## 3.1 Set up

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### Model setup and timing
- Three periods: t0, t1, t2.
- Single bank lends L to a firm at t0 for a project with cash flows L(1 + P) at t2; firm promises to pay L(1 + μ) at t2. μ is the (fixed) interest rate; L and μ determined in equilibrium.
- Bank finances with two short-term loans from investors:
  - borrows L at t0 and must pay L(1 + S0) at t1 with normalization S0 = 0.
  - borrows L at t1 and promises to pay L(1 + S1) at t2.
- Random variables and covariances:
  - P ∼ N(μP, σP^2).
  - S1 ∼ N(μS, σS^2).
  - R1 ∼ N(μR, σR^2) with μR = 0.
  - Cov(S1, R1) = ρ > 0; Cov(P, R1) = π. Baseline case often π = 0.
- Assumption: μP > μS so the project is worth funding in expectation.
- Timeline summary: investors → bank at t0; bank → investors at t1; bank borrows again at t1; final payments at t2 include project returns and loan repayments.
- Example numeric illustration retained verbatim: "For example, the bank might lend $10 MM to the firm at an interest rate of 500 basis points, or 5%."

### Key frictions (Section 3.2)
- Three frictions generating funding risk and motivating hedging:
  1. Short-term funding must be used (maturity transformation): banks do not fully match maturities of assets and liabilities.
  2. S1 is not contractible: contracts referring directly to an individual bank’s funding cost are infeasible; reference rates serve as standardized, publicly available, contractible proxies.
  3. S1 is not known until t1, and financial frictions prevent the bank from being risk neutral: effective risk aversion arises because risky payoffs affect future investable funds when investment opportunities are concave and external financing is costly.
- Mechanism generating effective risk aversion:
  - If investment opportunities F(I) are concave and internal funds w are stochastic, the profit function P(w) inherits concavity ⇒ effective risk aversion.
  - Costly external finance (convex cost C(e)) preserves concavity: Pww ∝ −FII Cee (Equation 1).
  - Townsend (1979) costly state verification framework is used to generate concavity of the production function.

### Reduced-form CARA representation
- Optimal allocation for adding θ units of a Normally distributed payoff X:
  - θ* = μX / (A σX^2) with A = −EPww / EPw (Equation 2).
- Bank and firm directly modeled as maximizing CARA utility with exogenous coefficients AB and AF; first-order condition for CARA gives μX / (A σX^2) (Equation 3).
- Aggregate behavior of N CARA agents is equivalent to single agent with risk aversion ĀN (harmonic mean).

### Basic contract forms and equilibrium with fixed rates (start of Section 4)
- In absence of reference rate: only fixed-rate loan available. Per unit borrowed:
  - Firm payoff: P − μ.
  - Bank payoff: μ − S1.
- Equilibrium μ* equates demand and supply (Equation 4):
  - D(μ*) = (μP − μ*) / (AF σP^2) = μ* − μS / (AB σS^2) = L(μ*).
- Define total utility cost of risk:
  - Φ = AF σP^2 + AB σS^2.

### Reference rates: two principal uses (overview)
- Two ways to reallocate funding risk when reference rate R1 is available:
  1. Floating interest rates: loan rate R1 + ν (e.g., LIBOR + 300 bps), ν is fixed premium. No direct cost to writing floating-rate contract assumed.
  2. Interest-rate derivatives (swaps): allow hedging and transfer of risk to broader market at explicit cost.
- Both are contracts contingent on realized R1; facilitating these contracts is the basic purpose of reference rates.
- If used independently, derivatives are welfare-superior to floating rates (Section 4 conclusion).

### Floating-rate loans (Section 4.1)
- Floating loan interest rate: R1 + ν.
- Variance effects when reference rate correlated with funding costs:
  - Bank variance reduces from Var(S1) to Var(R1 − S1) = σS^2 + σR^2 − 2ρ.
  - Firm variance increases from Var(P) to Var(P − R1) = σP^2 + σR^2 − 2π.
- New equilibrium ν* satisfies (Equation 5):
  - D′(ν*) = (μP − ν*) / (AF (σP^2 + σR^2 − 2π)) = (ν* − μS) / (AB (σS^2 + σR^2 − 2ρ)) = L′(ν*).
- Define total utility cost with floating rates:
  - Φ(R1) = AF (σP^2 + σR^2 − 2π) + AB (σS^2 + σR^2 − 2ρ).
- Welfare implications and Proposition 1 (exact statements preserved):
  - Floating rates transfer funding risk from the bank to the firm. If ρ > σR^2 / 2 and π < σR^2 / 2:
    - Interest rates fall:
      μ* > ν* ⇐⇒ (2ρ − σR^2) / σS^2 + σR^2 − 2π / σP^2 > 0 (Equation 6)
    - Welfare increases only if:
      Ω(R1) > Ω ⇐⇒ AB (2ρ − σR^2) > AF (σR^2 − 2π) (Equation 7)

