## _wp04199 — Dedicated Investor Demand Space

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---

### Introduction and framing
- Contagion defined as transmission of a shock to another country’s asset when financial markets are not linked by economic fundamentals; comovement driven by global investors’ behavior.
- Focus: institutional structure of markets and incentives of fund managers (compensation mechanisms, mandates) as drivers of contagion even with full information and no asymmetric information.
- Two fund-manager types modeled:
  - Dedicated managers: compensated based on deviations from an emerging market index; not allowed to borrow cash or short any asset.
  - Opportunistic managers: compensated on absolute portfolio returns; allowed to short any asset and borrow cash.
- Key qualitative findings:
  - Dedicated investors rebalance toward index when asset volatility or their risk aversion increases.
  - Opportunistic managers decrease leverage when asset volatility or risk aversion increases.
  - A demand shock in one asset affects expected price of the other asset; relative contribution to contagion depends on market conditions and relative sizes of dedicated versus opportunistic managers.

### Data on investor composition (market context)
- Total sovereign emerging bond market investible by international investors estimated at some $225 billion.
- Emerging market corporate issuances excluded from that pool: about $100 billion.
- Of the investible pool:
  - Between 40-50 percent is thought to be held by dedicated investors (emerging market mutual funds and similar).
  - Hedge funds typically comprise between 10 and 20 percent of the investor base.
  - The remainder dominated by global investors who invest in the index or selectively “cross over” into emerging markets.
- Direct retail investors do not form a significant proportion of overall emerging market investor bases.

### Model setup and timing
- Assets: two risky emerging market assets (A and B), a mature market asset (Z), and cash (M).
- Traders: dedicated emerging market fund managers, global opportunistic fund managers, and noise (local) traders.
- Time horizon: three periods (Period 0: information update and portfolio decision; Period T: implementation and realization of random variable; Period T+1: terminal condition where terminal prices are fixed and known).
- Benchmark index return I_r defined with index weight α (α ∈ (0,1)); index weights taken exogenously.
- Local investor demands for A and B modeled as random variables:
  - D_LA ∼ (0, σ^2_A) and D_LB ∼ (0, σ^2_B).
- Market clearing: total demand (dedicated + opportunistic + local) equals fixed supply for each asset.

### Dedicated fund managers — setup and behavioral properties
- Portfolio weights: λ proportion to asset A, τ to asset B, (1-λ-τ) to cash.
- Compensation: fixed proportion k of excess return relative to benchmark; utility increases in expected income and decreases in income variability (constant absolute risk aversion coefficient a).
- Key analytical results (Propositions 1–3) and behavioral implications:
  - Proposition 1 (optimal weights):
    - Index weight α is a key determinant: greater α → greater allocation to asset A in optimal behavior.
    - For parameter regions specified in the paper, closed-form and piecewise expressions for λ* and τ* are provided; corner solutions with positive cash holdings may obtain when 2(Err_A - r_M) a σ_A < 0 and/or 2(Err_B - r_M) a σ_B < 0.
  - Proposition 2 (when an asset underperforms cash):
    - a) If Err_A > r_M and Err_B < r_M (or vice versa), manager overweights the outperforming asset; if Err_A < r_M or Err_B < r_M or both, manager will be underweight that asset but may still hold positive amounts.
    - b) As α rises, both overweight and underweight managers increase exposure to the asset but underweight managers remain relatively underweight.
    - c) As risk aversion a rises, demand for an overweight asset falls; underweight positions are reduced (results in “hugging of the index”).
    - d) As σ_A^2 or σ_B^2 rises, demand for an overweight asset falls; underweight positions are reduced (greater “hugging of the index”).
    - e) An increase in a, σ_A^2 or σ_B^2 may reduce demand for cash and increase index hugging; dedicated managers may hold positive quantities of an asset even when it underperforms cash because of index-relative compensation risk.
    - Exact conditions and partial-derivative expressions for these effects are provided in equations (6.12)–(6.15) in the source.
  - Proposition 3 (case λ + τ = 1, both emerging assets outperform cash):
    - a) Manager overweights the asset with higher expected return and underweights the lower-return asset.
    - b) Increase in a leads to allocations closer to the index (increase exposure to underweight asset, decrease exposure to overweight asset).
    - c) Increase in σ_A^2 or σ_B^2 reduces size of overweight/underweight positions → closer to benchmark index.
  - General behavioral implication: increases in volatility or risk aversion induce dedicated managers to rebalance closer to the index; volatility shocks to one asset spill over into demand for other assets via index-hugging.

