## Section 5 solves the optimal decisions under asymmetric information, and discusses the pricing of

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### Related literature
- Empirical evidence on green bond pricing is mixed:
  - IMF (2019), Flammer (2021), Larcker and Watts (2020): no noticeable difference between yields of green bonds and brown bonds.
  - Sebastian and Karim (2020), Zerbib (2019), Baker et al. (2018): green bonds tend to be priced at a premium (lower yields).
  - MacAskill et al. (2021): somewhat greater number of studies in favor of a small greenium for government-issued, investment grade, and well-governed green bonds.
- Theoretical explanations often rely on non-financial green preferences (Pedersen, Fitzgibbons and Pomorski (2021); Pástor, Stambaugh and Taylor (2021); Baker et al. (2018); Agliardi and Agliardi (2021)).
- This paper’s model:
  - Builds on Flammer (2021) signaling logic but directs signals to bond investors.
  - Explains both positive greenium and no premium without assuming non-financial green preferences.
  - Attributes greenium to interaction of transition risks, green bond issuing costs, and greenwashing costs.
- Transition risks and policy relevance:
  - Delayed and large carbon taxes can increase instability and greenwashing.
  - A swift and gradual introduction of a carbon tax supports green firm growth, reduces greenwashing, aggregate emissions, and financial stability risks.
- On subsidies:
  - General subsidies for green bonds raise greenwashing and can undermine emissions reductions unless combined with strong disclosures, regulations, and enforcement.

### Model setup — core assumptions and primitives
- One-period model with risk-neutral firms and investors; no non-financial green preference.
- Two bond types: green bonds and brown bonds; firms can issue only one type.
- Transition risk via a possible carbon tax introduced after issuance:
  - Carbon tax τ per unit of emission with probability p; no carbon tax with probability (1−p).
  - τ ∈ (0,1), p ∈ (0,1).
- Firms:
  - N firms; green firms share α, brown firms share (1−α).
  - Firm type is private information.
  - Brown firm i has emission intensity e_i ∼ U(0,1].
  - Production: y_j = A_d k_j for j ∈ {g,b}.
  - Green firms: zero emissions; Brown firms: strictly positive emissions.
  - Firms have limited liability and borrow to produce.
- Bonds:
  - Normalized prices to one; coupon rates r_b (brown, exogenous) and r_g (green, endogenous).
  - Green issuing costs: ̄C for green firms; brown firms issuing green bonds pay ̄C + f(e_i).
  - Greenwashing cost function: f(e_i) increasing, strictly convex, twice continuously differentiable; f(e_i=0)=0; f(e_i=1)→∞.
  - Specific form: f(e_i) = −σ ln(1−e_i) (σ interpretable as completeness of regulation).
- Investors:
  - Many identical, risk neutral investors with one unit each.
  - Investors observe bond type but not firm type; know aggregate greenwashing in green bond market.
  - Expected returns of green and brown bonds must be equal (no arbitrage).
- Timeline:
  1. Firms choose bond type.
  2. Investors choose bonds to buy.
  3. Transition risk (carbon tax realization) occurs.
  4. Bonds repaid or firms default.

### Benchmark: model with perfect information
- If firm types observable, firms issue conventional (brown) bonds only.
- Without carbon tax: no defaults; investors require coupon rate = productivity A_d; firms earn zero profit.
- With known carbon tax τ before issuance:
  - Green firms: coupon A_d; brown firms: coupon (A_d − τ).
  - Investors prefer green firms; brown firms exit for any substantial small positive τ.
- With transition risk (tax implemented with probability p):
  - Brown firms offer expected coupon (A_d − pτ); green firms offer A_d; brown firms fail to attract funding and exit for any substantial small positive τ.

### Model with asymmetric information — overview
- Investors observe bond type but not firm type; firms choose bond type to maximize expected profits; investors equalize expected returns across bond types.
- Green firms:
  - E(π_ĝg) = y_g − r_g k_g − ̄C k_g. (equation (3))
  - E(π_ĝb) = y_g − r_b k_g. (equation (4))
  - Assumption: green firms always prefer green bonds, i.e., r_g + ̄C ≤ r_b. (inequality (5))
- Brown firms (firm i with emission intensity e_i):
  - π_b̂gt(e_i) = max{ y_b − r_g k_b − (̄C + f(e_i)) k_b − τ e_i y_b, 0 }. (equation (6))
  - π_b̂bt(e_i) = max{ y_b − r_b k_b − τ e_i y_b, 0 }. (equation (7))
  - π_b̂g0(e_i) = max{ y_b − r_g k_b − (̄C + f(e_i)) k_b, 0 }. (equation (8))
  - π_b̂b0(e_i) = y_b − r_b k_b. (equation (9))
  - Turning points e_g, e_gt, e_bt solve:
    - y_b − r_g k_b − (̄C + f(e_g)) k_b = 0. (equation (10))
    - y_b − r_g k_b − (̄C + f(e_gt)) k_b − τ e_gt y_b = 0. (equation (11))
    - y_b − r_b k_b − τ e_bt y_b = 0. (equation (12))
  - Overall expected profits:
    - E(π_b̂g(e_i)) = (1−p) π_b̂g0(e_i) + p π_b̂gt(e_i). (equation (13))
    - E(π_b̂b(e_i)) = (1−p) π_b̂b0(e_i) + p π_b̂bt(e_i). (equation (14))

