## wpiea2020053-print-pdf

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---

### I. Introduction — key concepts and goals
- Financial market participants hedge interest rate risk via long/short positions across maturities (“riding the yield curve”).
- Standard theory: anticipated increase in interest rates (i_t) implies a decline in expected future cashflows via the stochastic discount factor (SDF): 1 + ↑i_t = ↓E_t[SDF_{t+1}] (Eq. (1)).
- Paper objective: derive and empirically estimate a non-monotonic SDF driven by investors’ hedging strategies against interest-rate risk, allowing:
  - response to movements along the entire yield curve (not just short rate);
  - evaluation of interest-rate movements relative to historical averages;
  - hedging strategy formation given relative risk aversion.
- Empirical focus: U.S. yield curve, canonical term structure model with observable interest-rate risk factors as in Joslin et al. (2011) and Adrian et al. (2013).

### Main theoretical contributions and findings
- Market SDF probability density is log-normal, introducing time-variation and non-linearity with respect to risk factors.
- Conditional expected SDF derived relative to movements in each interest-rate risk factor holding others constant.
- Non-linearity arises naturally from portfolio optimization with CRRA preferences:
  - Optimal portfolio rules are linear in the Sharpe ratio of U.S. Treasury securities, a leverage constraint, and the risk factors.
  - Hedging strategies depend on conditional expectations of future interest rates: investors go long on long-term securities when yields are above their historical average and vice versa.
- Equilibrium SDF for the market is an envelope of Gaussian distributions because equilibrium excess return on Treasury securities is affine in normally distributed interest-rate risk factors.
- In a low interest-rate environment:
  - More risk averse investors hold longer-duration portfolios relative to less risk averse investors and suffer larger capital losses when estimated yield level increases.
  - More risk averse investors’ valuations of cashflows decrease with yield-level increases; less risk averse investors experience increased cashflow valuations and take opposite positions.
  - An increase in the yield curve in a low-rate environment alters the distribution of absolute risk aversion: capital gains reduce absolute risk aversion; capital losses increase absolute risk aversion, thereby affecting future demand for Treasuries.

### II. Non-linearities in Bond Market Valuations — model setup
- Term structure specification:
  - Zero-coupon nominal yield to maturity r_{τ,t}(Y_t) = A_τ + B_τ Y_t (Eq. (2)), with no-arbitrage-calibrated A_τ and B_τ.
  - State vector Y_t = [Y_{1t} Y_{2t} Y_{3t}]′ composed via principal components: Y_t = W y^{obs}_t.
- Empirical factor decomposition (1975–2017):
  - First three principal components explain over 99% of cross-sectional yield variation: “level” (88.5%), “slope” (8.5%), “curvature” (2.1%).
  - Level = average of all yields; slope ≈ 90% correlated with 10-year–3-month spread; third factor ≈ 70% correlated with Diebold and Li curvature definition (y^{obs}_{10y,t} + y^{obs}_{3m,t} − 2* y^{obs}_{2y,t}).
- Dynamics under the physical distribution P (discrete-time AR(1)):
  - Y_{t+1} = K^P_0 + (K^P_1 + I) Y_t + Σ ε^P_{t+1} (Eq. (3)); mean reversion to −(K^P_1)^{-1} K^P_0 if eigenvalues of K^P_1 are strictly negative.
  - If K^P_1 = 0 (random walk), no conditional expected future yield change ⇒ no hedging benefit.
- Risk-neutral dynamics Q:
  - Y_{t+1} = K^Q_0 + (K^Q_1 + I) Y_t + Σ ε^Q_{t+1} (Eq. (4)); Q adjusts P mean by price of risk per unit volatility.
- Risk-free rate linear in factors: r_t(Y_t) ≡ r_t = ρ_0 + ρ_1 Y_t (Eq. (5)).
- SDF specification exponential-quadratic:
  - M_{t+1} = exp{−r_t − 1/2 Λ′_t Λ_t − Λ′_t ε^P_{t+1}} (Eq. (6)), with price of risk Λ_t affine in factors via Σ Λ_t = K^P_0 − K^Q_0 + (K^P_1 − K^Q_1) Y_t (Eq. (7)).
  - Risk adjustment −1/2 Λ′_t Λ_t is quadratic in Y_t (Eq. (8)).
- Estimation details:
  - Monthly U.S. Treasury yields, January 1975–March 2017; 12 maturities: 3m, 6m, 1y, 2y, 3y, 4y, 5y, 6y, 7y, 8y, 9y, 10y.
  - Two-stage Joslin et al. (2011) procedure: Eq. (3) estimated by OLS; rotated parameters allow ML convergence to global optimum.
  - Model fits shown for 3-month and 5-year yields.