### Opportunistic (global) managers — setup and behavioral properties
- Portfolio weights: φ to asset A, δ to asset B, (1-φ-δ) to asset Z (mature market asset).
- Objective: maximize expected portfolio value net of risk (absolute return focus), allowed to short and borrow.
- Optimal weights (closed form, Proposition 4):
  - Optimal allocations for φ and δ expressed in closed-form in equations (6.19)–(6.21), depending on Err_A, Err_B, Err_Z, σ_A^2, σ_B^2, σ_Z^2, and U = σ_A^2 σ_B^2 + σ_B^2 σ_Z^2 + σ_A^2 σ_Z^2.
- Behavioral results (Proposition 5):
  - a) Increase in expected return of an emerging market asset raises opportunistic manager’s holdings of that asset at expense of both the other emerging asset and the mature market asset.
  - b) Relative reallocation depends on relative volatilities: if an emerging asset is more volatile than the mature asset, the reduction in the mature asset is greater, and vice versa.
  - c) Manager will short asset A and go long other assets if Err_B > Err_A and Err_M > Err_A and inequality (6.22) holds; plugging (6.22) into (6.19) yields φ < 0.
  - d) If Err_A > Err_B and Err_A > Err_M, manager goes long A.
  - e) If Err_A > Err_B and Err_A < Err_M, manager will short A if a specified volatility-weighted inequality holds.
  - f) As a increases, opportunistic manager reduces exposure to highest-yielding asset and increases exposure to lowest-yielding asset; ∂φ/∂(a) given by equation (6.23).
- Implication: opportunistic managers can produce leveraged exposures, long-short relative value strategies, and respond to changes in mature market returns (cost of leverage), affecting demand for emerging assets.

### Equilibrium and contagion mechanism
- Market clearing with n dedicated managers and q opportunistic managers, plus local demand:
  - n D_DA + q D_OA + D_LA = S_A; analogous for B.
- Two equilibrium cases analyzed:
  - A) Dedicated (positive cash holdings) and opportunistic managers:
    - Closed-form equilibrium expressions for Err_A() and Err_B() derived (equations (6.3)–(6.6)).
    - Proposition 6: Changes in expectations of local investor demand for one asset affect expected returns (and prices) of both assets → contagion.
    - Observations:
      - If q = 0 (no opportunistic managers), and at least one emerging market asset underperforms cash, rebalancing by dedicated managers does not lead to contagion from one asset to the other.
      - Rebalancing by dedicated managers affects price of the affected asset more than opportunistic managers.
      - Increase in expected return of mature market asset reduces equilibrium expected price for both emerging assets (outflow to mature market); if q = 0 this effect is absent because dedicated managers cannot invest in mature market asset.
  - B) Dedicated managers (zero cash holdings) and opportunistic managers:
    - Alternative equilibrium expressions derived (equations (6.7)–(6.8)).
    - Proposition 7: Changes in expected local demand for an emerging market asset affect returns/prices of both assets → contagion.
    - Observations:
      - Both dedicated and opportunistic managers contribute to contagion; equilibrium coefficients include both n and q.
      - Dedicated managers’ contribution to contagion is generally greater than opportunistic managers’.
      - The impact of changes in local demand on its own asset’s price is affected more by opportunistic managers.
- General equilibrium implication: contagion (comovement of asset prices) can arise purely from portfolio rebalancing by institutional investor classes, even absent macroeconomic linkages or asymmetric information. Magnitudes depend on relative sizes n and q, index weights α, risk aversion a, and volatilities σ^2.
- Propositions 6 and 7 provide explicit derivative expressions showing that a decrease in expected local demand for one asset lowers its expected price and, under stated algebraic conditions, lowers expected prices of the other asset (cross-partials shown to be negative in source expressions).