### Interest-rate derivatives (swaps) (Section 4.2)
- Swap contract at t0: buyer pays fixed rate λ in return for R1 at t2 (per dollar notional). Hedge ratio α is notional per unit of lending. Unit cost of hedging λ initially taken exogenous.
- Bank’s joint choice of credit supply and hedging maximizes EU[W0 + θ(κ − S1 + α R1)] (Equation 8), κ is fixed rate charged when swaps available.
- First-order conditions (Equations 9 and 10):
  - L′′(κ) = θ* = (κ − μS − λ α*) / (AB (σS^2 − ρ^2 σR^2))
  - α*(κ, λ) = ρ / σR^2 ︸︷︷︸ α* − λ / (AB θ* σR^2)
- Zero NPV swap case (λ = 0): optimal hedging ratio α* = ρ / σR^2. Residual funding risk after optimal hedging:
  - Var(−S1 + α* R1) = σS^2 − ρ^2 σR^2.
- If swap is costly (λ > 0) bank reduces α* but demand for loans remains same as when optimal hedge purchased (Appendix C.1): bank keeps some risk in exchange for expected return λ.
- Equilibrium κ determined from L′′(κ) = D(κ).
- Welfare and cost-of-hedging considerations:
  - Total cost of hedging λα* thought of as fraction of initial surplus μP − μS.
  - If cost not too high, reduction in bank funding risk lowers interest rates and increases welfare relative to fixed rates.
  - Compared to floating rates, two opposing effects:
    - Lower residual risk with derivatives (unless α* = 1) pushes rates lower.
    - Firms no longer bear risk (risk shifted to broader market) pushes rates higher.
  - Both effects imply welfare is higher with optimal hedging than with floating rates.
- Proposition 2 (exact statements preserved):
  - If interest rate swaps are zero NPV transactions (λ = 0)
    - The effect on interest rates relative to floating rates is ambiguous:
      ν* > κ* ⇐⇒ σR^2 (α* − 1)^2 σS^2 − ρ^2 σR^2 > σR^2 − 2π / σP^2 (Equation 11)
    - Welfare is improved relative to floating rates:
      Ω(R1, λ = 0) > Ω(R1) ⇐⇒ AB σR^2 (α* − 1)^2 + AF (σR^2 − 2π) > 0 (Equation 12)
  - If interest rate swaps are costly, welfare is still increased as long as
    λα* ≤ C̄ = (μP − μS) (1 − √(Φ(R1) / Φ(R1))) (Equation 13)

### Interpretation and extensions (Section 5 summary)
- Swaps are a more effective method than floating rates for banks to reduce funding risk, provided swaps are not too costly.
- Empirical puzzle: floating rates remain widespread and firms often hedge the floating exposure using swaps themselves.
- Sections foreshadowed:
  - Section 5.1 connects cost of hedging to aggregate risk tolerance for funding risk.
  - Section 5.2 shows combined use of floating rates and firm swap positions broadens market participants bearing funding risk, lowering hedging costs.
  - Section 5.3 cautions analysis requires sufficiently competitive derivatives markets.
  - Section 5.4 notes that manipulation increasing reference rate volatility reduces welfare when swaps are not too costly.

---

### Cost of hedging (Section 5.1)
- Equilibrium pricing and why swaps can be costly:
  - If both parties to a swap are risk averse, hedging should be costly (Demsetz (1969) point).
  - Equilibrium cost of hedging λ∗ satisfies:
    - L′′(κ) α∗ − λ∗ A_B σ^2_R = λ∗ A_D σ^2_R  (Equation 14)
  - Combined market risk tolerance of swap dealers and the bank:
    - T_D,B = 1/A_D + 1/A_B  (Equation 15)
  - Rearranged:
    - λ∗ α∗ = 1/T_D,B L′′(κ) ρ^2 σ^2_R  (Equation 16)
- Intuition: cost of hedging depends on market risk tolerance (T_D,B), size of credit markets (L′′(κ)), and how much swaps reduce risk (ρ^2 σ^2_R).

### Key statistics and exact expressions (Section 5.1)
- Equation 14: L′′(κ) α∗ − λ∗ A_B σ^2_R = λ∗ A_D σ^2_R
- Equation 15: T_D,B = 1/A_D + 1/A_B
- Equation 16: λ∗ α∗ = 1/T_D,B L′′(κ) ρ^2 σ^2_R

---

### Complementarity between floating rates and swaps (Section 5.2)
- Firm optimal hedging ratio:
  - β∗(κ,λ) = −π/σ^2_R − λ/A_F D(κ) σ^2_R  (Equation 17)
  - π enters with opposite sign because the firm’s problem is mirror image of the bank’s.
- With floating-rate obligations, per unit of credit the floating rate transfers one unit of exposure to the reference rate from the bank to the firm; the firm’s desired hedging ratio increases by one and the bank’s desired hedging ratio decreases by one.
- Baseline case: π = 0 implies β∗ = 0. With floating rates, the firm’s desired hedging ratio is 1, and the bank would like α∗ − 1.
- Modified market-cost equilibrium when firms also bear some risk:
  - L′′(ω)[1 + (α∗ − 1)] − λ∗ A_B σ^2_R − λ∗ A_F σ^2_R = λ∗ A_D σ^2_R  (Equation 18)
- Aggregate market risk tolerance with firms included:
  - T_D,B,F = 1/A_D + 1/A_B + 1/A_F  > T_D,B  (Equation 19)
- Cost of hedging when firms participate:
  - λ∗ α∗ = 1/T_D,B,F L′′(κ) ρ^2 σ^2_R  (Equation 20)
- Additional effect: if reference rates are positively correlated with project outcomes (Cov(R1, S1) = ρ > 0), firms may elect to keep more exposure, further reducing the cost of hedging.