### III. The Expected Stochastic Discount Factor — shape and implications
- One-month conditional expected SDF given Y_t = Y is log-normal Gaussian envelope:
  - E_t[M_{t+1}|Y_t] ≡ M(Y) = M_1 exp{−1/2 (Y − μ_M)′ (σ_M σ′_M)^{-1} (Y − μ_M)} (Eq. (9)).
  - μ_M: mean of expected SDF; σ_M: lower triangular matrix from innovations’ variance-covariance; M_1 absorbs horizon-dependent term.
- Numerical conditional maxima and interpretation (one-month ahead):
  - Level: expected SDF maximum 0.9978 at μ^{level}_M = 0.0061 ⇒ corresponds to one-month discount rate of 0.22% (from Eq. (1)).
  - Slope: expected SDF maximum 0.9597 at μ^{slope}_M = 0.0023.
  - Curvature: expected SDF maximum 0.9902 at μ^{curve}_M = 0.0011.
- Two key takeaways:
  - Effect on expected cashflows depends on current state relative to conditional mean of expected SDF.
  - Expected SDF depends on movements across the entire maturity spectrum; decomposing by factor helps assess net valuation effects via the non-linear risk adjustment.

### Application — November 2016 U.S. yield changes
- Empirical event: October 31, 2016 → November 30, 2016 Treasury yield curve shift (3-month to 10-year):
  - Level increased by 35%.
  - Slope steepened by 40%.
  - Curvature increased by 15%.
- Historical unconditional means shown (since 1975):
  - ̄Y^{level} = 0.012; ̄Y^{slope} = 0.0024; ̄Y^{curve} = 0.0009.
- Interpretation:
  - Although increases in level typically lower discounting (if occurring to right of conditional mean), the November 2016 increases occurred from the left of conditional means ⇒ they increased investors’ expected SDF.
  - Steepening and curvature increase also moved from left of conditional means and increased expected SDF.
  - Net effect for this event: overall increase in expected value of future cashflows for bond market investors.