### Conclusions and policy implications
- Aggregation of fund managers’ optimal micro-level behavior can produce market equilibria that deviate from efficient-market predictions even without asymmetric information or regulatory distortions.
- Key conclusions:
  - Different investment-manager objectives generate differential effects on price dynamics.
  - Managerial incentive structures can create systematic deviations of prices from long-term fundamentals, potentially persistent.
  - Presence of leveraged investors (able to go long and short) significantly impacts market valuations and price dynamics; cost of leverage affects exposures (higher cost reduces leverage → emerging market selloffs).
  - Noise trading in one country can spill over to unrelated countries through international investors’ optimal behavior → “pure contagion.”
  - No single manager type always dominates contagion; both dedicated and opportunistic managers may contribute.
- Policy responses that improve market efficiency, transparency, and coping with volatility will mitigate but may not eliminate contagion.
- Suggested areas for future research:
  - Optimal incentive contracts for different classes of fund managers.
  - Optimal construction of market indices as benchmarks for managerial compensation.

*Source: IMF Working Paper section "2. Dedicated Investor Demand Space" (_wp04199).*

### References..............................................................................................................

### _wp04199 - References

### References
- References................................................................................................................................25

### Appendix: Proofs of Propositions
- Appendix: Proofs of Propositions............................................................................................26

### Figures
- 1. The Timeline ..........................................................................................................................8

*Source: _wp04199 - References..............................................................................................................*

### 2. Dedicated Investor Demand Space .....................................................................................

### 2. Dedicated Investor Demand Space

### Introduction and framing
- Contagion defined as transmission of a shock to another country’s asset when financial markets are not linked by economic fundamentals; comovement driven by global investors’ behavior.
- Focus: institutional structure of markets and incentives of fund managers (compensation mechanisms, mandates) as drivers of contagion even with full information and no asymmetric information.
- Two fund-manager types modeled:
  - Dedicated managers: compensated based on deviations from an emerging market index; not allowed to borrow cash or short any asset.
  - Opportunistic managers: compensated on absolute portfolio returns; allowed to short any asset and borrow cash.
- Key qualitative findings previewed:
  - Dedicated investors rebalance toward index when asset volatility or their risk aversion increases.
  - Opportunistic managers decrease leverage when asset volatility or risk aversion increases.
  - Demand shock in one asset affects expected price of the other asset; relative contribution to contagion depends on market conditions and relative sizes of dedicated versus opportunistic managers.

### Data on investor composition (market context)
- Total sovereign emerging bond market investible by international investors estimated at some $225 billion.
- Emerging market corporate issuances excluded from that pool: about $100 billion.
- Of the investible pool:
  - Between 40-50 percent is thought to be held by dedicated investors (emerging market mutual funds and similar).
  - Hedge funds typically comprise between 10 and 20 percent of the investor base.
  - The remainder dominated by global investors who invest in the index or selectively “cross over” into emerging markets.
- Direct retail investors do not form a significant proportion of overall emerging market investor bases.

### Model setup and timing
- Assets: two risky emerging market assets (A and B), a mature market asset (Z), and cash (M).
- Traders: dedicated emerging market fund managers, global opportunistic fund managers, and noise (local) traders.
- Time horizon: three periods (Period 0: information update and portfolio decision; Period T: implementation and realization of random variable; Period T+1: terminal condition where terminal prices are fixed and known).
- Benchmark index return I_r defined with index weight α (α ∈ (0,1)); index weights taken exogenously.

### Local investors as uncertainty
- Local investor demands for A and B modeled as random variables:
  - D_LA ∼ (0, σ^2_A) and D_LB ∼ (0, σ^2_B) (For simplicity both assets share same distribution properties though not necessarily same parameters).
- Market clearing conditions: total demand (dedicated + opportunistic + local) equals fixed supply for each asset.