### Greenwashing threshold (Lemma 1) and interpretation
- Lemma 1: brown firm i prefers green bonds (greenwashing) when e_i ∈ (0, ˆe). The threshold ˆe satisfies:
  - r_g + ̄C + f(ˆe) = r_b. (equation (20))
- With f(e) = −σ ln(1−e), solve ˆe:
  - ˆe = 1 − e^(−(1/σ)(r_b − r_g − ̄C)).

### Brown-firm strategy regions and three carbon-tax cases (overview)
- Three cases depending on τ relative to baseline ˆτ:
  1. τ = ˆτ (baseline): turning points coincide; brown firms with e_i < ˆe_1 greenwash and do not default; brown firms with e_i ≥ ˆe_1 issue brown bonds and default under tax.
  2. τ < ˆτ (lower tax): ˆe_2 < e_bt2; some intermediate brown firms issue brown bonds and do not default; greenwashing region shrinks.
  3. τ > ˆτ (higher tax): e_gt3 < ˆe_3; some green bonds issued by brown firms default under tax; greenwashing region and default patterns change non-monotonically.

### Green bond pricing and greenwashing — equilibrium characterizations (key equations)
- No-arbitrage: E(R_g) = E(R_b) in equilibrium.
- Case 1 — Baseline τ = ˆτ:
  - τ = (y_b − r_b k_b) / (ˆe_1 y_b). (equation (23))
  - E(R_g1) = r_g1. (equation (24))
  - E(R_b1) = r_b − (1/2) p ( y_b / k_b − r_b ) (1/ˆe_1 − 1). (equation (25) rearranged)
  - No-arbitrage F1: r_g1 − r_b + (1/2) p ( y_b k_b − r_b ) (1/ˆe_1 − 1) = 0. (equations (27),(28))
  - ˆe_*1 = 1 − e^(−(1/σ)(r_b − r_g*1 − ̄C)). (equation (29))
  - ˆτ = (y_b − r_b k_b) / (ˆe_*1 y_b). (equation (30))
- Case 2 — Lower τ < ˆτ:
  - E(R_g2) = r_g2. (equation (31))
  - E(R_b2) = r_b − (1/2) p ( y_b k_b − r_b ) (1−e_bt2)^2 e_bt2 (1−ˆe2)^{-1}. (equation (32))
  - No-arbitrage F2: r_g2 − r_b + (1/2) p ( y_b k_b − r_b ) (1−e_bt2)^2 / ( e_bt2 (1−ˆe2) ) = 0. (equations (33),(34))
  - ˆe_*2 = 1 − e^(−(1/σ)(r_b − r_g*2 − ̄C)). (equation (35))
  - Lemma 2: For τ < ˆτ, keeping r_b fixed:
    - r_g*2 > r_g*1.
    - ˆe_*2 < ˆe_*1.
- Case 3 — Higher τ > ˆτ:
  - E(R_g3) accounts for defaults among green bonds issued by brown firms. (equation (36))
  - E(R_b3) = r_b − p ( r_b − ( (1−τ) ˆe_3 + 1/2 ) y_b k_b ). (equation (37))
  - No-arbitrage F3 yields equilibrium condition (equations (38),(39)).
  - ˆe_*3 = 1 − e^(−(1/σ)(r_b − r_g*3 − ̄C)). (equation (40))
  - Lemma 3: For τ > ˆτ, keeping r_b fixed, change in r_g and greenwashing relative to baseline is parameter-dependent and ambiguous.