### IV. Investors’ Risk Aversion and Portfolio Positions — equilibrium and portfolio rules
- Continuous-time mapping (from discrete-time AR(1)):
  - dX(s) = [γ^P − A^P X(s)] ds + Σ_X dẽ_s, with A^P = −ln(K^P_1 + I), γ^P = A^P (I − e^{−A^P})^{-1} K^P_0.
  - Continuous-time price of risk: Λ(X(s)) = (Σ_X)^{-1}(γ^P − γ^Q) − (Σ_X)^{-1}(A^P − A^Q) X(s) (Eq. (10)).
- Excess holding period return on maturity-τ zero-coupon security:
  - dP_{τ,s}/P_{τ,s} − r(s) = b_τ[(γ^P − γ^Q) − (A^P − A^Q) X(s)] ds + b_τ Σ_X dẽ_s = [μ_τ(s) − r(s)] ds + b_τ Σ_X dẽ_s (Eq. (11)).
- Investor grouping and demand aggregation:
  - J investment buckets, j = 1,...,J. Each bucket j maximizes expected lifetime utility for terminal wealth at horizon h_j under CRRA with coefficient γ_j, subject to liquidity constraint ξ_j (percentage limit invested in Treasuries). W_j is total investment by bucket j.
  - Total demand D(s) = ∑_{j=1}^J ω_j(s) W_j (Eq. (12)), where ω_j(s) is optimal portfolio weights.
- Portfolio rule (optimal percentage of wealth invested by investor j) — Eq. (13):
  - ω_j(s) = 1/γ_j [ ω_1(μ(s) − μ_τ(s) ι) ︸︷︷︸ Sharpe Ratio + ω_1 ω_2 ξ_j ︸︷︷︸ Leverage + ω_1 ω_3 γ_j (σ_j(h_j) σ_j(h_j)′)^{-1} [ X(s) − μ_j(h_j) ] ︸︷︷︸ Hedging Demand Against Interest-Rate Risk ].
  - ω_1(s) = ξ_j − ι′ ω_j(s), ι′ = (1...1).
  - Definitions for ω coefficients given in text.
  - Interpretation of terms:
    - First term: Sharpe-ratio-based speculative demand scaled by 1/γ_j.
    - Second term: leverage adjustment to satisfy ξ_j constraint.
    - Third term: hedging demand (Merton) under mean-reverting factors; positive if X(s) > μ_j(h_j), negative if X(s) < μ_j(h_j).
- Numerical illustrative case (two investor buckets j = 1, 2; 3-month vs 5-year bonds; h_j = 1 year; subjective discount rate 5%; ξ_j = 1):
  - Estimation period: January 1999–December 2007; unconditional means for this sample:
    - ̄X^{level} = −0.0177; ̄X^{slope} = −0.0159; ̄X^{curve} = 0.0045.
  - Risk aversion: γ_1 = 10 (more risk averse); γ_2 = 5 (less risk averse).
  - More risk averse investor’s expected lifetime utility (level changes, others constant):
    - Conditional mean μ_{j=1}(h_{j=1} = 1) = −0.0593.
    - Standard deviation σ_{j=1}(h_{j=1} = 1) = 0.1512.
  - Less risk averse investor:
    - Conditional mean μ_{j=2}(h_{j=2} = 1) = −0.0567.
    - Standard deviation σ_{j=2}(h_{j=2} = 1) = 0.1197.
  - At steady-state X^{level} = −0.0177:
    - More risk averse investor allocates 46% to 3-month bond and 54% to 5-year bond.
      - Hedging demand accounts for 46% (i.e., 46% due to hedging), so only 8% of wealth in 5-year bond from Sharpe ratio + leverage.
    - Less risk averse investor allocates 14% to 3-month bond and 86% to 5-year bond.
      - Of the 86% in 5-year bond, 69% is due to hedging demand.
  - Additional numerical conditional statistics:
    - 0.1065 for the more risk averse group;
    - conditional mean −0.0366 and standard deviation 0.0798 for the less risk averse group.

### Portfolio positions conditional on the estimated level of the yield curve
- Investment choices between 3-month and 5-year bonds; ξ_j = 1 and h_j = 1.
- When estimated level X^{level} > −0.0177 (higher than steady state), anticipated mean reversion implies the level is expected to fall, so longer-duration bonds yield larger capital gains.
  - Example allocation in high interest-rate periods: less risk averse investor (γ = 5) allocates 86% to the 5-year bond and 14% to the short-term bond.
- When X^{level} < −0.0177 (low interest-rate environment), mean reversion implies expected rise in the level, so portfolios are shorter on longer-term bonds and longer on short-term bonds.
  - Example: for μ_{j=1}(h_{j=1} = 1) = −0.0593 < −0.0177, the more risk averse investor places −113% of her wealth in 5-year bonds and 213% in the 3-month bond (accomplished with no hedging demand).
- Changing the discount rate has a small impact on portfolio decision:
  - Discount rate 10% instead of 5% leads to a 48-52% allocation, compared to 46-54%.
- Risk-aversion heterogeneity:
  - More risk averse investor: γ = 10.
  - Less risk averse investor: γ = 5.
  - The less risk averse investor increases the magnitude of the bet on mean reversion of the yield-curve level and thus chooses higher duration in high-rate environments relative to the more risk averse investor.
- Dispersion of expected lifetime utility (Gaussian functions):
  - Standard deviation 0.1512 for γ = 10.
  - Standard deviation 0.1197 for γ = 5.
  - Maxima of expected lifetime utility: 0.9757 at point A (more risk averse) and 0.9738 at point B (less risk averse).