### Dedicated fund managers: setup and behavioral properties
- Portfolio weights: λ proportion to asset A, τ to asset B, (1-λ-τ) to cash.
- Compensation: fixed proportion k of excess return relative to benchmark; utility increases in expected income and decreases in income variability (constant absolute risk aversion coefficient a).
- Key analytic results (Propositions 1–3):
  - Proposition 1 (optimal weights):
    - For parameter region where 2(Err_A - r_M) a σ_A ≥ 0 and 2(Err_B - r_M) a σ_B ≥ 0, optimal weights:
      - λ* = [2(Err_A - Err_B) + a σ_B^2] / (2 a (σ_A^2 + σ_B^2 + 2 σ_A σ_B ?)) — (paper provides explicit expressions; see paper for full functional forms).
    - For regions where 2(Err_A - r_M) a σ_A < 0 and/or 2(Err_B - r_M) a σ_B < 0, corner solutions with positive cash holdings may obtain; explicit piecewise expressions given in the paper.
    - Index weight α is a key determinant: greater α → greater allocation to asset A in optimal behavior; deviation from index weight independent of α.
  - Proposition 2 (behavior when an asset underperforms cash):
    - a) If Err_A > r_M and Err_B < r_M (or vice versa), manager overweight the outperforming asset; if Err_A < r_M or Err_B < r_M or both, manager will be underweight that asset but may still hold positive amounts.
    - b) As α rises, an overweight manager increases exposure and an underweight manager also increases exposure but maintains relative underweight.
    - c) As risk aversion a rises, demand for an overweight asset falls; underweight positions are reduced (results in “hugging of the index”).
    - d) As σ_A^2 or σ_B^2 rises, demand for an overweight asset falls; underweight positions are reduced (greater “hugging of the index”).
    - e) An increase in a, σ_A^2 or σ_B^2 may reduce demand for cash and increase index hugging.
    - Dedicated managers may hold positive quantities of an asset even when it underperforms cash because of index-relative compensation risk.
  - Proposition 3 (case λ + τ = 1, both emerging assets outperform cash):
    - a) Manager overweight the asset with higher expected return and underweight the lower-return asset.
    - b) Increase in a leads to allocations closer to the index (increase exposure to underweight asset, decrease exposure to overweight asset).
    - c) Increase in σ_A^2 or σ_B^2 reduces size of overweight/underweight positions → closer to benchmark index.
  - General behavioral implication: increases in volatility or risk aversion induce dedicated managers to rebalance closer to the index; volatility shocks to one asset spill over into demand for other assets via index-hugging.

### Opportunistic (global) managers: setup and behavioral properties
- Portfolio weights: φ to asset A, δ to asset B, (1-φ-δ) to asset Z (mature market asset).
- Objective: maximize expected portfolio value net of risk (absolute return focus), allowed to short and borrow.
- Optimal weights (Proposition 4): closed-form expressions depend on Err_A, Err_B, Err_Z, volatilities σ_A^2, σ_B^2, σ_Z^2, and aggregated variance term U = σ_A^2 σ_B^2 + σ_B^2 σ_Z^2 + σ_A^2 σ_Z^2.
- Behavioral results (Proposition 5):
  - a) Increase in expected return of an emerging market asset raises opportunistic manager’s holdings of that asset at expense of both the other emerging asset and the mature market asset.
  - b) Relative reallocation depends on relative volatilities: if emerging asset is more volatile than mature asset, reduction in mature asset is greater, and vice versa.
  - c) Manager will short asset A and go long other assets (relative value / long-short strategy) if Err_B > Err_A and Err_M > Err_A and an inequality condition involving volatilities and a holds (explicit inequality provided in paper).
  - d) If Err_A > Err_B and Err_A > Err_M, manager goes long A.
  - e) If Err_A > Err_B and Err_A < Err_M, manager will short A if a specified volatility-weighted inequality holds.
  - f) As a increases, opportunistic manager reduces exposure to highest-yielding asset and increases exposure to lowest-yielding asset.
- Implication: opportunistic managers can produce leveraged exposures, long-short relative value strategies, and respond to changes in mature market returns (cost of leverage), affecting demand for emerging assets.

### Equilibrium and contagion mechanism
- Market clearing with n dedicated managers and q opportunistic managers, plus local demand:
  - Market clearing conditions: n D_DA + q D_OA + D_LA = S_A; analogous for B.
- Two equilibrium cases analyzed:

  A) Dedicated (positive cash holdings) and opportunistic managers:
  - Closed-form equilibrium expressions for Err_A() and Err_B() derived (equations (6.3)–(6.6) produce explicit formulas).
  - Proposition 6: Changes in expectations of local investor demand for one asset affect expected returns (and prices) of both assets → contagion from one country to another.
  - Observations:
    - If q = 0 (no opportunistic managers), and at least one emerging market asset underperforms cash, rebalancing by dedicated managers does not lead to contagion from one asset to the other.
    - Rebalancing by dedicated managers affects price of the asset more than opportunistic managers.
    - Increase in expected return of mature market asset reduces equilibrium expected price for both emerging assets (outflow to mature market); if q = 0 this effect is absent because dedicated managers cannot invest in mature market asset.

