Optimal Asset Allocation and Risk Shifting in Money Management Suleyman Basak (LBS), Anna Pavlova (LBS), and Alex Shapiro (Azimuth Trust) (1) Motivation and Objective (2) Model (3) Empirical Analysis (4) Costs of Active Management to Investors
1. Motivation and Objective Page 1 1. Motivation and Objective • Mutual fund managers’ compensation is linked to the value of assets under management • Implicit incentives due to fund fl ows to performance relationship • The fl ow-performance relationship is – positive – exhibits convexities • Question: How does a fund manager respond to these incentives?
1. Motivation and Objective Page 2 Fund Flow - Performance Relationship (Chevalier and Ellison (1997))
1. Motivation and Objective Page 3 Summary of Main Testable Implications • Taking risk � = increasing volatility of portfolio • Gambling entails either an increase or a decrease in portfolio volatility – a suf fi ciently risk averse manager decreases volatility – manager manipulates systematic risk rather than idiosyncratic – gambling intensi fi es towards year-end • Incentives to gamble are state-dependent. For example, the manager’s risk-taking incentives are year-end fl ow f H f L ❨ 0 relative return ✻ Highest Lowest
1. Motivation and Objective Page 4 Related Literature • Risk-taking of fund managers in response to fund fl ows: Chevalier and Ellison (1997) • Managerial incentives and portfolio choice: Brennan (1993), Carpenter (2000), Cuoco and Kaniel (2000), Berk and Green (2005), Hugonnier and Kaniel (2002), Gomez and Zapatero (2003), Ross (2003) • Empirical literature on risk-taking by mutual fund managers: Brown, Harlow, and Starks (1996), Busse (2001), Reed and Wu (2005)
2. Model Page 5 2. Model • Finite horizon, [0 , T ] , Black-Scholes economy • Assets: – Money market account with rate r – Stock follows dS t = µS t dt + σS t dw t • Fund manager: – evaluated relative to the index Y t (fraction β in stock) – receives fl ows at T at rate f T – chooses a trading strategy θ and terminal portfolio value W T θ, W T E [ u ( W T f T )] = E ( W T f T ) 1 − γ max 1 − γ subject to dW t = [ r + θ t ( µ − r )] W t dt + θ t σW t dw t
2. Model Page 6 Flow-Performance Relationship (Simplest) f T f H f L 0 relative return η How does one measure risk-taking incentives? • Conventional view: ∂V ( σ W ; R W − R Y t ) – sensitivity of the payoff’s value to volatility (vega): t t ∂σ W t • This paper: ∂V ( σ W ; R W − R Y t = ˆ t ) σ W σ W – optimal volatility ˆ θ t σ . That is, = 0 ˆ t . t t ⇒ ∂σ W t
2. Model Page 7 Manager’s Optimal Risk Exposure ˆ ˆ θ t θ t θ Y 1 4 0.8 3 θ N θ N 0.6 2 0.4 0.2 θ Y 1 -0.3 -0.2 -0.1 0.1 0.2 0.3 η R W − R Y η -0.2 t t -1.5 -1 -0.5 0.5 R W − R Y t t (a) Economies with θ N > θ Y (b) Economies with θ N < θ Y θ N : risk exposure in Merton’s problem, θ Y : risk exposure of the index
2. Model Page 8 An Alternative Flow-Performance Relationship (Collar-Type) f T f H f L 0 relative return η L η H Can also be reinterpreted as an 80/120 annual bonus plan.
2. Model Page 9 Manager’s Optimal Risk Exposure (Collar-Type) ˆ ˆ θ t θ t θ Y 1 4 0.8 3 θ N θ N 0.6 2 0.4 0.2 θ Y 1 -0.3 -0.2 -0.1 0.1 0.2 0.3 η H R W − R Y η H -0.2 t t -1.5 -1 -0.5 0.5 R W − R Y t t (a) Economies with θ N > θ Y (b) Economies with θ N < θ Y θ N : risk exposure in Merton’s problem, θ Y : risk exposure of the index
2. Model Page 10 Further Alternative Flow-Performance Speci fi cations • Linear-convex (Sirri and Tufano (1998)) year-end fl ow f L 0 relative return • Linear-linear (asymmetric fee structure) year-end fl ow f L 0 relative return
2. Model Page 11 Manager’s Optimal Risk Exposure: Dynamics ˆ ˆ θ t θ t 9 θ Y 1 T - t low 7 0.8 T - t high T - t med 0.6 θ N 5 0.4 ✸ θ N 3 T - t low T - t med 0.2 T - t high θ Y 1 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 η R W − R Y η -0.2 t t -1.5 -1 -0.5 0.5 R W − R Y t t (a) Economies with θ N > θ Y (b) Economies with θ N < θ Y • Manager engages in risk shifting well before the year-end • Risk shifting more pronounced as the year-end approaches
2. Model Page 12 Multiple Stocks Stock 1 Stock 2 ˆ ˆ θ 1 t θ 2 t 3 3 θ N θ N 1 2 2 2 1 1 θ Y θ Y 1 2 η η - 1 -0 . 5 0 . 5 R W − R Y - 1 -0 . 5 0 . 5 R W − R Y t t t t (a) Economies with θ N 1 > θ Y 1 and θ N 2 > θ Y 2 Stock 1 Stock 2 ˆ ˆ θ 1 t θ 2 t 4 0 . 6 θ Y 1 3 0 . 4 θ N 1 θ N 2 0 . 2 2 1 - 1 -0 . 5 η 0 . 5 θ Y R W − R Y 2 t t -0 .2 η - 1 -0 . 5 0 . 5 R W − R Y t t (b) Economies with θ N 1 < θ Y 1 and θ N 2 > θ Y 2
2. Model Page 13 Idiosyncratic versus Systematic Risk Economic setup: = µ 1 S 1 t dt + σ 11 S 1 t dw 1 t + σ 12 S 1 t dw 2 t dS 1 t = µ 2 S 2 t dt + σ 21 S 2 t dw 1 t + σ 22 S 2 t dw 2 t dS 2 t 0 1 0 1 0 @ µ 1 @ σ 11 A , A , µ = σ = Y = S 1 0 r σ 22 ˆ ˆ θ 1 t θ 2 t 4 0 . 6 3 0 . 4 θ N 1 2 0 . 2 θ N θ Y 2 1 1 - 1 -0 . 5 0 . 5 η R W − R Y t t -0 .2 - 1 -0 . 5 η 0 . 5 R W − R Y t t
3. Empirical Analysis Page 14 3. Empirical Analysis Existing Work • Brown, Harlow, and Starks (1996) – fi nd that underperforming managers increase volatility towards the year-end • Busse (2001) – shows that the above test fails on daily data • Chevalier and Ellison (1997) – look at σ ( R W − R Y ) towards the year-end; fi nd an increase – use monthly data • Reed and Wu (2005) – test the results of this paper on daily data – distinguish between tournaments- vs. benchmarking-induced incentives
3. Empirical Analysis Page 15 Data • Daily mutual fund returns from Will Goetzmann and Geert Rouwenhorst (International Center for Finance at Yale SOM). • Data range: 1970 through 1998 (comparable with Chevalier-Ellison, Sirri-Tufano). • Merged with CRSP to fi nd out mutual funds objective codes – left only actively managed US equity mutual funds in the aggressive growth, growth and income, and long-term growth categories. • Used the S&P 500 index as the benchmark.