### Proposition 3 (summary)
- If T_D = ∞ (⇒ λ = 0) and π = 0:
  - Fixed rates with bank hedging and floating rates with hedging on both sides are equivalent.
  - Compared to fixed rates without hedging, both lower interest rates in proportion to the reduction in risk (ρ^2 σ^2_R) and increase welfare in proportion to the utility benefit of this reduction in risk (A_B ρ^2 σ^2_R).
- If T_D < ∞:
  - Welfare is higher with floating rates than with fixed rates as firms can bear some of the risk.
  - If π ∈ (0, ρ), firm optimal hedging ratio is 1 − π/σ^2_R < 1; the cost of hedging is decreasing in π and welfare is increasing in π.

---

### Competition in derivatives markets (Section 5.3)
- Market concentration raises hedging costs:
  - Concentration statistic: top six bank holding companies account for about 95% of the total for swaps and for derivatives (textual statistic).
- Oligopolistic dealers, symmetric Cournot equilibrium:
  - λ′ α∗ = 1 + T^N_D / (T_B,F T_D,B,F + T^N_D) L′′(ω) ρ^2 σ^2_R  > λ∗ α∗  (Equation 21)
- Compared with competitive case (Equation 20), cost of hedging is higher under oligopoly.
- Numerical illustration: if six large dealers account for even 80% of the total risk tolerance in the market for swaps, Equation 21 implies a cost of hedging 50% higher than the competitive situation described in Equation 20.
- Consequence: high concentration can prevent derivatives from being superior to floating rates or fixed rates without hedging.
- Alternative motive: universal banks that both lend and act as dealers may use swaps business to generate profitable derivatives activity, potentially subsidizing lending.

---

### Welfare and the cost of manipulation (Section 5.4)
- Manipulation modeled as added noise to the reference rate:
  - R̃1 = R1 + √(K−1) Z  (Equation 22)
  - Z ~ Normal(0, σ^2_Z) independent of R1 and other risks, with σ^2_Z = σ^2_R. Var(R̃1) = K σ^2_R.
  - Cov(R̃1, S1) = Cov(R1, S1); Cov(R̃1, P) = Cov(R1, P).
- Welfare effects depending on aggregate risk tolerance:
  - If hedging were zero NPV, added noise reduces welfare because institutions bear more risk.
  - When hedging is costly, added noise can induce institutions to keep more underlying risk themselves, which can lower the cost of hedging — net welfare effect ambiguous a priori.
- Proposition 4 (exact limit expression):
  - For aggregate risk tolerance T high enough, welfare is decreasing in added noise. In the limit as T → ∞:
    - lim_{T→∞} ∂Ω(K)/∂K = −(μ_P − μ_S)^2 Φ′(K)^2 / (2 Φ^2(K)) < 0  (Equation 23)
- Additional qualitative points:
  - Pure change in level of reference rate does not affect welfare in the framework; large persistent level shifts could matter in practice.
  - Manipulation that increases noise makes reference rates less useful for hedging.
  - Trade-off for reference-rate design: rates less manipulable may also be less useful for hedging credit-market funding risk if they omit credit-risk components.

---

### Key policy-relevant implications and observations
- The usefulness of swaps versus floating-rate loans depends critically on the aggregate risk tolerance of entities bearing interest-rate/funding risk (T_D,B and T_D,B,F).
- Drawing firms into derivatives markets via floating-rate borrowing can increase aggregate market risk tolerance and reduce the equilibrium cost of hedging (Equation 20).
- Market concentration among swap dealers can substantially raise the cost of hedging (Equation 21). Example: high concentration (six dealers holding a large share) can raise hedging costs materially (text example: 80% share ⇒ 50% higher cost).
- Manipulation that increases the volatility (noise) of reference rates (K > 1) harms welfare when aggregate risk tolerance is sufficiently high (Proposition 4 and Equation 23).
- Reference-rate design trade-offs: rates that better capture bank funding costs (including bank credit risk) are more useful for hedging in credit markets but may be more manipulable; alternatives less prone to manipulation may be less useful for hedging funding risk.

