms’ investment, merger and IPO decisions. We introduce asset managers into an otherwise standard asset pricing model and show that firms that are part of the benchmark are effectively subsidized by the asset managers. This “benchmark inclusion subsidy” arises because asset managers have incentives to hold some of the equity of firms in the benchmark regardless of the risk characteristics of these firms. Due to the benchmark inclusion subsidy, a firm inside the benchmark would accept some projects that an identical one outside the benchmark would decline. This finding is in contrast to the usual presumption in corporate finance that the value of an investment project is governed solely by its own cash-flow risk. The incentives of the firms inside the benchmark to undertake mergers or spinoffs also differ. Additionally, the presence of the benchmark inclusion subsidy can alter a decision to take a firm public. We show that the higher the cash-flow risk of an investment and the more correlated the existing and new cash flows are, the larger the benchmark inclusion subsidy; the subsidy is zero for safe projects. Benchmarking also leads fundamental firm-level cash-flow correlations to rise. We review a host of empirical evidence that is consistent with the implications of the model. JEL Codes: G11, G12, G23, G32, G Keywords: Project Valuation, Investment, Mergers, Asset Management, Bench- mark, Index
We have benefited from suggestions and comments from Andrea Buffa, Hui Chen, Ralph Koijen, Felix Kubler, Shohini Kundu, Ailsa Roell, Jeremy Stein, Dimitri Vayanos, Rob Vishny, and seminar partici- pants at Chicago Booth, LBS, Philadelphia Fed, University of Zurich, Fall 2018 NBER Corporate Finance Program Meeting, the 2019 European Winter Finance conference, the Macro Financial Modeling (MFM) Winter 2019 Meeting, and the 2019 UBC Winter Finance conference. This research has been supported by a grant from the Alfred P. Sloan Foundation to the MFM project at the University of Chicago. The views expressed here are ours only and not necessarily those of the institutions with which we are affiliated, and all mistakes are our own. ∗Booth School of Business, University of Chicago, National Bureau of Economic Research, Centre for Economic Policy Research and Bank of England. Email: anil.kashyap@chicagobooth.edu. †Department of Economics, Arizona State University. Email: natalia.kovrijnykh@asu.edu. ‡Department of Economics, University of Chicago. Email: lijian@uchicago.edu. §London Business School and Centre for Economic Policy Research. Email: apavlova@london.edu.
1 Introduction
The asset management industry is estimated to control more than $85 trillion worldwide. Most of this money is managed against benchmarks. For instance, S&P Global reports that as of the end of 2017, there was just under $10 trillion managed against the S&P 500 alone.^1 Existing research related to benchmarks has largely focused on asset pricing implications of benchmarking. Instead, we look at the implications of benchmarking for corporate decisions. We argue that firms included in a benchmark are effectively subsidized by asset managers and so should evaluate investment opportunities differently. Our analysis runs counter to the received wisdom regarding investment decisions. Usu- ally in corporate finance, the appropriate cost of capital depends purely on the character- istics of a project and not on the entity that is considering investing in it. More precisely, the “asset beta” computed by the capital asset pricing model (CAPM) is presumed to be the correct anchor for computing the discount factor used in evaluating a project’s risk. We show that when asset managers are present and their performance is measured against a benchmark, the correct discount factor depends on more than just the asset beta. Instead, discount rates will differ for firms that are inside the benchmark relative to similar firms that are outside. To be specific, when a firm adds risky cash flows, say, because of an acquisition or by investing in a new project, the increase in the stockholder value is larger if the firm is inside the benchmark. Hence, a firm in the benchmark would accept cash flows with lower mean and/or larger variance than an otherwise identical non-benchmark firm would. The underlying reason for this result is that when a firm is part of a widely-held bench- mark, asset managers are compelled to hold some shares of that firm’s equity regardless of the characteristics of the firm’s cash flows. So when a firm adds risky cash flows, the market demand for them is higher and hence the increase in the stockholder value is also higher if the firm is inside the benchmark rather than outside. We call this the “benchmark inclusion subsidy.” The firm, therefore, should take this consideration into account in deciding on its investments, acquisitions and spinoffs. Here is how our the model works. We take a standard asset pricing model and allow for heterogeneity, where some investors manage their own portfolios and others use asset
(^1) As of November 2017, Morgan Stanley Capital International reports that $3.2 trillion was bench- marked against its All Country World Index and $1.9 trillion was managed against its Europe, Australasia and Far East index. Across various markets, FTSE-Russell reports that at the end of 2016 $8.6 trillion was benchmarked to its indices. CRSP reports that assets linked to its indices exceed $1.3 trillion as of September 2018.