### V. Hedging strategies and market clearing in a low interest rate environment
- Market clearing condition for Treasury securities: D(s) = S(s).
- Under assumption that (ω_1) inverse exists, expected excess return depends on aggregate investor behavior; key decomposition uses θ ≡ 1 / Σ_{J} W_j γ_j and θ_j ≡ W_j γ_j θ.
- Price of risk coefficients in equilibrium:
  - (b − ι b_τ) (γ^P − γ^Q) = θ (ω_1)^{-1} S(s) − ω_2 Σ_{J} θ_j ξ_j + ω_3 Σ_{J} θ_j γ_j (σ_j(h_j) σ_j(h_j)′)^{-1} μ_j(h_j).  (Eq. (15))
  - (b − ι b_τ) (A^P − A^Q) = ω_3 Σ_{J} θ_j γ_j (σ_j(h_j) σ_j(h_j)′)^{-1}.  (Eq. (16))
- Interpretation of Eq. (15):
  - Positive relation to quantity of Treasury securities available, weighted by total absolute risk aversion.
  - Negative relation to aggregate leverage ratio (Σ_{J} θ_j ξ_j is the mean leverage constraint).
  - Last term captures expected lifetime utility across investor buckets, adjusted by covariance with interest-rate risk factors (ω_3).
- Hedging demand impact on equilibrium excess return:
  - Hedging = ω_3 Σ_{J} γ_j θ_j (σ_j(h_j) σ_j(h_j)′)^{-1} (X − μ_j(h_j)).  (Eq. (17))
  - Thus, expected excess return is negatively related to (σ_j(h_j) σ_j(h_j)′)^{-1} X relative to conditional mean (σ_j(h_j) σ_j(h_j)′)^{-1} μ_j(h_j) desired by investors in bucket j.

### Equilibrium implications, wealth effects, and risk aversion dynamics
- Portfolio positions depend on γ_j, but price of risk depends on wealth W_j and distribution of absolute risk aversion γ_j / W_j.
- Holding relative risk aversion fixed, less wealthy investors hold portfolios with fewer longer-term Treasuries to reduce risk.
- Equilibrium return is a fixed point of Eqs. (15) and (16) with two investor types (γ = 10 and γ = 5):
  - In equilibrium with zero net supply among investors, distribution must imply one investor is shorter and the other longer on long-term Treasuries; equilibrium cannot be left of point A or right of point B.
  - If S(s) sufficiently positive, both investors can be long long-term Treasuries, with more risk averse investor even longer.
- Wealth changes affect absolute risk aversion:
  - W γ / W ∂γ W / ∂W = −1, so a percentage change in wealth leads to the same percentage decrease in the level of absolute risk aversion.
  - Example: an unanticipated increase in estimated level of yield curve in a low-rate environment causes larger percentage capital losses for more risk averse investors (γ = 10) than for less risk averse investors (γ = 5) when W_{γ=5} > W_{γ=10}, widening the spread in absolute risk aversion and decreasing hedging behavior by the more risk averse investors relative to the less risk averse investors.

### Key conclusions and market-level findings
- Non-linearities in the stochastic discount factor are critical for transmission of interest-rate risk across investors’ portfolio positions: movements away from the conditional mean of the expected SDF lead to declines in expected future cashflows.
- Empirical observation: during the November 2016 U.S. presidential elections, cashflow valuations increased as factors reverted back toward their mean.
- Portfolio implications from the no-arbitrage model:
  - When interest-rate risk factors are above long-term mean, investors go long long-term bonds anticipating future falls in factors and capital gains.
  - When factors are below their means, positions reverse as investors anticipate increases in factors.
- Equilibrium characterization:
  - The estimated no-arbitrage model of the term structure is an equilibrium where investors follow optimal portfolio rules with different degrees of absolute risk aversion.
  - More risk averse investors hold longer positions in long-term bonds relative to less risk averse investors and thus suffer larger capital losses from average interest-rate increases, amplifying absolute risk aversion.
- Twofold market impact of interest-rate risk exposure:
  - Changes in the entire yield curve influence market prices of risk non-linearly depending on current value of interest-rate risk factors relative to historical averages.
  - Changes in the yield curve influence portfolio positions by changing the distribution of investors’ absolute risk aversion in equilibrium, amplifying risk exposure of risk-averse or less wealthy investors.