*Source: IMF Working Paper wp1713, sections 3.1 and 5.1–5.4 (content unit).*

### 3.1    Set up

### wp1713 - 3.1 Set up

### Model setup and timing
- Three periods: t0, t1, t2.
- Single bank lends L to a firm at t0 for a project with cash flows L(1 + P) at t2; firm promises to pay L(1 + μ) at t2. μ is the (fixed) interest rate; L and μ determined in equilibrium.
- Bank finances with two short-term loans from investors:
  - borrows L at t0 and must pay L(1 + S0) at t1 with normalization S0 = 0.
  - borrows L at t1 and promises to pay L(1 + S1) at t2.
- Figure 2 timeline (per dollar of lending) summary of payments: investors → bank at t0; bank → investors at t1; bank borrows again at t1; final payments at t2 include project returns and loan repayments.
- Random variables:
  - Project outcomes P ∼ N(μP, σP^2).
  - Short-term funding costs S1 ∼ N(μS, σS^2).
  - Reference rate R1 ∼ N(μR, σR^2) with μR = 0 assumed for notation simplification.
  - Covariances: Cov(S1, R1) = ρ > 0; Cov(P, R1) = π. Baseline case to keep in mind is π = 0.
- Assumption μP > μS so the project is worth funding in expectation.

### Key frictions (Section 3.2)
- Three frictions that generate funding risk and motivate hedging:
  1. Short-term funding must be used (maturity transformation): banks do not fully match maturities of assets and liabilities; substantial majority of commercial bank liabilities are interest-bearing deposits.
  2. S1 is not contractible: contracts referring directly to an individual bank’s funding cost are infeasible; reference rates serve as standardized, publicly available, contractible proxies.
  3. S1 is not known until t1, and financial frictions prevent the bank from being risk neutral: effective risk aversion arises because risky payoffs affect future investable funds when investment opportunities are concave and external financing is costly.
- Mechanism generating effective risk aversion:
  - If investment opportunities F(I) are concave and internal funds w are stochastic, the profit function P(w) inherits concavity ⇒ effective risk aversion.
  - Costly external finance (convex cost C(e)) preserves concavity: Pww ∝ −FII Cee (Equation 1).
  - Townsend (1979) costly state verification framework is used to generate concavity of the production function.
- Reduced-form CARA representation:
  - Optimal allocation for adding θ units of a Normally distributed payoff X yields θ* = μX / (A σX^2) with A = −EPww / EPw (Equation 2).
  - Directly model bank and firm as maximizing CARA utility with exogenous coefficients AB and AF. First-order condition for CARA gives μX / (A σX^2) (Equation 3).
  - Aggregate behavior of N CARA agents is equivalent to single agent with risk aversion ĀN (harmonic mean).

### Basic contract forms and equilibrium with fixed rates (start of Section 4)
- In absence of reference rate: only fixed-rate loan available. Per unit borrowed:
  - Firm payoff: P − μ.
  - Bank payoff: μ − S1.
- Equilibrium μ* equates demand and supply (Equation 4):
  - D(μ*) = (μP − μ*) / (AF σP^2) = μ* − μS / (AB σS^2) = L(μ*).
- Example numeric illustration preserved from source:
  - "For example, the bank might lend $10 MM to the firm at an interest rate of 500 basis points, or 5%."
- Define total utility cost of risk: Φ = AF σP^2 + AB σS^2.

### Reference rates: two principal uses (overview)
- Two ways to reallocate funding risk when reference rate R1 is available:
  1. Floating interest rates: loan rate R1 + ν (e.g., LIBOR + 300 bps), ν is fixed premium. No direct cost to writing floating-rate contract assumed.
  2. Interest-rate derivatives (swaps): allow hedging and transfer of risk to broader market at explicit cost.
- Both are contracts contingent on realized R1; facilitating these contracts is the basic purpose of reference rates.
- If used independently, derivatives are welfare-superior to floating rates (Section 4 conclusion).

### Floating-rate loans (Section 4.1)
- Floating loan interest rate: R1 + ν. Timeline shown in Figure 3.
- Variance effects when reference rate correlated with funding costs:
  - Bank variance reduces from Var(S1) to Var(R1 − S1) = σS^2 + σR^2 − 2ρ.
  - Firm variance increases from Var(P) to Var(P − R1) = σP^2 + σR^2 − 2π.
  - Baseline case often π = 0.
- New equilibrium ν* satisfies (Equation 5):
  - D′(ν*) = (μP − ν*) / (AF (σP^2 + σR^2 − 2π)) = (ν* − μS) / (AB (σS^2 + σR^2 − 2ρ)) = L′(ν*).
- Define total utility cost with floating rates: Φ(R1) = AF (σP^2 + σR^2 − 2π) + AB (σS^2 + σR^2 − 2ρ).
- Welfare implications:
  - Floating rates transfer funding risk from bank to firm.
  - Interest rates fall relative to fixed rates under condition involving covariances (Proposition 1, Equation 6).
  - Welfare increases only if utility benefit to banks outweighs utility cost to firms (Proposition 1, Equation 7).
- Proposition 1 (preserving exact statements):
  - Floating rates transfer funding risk from the bank to the firm. If ρ > σR^2 / 2 and π < σR^2 / 2:
    - Interest rates fall:
      μ* > ν* ⇐⇒ (2ρ − σR^2) / σS^2 + σR^2 − 2π / σP^2 > 0 (Equation 6)
    - Welfare increases only if:
      Ω(R1) > Ω ⇐⇒ AB (2ρ − σR^2) > AF (σR^2 − 2π) (Equation 7)
  - Proofs referenced in Appendix B.1.