over. The extended model also features an additional channel, owing to the correlation of a target’s (or a project’s) cash flows with the firm’s assets-in-place. We demonstrate that if this correlation is positive, the value of the new asset to the benchmark firm exceeds the asset’s value were it to join the benchmark as a standalone. We derive a closed-form expression for the benchmark inclusion subsidy, which turns out to be very simple, and study the variables that influence its size. We show that the higher the cash-flow risk of an investment, the larger the benchmark inclusion subsidy. Furthermore, the benchmark inclusion subsidy is increasing in the correlation of a project’s cash flows with the existing assets; in particular, the subsidy is the largest for projects that are clones of a firm’s existing assets. Finally, the size of the subsidy rises with the asset management sector’s size. The ability to characterize the exact determinants of the subsidy allows us to predict the situations when benchmarking is the most and least important. To the best of our knowledge, other theories do not deliver such cross-sectional predictions. We are able to tie the size of the subsidy to characteristics of firms, investment opportunities, or potential acquisitions or divestitures. The model also implies that benchmarking alters payoffs so that the benchmark becomes a factor that explains expected returns. Hence, in our model, both the benchmark and the usual market portfolio matter for pricing assets. The right model for the cost of capital in our environment is, therefore, not the CAPM that is typically used in corporate finance, but its two-factor modification that accounts for the presence of asset managers. Discussions about benchmarking often revolve around the possibility that it leads to more correlation in risk exposures for the people hiring asset managers. Our model points to an additional source of potential correlation generated by benchmarking. Because bench- marking leads to higher valuations of stocks that are correlated with the benchmark, it induces firms—both inside and outside the benchmark—to take on more fundamental risk that is correlated with the benchmark (relative to the economy without benchmarking). Thus, our model predicts that cash flows in the economy with asset managers endogenously become more homogeneous/correlated with each other. Finally, it is worth noting that our model applies to both active and passive asset management. We show that the benchmark inclusion subsidy is larger when more asset managers are passive rather than active. We review existing empirical work that relates to the model’s predictions. Past research confirms, to varying degrees, the predictions regarding the propensity to invest and engage in acquisitions for benchmark vs. non-benchmark firms, the factor structure of returns, as
well as the size of the benchmark inclusion subsidy increasing in assets-under-management. The remainder of the paper is organized as follows. In the next section, we explain how our perspective compares to previous work. Section 3 presents the example, and Section 4 analyzes the general model. Section 5 reviews related empirical evidence. Section 6 presents our conclusions and suggestions for future areas of promising research. Omitted proofs are in the appendix.
2 Related Literature
The empirical motivation for our work comes from the index additions and deletions liter- ature. Harris and Gurel (1986) and Shleifer (1986) were the first to document that when stocks are added to the S&P 500 index, their prices rise. Subsequent papers have also shown that firms that are deleted experience a decline in price. The findings have been confirmed across many studies and for many markets, so that financial economists consider these patterns to be stylized facts.^2 The estimated magnitudes of the index effect vary across studies, and typically most of the effect is permanent. For example, Chen, Noronha, and Singal (2004) find the cumulative abnormal returns of stocks added to the S&P 500 during 1989-2000, measured over two months post announcement, to be 6.2%.^3 Several theories have been used to interpret the index effect. The first is the investor awareness theory of Merton (1987). Merton posits that some investors become aware of and invest in a stock only when it gets included in a popular index. It is unclear why investor awareness declines for index deletions, although there is evidence of a decrease in analyst coverage. The second theory posits that index inclusions convey information about a firm’s improved prospects. This theory has difficulty explaining the presence of index effects around mechanical index recompositions (see, e.g., Boyer, 2011, among others). The third theory is that index inclusion leads to improved liquidity, and this in turn boosts stock prices. This theory, however, does not explain increased correlations with other index stocks (documented in, e.g., Barberis, Shleifer, and Wurgler, 2005 and Boyer, 2011). The final theory can be broadly described as the price pressure theory, proposed by Scholes (1972). Scholes’ prediction is that prices of included stocks should rise temporarily, (^2) See, e.g, Beneish and Whaley (1996), Lynch and Mendenhall (1997), Wurgler and Zhuravskaya (2002), Chen, Noronha, and Singal (2004), Petajisto (2011), and Hacibedel (2018). (^3) Calomiris, Larrain, Schmukler, and Williams (2018) find an index effect for emerging market corporate bonds. They trace the rise of the JP Morgan Corporate Emerging Market Bond Index and show how firms in countries that became eligible change issuance patterns (to qualify for index inclusion) and receive lower yields on qualifying bond issues.