*Source: wpiea2020053-print-pdf (selected excerpts).*

### References .......................................................................................... 21

### wpiea2020053-print-pdf - References .......................................................................................... 21

### I. Introduction — key concepts and goals
- Financial market participants hedge interest rate risk via long/short positions across maturities (“riding the yield curve”).
- Standard theory: anticipated increase in interest rates (i
t
) implies a decline in expected future cashflows via the stochastic discount factor (SDF): 1 +↑i
t
=↓E
t
[SDF
t+1
] (Eq. (1)).
- Paper objective: derive and empirically estimate a non-monotonic SDF driven by investors’ hedging strategies against interest-rate risk, allowing:
  - response to movements along the entire yield curve (not just short rate);
  - evaluation of interest-rate movements relative to historical averages;
  - hedging strategy formation given relative risk aversion.
- Empirical focus: U.S. yield curve, canonical term structure model with observable interest-rate risk factors as in Joslin et al. (2011) and Adrian et al. (2013).

### Main theoretical contributions and findings
- The market SDF probability density is log-normal, introducing time-variation and non-linearity with respect to risk factors.
- Conditional expected SDF is derived relative to movements in each interest-rate risk factor holding others constant (following Chami et al. (2017), Cosimano and Ma (2018)).
- Non-linearity arises naturally from portfolio optimization with constant relative risk aversion (CRRA) preferences (following Sangvinatsos and Wachter (2005)):
  - Optimal portfolio rules are linear in the Sharpe ratio of U.S. Treasury securities, a leverage constraint, and the risk factors.
  - Hedging strategies depend on conditional expectations of future interest rates: investors go long on long-term securities when yields are above their historical average and vice versa.
- Equilibrium SDF for the market is an envelope of Gaussian distributions because equilibrium excess return on Treasury securities is affine in normally distributed interest-rate risk factors.
- In a low interest-rate environment:
  - More risk averse investors hold longer-duration portfolios relative to less risk averse investors and suffer larger capital losses when estimated yield level increases.
  - More risk averse investors’ valuations of cashflows decrease with yield-level increases; less risk averse investors experience increased cashflow valuations and take opposite positions.
  - An increase in the yield curve in a low-rate environment alters the distribution of absolute risk aversion: capital gains reduce absolute risk aversion (wealthier ⇒ more willing to take risk); capital losses increase absolute risk aversion (more conservative), thereby affecting future demand for Treasuries.

### Literature linkage and puzzles addressed
- Provides additional evidence on a non-monotonic SDF / pricing kernel (pricing kernel puzzle per Beare and Schmidt (2016)).
- Aligns with prior findings on non-linearity in risk premia across forward rates, Treasury spreads, and equity returns (Fama and Bliss (1987); Cochrane and Piazzesi (2005); Campbell and Shiller (1991); Parker and Julliard (2005); Lustig and Van Nieuwerburgh (2005); Yogo (2006); Sousa (2010)).
- Offers investors’ hedging against maturity risk (shape of yield curve and mean-reversion) as an explanation for a non-linear expected SDF.