### Interest-rate derivatives (swaps) (Section 4.2)
- Swaps modeled as contract at t0 obligating buyer to pay fixed rate λ in return for R1 at t2 (per dollar notional). Hedge ratio α is notional per unit of lending. Unit cost of hedging λ initially taken exogenous.
- Bank’s joint choice of credit supply and hedging: maximize EU[W0 + θ(κ − S1 + α R1)] (Equation 8), where κ is fixed rate charged when swaps available.
- First-order conditions give (Equations 9 and 10):
  - L′′(κ) = θ* = (κ − μS − λ α*) / (AB (σS^2 − ρ^2 σR^2))
  - α*(κ, λ) = ρ / σR^2 ︸︷︷︸ α* − λ / (AB θ* σR^2)
- Zero NPV swap case (λ = 0): optimal hedging ratio α* = ρ / σR^2, the beta of S1 with respect to R1. Residual funding risk after optimal hedging: Var(−S1 + α* R1) = σS^2 − ρ^2 σR^2.
- If swap is costly (λ > 0) bank reduces α* but demand for loans remains same as when optimal hedge purchased (Appendix C.1): bank keeps some risk in exchange for expected return λ.
- Equilibrium κ determined from L′′(κ) = D(κ). Figure 6 conceptual: with optimal hedging via derivatives, interest rates fall relative to baseline but may be higher or lower than with floating rates.
- Welfare and cost-of-hedging considerations:
  - Total cost of hedging λα* thought of as fraction of initial surplus μP − μS.
  - If cost not too high, reduction in bank funding risk lowers interest rates and increases welfare relative to fixed rates.
  - Compared to floating rates, two opposing effects:
    - Lower residual risk with derivatives (unless α* = 1) pushes rates lower.
    - Firms no longer bear risk (risk shifted to broader market) pushes rates higher.
  - Both effects imply welfare is higher with optimal hedging than with floating rates.

- Proposition 2 (preserving exact statements):
  - If interest rate swaps are zero NPV transactions (λ = 0)
    - The effect on interest rates relative to floating rates is ambiguous:
      ν* > κ* ⇐⇒ σR^2 (α* − 1)^2 σS^2 − ρ^2 σR^2 > σR^2 − 2π / σP^2 (Equation 11)
    - Welfare is improved relative to floating rates:
      Ω(R1, λ = 0) > Ω(R1) ⇐⇒ AB σR^2 (α* − 1)^2 + AF (σR^2 − 2π) > 0 (Equation 12)
  - If interest rate swaps are costly, welfare is still increased as long as
    λα* ≤ C̄ = (μP − μS) (1 − √(Φ(R1) / Φ(R1))) (Equation 13)
  - Proofs referenced in Appendix B.2.

### Interpretation and extensions (Section 5 summary)
- Swaps are a more effective method than floating rates for banks to reduce funding risk, provided swaps are not too costly.
- Empirical puzzle: floating rates remain widespread and firms often hedge the floating exposure using swaps themselves.
- Sections foreshadowed:
  - Section 5.1 connects cost of hedging to aggregate risk tolerance for funding risk.
  - Section 5.2 shows combined use of floating rates and firm swap positions broadens market participants bearing funding risk, lowering hedging costs.
  - Section 5.3 cautions analysis requires sufficiently competitive derivatives markets.
  - Section 5.4 notes that manipulation increasing reference rate volatility reduces welfare when swaps are not too costly.

*Source: wp1713 - 3.1 Set up (IMF working paper chapter content).*

### 5.1    Cost of hedging

### 5.1 Cost of hedging

### Equilibrium pricing and why swaps can be costly
- Standard swap pricing assumes initial contractual terms are zero NPV, but if the market in aggregate is risk averse about funding risk, insuring it should not be a zero NPV proposition.
- If both parties to a swap are risk averse, hedging should be costly (Demsetz (1969) point).
- Suppose swap counterparties are dealers with aggregate risk aversion A_D providing a supply function for swaps. The equilibrium cost of hedging, λ∗, satisfies:
  - L′′(κ) α∗ − λ∗ A_B σ^2_R = λ∗ A_D σ^2_R  (Equation 14)
  - Recall: L′′(κ) is the amount of credit supplied, and α∗ is the optimal hedging ratio.
- Define combined market risk tolerance of swap dealers and the bank:
  - T_D,B = 1/A_D + 1/A_B  (Equation 15)
- Rearranging gives:
  - λ∗ α∗ = 1/T_D,B L′′(κ) ρ^2 σ^2_R  (Equation 16)
- Intuition: the cost of hedging depends on
  - how much risk tolerance there is (T_D,B),
  - the size of underlying credit markets (L′′(κ)),
  - how much swaps reduce risk with optimal hedging (ρ^2 σ^2_R).
- Proposition 2 implication: swaps are more effective than floating rates when T_D,B is high enough (λα∗ < C̄).