decisions is critical, and those that are short-term oriented will potentially respond to mispricing if it is big enough. In our model, all managers of firms in the benchmark should account for the subsidy (for as long as the firm remains in the benchmark). Stein’s paper led to a number of follow-on studies that look at other potential behavioral effects that could be associated with inclusion in a benchmark (see Baker and Wurgler, 2013 for a survey). These papers contain much of the empirical work that we cite in favor of our model. While we share several predictions with Stein (1996) there are some notable differences. For instance, Stein’s model connects managerial time horizons and financial constraints to capital budgeting decisions. Our model has nothing to say about these considerations. However, we also have many implications that are distinct from his. For instance, our closed-form expression for the benchmark inclusion subsidy generates a number of predictions about which factors should lead firms in the benchmark to make different decisions than ones outside. On the whole, we see the behavioral theories and ours complementing each other. Finally, there is recent literature on mistakes that managers make in project valuation. Survey evidence from Graham and Harvey (2001) shows that a large percentage of publicly traded companies use the CAPM to calculate the cost of capital. In addition, they seem to use the same cost of capital for all projects. Krüger, Landier, and Thesmar (2015) document that this tendency appears to distort investments by diversified firms. In particular, they appear to make investment decisions in non-core businesses by using the discount rate from their core business. Interestingly, in our model, the benchmark inclusion subsidy applies to the entire firm so there is a basis for having part of the cost of capital depend on that firm-wide characteristic.
3 Example
To illustrate the main mechanism, we begin with a simple example with three firms with uncorrelated cashflows. We first consider an economy populated by identical investors in these firms who manage their own portfolios. We then modify the economy by introducing another group of investors who hire asset managers to run their portfolios. Asset managers’ performance is evaluated based on a comparison with a benchmark. We show that the presence of asset managers invalidates the standard approach to corporate valuation.
3.1 Baseline Economy
Consider the following environment. There are two periods, t = 0, 1. Investment opportuni- ties are represented by three risky assets denoted by 1 , 2 , and y, and one risk-free bond. The risky assets are claims to cash flows Di realized at t = 1, where Di ∼ N (μi, σ^2 i ), i = 1, 2 , y, and these cash flows are uncorrelated. We think of these assets as stocks of all equity firms. The risk-free bond pays an interest rate that is normalized to zero. Each of the risky assets is available in a fixed supply that is normalized to one. The bond is in infinite net supply. Let Si denote the price of asset i = 1, 2 , y. There is measure one of identical agents who invest their own funds. Each investor has a constant absolute risk aversion (CARA) utility function over final wealth W , U (W ) = −e−αW^ , where α > 0 is the coefficient of absolute risk aversion. All investors are endowed with one share of each stock and no bonds. At t = 0, each investor chooses a portfolio of stocks x = (x 1 , x 2 , xy)>^ and the bond holdings to maximize his utility, with W (x) = ∑ i=1, 2 ,y Si^ +^ xi(Di^ −^ Si). As is well-known in this kind of setup, the demand xi for risky asset i and the corre- sponding equilibrium price Si will be
xi = μi^ −^ Si ασ^2 i
Si = μi − ασ^2 i
for i = 1, 2 , y, where the second equation follows from setting the number of shares de- manded equal to the supply (which is 1).^4 When firms i ∈ { 1 , 2 } and y merge into a single firm, the demand for the combined firm’s stock and the corresponding equilibrium stock price are
x′ i = μi^ +^ μy^ −^ S i′ α(σ^2 i + σ^2 y ) , S i′ = μi + μy − α(σ^2 i + σ y^2 ) = Si + Sy.