3. Empirical Analysis Page 16 Tracking error and standard deviation tests Hypothesis 1: Tracking error variance is higher for underperforming managers. LHS: σ m ( R W i, t − R Y LHS: σ m ( R W t ) i, t ) Point Estimate t-Statistic Point Estimate t-Statistic OVER i,m × 10 3 -0.1819 -4.73 0.1058 2.38 Fund-year FE Yes Yes R 2 0.39 0.36 N 40721 40721 σ m ( R W i, t − R Y t ) – standard deviation of tracking error for month m ; Y is S&P 500 σ m ( R W i, t ) – standard deviation of fund returns for month m OVER i,m – relative performance indicator prior to month m
3. Empirical Analysis Page 17 Beta tests Hypothesis 2: Suf fi ciently risk-averse managers decrease their portfolio betas when underperforming the market. R W i, t − R F t = a + ( b F und − Y ear 1 F Y + b Month 1 M + b U UNDER i,w )( R Y t − R F t ) + ε i, t Dependent Variable: R W i, t − R F t Beta T below 1 Beta T − 1 below 1 (1) (2) (3) (4) UNDER i,w × ( R Y t − R F t ) -0.017 -0.020 -0.018 -0.021 (-6.61) (-7.72) (-6.99) (-8.22) Month fi xed effects No Yes No Yes Fund-year fi xed effects Yes Yes Yes Yes R 2 0.37 0.37 0.37 0.37 Number of observations 808642 816783
3. Empirical Analysis Page 18 Robustness • Tried several alternative de fi nitions of the OVER/UNDER indicator • Included lagged dependent variables to deal with autocorrelation • Clustered errors by month/day and fund objective code
4. Costs of Active Management to Investors Page 19 4. Costs of Active Management to Investors • De fi ne a measure of gain/loss, ˆ λ , in units of investor’s initial wealth: I ((1 + ˆ λ ) W 0 ) = ˆ V ( W 0 ) V – V I ( · ) is investor’s indirect utility under θ I – ˆ V ( · ) is investor’s indirect utility under delegation λ = (1 + λ N )(1 + λ Y ) • Decompose ˆ λ into two components: 1 + ˆ – λ N : gain/loss due to explicit incentives, solves V I ((1 + λ N ) W 0 ) = ˆ V ( W 0 ; f T = 1) – λ Y : gain/loss due to implicit incentives
4. Costs of Active Management to Investors Page 20 Costs of Active Management in Economies (a) ( θ N > θ Y ) Fixed parameter values: γ = 1 , γ I = 2 , f L = 0 . 8 , f H = 1 . 5 , f L + f H = 2 . 3 , β = 1 , η L = − 0 . 08 , η H = 0 . 08 , η L + η H = 0 , µ = 0 . 06 , r = 0 . 02 , σ = 0 . 29 , W 0 = 1 , T = 1 . Cost-bene fi t measures λ Y , λ N Effects of ˆ λ (%) Managerial risk 0.5 1.0 2.0 3.0 4.0 γ aversion -8.13, -4.19 -5.12, -0.47 -3.31, 0.00 -2.56, -0.05 -2.15, -0.11 -11.98 -5.61 -3.31 -2.61 -2.27 Implicit reward f H - f L 0.3 0.5 0.7 0.9 1.1 for outperformance -3.33, -0.47 -4.29, -0.47 -5.12, -0.47 -6.01, -0.47 -6.88, -0.47 -3.79 -4.74 -5.61 -6.46 -7.32 θ Y Risk exposure 0.50 0.75 1.00 1.25 1.50 of the benchmark -5.43, -0.47 -4.63, -0.47 -5.12, -0.47 -6.69, -0.47 -8.45, -0.47 -5.88 -5.08 -5.61 -7.13 -8.88 Flow threshold η H - η L 0.08 0.12 0.16 0.20 0.24 differential -4.33, -0.47 -4.70, -0.47 -5.12, -0.47 -5.67, -0.47 -6.21, -0.47 -4.78 -5.15 -5.61 -6.12 -6.65
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