### Key statistics and exact expressions
- Equation 14: L′′(κ) α∗ − λ∗ A_B σ^2_R = λ∗ A_D σ^2_R
- Equation 15: T_D,B = 1/A_D + 1/A_B
- Equation 16: λ∗ α∗ = 1/T_D,B L′′(κ) ρ^2 σ^2_R

*Italic source attribution: IMF Working Paper wp1713, section 5.1.*

### 5.2 Complementarity between floating rates and swaps

### How floating rates draw firms into hedging and change market tolerance
- Firms can themselves have hedging demand; firm optimal hedging ratio:
  - β∗(κ,λ) = −π/σ^2_R − λ/A_F D(κ) σ^2_R  (Equation 17)
  - Note: π enters with opposite sign because the firm’s problem is a mirror image of the bank’s.
- With floating-rate obligations, per unit of credit the floating rate transfers one unit of exposure to the reference rate from the bank to the firm; the firm’s desired hedging ratio increases by one and the bank’s desired hedging ratio decreases by one.
- Baseline case: π = 0 implies β∗ = 0. With floating rates, the firm’s desired hedging ratio is 1, and the bank would like α∗ − 1.
- Modified market-cost equilibrium when firms also bear some risk:
  - L′′(ω)[1 + (α∗ − 1)] − λ∗ A_B σ^2_R − λ∗ A_F σ^2_R = λ∗ A_D σ^2_R  (Equation 18)
- Aggregate market risk tolerance with firms included:
  - T_D,B,F = 1/A_D + 1/A_B + 1/A_F  > T_D,B  (Equation 19)
- Cost of hedging when firms participate:
  - λ∗ α∗ = 1/T_D,B,F L′′(κ) ρ^2 σ^2_R  (Equation 20)
- Additional effect: if reference rates are positively correlated with project outcomes (Cov(R1, S1) = ρ > 0), firms may elect to keep more exposure, further reducing the cost of hedging.

### Proposition 3 (summary of comparative welfare and effectiveness)
- If T_D = ∞ (⇒ λ = 0) and π = 0:
  - Fixed rates with bank hedging and floating rates with hedging on both sides are equivalent.
  - Compared to fixed rates without hedging, both lower interest rates in proportion to the reduction in risk (ρ^2 σ^2_R) and increase welfare in proportion to the utility benefit of this reduction in risk (A_B ρ^2 σ^2_R).
- If T_D < ∞:
  - Welfare is higher with floating rates than with fixed rates as firms can bear some of the risk.
  - If π ∈ (0, ρ), firm optimal hedging ratio is 1 − π/σ^2_R < 1; the cost of hedging is decreasing in π and welfare is increasing in π.

### Exact equations and expressions referenced
- Equation 17, 18, 19, 20 (see above)
- Firm desired hedging ratio when π = 0: β∗ = 0; with floating rates: β∗ = 1 (intuitive statement in text)

*Italic source attribution: IMF Working Paper wp1713, section 5.2.*

### 5.3 Competition in derivatives markets

### Market concentration raises hedging costs
- Derivatives markets are very concentrated: US bank regulatory data shows the top six bank holding companies account for about 95% of the total for swaps and for derivatives (textual statistic).
- Suppose there are N oligopolistic dealers, each with symmetric risk tolerance T^N_D. In a symmetric Cournot equilibrium, the cost of hedging is:
  - λ′ α∗ = 1 + T^N_D / (T_B,F T_D,B,F + T^N_D) L′′(ω) ρ^2 σ^2_R  > λ∗ α∗  (Equation 21)
- Compared with the competitive case (Equation 20), the cost of hedging is higher under oligopoly.
- Numerical illustration from text: if six large dealers account for even 80% of the total risk tolerance in the market for swaps, Equation 21 implies a cost of hedging 50% higher than the competitive situation described in Equation 20.
- Consequence: if swap markets are sufficiently concentrated, even adding firms to the set of players that bear funding risk may not reduce the cost of hedging enough for derivatives to be superior to floating rates or fixed rates without hedging.
- Alternative motive for banks using floating rates and swaps: universal banks that both lend and act as dealers may use swaps business to generate profitable derivatives activity, potentially subsidizing lending.

### Exact expressions and statistics
- Concentration statistic: top six bank holding companies account for about 95% of total for swaps and derivatives (text).
- Equation 21: λ′ α∗ = 1 + T^N_D / (T_B,F T_D,B,F + T^N_D) L′′(ω) ρ^2 σ^2_R
- Example implication: 80% share by six dealers ⇒ cost of hedging 50% higher than competitive case (textual statement).

*Italic source attribution: IMF Working Paper wp1713, section 5.3.*

### 5.4 Welfare and the cost of manipulation

### Manipulation modeled as added noise to the reference rate
- Manipulation that adds pure noise to the reference rate is parametrized by:
  - R̃1 = R1 + √(K−1) Z  (Equation 22)
  - Z ~ Normal(0, σ^2_Z) independent of R1 and other risks, with σ^2_Z = σ^2_R. K is a parameter capturing the extent of manipulation.
- Under this parametrization:
  - Cov(R̃1, S1) = Cov(R1, S1)
  - Cov(R̃1, P) = Cov(R1, P)
  - Var(R̃1) = K σ^2_R
- The analysis considers the effect of slightly increasing K from 1 (small added noise).