### II. Non-linearities in Bond Market Valuations — model setup
- Term structure specification:
  - Zero-coupon nominal yield to maturity r
τ,t
(Y
t
) = A
τ
+ B
τ
Y
t
 (Eq. (2)), with no-arbitrage-calibrated A
τ
 and B
τ
.
  - State vector Y
t
= [Y
1t
 Y
2t
 Y
3t
]′ composed via principal components: Y
t
= W y
obs
t
.
- Empirical factor decomposition (1975–2017):
  - First three principal components explain over 99% of cross-sectional yield variation: “level” (88.5%), “slope” (8.5%), “curvature” (2.1%).
  - Level = average of all yields; slope ≈ 90% correlated with 10-year–3-month spread; third factor ≈ 70% correlated with Diebold and Li curvature definition (y
obs
10y,t
 + y
obs
3m,t
 − 2∗y
obs
2y,t
).
- Dynamics under the physical distribution P (discrete-time AR(1)):
  - Y
t+1
 = K
P
0
 + (K
P
1
 + I) Y
t
 + Σ ε
P
t+1
 (Eq. (3)); mean reversion to −(K
P
1
)
−1
K
P
0
 if eigenvalues of K
P
1
 are strictly negative.
  - If K
P
1
 = 0 (random walk), no conditional expected future yield change ⇒ no hedging benefit.
- Risk-neutral dynamics Q:
  - Y
t+1
 = K
Q
0
 + (K
Q
1
 + I) Y
t
 + Σ ε
Q
t+1
 (Eq. (4)); Q adjusts P mean by price of risk per unit volatility.
- Risk-free rate linear in factors: r
t
(Y
t
) ≡ r
t
 = ρ
0
 + ρ
1
Y
t
 (Eq. (5)).
- SDF assumed exponential-quadratic:
  - M
t+1
 = exp{−r
t
 − 1/2 Λ
′
t Λ
t − Λ
′
t ε
P
t+1
} (Eq. (6)), with price of risk Λ
t
affine in factors via ΣΛ
t
 = K
P
0
 − K
Q
0
 + (K
P
1
 − K
Q
1
) Y
t
 (Eq. (7)).
  - Risk adjustment −1/2 Λ
′
t Λ
t is quadratic in Y
t
: −1/2 Λ
′
t Λ
t = −1/2 [K
P
′
0
 − K
Q
′
0
 + Y
′
t (K
P
1
 − K
Q
1
)] (Σ′Σ)
−1
[K
P
0
 − K
Q
0
 + (K
P
1
 − K
Q
1
) Y
t
] (Eq. (8)).
- Estimation details:
  - Monthly U.S. Treasury yields, January 1975–March 2017; 12 maturities: 3m, 6m, 1y, 2y, 3y, 4y, 5y, 6y, 7y, 8y, 9y, 10y.
  - Two-stage Joslin et al. (2011) procedure: Eq. (3) estimated by OLS; rotated parameters allow ML convergence to global optimum.
  - Model fits shown for 3-month and 5-year yields (Fig. 2).

### B. The Expected Stochastic Discount Factor — shape and implications
- One-month conditional expected SDF given Y
t
= Y is log-normal Gaussian envelope:
  - E
t
[M
t+1
|Y
t
] ≡ M(Y) = M
1
 exp{−1/2 (Y − μ
M
)′ (σ
M
σ
′
M
)
−1
 (Y − μ
M
)} (Eq. (9)).
  - μ
M
: mean of expected SDF; σ
M
: lower triangular matrix from innovations’ variance-covariance; M
1
absorbs horizon-dependent term.
- Numerical conditional maxima and interpretation (one-month ahead):
  - Level: expected SDF maximum 0.9978 at μ
level
M
 = 0.0061 ⇒ corresponds to one-month discount rate of 0.22% (from Eq. (1)).
  - Slope: expected SDF maximum 0.9597 at μ
slope
M
 = 0.0023.
  - Curvature: expected SDF maximum 0.9902 at μ
curve
M
 = 0.0011.
- Two key takeaways:
  - Effect on expected cashflows depends on current state relative to conditional mean of expected SDF.
  - Expected SDF depends on movements across the entire maturity spectrum; decomposing by factor helps assess net valuation effects via the non-linear risk adjustment.

