Asset Management Contracts and Equilibrium Prices

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1 Asset Management Contracts and Equilibrium Prices ANDREA M. BUFFA DIMITRI VAYANOS PAUL WOOLLEY Boston University London School of Economics London School of Economics September, 2013

2 Abstract We study the joint determination of fund managers contracts and equilibrium asset prices. Because of agency frictions, fund investors make managers fees more sensitive to performance and benchmark performance against a market index. In equilibrium, managers underweight expensive stocks that are in high demand by other traders and have endogenously high market betas, and overweight low-demand stocks. Since benchmarking makes it risky for managers to deviate from the index, it exacerbates cross-sectional differences in alphas and the negative relationship between alpha and beta. Moreover, because underweighting the expensive stocks is riskier than overweighting the cheaper ones, benchmarking raises the price of the aggregate stock market and can increase its volatility. Socially optimal contracts involve less benchmarking than privately optimal ones and attenuate the asset-pricing effects of agency frictions.

3 Motivation Asset management and benchmarking. Most wealth invested in financial assets is managed professionally. Individual investors held only 21.5% of US stocks in 2007 (French 2008). Asset managers risk and return is measured against benchmarks. Questions. How does delegation of asset management affect equilibrium prices? What are consequences of benchmarking? Example: Dot-com bubble. Benchmarking concerns Fund managers bought tech stocks they considered overvalued Bubble was amplified. 3 / 35

4 This Paper Study joint determination of asset management contracts and equilibrium prices. Prices clear markets given contracts. Contracts (e.g., extent of benchmarking) are not exogenous, but are optimal, within a parametrized class, given prices. Advantages of endogenizing contracts in equilibrium setting: Sharpen analysis of equilibrium effects of delegation. Tighter link between agency frictions and asset prices, through contracts. Address public policy questions. Are privately optimal contracts socially optimal? Do they generate excessive mispricings or volatility? Is there scope for regulation? Continuous-time, infinite-horizon model with multiple stocks. 4 / 35

5 Main Results: Positive Contracts. Prices. When agency frictions become more severe: Managers compensation becomes more sensitive to performance. There is more use of benchmarking. Betting-against-beta anomaly arises even without agency frictions. Only assumption: Some investors do not hold market portfolio. When agency frictions become more severe: Betting-against-beta anomaly is exacerbated. Beta of low-beta stocks decreases and their positive alpha increases. Beta of high-beta stocks increases and their negative alpha decreases (becomes more negative). More generally, cross-sectional dispersion in alphas increases. Expected return of aggregate market decreases. Cross-sectional effect is asymmetric. Bubbles. Volatility of aggregate market can increase. 5 / 35

6 Main Results: Normative Privately optimal contracts are not socially optimal. Socially optimal contracts: Are less sensitive to performance and involve less benchmarking. Attenuate asset-pricing effects of agency frictions. 6 / 35

7 Related Literature Delegated asset management and equilibrium prices. Benchmarks and tracking error: Roll (1992), Brennan (1993), Cuoco-Kaniel (2011), Basak-Pavlova (2013). Flow-performance: Shleifer-Vishny (1997), Berk-Green (2004), Vayanos (2004), He- Krishnamurthy (2012, 2013), Kaniel-Kondor (2013), Vayanos-Woolley (2013). Reputation concerns: Dasgupta-Prat (2008), Dasgupta-Prat-Verardo (2011), Guerrieri-Kondor (2012), Malliaris-Yan (2012). Alternative explanations for asset bubbles. Differences in beliefs and short-sale constraints: Miller (1977), Harrison-Kreps (1978), Scheinkman-Xiong (2003), Hong-Scheinkman-Xiong (2006). Investor sentiment and limited arbitrage: Shleifer-Vishny (1997), Abreu-Brunnermeier (2002, 2003). Leverage constraints: Allen-Gale (2000). Betting against beta. Differences in beliefs and short-sale constraints: Hong-Sraer (2013). Leverage constraints: Black (1972), Frazzini-Pedersen (2013). 7 / 35

8 Model

9 Assets Continuous time, infinite horizon. Riskless asset: Exogenous return r. Risky assets (stocks): Dividend flow D it per share of stock i = 1,.., N at time t [0, ) is D it = b i f t + e it, where df t = κ f ( f ft ) dt + σf ft dw ft, de it = κ i (ē i e it ) dt + σ i eit dw it, and (w ft, w 1t,.., w Nt ) are independent Brownian motions. Systematic and idiosyncratic component. Volatility increases with level. (Key) Share price S it of stock i at time t is endogenous: Excess return per share of stock i at time t is dr it D it dt + ds it rs it dt. 9 / 35