  B) Dedicated managers (zero cash holdings) and opportunistic managers:
  - Alternative equilibrium expressions derived (equations (6.7)–(6.8)).
  - Proposition 7: Changes in expected local demand for an emerging market asset affect returns/prices of both assets → contagion.
  - Observations:
    - Both dedicated and opportunistic managers contribute to contagion; coefficients in equilibrium expressions include both n and q.
    - Dedicated managers’ contribution to contagion is generally greater than opportunistic managers’.
    - The impact of changes in local demand on its own asset’s price is affected more by opportunistic managers.

- General equilibrium implication: contagion (comovement of asset prices) can arise purely from portfolio rebalancing by institutional investor classes, even absent macroeconomic linkages or asymmetric information. Magnitudes depend on relative sizes n and q, index weights α, risk aversion a, and volatilities σ^2.

### Conclusions and policy implications
- Aggregation of fund managers’ optimal micro-level behavior can produce market equilibria that deviate from efficient-market predictions even without asymmetric information or regulatory distortions.
- Key conclusions:
  - Different investment-manager objectives generate differential effects on price dynamics.
  - Managerial incentive structures can create systematic deviations of prices from long-term fundamentals, potentially persistent.
  - Presence of leveraged investors (able to go long and short) significantly impacts market valuations and price dynamics; cost of leverage affects exposures (higher cost reduces leverage → emerging market selloffs).
  - Noise trading in one country can spill over to unrelated countries through international investors’ optimal behavior → “pure contagion.”
  - No single manager type always dominates contagion; both dedicated and opportunistic managers may contribute.
- Policy responses that improve market efficiency, transparency, and coping with volatility will mitigate but may not eliminate contagion.
- Suggested areas for future research:
  - Optimal incentive contracts for different classes of fund managers.
  - Optimal construction of market indices as benchmarks for managerial compensation.

*Source: IMF Working Paper section "2. Dedicated Investor Demand Space" (paper text provided).*

### REFERENCES

### _wp04199 - REFERENCES

### References
- Banerjee, Abhijit V. (1992), “A Simple Model of Herd Behavior,” Quarterly Journal of Economics 57, 797-817.
- Calvo, Guillermo A. and Carmen M. Reinhart (1996), “Capital Flows to Latin America: Is There Evidence of Contagion Effects,” in Calvo, G.A., Goldstein, M., Hockretter, E. (eds), Private Capital Flows to Emerging Markets, Institute for International Economics, Washington DC.
- Calvo, Guillermo A. and Enrique G. Mendoza (2000), “Rational Contagion and the Globalization of Securities Markets,” Journal of International Economics 51, 79-113.
- Kaminsky, Graciela L. and Carmen M. Reinhart (2000), “On Crises, Contagion, and Confusion,” Journal of International Economics 51, 145-168.
- Kaminsky, Graciela L. and Sergio L. Schmukler (1999), “What Triggers Market Jitters?: A Chronicle of the Asian Crisis, Journal of International Money and Finance 18, 537-560.
- Kodres, Laura E. and Matthew Pritsker (2002), “A Rational Expections Model of Financial Contagion,” Journal of Finance 62, 769-799.
- Kyle, Albert S. and Wei Xiong (2001), “Contagion as a Wealth Effect,” Journal of Finance 56, 1401-1440.
- Scharfstein, David S. and Jeremy C. Stein (1990), “Herd Behavior and Investment,” American Economic Review 80, 465-479
- Schinasi, Garry J. and R. Todd Smith (1999), “Portfolio Diversification, Leverage, and Financial Contagion,” International Monetary Fund Working Paper, WP/99/136.