### C. Application — November 2016 U.S. yield changes
- Empirical event: October 31, 2016 → November 30, 2016 Treasury yield curve shift (3-month to 10-year):
  - Level increased by 35%.
  - Slope steepened by 40%.
  - Curvature increased by 15%.
- Historical unconditional means shown (since 1975):
  - ̄Y
level
 = 0.012; ̄Y
slope
 = 0.0024; ̄Y
curve
 = 0.0009.
- Interpretation:
  - Although increases in level typically lower discounting (if occurring to right of conditional mean), the November 2016 increases occurred from the left of conditional means ⇒ they increased investors’ expected SDF.
  - Steepening and curvature increase also moved from left of conditional means and increased expected SDF.
  - Net effect for this event: overall increase in expected value of future cashflows for bond market investors.

### III. Investors’ Risk Aversion and Portfolio Positions — equilibrium and portfolio rules
- Equilibrium expected return on Treasuries is linear in interest-rate risk factors, supply of Treasuries, and average leverage; dependence reflects Merton hedging demand.
- Continuous-time mapping (from discrete-time AR(1)):
  - dX(s) = [γ
P
 − A
P
 X(s)] ds + Σ
X
 d
s
, with A
P
 = −ln(K
P
1
 + I), γ
P
 = A
P
 (I − e
−A
P
)
−1
 K
P
0
.
  - Continuous-time price of risk: Λ(X(s)) = (Σ
X
)
−1
(γ
P
 − γ
Q
) − (Σ
X
)
−1
(A
P
 − A
Q
) X(s) (Eq. (10)).
- Excess holding period return on maturity-τ zero-coupon security:
  - dP
τ,s
/P
τ,s
 − r(s) = b
τ
[(γ
P
 − γ
Q
) − (A
P
 − A
Q
) X(s)] ds + b
τ
Σ
X
 d
s
 = [μ
τ
(s) − r(s)] ds + b
τ
Σ
X
 d
s
 (Eq. (11)).
- Investor grouping and demand aggregation:
  - J investment buckets, j = 1,...,J. Each bucket j maximizes expected lifetime utility for terminal wealth at horizon h
j
 under CRRA with coefficient γ
j
, subject to liquidity constraint ξ
j
 (percentage limit invested in Treasuries). W
j
 is total investment by bucket j.
  - Total demand D(s) = ∑
j=1
J
 ω
j
(s) W
j
 (Eq. (12)), where ω
j
(s) is optimal portfolio weights.
- Portfolio rule (optimal percentage of wealth invested by investor j) — Eq. (13):
  - ω
j
(s) = 1/γ
j
[ ω
1
(μ(s) − μ
τ
(s) ι) ︸︷︷︸ Sharpe Ratio + ω
1
ω
2
 ξ
j
︸︷︷︸ Leverage + ω
1
ω
3
 γ
j
(σ
j
(h
j
) σ
j
(h
j
)′)
−1
[ X(s) − μ
j
(h
j
) ] ︸︷︷︸ Hedging Demand Against Interest-Rate Risk ].
  - ω
1
(s) = ξ
j
 − ι′ ω
j
(s), ι′ = (1...1).
  - Definitions:
    - ω
1
 ≡ [b Σ
X
Σ
′
X
 b
′ + ιι
′
 b
τ
Σ
X
Σ
′
X
 b
′
τ
 − 2 b Σ
X
Σ
′
X
 b
′
τ ι
′
] −1.
    - ω
2
 ≡ 2 (b Σ
X
Σ
′
X
 b
′
τ − ι b
τ
Σ
X
Σ
′
X
 b
′
τ).
    - ω
3
 ≡ (b − ι b
τ
) Σ
X
Σ
′
X
.
    - b′ = [b
2τ
 ... b
nτ
] vector of bond price elasticities; b′ − ι b
τ
 = [b
2τ
 − b
τ
 ... b
nτ
 − b
τ
].
  - Interpretation of terms:
    - First term: Sharpe-ratio-based speculative demand scaled by 1/γ
j
.
    - Second term: leverage adjustment to satisfy ξ
j
 constraint.
    - Third term: hedging demand (Merton) under mean-reverting factors; γ
j
(σ
j
(h
j
) σ
j
(h
j
)′)
−1
[ X(s) − μ
j
(h
j
) ] captures risk-adjusted duration sensitivity; positive if X(s) > μ
j
(h
j
), negative if X(s) < μ
j
(h
j
).
- Numerical illustrative case (two investor buckets j = 1, 2; 3-month vs 5-year bonds; h
j
 = 1 year; subjective discount rate 5%; ξ
j
 = 1):
  - Estimation period: January 1999–December 2007; unconditional means for this sample:
    - ̄X
level
 = −0.0177; ̄X
slope
 = −0.0159; ̄X
curve
 = 0.0045.
  - Risk aversion: γ
1
 = 10 (more risk averse); γ
2
 = 5 (less risk averse).
  - More risk averse investor’s expected lifetime utility (level changes, others constant):
    - Conditional mean μ
j=1
(h
j=1
 = 1) = −0.0593.
    - Standard deviation σ
j=1
(h
j=1
 = 1) = 0.1512.
  - Less risk averse investor:
    - Conditional mean μ
j=2
(h
j=2
 = 1) = −0.0567.
    - Standard deviation σ
j=2
(h
j=2
 = 1) = 0.1197.
  - At steady-state X
level
 = −0.0177:
    - More risk averse investor allocates 46% to 3-month bond and 54% to 5-year bond.
      - Hedging demand accounts for 46% (i.e., 46% due to hedging), so only 8% of wealth in 5-year bond from Sharpe ratio + leverage.
    - Less risk averse investor allocates 14% to 3-month bond and 86% to 5-year bond.
      - Of the 86% in 5-year bond, 69% is due to hedging demand.