### Key mechanisms and comparative statics
- Greenium arises when brown bonds carry transition risk and green bonds carry lower expected defaults.
- Equilibrium gaps:
  - r_b − r_g*1 = p [ r_b − ( (1−τ) ˆe_1 + 1/2 ) y_b k_b ]. (equation (41))
  - r_b − r_g*2 = p (1−e_bt2)/(1−ˆe2) [ r_b − ( (1−τ) e_bt2 + 1/2 ) y_b k_b ]. (equation (42))
  - r_b − r_g*3 = p [ r_b − ( (1−τ) ˆe_3 + 1/2 ) y_b k_b ] − p A [ r_g3 − ( (1−τ) e_gt3 + ˆe_3 / 2 ) y_b k_b − ̄C − E(f(e)) ]. (equation (43))
- Presence or absence of a greenium depends on p, τ, ̄C, f(·) (σ), distribution of e_i, r_b, y_b, k_b.
- Comparative insights:
  - Lower τ (case 2) raises r_g and reduces greenwashing.
  - Higher τ (case 3) produces ambiguous effects because defaults can affect green bonds issued by brown firms.

### Default risk premium decomposition (summary of Table 1)
- Default risk premium = difference between default losses of green and brown bonds = spread (yields difference).
- Decomposed into probability of default and average loss given default.
- Green bonds — Probability of default and loss given default by case:
  - Case 1 (τ = ˆτ): Probability of default = 0; Loss given default = 0.
  - Case 2 (τ < ˆτ): Probability of default = 0; Loss given default = 0.
  - Case 3 (τ > ˆτ): Probability of default = ˆe^3 − e_gt^3; Loss given default = r_g^3 − [1−τ e_gt^3 + ˆe^3 / 2] y_b k_b + ̄C + E(f(e)).
- Brown bonds — Probability of default and loss given default by case:
  - Case 1 (τ = ˆτ): Probability of default = 1; Loss given default = r_b − [1−τ ˆe^1 + 1/2] y_b k_b r_b − [1−τ e_bt^2 + 1/2] y_b k_b r_b − [1−τ ˆe^3 + 1/2] y_b k_b r_b.
  - Case 2 (τ < ˆτ): Probability of default = 1−e_bt^2; Loss given default = r_b − [1−τ e_bt^2 + 1/2] y_b k_b r_b.
  - Case 3 (τ > ˆτ): Probability of default = 1−ˆe^2; Loss given default = r_b − [1−τ ˆe^3 + 1/2] y_b k_b r_b.
- Interpretation:
  - When brown bonds have higher default probability or higher average loss than green bonds, default risk premium is positive (greenium).
  - Brown bond market concentrates heavily-polluting firms; green bond market includes green firms and brown firms with low e_i.
  - High-emission brown firms avoid issuing green bonds because f(e) increases with e.

### Special cases with equal green and brown bond prices (r_b = r_g∗)
- Case 1: No transition risk: p = 0 or p = 1.
  - If p = 0: no carbon taxation before maturity; by (41),(42),(43), r_b = r_g∗; greenium absent; typically ̄C > 0 so firms will not issue green bonds.
  - If p = 1: tax implemented in the period; brown firms anticipate default and will not enter; yields equal by no arbitrage.
- Case 2: No greenwashing cost: ∀e_i ∈ (0,1], f(e_i) = 0 implying ˆe = 1.
  - Setting f(e_i) = 0 in relevant equations yields r_b = r_g∗.
  - If r_g + ̄C < r_b both brown and green firms issue green bonds — no brown bond market.
  - If r_g + ̄C > r_b both types issue brown bonds — no green bond market.
  - If r_g + ̄C = r_b both types indifferent; coupon rates equal and ̄C must be zero.
  - Intuition: free greenwashing eliminates signaling value.
- Case 3: No information asymmetry (proof in Section IV).

### Impact of transition risk on green bond pricing and greenwashing (Lemma 4)
- Baseline solutions: (r_g∗_1, ˆe∗_1) with baseline carbon tax ˆτ.
- If τ < ˆτ:
  - Green bond coupon rate r_g > baseline r_g∗_1.
  - Greenwashing ˆe < baseline ˆe∗_1.
- If τ > ˆτ:
  - Direction of change in r_g and greenwashing is uncertain.
- Mechanism:
  - Greenium (r_b − r_g) is default risk premium (difference in default probabilities × loss given default).
  - At ˆτ difference of default probabilities is 1.
  - If τ < ˆτ difference in default probability and loss given default both fall → default risk premium smaller → spread smaller → r_g higher.
  - If τ > ˆτ difference in default probability falls but difference in loss given default rises → net effect ambiguous.
  - Greenwashing threshold determined by spread (r_b − r_g), ̄C, f(·). Lower τ reduces spread and greenwashing; higher τ yields ambiguous greenwashing change.