### Welfare effects depending on aggregate risk tolerance
- If hedging were zero NPV, added noise reduces welfare because institutions bear more risk as the reference rate moves.
- When hedging is costly, added noise can induce institutions to keep more underlying risk themselves, which can lower the cost of hedging — the net welfare effect is ambiguous a priori.
- Proposition 4: For aggregate risk tolerance T high enough, welfare is decreasing in added noise. In the limit as T → ∞:
  - lim_{T→∞} ∂Ω(K)/∂K = −(μ_P − μ_S)^2 Φ′(K)^2 / (2 Φ^2(K)) < 0  (Equation 23)
  - Where Φ′(K) is the added utility cost of risk participants bear as the reference rate becomes more volatile.

### Additional qualitative points
- A pure change in the level of the reference rate does not affect welfare according to the framework; however, large persistent level shifts could matter in practice because reference rates feed other functions (gauge of financial health, discounting).
- Manipulation driven by variable portfolio incentives (e.g., bank exposure on swap settlement days) can add noise and make reference rates less useful for hedging.
- Trade-off for reference-rate design: rates less manipulable may also be less useful for hedging credit-market funding risk if they omit credit-risk components.

### Exact expressions and parameters
- Manipulation model: Equation 22 (R̃1 = R1 + √(K−1) Z)
- Var(R̃1) = K σ^2_R
- Proposition 4 limit expression: lim_{T→∞} ∂Ω(K)/∂K = −(μ_P − μ_S)^2 Φ′(K)^2 / (2 Φ^2(K)) < 0

*Italic source attribution: IMF Working Paper wp1713, section 5.4.*

### Key policy-relevant implications and observations

- The usefulness of swaps versus floating-rate loans depends critically on the aggregate risk tolerance of entities bearing interest-rate/funding risk (T_D,B and T_D,B,F).
- Drawing firms into derivatives markets via floating-rate borrowing can increase aggregate market risk tolerance and reduce the equilibrium cost of hedging (Equation 20).
- Market concentration among swap dealers can substantially raise the cost of hedging (Equation 21). Example: high concentration (six dealers holding a large share) can raise hedging costs materially (text example: 80% share ⇒ 50% higher cost).
- Manipulation that increases the volatility (noise) of reference rates (K > 1) harms welfare when aggregate risk tolerance is sufficiently high (Proposition 4 and Equation 23).
- Reference-rate design trade-offs: rates that better capture bank funding costs (including bank credit risk) are more useful for hedging in credit markets but may be more manipulable; alternatives less prone to manipulation may be less useful for hedging funding risk.

*Italic source attribution: IMF Working Paper wp1713, sections 5.1–5.4.*

### 33. Rearrangements show that the change in interest rates is

### wp1713 - 33. Rearrangements show that the change in interest rates is

### Rearrangement and sign conditions (Proposition 1)
- Change in interest rates expression:
  - μ∗ − ν∗ = (μP − μS) (YD XS − XD YS) / (XD + XS) (YD + YS) (Equation 34)
- Equivalent sign condition:
  - Sgn(μ∗ − ν∗) = Sgn( XS − YS / XS − XD − YD / XD ) (Equation 35)
- Sign of welfare difference:
  - Sgn (Ω(R1) − Ω) = Sgn((XD − YD) + (XS − YS)) (Equation 36)
- Conclusion: These rearrangements prove Proposition 1.

### Proof of Proposition 2 — threshold cost of optimal hedging
- Established relation: Ω(R1,0) > Ω(R1).
- Define utility costs of risk with swaps: ZD and ZS.
- Threshold cost equation (equating welfare levels):
  - Ω(R1, C̄) = Ω(R1)
  - 1/2 (μP − μS)2 (YD + YS) = 1/2 (μP − μS − C̄)2 (ZD + ZS)
- Smaller root selected from the quadratic:
  - C̄ = (μP − μS) (1 − √(ZD + ZS / YD + YS))
  - Note: maximal cost must be smaller than the surplus μP − μS.
- Total utility costs of risk:
  - Φ(R1) = ZD + ZS and Φ(R1) = YD + YS.
- Conclusion: This proves Proposition 2.

### Proof of Proposition 3 — hedging, risk tolerance, and welfare
- When hedging is a zero NPV transaction and π = 0, fixed rates with bank hedging via swaps and floating rates with both firm and bank hedging via swaps are equivalent.
- Welfare as a function of risk tolerance in the market for swaps (π = 0):
  - Ω(T) = 1/2 (μP − μS − λ(T)α∗)2 Φ(ρ) (Equation 37)
  - Substituting cost of hedging from Equation 16:
    - λ(T)α∗ = (μP − μS) / (T Φ(ρ)) ρα∗
    - ⇒ Ω(T) = 1/2 (μP − μS)2 Φ(ρ) (1 − 1 / (T Φ(ρ)) ρ2 σ2R )2 (Equation 38)
- When π 6= 0:
  - Ω(T) = 1/2 (μP − μS)2 Φ(ρ,π) (1 − 1 / (T Φ(ρ,π)) (ρ − π)2 σ2R )2 (Equation 39)
- For π ∈ (0,ρ): the cost of hedging is decreasing in π and welfare is increasing in π.
- Conclusion: This proves Proposition 3.