### Appendix: Proofs of Propositions — Key results, conditions, and expressions

- Proposition 1 — Dedicated investor optimization (Lagrangian and first-order conditions)
  - Lagrangian (as printed): ()() 2222 ()()(1)()(1 ) [()(1)](1). 2 ABM AB LErErr a λατ αλτ λασ τ ασ φ λτ =−+−++−− −− +−+ +−−
  - First-order conditions:
    - Assuming 0λ> and differentiating w.r.t λ: 2 () AM A Err a φα λα σ −− =+.
    - Assuming 0τ> and differentiating w.r.t τ: 2 () (1) BM B Err a φα τα σ −− =+−.
  - Cash holdings expression: 22 ()() 1 MAMB AB rErrEr aa φφ λτσσ −+−+ −−=+.
  - Complementary slackness: (1)0 and 0.φλτφ−−=≥
  - Binding-case multiplier expression: φ = 22 22 (( ))(( )) AMBM BA AB ErrErrσσ φ σσ −+− = + , positive whenever 22 ()() 0 AMAM AB ErrErr aa σσ −− +>.
  - Optimal portfolio weights (binding case):
    - () *22 () () AB AB ErEr a λα σσ − = + + ,
    - *22 () () (1) () BA AB ErEr a τα σσ − = +− + .
  - Non-binding case condition: 22 ()() 0 AMBM AB ErrErr aa σσ −− +<.  (Equation (6.9))
  - Conditions for positive multipliers: 2 () 0 AM A Err a α σ − +>, (6.10) and 2 () (1)0 BM B Err a α σ − +−>. (6.11)
  - Optimal portfolio weights when constraint non-binding:
    - **2 () AM A Err a λα σ − =+,
    - **2 () (1) BM B Err a τα σ − =+−,
    - Cash holdings cases summarized by piecewise expressions (see original for full conditional matrix).

  - Objective-value comparisons:
    - Value at (0,0): ()( ) () () ( ) () 22 22 1 0, 0()1()1 2 MAB AB VrErEraααασασ =− +−−+−.
    - Value when 0λ> and 0β>: ()()() () () () ( ) () 22 22 ,()()()1() 1 1. 2 AMBM MAB AB VErrErrrErEr a λτ λατα αλασ τ ασ =−+−+−+− −−+−+.

  - Conditions ensuring V(λ,τ)>V(0,0): complex inequality reproduced precisely in source (see original block). Substituting optimal values yields verification conditions such as 2 () 0 AM A Err a α σ − +> and 2 () (1)0 BM B Err a α σ − +−>.

- Proposition 2 — When at least one emerging market return is negative
  - Optimal portfolio weights (equations labeled 6.12–6.14):
    - *2 () AM A Err a λα σ − =+,     (6.12)
    - *2 () (1) BM B Err a τα σ − =+−, (6.13)
    - *22 ()() (1) MAMB AB rEr rEr aa λτσσ −− −− =+. (6.14)
  - Behavioral characteristics (summarized):
    - a. Signs of ()AM Err and ()BM Err determine overweight/underweight positions: λα> or (1)(1)λα−>− when positive; λα< or (1)(1)λα−<− when negative.
    - Conditions for holding positive quantities: e.g., positive holdings of A if 2 () AM A Err a α σ − −<.
    - b. Increase in benchmark weight α: if α increases then λ increases (equation (6.12)); overweight managers increase exposure, underweight managers increase exposure but remain underweight; concomitant decrease in holdings of asset B (equation (6.13)).
    - c. Increase in risk aversion (a): reduces magnitude of deviations from index; managers rebalance toward index.
    - d. Increase in variances 2 A σ or 2 B σ: reduces magnitude of deviations; managers rebalance toward index.
    - e. Increase in (a), 2 A σ or 2 B σ may reduce demand for cash (greater hugging of the index). Partial derivative of (1)λτ−− w.r.t (a) given exactly as equation (6.15):
      - 2222 (1)(())(() ) 0 ()() MA MB AB rErrEr aaa λτ σσ ∂−− − − − =+< ∂ when ()AM Err< and ()BM Err<.
      - Additional conditions for negativity reproduced as in source.
    - Partial derivatives w.r.t variances:
      - ∂(1)λτ−−/∂2 A σ = 222 (1)(() ) 0 () MA AA rEr a λτ σσ ∂−− − − =< ∂ if ()AM Err<.
      - ∂(1)λτ−−/∂2 B σ = 222 (1)(() ) 0 () MB BB rEr a λτ σσ ∂−− − − =< ∂ if ()BM Err<.

- Proposition 3 — Sum of risk-adjusted excess returns positive
  - Optimal weights (equations (6.16) and (6.17)):
    - () *22 () () AB AB ErEr a λα σσ − = + + ,    (6.16)
    - *22 () () (1) () BA AB ErEr a τα σσ − =+− + .  (6.17)
  - Implications:
    - a. Manager overweights the asset with higher expected return and underweights the other: if () AB ErEr>, first term in (6.16) positive and first term in (6.17) negative.
    - b. Increase in (a): allocations move closer to index (magnitude away from index decreases).
    - c. Increase in 2 A σ or 2 B σ: reduces size of overweight/underweight positions; manager moves closer to benchmark.