### Policy implications
- A. The Need for Carbon Pricing and Climate Financial Regulations
  - Without transition risk, green bond market unlikely to scale: investors lack monetary incentive absent non-financial preferences.
  - Green finance requires greenwashing to be costly; practical measures: guidelines, disclosures, fines for false information, effective validation and screening.
  - Current status: guidelines mostly non-binding; firm-level environmental disclosures, reporting on additionality, and ex post monitoring need improvement.
  - Regulators recommend detailed reporting and external reviewer assurances; taxonomies help reduce greenwashing.
- B. The Need for Swift Action on Carbon Pricing
  - Delay leads to very high eventual carbon price and risk of a “climate Minsky moment.”
  - Swift and gradual carbon pricing reduces disorderly adjustments and financial instability.
  - One-period model implications:
    - Lemma 2: small carbon tax (τ < ˆτ) → small greenium and low greenwashing.
    - Lemma 3: large carbon tax (τ > ˆτ) → uncertain greenium and greenwashing, adding market uncertainty.
  - Multi-period extension: gradual carbon pricing increases green firms’ market share over time; delayed final large tax causes bankruptcies and instability.
- C. The Need for Strong Disclosure Requirements Attached to Subsidy Programs
  - Green bond issuance more costly due to reporting, consultancy, external verification; subsidies (e.g., reimburse external review) lower ̄C.
  - Without safeguards, subsidies can encourage brown firms to issue green bonds, increasing greenwashing.
  - Strong disclosure and regulation steepen f(·), reducing greenwashing and brown issuance of green bonds.
  - Conclusion: subsidies should be tied to strong disclosure, regulation, and screening.

### Conclusion (summary)
- Model: adverse selection framework solving equilibrium prices in green bond market and extent of greenwashing.
- Key findings:
  - A greenium exists when there is asymmetric information, transition risk from carbon pricing, and costly greenwashing.
  - Impact of carbon pricing on greenium and greenwashing depends on speed of introduction:
    - Swift but gradual implementation → small greenium and low greenwashing.
    - Delayed and large carbon pricing → ambiguous effects and additional market uncertainty.
- The model does not assume exogenous investor green preference; green bond pricing can arise from valuable signaling.
- Policy recommendations reiterated:
  - Transition risk via carbon pricing is essential absent innate green preference.
  - Swift action on carbon pricing preferable to delayed large steps.
  - Strong supervision and disclosure/reporting requirements increase greenwashing costs and reduce greenwashing.
  - Government subsidies should be tied to strong information disclosure and compliance measures.

*Official source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022246-print-pdf.pdf*

### Section 5 solves the optimal decisions under asymmetric information, and discusses the pricing of

### wpiea2022246-print-pdf - Section 5 solves the optimal decisions under asymmetric information, and discusses the pricing of

### Related literature
- Empirical evidence on green bond pricing is mixed:
  - IMF (2019), Flammer (2021), Larcker and Watts (2020): no noticeable difference between yields of green bonds and brown bonds.
  - Sebastian and Karim (2020), Zerbib (2019), Baker et al. (2018): green bonds tend to be priced at a premium (lower yields).
  - MacAskill et al. (2021): somewhat greater number of studies in favor of a small greenium for government-issued, investment grade, and well-governed green bonds.
- Theoretical asset-pricing explanations for a greenium often rely on non-financial green preferences (Pedersen, Fitzgibbons and Pomorski (2021); Pástor, Stambaugh and Taylor (2021); Baker et al. (2018); Agliardi and Agliardi (2021)).
- This paper’s model:
  - Builds on Flammer (2021) signaling logic but signals are directed to bond investors (not equity investors).
  - Explains both positive greenium and no premium without assuming non-financial green preferences.
  - Attributes greenium to the interaction of transition risks, green bond issuing costs, and greenwashing costs.
- Transition risks and policy relevance:
  - Transition risk can produce financial instability; delayed and large carbon taxes can increase instability and greenwashing.
  - A swift and gradual introduction of a carbon tax supports green firm growth, reduces greenwashing, aggregate emissions, and financial stability risks.
- On subsidies:
  - General subsidies for green bonds raise greenwashing and can undermine emissions reductions unless combined with strong disclosures, regulations, and enforcement.