Notice that the combined value of either firm is exactly equal to the sum of its initial value plus the value of y. This is a standard corporate valuation result that says that the owner of a firm does not determine its value. Instead, the value arises from the cash flows (and
(^4) We omit derivations for this simple example, but the analysis of our main model contains all proofs for the general case.
risk aversion α. Asset managers’ portfolio choices differ from those of the conventional investors in two ways. First, they hold a scaled version of the same mean-variance portfolio as the one held by the conventional investors. The reason for the scaling is that, as we can see from the first term in (1), for each share that the asset manager holds, she gets a fraction a + b of the total return. Thus the asset manager scales her asset holdings by 1 /(a + b) relative to those of a conventional investor. Second, and more importantly, the asset managers are penalized by b for underperform- ing the benchmark. Because of this penalty, the manager always holds b/(a + b) shares of stock 1 (or more generally whatever is in the benchmark). This consideration explains the second term in (2). This mechanical demand for the benchmark will be critical for all of our results. In particular, the asset managers’ incentive to hold the benchmark portfolio (regardless of the risk characteristics of its constituents) creates an asymmetry between stocks in the benchmark and all other stocks. The second implication is very general and extends beyond our model with CARA preferences. Having a relative performance component as part of her compensation exposes the manager to an additional source of risk—fluctuations in the benchmark—which she optimally decides to hedge. The manager would, therefore, hold a hedging portfolio that is (perfectly) correlated with the benchmark, i.e., the benchmark itself. Given the demands, we can now solve for the equilibrium prices. Using the market- clearing condition for stocks, λAM xAMi + λC xCi = 1, i = 1, 2 , y, we find
S 1 = μ 1 − αΛσ 12
1 − λAMa +^ b b
S 2 = μ 2 − αΛσ 22 , (4) Sy = μy − αΛσ^2 y , (5)
where Λ = [λAM /(a + b) + λC ]−^1 modifies the market’s effective risk aversion. For concreteness, suppose that μ 1 = μ 2 and σ 1 = σ 2 so that the return and risks of stocks 1 and 2 are identical. Our first noteworthy finding is that the share price of firm 1 that is inside the benchmark is higher than that of its twin that is not. This happens because asset managers automatically tilt their demand towards the benchmark, effectively reducing the supply of this stock by b/(b + a). The lower the supply of the stock (all else equal), the higher must be its equilibrium price. Another way to understand the result is that the asset managers’ mechanical demand for the benchmark means that the adverse effects of variance that typically reduce the demand for any stock, are less relevant for the
assets in the benchmark.^8 Next, consider potential mergers. Suppose first that y merges with the non-benchmark firm (firm 2 ). The new demands of conventional investors and asset managers for the stock of firm 2 are
x′ 2 C = μ^2 +^ μy^ −^ S 2 ′ α
σ^22 + σ^2 y
x′ 2 AM = (^) a +^1 bμ^2 +^ μy^ −^ S 2 ′ α
σ^22 + σ^2 y
The new equilibrium price of firm 2’s stock is
S 2 ′ = μ 2 + μy − αΛ
σ^22 + σ^2 y
= S 2 + Sy.
As before, the combined value of firm 2, continues to be the sum of the initial value plus the value of y. Suppose instead that asset y is acquired by firm 1 , which is in the benchmark. Re- normalizing the combined number of shares of firm 1 to one, the demands for the stock of the combined firm are
x′ 1 C = μ^1 +^ μy^ −^ S
′ 1 α
σ 12 + σ^2 y
x′ 1 AM = (^) a +^1 bμ^1 +^ μy^ −^ S
′ 1 α
σ 12 + σ^2 y ) (^) + b a + b.
Our next major finding is that there is a benchmark inclusion subsidy. Specifically, the new price of firm 1’s shares is
S 1 ′ = μ 1 + μy − αΛ
σ^21 + σ y^2
1 − λAM^ b a + b
= S 1 + Sy + αΛσ y^2 λAM^ b a + b
which is strictly larger than the sum of S 1 and Sy. So when a firm inside the benchmark
(^8) Notice that in this model the asymmetry between benchmark and non-benchmark stocks cannot be arbitraged away. The conventional investors are unrestricted in their portfolio choice and therefore can engage in any arbitrage activity. However, as the asset managers permanently reduce the supply of the benchmark stock, conventional investors simply reduce their holdings of the benchmark stock and hold more of the non-benchmark stock. These implications are similar to the effects of quantitative easing in bond markets, whereby a central bank buys a significant fraction of outstanding bonds. As long as asset managers represent a meaningful fraction of the market (i.e., λAM is non-negligible), there are always differences in prices of stocks inside and outside the benchmark.