### Proof of Proposition 4 — effect of manipulation K on welfare
- Welfare as function of manipulation K:
  - Ω(K) = 1/2 (μP − μS)2 Φ(K) (1 − 1 / (T Φ(K)) (ρ − π)2 K σ2R )2 (Equation 40)
- Derivative with respect to K:
  - ∂Ω(K)/∂K = (1 − 1 / (T Φ(K)) (ρ − π)2 K σ2R )
    × [ 1/T (μP − μS)2 (ρ − π)2 / (2 Φ2(K)) K σ2R (2/K + 3 Φ′(K)/Φ(K)) − (μP − μS)2 Φ′(K) / (2 Φ2(K)) ] (Equation 41)
  - The two marked terms are positive because Φ′(K) > 0.
- Limit as T → ∞:
  - lim_{T→∞} ∂Ω(K)/∂K = − (μP − μS)2 Φ′(K) / (2 Φ2(K)) < 0 (Equation 42)
- Interpretation: Added risk (higher K) increases risk borne by lenders and borrowers; when T is sufficiently large the net effect on welfare is negative.
- Conclusion: This proves Proposition 4.

### Appendix C.1 — Optimal hedging with costly derivatives (notation and identity)
- Simplified notation: H = κ − μS, σ2R = 1, and σ2S = σ2.
- Target identity to show:
  - κ − μS − λ α∗(κ,λ) A B Var(−S1 + α∗(κ,λ) R1) = κ − μS − λ α∗ A B Var(−S1 + α∗ R1)
  - ⇐⇒ H − λ (ρ − λ / (A θ)) A (σ2 + (ρ − λ / (A θ))2 − 2 (ρ − λ / (A θ)) ρ) = H − λ ρ A (σ2 − ρ2)
- Verification equality (as in text):
  - H − λ (ρ − λ / (A θ)) A (σ2 + (ρ − λ / (A θ))2 − 2 (ρ − λ / (A θ)) ρ)
    = A θ (H − λ ρ) − λ2 / A2 θ (σ2 − ρ2) + λ2 θ
    = A θ (H − λ ρ) / (A2 θ (σ2 − ρ2)) = X
    and with Y defined analogously leads to θ where final equality follows from θ (Y − X) = λ2.

### Appendix C.2 — Optimal contracting: pointwise maximization and linearity conditions
- Simplified problem: P, S are functions of random variable R; consider payment f(R) from firm to bank with bank participation constraint; pdf g(R).
- Lagrangian:
  - L = max_{f(R)} ∫ UF[P(R) − f(R)] g(R) dR + λ ( Ū − ∫ UB[f(R) − S(R)] g(R) dR ) (Equation 43)
- First-order/Borsch rule (marginal utilities):
  - − UF_W(P(R) − f(R)) / UB_W(f(R) − S(R)) = λ (Equation 44)
- Implicit differentiation yields:
  - [ − UF_WW / UF_W A_F ( df∗(R)/dR − dP/dR ) ] + [ − UB_WW / UB_W A_B ( df∗(R)/dR − dS/dR ) ] = 0 (Equation 45)
  - Hence:
    - df∗(R)/dR = (A_F dP/dR + A_B dS/dR) / (A_F + A_B) (Equation 46)
- Conclusion: f is linear only if P and S are linear functions of R.

### Appendix C.3 — Competition in derivatives markets and equilibrium hedging cost
- Dealer inverse demand (Equation 18 rearranged) as function of total swaps demand Q:
  - λ(Q) = (L′′(ω) α∗ − Q) T_{B,F} / σ2R (Equation 47)
- Dealer i maximization problem (q_i, holding r = ∑_{j≠i} q_j fixed):
  - max_{q_i} q_i λ(q_i + r) − 1/2 A_i q_i2 σ2R (Equation 48)
- First-order condition gives:
  - q_i = T_{B,F} (L′′(ω) α∗ − r) / (2 T_{B,F} + A_i) (Equation 49)
- Symmetric equilibrium (r = (N − 1) q):
  - q∗ = T_{B,F} L′′(ω) α∗ / ( (N + 1) T_{B,F} + A_i ) (Equation 50)
- Equilibrium cost of hedging (Equation 21 form):
  - λ(N q∗) α∗ = 1 + T_D / (N T_{B,F}) T_{D,B,F} + T_D / N L′′(ω) ρ2 σ2R (Equation 51)
- As N → ∞, this approaches competitive cost of hedging λ∗ α∗ (Equation 20). The expression is always greater for finite N, shown by:
  - 1 + T_D / (N T_{B,F}) T_{D,B,F} + T_D / N = 1 / T_{D,B,F} ( 1 + T_D2 / T_{B,F} (T_D + N T_{D,B,F}) ) > 0 (Equation 52)

*Italic: Source — wp1713 - 33. Rearrangements show that the change in interest rates is (IMF working paper content unit).*

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_Source: https://www.imf.org/-/media/files/publications/wp/wp1713.pdf_