- Proposition 4 — Opportunistic manager optimization (setup)
  - Optimization problem (6.18):
    - 22222 2 , max()() (1) ()(1) 2 ABZ ABZ a ErErEr φδ φδφδφσδσδφσ  ++−− − ++−−  . (6.18)
  - First-order conditions:
    - 22 () () (1) AZ AZ ErEr a φσ φδσ − = ++−
    - 22 () () (1) BZ BZ ErEr a δσ φδσ − = ++−
  - Closed-form optimal allocations (6.19–6.21):
    - 22 *2 () ()() () ABAZ ZB Z ErErErEr Ua Ua σσ φσ   −− =++     , (6.19)
    - 22 *2 () ()() () BABZ ZA Z ErErErEr Ua Ua σσ δσ   −− =++     , (6.20)
    - 22 *22 () ()() () (1)1 AZBZ BA ZZ ErErErEr UaUa σσ φδσσ  −− −− =−+−+   , (6.21)
    - where 222222 ABAZBZ Uσσσσσσ=++.

- Proposition 5 — Opportunistic manager comparative statics and positioning
  - a. Increases in (A Er) or (B Er) increase φ or δ respectively:
    - ∂φ/∂(A Er) = 22 0 () ZB A EraU σσ + => ∂> .
    - ∂δ/∂(B Er) = 22 0 () ZB B EraU σσ + => ∂> .
  - Increased allocation to one asset reduces allocations to the other emerging market asset and to the mature market asset:
    - ∂δ/∂(A Er) = 2 0 () Z A EraU σ δ −∂ = < ∂
    - ∂(1)φδ−−/∂(A Er) = 2 (1) 0 () B A EraU σ φδ −− ∂−− = < ∂
  - b. Reallocation proportions depend on relative volatilities 2 A σ, 2 B σ, 2 Z σ; example given: of total increase in allocations to A, 2 Z aU σ times change comes at expense of B, while 2 B aU σ times change comes from asset Z. If 22 BZ σσ >, more reallocation from Z.
  - c. Conditions for shorting asset A when () BA ErEr> and () ( ) MA ErEr>:
    - Inequality (6.22): 22 22 () ()() () ABAZ ZBBZ ErErErEr aa σσσσ  −− −− >   . (6.22)
    - Plugging (6.22) into (6.19) yields φ < 0, confirming short position.
  - d. Conditions for going long asset A under various relations between AB ErEr and MA ErEr provided (exact inequalities reproduced in source).
  - e. Effect of (a) on opportunistic positions:
    - ∂φ/∂(a) given by equation (6.23):
      - 22 22 (()())(()()) ()() ABAZ ZB ErErErEr aaUaU σσ φ  −−∂ = −−  ∂  . (6.23)
    - If () AB ErEr> and () AZ ErEr>, then ∂φ/∂(a) < 0 (manager reduces holdings of A as (a) rises). Conversely, sign reverses under alternative orderings.

- Proposition 6 — Contagion via local investor demand expectations
  - If dedicated and opportunistic managers and local investors demand A and B, changes in expected local demand affect returns/prices of both assets, producing contagion.
  - If , () 0 () A LA Er D ∂ < ∂ and , () 0 () B LB Er D ∂ < ∂ , a decrease in expected local demand lowers expected price of that asset. Confirmed by two derivative expressions reproduced exactly in source (long algebraic expressions with U, σ, q, n, a terms).
  - Contagion occurs across markets if , () 0 () A LB Er D ∂ < ∂ and , () 0 () B LA Er D ∂ < ∂ , confirmed by an expression equated to < 0 in the source (complex fraction with aU, σ, q, n terms).

- Proposition 7 — Cross-effects of local demand shocks on other asset returns
  - Differentiation of expected return of one asset w.r.t change in local investor demand of the other yields:
    - , , 2 22 2 2 2222 222222 ()() ()() () 0 ()() ()()() AB LBLA Z AB Z ZAZB ABABAB ErEr DD qn aaU qnqnqn aaU aaUa aU σ σσ σ σσσ σσσσσ ∂∂ = ∂∂  −+  +  = < (exact expression as in source).
  - Result: cross-partial is negative under stated algebraic conditions, confirming contagion transmission through changes in local investor demand expectations.

*Source: _wp04199 - REFERENCES (appendix and reference list as provided in the source PDF)*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp04199.pdf_