### Model setup — core assumptions and primitives
- One-period model with risk-neutral firms and investors; no non-financial green preference.
- Two bond types: green bonds and brown bonds. Firms can issue only one type.
- Transition risk only via a possible carbon tax introduced after issuance:
  - Carbon tax τ per unit of emission with probability p; no carbon tax with probability (1−p).
  - τ∈(0,1), p∈(0,1).
- Firms:
  - N firms; green firms share α, brown firms share (1−α).
  - Firm type (green or brown) is private information.
  - Brown firm i has emission intensity e_i ∼ U(0,1].
  - Production: y_j = A_d k_j for type j∈{g,b}.
  - Green firms: zero emissions. Brown firms: strictly positive emissions.
  - Firms have limited liability and borrow to produce.
- Bonds:
  - Normalized prices to one; coupon rates r_b (brown, exogenous) and r_g (green, endogenous).
  - Green issuing costs: for green firms, additional issuing cost per bond ̄C; brown firms issuing green bonds pay ̄C plus greenwashing cost f(e_i).
  - Greenwashing cost function requirements: f(e_i) increasing, strictly convex, twice continuously differentiable; f(e_i=0)=0; f(e_i=1)→∞.
  - Specific functional form used: f(e_i) = −σ ln(1−e_i), where σ interpretable as completeness of regulation.
- Investors:
  - Many identical, risk neutral investors with one unit of money each.
  - Investors observe bond type (green/brown) but not firm type; know aggregate greenwashing level in green bond market.
  - Expected returns of green and brown bonds must be equal (no arbitrage).
- Timeline (one period):
  1. Firms choose bond type to issue.
  2. Investors choose which bonds to buy.
  3. Transition risk (carbon tax realization) occurs.
  4. Bonds repaid or firms default depending on emissions and tax realization.

### Benchmark: model with perfect information
- If firm types observable, no signaling benefit; firms issue conventional (brown) bonds only.
- Without carbon tax: no defaults; investors require coupon rate equal to productivity A_d; firms earn zero profit in competitive market.
- With a known carbon tax τ before issuance:
  - Green firms offer coupon rate A_d; brown firms offer (A_d − τ).
  - Investors prefer green firms; brown firms exit for any substantial small positive τ.
- With transition risk (tax implemented with probability p):
  - Brown firms offer expected coupon (A_d − pτ); green firms offer A_d; brown firms fail to attract funding and exit for any substantial small positive τ.

### Model with asymmetric information — overview
- Investors observe bond type but not firm type; firms choose bond type to maximize expected profits.
- Investors equalize expected returns across green and brown bonds.
- Green firms:
  - Expected profit issuing green bonds: E(π_ĝg) = y_g − r_g k_g − ̄C k_g.  (equation (3))
  - Expected profit issuing brown bonds: E(π_ĝb) = y_g − r_b k_g. (equation (4))
  - Assumption: green firms always prefer green bonds, i.e., r_g + ̄C ≤ r_b. (inequality (5))
- Brown firms:
  - Brown firm i with emission intensity e_i:
    - Profit under carbon tax when issuing green bonds: π_b̂gt(e_i) = max{ y_b − r_g k_b − (̄C + f(e_i)) k_b − τ e_i y_b, 0 }. (equation (6))
    - Profit under carbon tax when issuing brown bonds: π_b̂bt(e_i) = max{ y_b − r_b k_b − τ e_i y_b, 0 }. (equation (7))
    - Profit under no tax when issuing green bonds: π_b̂g0(e_i) = max{ y_b − r_g k_b − (̄C + f(e_i)) k_b, 0 }. (equation (8))
    - Profit under no tax when issuing brown bonds: π_b̂b0(e_i) = y_b − r_b k_b. (equation (9))
  - Define three turning points e_g, e_gt, e_bt solving:
    - π_b̂g0(e_g) = y_b − r_g k_b − (̄C + f(e_g)) k_b = 0. (equation (10))
    - π_b̂gt(e_gt) = y_b − r_g k_b − (̄C + f(e_gt)) k_b − τ e_gt y_b = 0. (equation (11))
    - π_b̂bt(e_bt) = y_b − r_b k_b − τ e_bt y_b = 0. (equation (12))
  - Overall expected profits:
    - E(π_b̂g(e_i)) = (1−p) π_b̂g0(e_i) + p π_b̂gt(e_i). (equation (13))
    - E(π_b̂b(e_i)) = (1−p) π_b̂b0(e_i) + p π_b̂bt(e_i). (equation (14))

### Greenwashing threshold (Lemma 1) and interpretation
- Lemma 1: When e_i ∈ (0, ˆe), brown firm i prefers green bonds (greenwashing). The greenwashing threshold ˆe satisfies:
  - r_g + ̄C + f(ˆe) = r_b. (equation (20))
- Interpretation:
  - Brown firm issues green bonds if the total cost of issuing green (r_g + ̄C + f(e)) is less than the cost of issuing brown (r_b).
  - With f(e) = −σ ln(1−e), solve ˆe from ˆe = 1 − e^(−(1/σ)(r_b − r_g − ̄C)) in subsequent equilibrium characterizations.

### Brown-firm strategy regions and three carbon-tax cases (Figure 3 overview)
- Three cases depending on τ relative to baseline ˆτ:
  1. τ = ˆτ (baseline): turning points coincide; brown firms with e_i < ˆe_1 engage in greenwashing and do not default; brown firms with e_i ≥ ˆe_1 issue brown bonds and default under tax.
  2. τ < ˆτ (lower tax): ˆe_2 < e_bt2; some intermediate brown firms issue brown bonds and do not default; greenwashing region shrinks.
  3. τ > ˆτ (higher tax): e_gt3 < ˆe_3; green bonds issued by some brown firms default under tax; greenwashing region and default patterns change non-monotonically.