valuations of spinoffs and divestitures. If y had been part of a firm inside the benchmark and is sold to a non-benchmark firm, the value of y would drop when it is transferred. In the next section, we consider a richer version of the setup that allows us to analyze several additional questions. Based just on this extremely simplified example, however, we already have seen two empirical predictions. First, consistent with the existing literature on index inclusions, we see that there should be an increase in a firm’s share price when it is added to the benchmark. We view this as a necessary condition for the existence of the benchmark inclusion subsidy. In our framework, the stock price increase would remain present for as long as the firm is part of the benchmark. The other prediction is related to acquisitions (and spinoffs) and is the one we would like to stress. If a firm that has not previously been part of the benchmark is acquired by a benchmark firm, its value should go up purely from moving into the benchmark. This breaks the usual valuation result which presumes that an asset purchase that does not alter any cash flows (of either the target or acquirer) should not create any value. Alternatively, if a firm were spun-off so that it moves from being part of the benchmark to no longer belonging to the benchmark, its value should drop even though its cash flows are unchanged. The results in this section, and all the ones in the following section, depend on the com- pensation contract having a non-zero value of b. There is both direct empirical evidence and strong intuitive reasons for why this assumption should hold. For instance, since 2005 mutual funds in U.S. have been required to include a “Statement of Additional Information” in the prospectus that describes how portfolio managers are compensated. Ma, Tang, and Gómez (forthcoming) hand collect this information for 4500 mutual funds and find more than three quarters of the funds explicitly base compensation on performance relative to a benchmark (that they are able to identify). Bank for International Settlements (2003) presents survey-based evidence for a sample of other asset managers including sovereign wealth funds and pension funds, and also finds that performance evaluation relative to benchmarks is pervasive. To see why these results are expected, consider any asset man- ager that runs multiple funds with different characteristics, for instance, a bond fund and an equity fund. To compensate the portfolio managers of each fund, the simple returns cannot be meaningfully compared because of the differences in risk. However, if each fund’s performance is adjusted for a benchmark for its type, then the relative performances can be compared. So, it is hardly surprising that the use of benchmarks is so pervasive and our assumption concerning b should not be controversial.
4 The General Model
We now generalize the example studied in Section 3 in several directions. All results from the previous section hold in this richer model. To analyze a new implication for investment, we will assume that y is not traded initially, so that it can be interpreted as a potential project. We will only describe elements of the environment that differ from those described in the previous section. There are n risky stocks, whose total cash flows D = (D 1 ,... , Dn)>^ are jointly normally distributed, D ∼ N (μ, Σ) , where μ = (μ 1 ,... , μn)>, Σii = Var(Di) = σ i^2 , and Σij = Cov(Di, Dj ) = ρij σiσj. We assume that the matrix Σ is invertible. Stock prices are denoted by S = (S 1 ,... , Sn)>. For simplicity of exposition and for easier comparison to Section 3, we normalize the total number of shares of each asset to one. However, for generality, all of our proofs in the appendix are written for the case when asset i’s total number of shares is x¯i. Some stocks are part of a benchmark. We order them so that all shares of the first k stocks are in the benchmark, and none of the remaining n − k stocks are included. Thus, the ith element of the benchmark portfolio equals the total number of shares of asset i times (^1) i, where (^1) i = 1 if i ∈ { 1 ,... , k} and (^1) i = 0 if i ∈ {k + 1,... , n}. Denote further (^1) b = ( 11 ,... , (^1) n)>^ = (1 ︸,... , ︷︷ 1 ︸ k
n−k
We follow the convention in the literature (see, e.g., Buffa, Vayanos, and Woolley, 2014) by defining rx = x>(D − S) to be the performance of portfolio x = (x 1 ,... , xn)>^ and rb = 1 > b (D − S) to be the performance of the benchmark portfolio. Then the compensation of an asset manager with contract (a, b, c) is w = arx + b(rx − rb) + c.^10 Denote by xC^ = (xC 1 ,... , xCn )>^ and xAM^ = (xAM 1 ,... , xAMn )>^ the optimal portfolio choices of a conventional investor and an asset manager, respectively.
Lemma 1 (Portfolio Choice). Given asset prices S, the demands of a conventional investor and an asset manager are given by
xC^ = Σ−^1 μ^ −α^ S, (8)
xAM^ =
a + bΣ
− 1 μ^ −^ S α +^
b a + b^1 b.^ (9) (^10) In Appendix B we repeat all of the analysis for the case where a manager’s compensation is tied to the per-dollar return on the benchmark rather than the per-share return (performance) and confirm that our key results continue to hold.