10 Asset Supply and Market Index Effective supply of stocks: θ (θ 1,.., θ N ) shares. Supply available to fund manager. Market index: η (η 1,.., η N ) shares. Used as benchmark. If θ differs from η: Active fund adds value relative to investing in index. (θ dominates η under equilibrium prices since manager must be induced to hold θ.) Manager s tracking error is non-zero. Interpretations. Buy-and-hold investors. Supply of stocks is η. Exogenous buy-and-hold investors hold η θ. Effective supply left for manager is θ. Non-representative index. Supply of stocks is θ. Index does not have market weights. 10 / 35

11 Agents Fund investor. Invests his wealth with fund manager. Pays manager flow fee dφ t. Maximizes expected utility of intertemporal consumption [ ] E exp( αc t δt)dt. Fund manager. 0 Invests investor s wealth in riskless asset and in stocks. Stock portfolio: z t (z 1t,.., z Nt ) shares. Receives flow fee dφ t. Can invest personal wealth in riskless asset. Maximizes expected utility of intertemporal consumption [ ] E exp( ᾱ c t δt)dt / 35

12 Agency Frictions At time t manager can take an action m t that: Lowers fund return by 1 2 m2 t dt, Has private benefit Bm tdt, between t and t + dt. Interpretations (reduced form). Low m t is high effort to find more efficient portfolio. Low m t is high effort to reduce operating (e.g., transaction) costs. 12 / 35

13 Contracts Manager s fee dφ t at time t can depend on Excess return on fund: z tdr t 1 2 m2 t dt, Excess return on market index: ηdr t, with constant coefficients (chosen at t = 0): dφ t = β (z t dr t 12 ) m2t dt γηdr t + ζdt. Optimal contract within this class Optimal choice of (β, γ, ζ) 13 / 35

14 Manager s Optimization Problem Objective: Choice variables: Consumption c t. Fund s stock portfolio z t. Action m t. Constraints: Budget constraint [ ] E exp( ᾱ c t δt)dt. 0 d W t = r W tdt + dφ t + Bm tdt c tdt. Fee dφ t = β (z tdr t 12 ) m2t dt γηdr t + ζdt. 14 / 35

15 Investor s Optimization Problem Objective: Choice variables: Consumption c t. Contract (β, γ, ζ). Constraints: Budget constraint [ ] E exp( αc t δt)dt. 0 d W t = dw t = rw tdt + ( z tdr t 1 ) 2 m2 t dt dφ t c tdt. Fee dφ t = β (z tdr t 12 ) m2t dt γηdr t + ζdt. Incentive compatibility (IC) constraint. (z t, m t) solves manager s optimization problem. Individual rationality (IR) constraint. Manager s CE at t = 0 exceeds W 0 + Π, where Π is outside option. 15 / 35

16 Equilibrium

17 Equilibrium Concept Definition: Prices S t (S 1t,.., S Nt ) and contract (β, γ, ζ) form an equilibrium if: (i) Given (β, γ, ζ), S t is such that solution to manager s optimization problem is z t = θ (markets clear). (ii) Given S t, (β, γ, ζ) solves investor s optimization problem. 17 / 35

18 Asset Prices where S it = a 0i + a 1i f t + a 2i e it, a 0i = 1 r ( a1i κ f f + a2i κ i ē i ), b i a 1i =, (r + κf ) 2 + 2rᾱβ(θ γη)bσf 2 1 a 2i =. (r + κi ) 2 + 2rᾱβ(θ i γη i )σi 2 18 / 35

19 Contract Without Agency Frictions (B = 0). Optimal risk-sharing: β = No benchmarking: γ = 0. α α+ᾱ. With Agency Frictions (B > 0). Manager s fee becomes more sensitive to performance: β > Incentives for effort. Manager is exposed to more risk. Performance is benchmarked against market index: γ > 0. Insulate manager from risk he cannot control. Used to improve risk-sharing. α α+ᾱ. 19 / 35

20 Supply Effects: Intuition Price Level Price Volatility Stocks in Large Supply LOW LOW Stocks in Small Supply HIGH HIGH Stocks in large effective supply (high θ i ) are cheap (low S it ). Large risk premium to induce manager to absorb large supply. Stocks in large effective supply have less volatile prices (low Var t (ds it )). Volatility is caused by shocks to dividend flow. Following positive shock: 1. Expected dividends increase Price increases. 2. Dividend volatility increases Large increase in risk premium Attenuates price increase. Key property: Dividend volatility increases with dividend level. With normal distributions, supply (θ i ) does not affect volatility. 20 / 35