### Green bond pricing and greenwashing — equilibrium characterizations
- No-arbitrage: expected returns on green and brown bonds equal in equilibrium.
- Case 1 — Baseline τ = ˆτ:
  - Brown default threshold links τ and ˆe_1:
    - τ = (y_b − r_b k_b) / (ˆe_1 y_b). (equation (23))
  - Green bonds: no default; E(R_g1) = r_g1. (equation (24))
  - Brown bonds: E(R_b1) = r_b − (1/2) p ( y_b / k_b − r_b ) (1/ˆe_1 − 1). (equation (25) rearranged)
  - No-arbitrage F1 ≡ E(R_g1) − E(R_b1) = r_g1 − r_b + (1/2) p ( y_b k_b − r_b ) (1/ˆe_1 − 1) = 0. (equation (27) and (28))
  - Greenwashing threshold expressed with f(e) form:
    - ˆe_*1 = 1 − e^(−(1/σ)(r_b − r_g*1 − ̄C)). (equation (29))
  - Baseline τ expression: ˆτ = (y_b − r_b k_b) / (ˆe_*1 y_b). (equation (30))
- Case 2 — Lower τ < ˆτ:
  - Green bond expected return E(R_g2) = r_g2. (equation (31))
  - Brown bond expected return E(R_b2) = r_b − (1/2) p ( y_b k_b − r_b ) (1−e_bt2)^2 e_bt2 (1−ˆe2)^{-1} (expression in (32)).
  - No-arbitrage F2 = r_g2 − r_b + (1/2) p ( y_b k_b − r_b ) (1−e_bt2)^2 / ( e_bt2 (1−ˆe2) ) = 0. (equation (33) and (34))
  - Greenwashing threshold: ˆe_*2 = 1 − e^(−(1/σ)(r_b − r_g*2 − ̄C)). (equation (35))
  - Lemma 2: For τ < ˆτ, keeping r_b fixed:
    - r_g*2 > r_g*1 (green bond coupon rate is higher than baseline).
    - ˆe_*2 < ˆe_*1 (level of greenwashing is lower than baseline).
- Case 3 — Higher τ > ˆτ:
  - Green bond expected return E(R_g3) accounts for defaults among green bonds issued by brown firms (equation (36)).
  - Brown bond expected return E(R_b3) = r_b − p ( r_b − ( (1−τ) ˆe_3 + 1/2 ) y_b k_b ). (equation (37))
  - No-arbitrage F3 gives an equilibrium condition (equation (38) and (39)).
  - Greenwashing threshold: ˆe_*3 = 1 − e^(−(1/σ)(r_b − r_g*3 − ̄C)). (equation (40))
  - Lemma 3: For τ > ˆτ, keeping r_b fixed, the direction of change in r_g and in greenwashing relative to baseline is uncertain and depends on parameters and state variables.

### Key mechanisms and comparative statics
- Greenium (brown bond spread over green bonds) arises when brown bonds carry transition risk and green bonds carry lower expected defaults:
  - Expressions (41), (42), (43) reorganize equilibrium gaps:
    - r_b − r_g*1 = p [ r_b − ( (1−τ) ˆe_1 + 1/2 ) y_b k_b ]. (equation (41))
    - r_b − r_g*2 = p (1−e_bt2)/(1−ˆe2) [ r_b − ( (1−τ) e_bt2 + 1/2 ) y_b k_b ]. (equation (42))
    - r_b − r_g*3 = p [ r_b − ( (1−τ) ˆe_3 + 1/2 ) y_b k_b ] − p A [ r_g3 − ( (1−τ) e_gt3 + ˆe_3 / 2 ) y_b k_b − ̄C − E(f(e)) ]. (equation (43))
- Presence or absence of a greenium depends on:
  - Transition risk (p, τ),
  - Green bond issuing costs ̄C,
  - Greenwashing costs f(e) (σ parameter),
  - Distribution of emission intensities (e_i ∼ U(0,1]),
  - Brown coupon r_b and technology/productivity y_b, k_b.
- Comparative insights:
  - Lower τ (case 2) raises r_g (green coupon) and reduces greenwashing.
  - Higher τ (case 3) produces ambiguous effects: green bond pricing and greenwashing may increase or decrease depending on parameter constellation because defaults begin to affect green bonds issued by brown firms.