The benchmark portfolio emerges as a factor because asset managers are evaluated relative to it. Stocks that load positively on this factor have lower expected returns because asset managers overinvest in the benchmark, which drives down the expected returns on its components. Stocks outside the benchmark that covary positively with the benchmark also have lower expected returns because conventional investors (as well as asset managers through their mean-variance portfolio) who desire exposure to the benchmark buy these cheaper, non-benchmark stocks instead, pushing up their prices. Lemma 3 demonstrates this formally. The two-factor CAPM is not intended to be a fully credible asset pricing model. Our model certainly has its limitations because it does not account for the fact that in practice managers are evaluated relative to heterogeneous benchmarks, representing different invest- ment objectives (large cap, value, growth, etc.).^13 Rather, we emphasize that the prevailing corporate finance approach to valuation based on the “asset beta” coupled with the standard CAPM does not apply in our economy. Lemma 3 implies that the cost of capital for firms inside the benchmark is lower than for their identical twins that are outside. Therefore, the usual conclusion that the value of a project is independent of which firm adopts it does not hold.
4.1 Investment
Suppose there is a project with cash flows Y ∼ N (μy, σ^2 y ), and Corr(Y, Di) = ρiy for i = 1,... , n. Investing in this project requires spending I. If firm i (whose cash flows are
Di) invests, its cash flows in period 1 become Di + Y. Let S(i)^ =
S( 1 i ),... , S n(i)
denote the stock prices if firm i invests in the project. The firm finances investment by issuing equity.^14 That is, we assume that if firm i invests in the project, it issues δi additional shares to finance it, where δiS i( i) = I. If firm i is in the benchmark, then the additional shares enter the benchmark. To proceed, suppose firm i (and only firm i) invests in the project. Then the new cash flows are D(i)^ = D + (0, ..., 0 , D ︸︷︷︸y i
, 0 , ..., 0)>, distributed according to N
μ(i), Σ(i)
, where
μ(i)^ = μ + (0, ..., 0 , μ ︸︷︷︸y i
, 0 , ..., 0)>^ and
(^13) See Cremers, Petajisto, and Zitzewitz (2012) for a multi-benchmark model that explains the cross- section of mutual fund returns. (^14) As in the example, the main results that follow hold even if the firm uses some debt financing. For instance, the size of the benchmark inclusion subsidy is literally identical even if the firm has risk-free debt.
Σ(i)^ = Σ +
ρ 1 yσ 1 σy
ρ 1 yσ 1 σy ... σ y^2 + 2ρiyσiσy ... ρnyσnσy
ρnyσnσy 0
Denote I(i)^ = (0, ..., 0 , (^) ︸︷︷︸I i
Lemma 4 (Post-Investment Asset Prices). The equilibrium stock prices when firm i invests in the project are given by
S(i)^ = μ(i)^ − I(i)^ − αΛΣ(i)
1 − λAMa +^ b b (^1) b
The change in the stockholder value of the investing firm i, ∆Si ≡ S i( i)− Si, is
∆Si = μy − I − αΛ
σ^2 y + ρiyσiσy
1 − λAMa +^ b b (^1) i
−αΛ
∑^ n j=
ρjyσj σy
1 − λAM b a + b^1 j
The first two terms in the first line of (13) are the expected cash flows of the project net of the cost of investment, and the remaining terms reflect the penalty for risk. It is evident from (13) that this penalty differs if i is part of the benchmark, so it will turn out to be subject to the benchmark inclusion subsidy that we have already seen in the example in Section 3. Notice that the terms on second line of (13) are the same regardless of the identity of the investing firm. When any firm invests in a project positively correlated with the benchmark, this firm’s cash flows become more correlated with the benchmark. As we have seen from the two-factor CAPM, the presence of asset managers makes stocks that covary positively with the benchmark more expensive relative to what they would have been in the economy with only conventional investors. This happens because the investors substitute from the expensive stocks in the benchmark to stocks that are correlated with it. A similar logic applies to new investment projects that are correlated with the benchmark: relative to
the benchmark inclusion subsidy exceeds the index effect. The covariance subsidy is the largest when ρy = 1, so that y is a clone of the existing assets. Moreover, assuming that the correlation ρy is large enough and the variance of existing cash flows exceeds that of the new cash flows, i.e., σ > σy (both are empirically reasonable assumptions), the covariance subsidy exceeds the index effect. Keeping in mind that the benchmark inclusion subsidy arises from taking a difference-in- differences, we can further explain the terms that comprise it. The asset managers subsidize the variance of a benchmark firm’s post-investment cash flow, which is σ^2 i + σ^2 y + 2ρiyσiσy. The first term, σ^2 i , washes out of the first difference given by equation (13) because it is present for the benchmark firm pre- and post-investment. Furthermore, the subsidy includes only one covariance term ρiyσiσy, not two. This is because any firm, either inside the benchmark or not, receives a subsidy for the covariance with the benchmark (one can see this from the second line of (13), which is the same for all firms). That is, projects with a positive covariance with the benchmark are more valuable, even if a non-benchmark firm undertakes them. The reason is that since prices of benchmark stocks are inflated due to the mechanical demand from asset managers, conventional investors (and asset managers through their mean-variance portfolios) substitute into assets that provide exposure to the benchmark without being in the benchmark itself. Consequently, of the two covariances that enter the extra variance, one is subsidized regardless of which firm invests and the other is subsidized only when the investing firm is a benchmark firm. Hence, one of the two covariances drops out from the difference-in-differences. The presence of the benchmark inclusion subsidy translates into different investment rules for firms inside and outside the benchmark. We formalize this result in Proposition 1 below.