21 Betting Against Beta Betting against beta (Black 1972, Frazzini-Pedersen 2013). Low-beta stocks have positive alpha. High-beta stocks have negative alpha. Arises in our model even without agency (or other) frictions. Intuition: Stocks in large effective supply: Are cheap Positive alpha. Have low price volatility Account for small fraction of market movements Low beta. 21 / 35

22 Numerical Example Two symmetric stocks (N = 2). Same systematic risk (b 1 = b 2). Same idiosyncratic risk (σ 1 = σ 2, ē 1 = ē 2, κ 1 = κ 2). Market index includes one share of each (η 1 = η 2 = 1). Stock 1 is in larger effective supply (θ 1 = 0.8, θ 2 = 0.2). Manager is 50 times more risk averse than investor (ᾱ/α = 50). Graphs: As a function of the severity of the agency friction (horizontal axis). Stock 1: blue. Stock 2: red. Market index: black. Compute moments of dollar returns by dividing corresponding moments of share returns by average share price. 22 / 35

23 Optimal Contract Sensitivity to Performance Β Benchmarking Γ Manager s fee becomes more sensitive to performance. Performance is benchmarked against market index. 23 / 35

24 Prices: Level Stock in Large Supply PRICE STOCK Stock in Small Supply PRICE STOCK Benchmarking renders manager less willing to deviate from market index. Supply effects on price levels are amplified. Stock 1, in large effective supply, becomes cheaper. Stock 2, in small effective supply, becomes more expensive. 24 / 35

25 Prices: Volatility 7 Stock in Large Supply PRICE VOLATILITY STOCK 1 22 Stock in Small Supply PRICE VOLATILITY STOCK Supply effects on price volatilities are amplified. Consider positive shock to dividend flow of stock 2, in small effective supply. Dividend volatility increases Manager becomes more eager to buy stock 2 to reduce risk of deviating from market index Large decrease in risk premium Price increase is amplified. 25 / 35

26 Betting Against Beta CAPM BETA CAPM ALPHA Arises in our model even without agency (or other) frictions. Intuition. Stocks in large effective supply: Are cheap Positive alpha. Have low price volatility Account for small fraction of market movements Low beta. Supply effects on alpha and beta discrepancies are amplified. 26 / 35

27 Aggregate Market: Price Level PRICE MARKET 4.7 EXCESS RETURN MARKET Benchmarking renders manager less willing to deviate from market index. Stock 1 (large effective supply) becomes cheaper and less volatile. Stock 2 (small effective supply) becomes more expensive and more volatile. Underweighting stocks in small effective supply is riskier for manager than overweighting stocks in large effective supply. Former stocks account for larger fraction of market movements. Aggregate market goes up. (Bubbles) 27 / 35

28 Aggregate Market: Volatility RETURN VOLATILITY MARKET Volatility of aggregate market increases. However: Result reverses for large levels of systematic risk relative to idiosyncratic risk. 28 / 35

29 Asset Correlations ASSET CORRELATION Correlation of stock returns decreases. Idiosyncratic shocks become more relevant for asset prices. 29 / 35

30 Normative Analysis

31 Social Planner s Problem Are privately optimal contracts socially optimal? Social planner chooses contract (β, γ, ζ ): To maximize sum of certainty equivalents of investor and manager. Internalizing how contract affects prices. Contract chosen by social planner: (β, γ, ζ ) arg max CE 0 ( β, γ, St (β, γ) ) + CE 0 ( β, γ, St (β, γ) ). 31 / 35

32 Socially Optimal Contract Sensitivity to Performance Β Benchmarking Γ Relatively to privately optimal contracts, socially optimal ones: Are less sensitive to performance. Involve less benchmarking. 32 / 35

33 Prices: Level, Volatility and Correlation PRICE STOCK 1 PRICE STOCK PRICE VOLATILITY STOCK 1 PRICE VOLATILITY STOCK 2 ASSET CORRELATION Socially optimal contracts attenuate asset-pricing effects of agency frictions. 33 / 35

34 Aggregate Market: Level and Volatility PRICE MARKET 4.7 EXCESS RETURN MARKET RETURN VOLATILITY MARKET Socially optimal contracts attenuate asset-pricing effects of agency frictions. Smaller rise of aggregate market. 34 / 35

35 Concluding Remarks Study joint determination of asset management contracts and equilibrium prices. New asset-pricing results: Betting against beta. Asymmetric cross-sectional effect of benchmarking Aggregate market. Open door for public-policy analysis of asset management contracts. Do existing contracts have adverse price effects? Should they be improved (e.g., design of benchmarks, risk measures)? Recent proposals by G30, European Commission, etc. 35 / 35

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