### Policy-relevant implications (as presented)
- Transition policy design matters:
  - Delayed and large carbon taxes increase default risks for brown firms, raise aggregate financial stability risks, and can increase greenwashing incentives.
  - Swift and gradual carbon pricing supports green firm growth and reduces greenwashing and financial instability risks (dynamic extension argument).
- Subsidy and disclosure interplay:
  - Broad subsidies for green bonds without robust information disclosure, regulation, and enforcement can raise greenwashing and undermine emissions reduction.
  - Subsidies combined with strong disclosure and enforcement can alleviate greenwashing and reinforce green effects.

_Official source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022246-print-pdf.pdf_

### introduction of a carbon tax after bond issuance. The transition risk leads to default risk for some

### introduction of a carbon tax after bond issuance. The transition risk leads to default risk for some

### Default risk premium decomposition
- Default risk premium is defined as the difference between default losses of green and brown bonds and equals the spread (difference between yields on green and brown bonds).
- The default risk premium is decomposed into two components: the probability of default and the average loss given default.
- Using (11), (12), (24), (25), (31), (32), (36), (37), the probability of default and average loss given default are calculated for three carbon tax scenarios (Table 1).

- Table 1 (Default risk premium decomposition) — Green bonds
  - Cases: Case 1 (τ = ˆτ), Case 2 (τ < ˆτ), Case 3 (τ > ˆτ)
  - Probability of default:
    - Case 1: 0
    - Case 2: 0
    - Case 3: ˆe^3 − e_gt^3
  - Loss given default:
    - Case 1: 0
    - Case 2: 0
    - Case 3: r_g^3 − [1−τ e_gt^3 + ˆe^3 / 2] y_b k_b + ̄C + E(f(e))

- Table 1 (Default risk premium decomposition) — Brown bonds
  - Cases: Case 1 (τ = ˆτ), Case 2 (τ < ˆτ), Case 3 (τ > ˆτ)
  - Probability of default:
    - Case 1: 1
    - Case 2: 1−e_bt^2
    - Case 3: 1−ˆe^2
  - Loss given default:
    - Case 1: r_b − [1−τ ˆe^1 + 1/2] y_b k_b r_b − [1−τ e_bt^2 + 1/2] y_b k_b r_b − [1−τ ˆe^3 + 1/2] y_b k_b r_b

- Figure 4 (graphical expressions of Table 1):
  - Horizontal axis: level of carbon tax.
  - LHS: average probability of default.
  - RHS: average loss given default.

- Interpretation:
  - When brown bonds have a higher default probability or a higher average loss given default than green bonds, the default risk premium is positive (a greenium exists).
  - Brown bond market concentrates heavily-polluting firms; green bond market comprises all green firms and brown firms with low emission intensities.
  - Greenwashing cost is an increasing function of emission intensity; high-emission brown firms avoid issuing green bonds.

### Special cases with equal green and brown bond prices
- Three cases where r_b = r_g∗ (no greenium):
  1. No transition risk: p = 0 or p = 1.
     - If p = 0: firms and investors believe no carbon taxation before bond maturity; by (41), (42), (43), r_b = r_g∗; greenium does not exist; typically ̄C > 0 so firms will not issue green bonds.
     - If p = 1: carbon tax is implemented in the period; brown firms anticipate default after paying carbon tax and will not enter the market; no defaults in either market; yields equal by no arbitrage.
  2. No greenwashing cost: ∀e_i ∈ (0,1], f(e_i) = 0 implying ˆe = 1.
     - Setting f(e_i) = 0 in (18), (19), (15), (16), (17) shows no pricing difference: r_b = r_g∗.
     - When r_g + ̄C < r_b both brown and green firms issue green bonds — no brown bond market.
     - When r_g + ̄C > r_b both brown and green firms issue brown bonds — no green bond market.
     - When r_g + ̄C = r_b both types indifferent; coupon rates equal and ̄C must be zero.
     - Intuition: if greenwashing is free, green bonds have no signaling value.
  3. No information asymmetry (proof in Section IV).

### Impact of transition risk on green bond pricing and greenwashing
- Lemma 4 (summary):
  - Given state variables (r_b, k_g, k_b, p, ̄C, A_d), green bond coupon rate varies as carbon tax rate changes.
  - Baseline solutions: (r_g∗_1, ˆe∗_1) with baseline carbon tax rate ˆτ.
  - If carbon tax is lower than baseline ˆτ:
    - Green bond coupon rate is higher than the baseline level.
    - Greenwashing is lower than the baseline level.
  - If carbon tax is higher than baseline ˆτ:
    - Green bond coupon rate and greenwashing level are uncertain compared to baseline.