Proposition 1 (Project Valuation). A firm in the benchmark is more likely to invest in a project than a firm outside the benchmark if and only if Assumption 1 holds. More precisely, all else equal, a firm in the benchmark accepts projects with a lower mean μy, larger variance σ^2 y , and/or larger correlation ρy than an otherwise identical firm outside the benchmark if and only if Assumption 1 holds.
Proposition 1 is at odds with the textbook treatment of investment taught in basic corporate finance courses. The usual rule states that a project’s value is independent of which firm undertakes it, and is simply given by the project’s cash flows discounted at the project-specific (not firm-specific) cost of capital.^15 The usual rule presumes that the (^15) See for example Jacobs and Shivdasani (2012) or Berk and DeMarzo (2014), chapter 19.
correct way to evaluate the riskiness of a project is to use the CAPM. That is not true in our model. In our model, the compensation for risk is described by a two-factor CAPM (Lemma 3), which accounts for the incentives of asset managers. The reason why a project is worth more to a firm in the benchmark than to one outside it is because when the project is adopted by the benchmark firm, it is incrementally financed by asset managers regardless of its variance. So, the additional overall cash-flow variance that the project generates is penalized less for a firm inside the benchmark. To further understand the importance of the variance, consider a special case where the project is risk free, i.e., σ y^2 = 0. Then Assumption 1 fails and we can see that the project would be priced identically by all firms.
Remark 1 (Risk-Free Projects). If σ^2 y = 0, then a firm’s valuation of project y is independent of whether this firm is included in the benchmark or not.
We can build further intuition about the model by considering what happens when the inequality in Assumption 1 is reversed. This happens if the project is sufficiently negatively correlated with the assets, that is, if ρy ≤ −σy/σ. In this case, a project is a hedge because it reduces the variance of a firm’s cash flow. Firms inside the benchmark benefit less from this reduction because their cash-flow variance is subsidized by asset managers and they lose some of that subsidy.^16 Consequently, a benchmark firm will value such a project less than a non-benchmark firm would. Figure 1 uses a numerical example to display the investment regions for a benchmark- and a non-benchmark firm as a function of μy, σy, and ρy (for a fixed σ). In the left panel, ρy is held constant, and σy and μy vary along the axes. On the right panel, σy is kept constant, and ρy and μy vary along the axes. From the left panel we can see that holding everything else fixed, a benchmark firm will invest in projects with a lower mean, μy, and/or higher variance, σ y^2 , than a non-benchmark firm. The right panel illustrates that compared to a non-benchmark firm, a benchmark firm prefers to invest in projects that are more correlated with its existing cash flows. Finally, as we mentioned earlier, our model also implies that projects correlated with assets inside and outside the benchmark are valued differently (by firms both inside and outside the benchmark). Projects that are positively correlated with the benchmark pro- vide alternative cheaper exposure to the benchmark firms’ cash flows. This is reflected in equation (13), which shows that for any firm, investing in a project that is positively (^16) Notice that when −σy /σ ≤ ρy ≤ −σy /(2σ), although the investment reduces the firm’s cash-flow variance, the benchmark inclusion subsidy is positive.