- Mechanism:
  - Given r_b, greenium (r_b − r_g) is interpreted as default risk premium (difference between probabilities of default times loss given default for green and brown bonds).
  - At ˆτ: difference of probabilities of default is 1; difference in loss given default shown in RHS chart.
  - If τ < ˆτ: difference of probabilities of default < 1 and difference in loss given default < baseline → default risk premium smaller → spread smaller → green bond coupon rate higher than baseline.
  - If τ > ˆτ: difference in probabilities < 1 but difference in losses given default > baseline → unclear which effect dominates → change in default risk premium and green bond coupon rate uncertain.
  - Greenwashing threshold determined by spread (r_b − r_g), ̄C, and greenwashing cost f(·) (Lemma 1).
    - Lower τ than baseline → smaller spread → lower greenwashing.
    - Higher τ than baseline → greenwashing uncertain.

### Policy implications
- A. The Need for Carbon Pricing and Climate Financial Regulations
  - Without transition risk, green bond market unlikely to work at scale: investors lack monetary incentive to invest in green bonds unless driven by non-financial green preferences.
  - Green finance requires greenwashing to be costly; otherwise green financial products have no signaling value.
  - Practical measures to make greenwashing costly include guidelines, disclosures, fines for false information about green credentials and emissions, and effective validation and screening before and after issuance.
  - Current status: progress has been made, but guidelines for green financial products are mostly non-binding; firm-level environmental disclosures and reporting on environmental additionality and ex post monitoring need improvement.
  - Regulators recommend detailed reporting and external reviewer assurances; taxonomies of green and brown activities help reduce greenwashing.

- B. The Need for Swift Action on Carbon Pricing
  - Delay in action requires a very high carbon price eventually to meet climate targets (IMF Fiscal Monitor October 2019); risk of delayed large adjustments noted as a potential “climate Minsky moment” (Carney 2015).
  - Swift and gradual introduction of carbon pricing reduces risk of disorderly adjustments and financial instability.
  - In one-period model:
    - By Lemma 2, comparing case 2 to case 1 under a small carbon tax: greenium is small and greenwashing low.
    - By Lemma 3, comparing case 3 to case 1 under a high carbon tax: levels of greenium and greenwashing are uncertain, adding market uncertainty.
  - Multi-period extension (summary):
    - Firms’ types assumed constant; types become public after production each period.
    - Period 1: green firms share α, brown firms share (1−α).
    - End of period 1: green firms (α) remain; brown firms (1−α) exit; equal number of new firms (1−α) with unknown types enter; among new entrants, green share α.
    - After iterations, green firms’ market share increases: gradual transition to carbon-neutral production incentivized by carbon taxation.
    - If no carbon taxation until last period, market shares remain constant; a large final carbon tax causes brown firm bankruptcies, financial instability, sudden production drop.

- C. The Need for Strong Disclosure Requirements Attached to Subsidy Programs
  - Green bond issuance is more costly than conventional bonds due to reporting, consultancy fees, external verification.
  - Governments (e.g., Japan and Singapore) provide subsidies to defray issuance costs; such programs typically include reporting and external verification requirements.
  - Model implications:
    - Green bond financing cost components: green bond coupon rate r_g, green bond issuance cost ̄C, greenwashing cost f(e_i) (Lemma 1).
    - Government subsidies lower ̄C (e.g., reimburse external review and consultant fees), encouraging green bond issuance.
    - Without strong safeguards, subsidies can encourage brown firms to issue green bonds, increasing greenwashing.
    - Strong disclosure requirements and regulation steepen the greenwashing cost function, reducing greenwashing and issuance of green bonds by brown firms.
  - Conclusion: subsidies can support market development but must be accompanied by strong disclosure, regulation, and screening to avoid greenwashing.

### Conclusion (summary)
- Model: adverse selection framework to solve equilibrium prices in green bond market and extent of greenwashing.
- Key findings:
  - A greenium exists when there is asymmetric information about issuer type, transition risks from carbon pricing, and costly greenwashing.
  - Impact of carbon pricing on greenium and greenwashing depends on speed of introduction:
    - Swift but gradual implementation → small greenium and low greenwashing.
    - Delayed and large carbon pricing → ambiguous effect on greenium and greenwashing and additional market uncertainty.
- The model does not assume an exogenous investor green preference; green bond pricing can arise from valuable signaling.
- Policy implications reiterated:
  - Transition risk via carbon pricing is essential in a rational framework without innate green preference.
  - Swift action on carbon pricing preferable to delayed large steps.
  - Strong supervision and disclosure/reporting requirements increase greenwashing costs and reduce greenwashing.
  - Government subsidies should be tied to strong information disclosure and compliance measures.

*Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022246-print-pdf.pdf*

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*Green Bond Pricing and Greenwashing under Asymmetric Information — Working Paper No. WP/2022/246*

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_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022246-print-pdf.pdf_