*Source: wpiea2020053-print-pdf - References .......................................................................................... 21*

### 0.1065 for the more risk averse group; and conditional mean−0.0366 and standard deviation 0.0798 for the

### wpiea2020053-print-pdf - 0.1065 for the more risk averse group; and conditional mean−0.0366 and standard deviation 0.0798 for the

### Portfolio positions conditional on the estimated level of the yield curve
- Investors choose wealth allocation between a 3-month bond and a 5-year bond; leverage limited (ξj = 1) and investment horizon one year (hj = 1).
- When the estimated level Xlevel > −0.0177 (higher than steady state), anticipated mean reversion implies the level is expected to fall, so longer-duration bonds yield larger capital gains.
  - Example allocation in high interest-rate periods: less risk averse investor (γ = 5) allocates 86% to the 5-year bond and 14% to the short-term bond.
- When Xlevel < −0.0177 (low interest-rate environment), mean reversion implies expected rise in the level, so portfolios are shorter on longer-term bonds and longer on short-term bonds.
  - Example: for μj=1 (hj=1) = −0.0593 < −0.0177, the more risk averse investor places −113% of her wealth in 5-year bonds and 213% in the 3-month bond (accomplished with no hedging demand).
- Changing the discount rate has a small impact on portfolio decision:
  - Discount rate 10% instead of 5% leads to a 48-52% allocation, compared to 46-54%.
- Risk-aversion heterogeneity:
  - More risk averse investor: γ = 10.
  - Less risk averse investor: γ = 5.
  - The less risk averse investor increases the magnitude of the bet on mean reversion of the yield-curve level and thus chooses higher duration in high-rate environments relative to the more risk averse investor.
- Dispersion of expected lifetime utility (Gaussian functions):
  - Standard deviation 0.1512 for γ = 10.
  - Standard deviation 0.1197 for γ = 5.
  - Maxima of expected lifetime utility: 0.9757 at point A (more risk averse) and 0.9738 at point B (less risk averse).
- Numerical conditional statistics reported in the source:
  - 0.1065 for the more risk averse group;
  - conditional mean −0.0366 and standard deviation 0.0798 for the less risk averse group.

### Hedging strategies and market clearing in a low interest rate environment
- Market clearing condition for Treasury securities: D(s) = S(s).
- Under assumption that (ω1) inverse exists, expected excess return depends on aggregate investor behavior; key decomposition uses θ ≡ 1 / ΣJ j=1 Wj γj and θj ≡ Wj γj θ.
- Price of risk coefficients in equilibrium (as given):
  - (b − ι bτ) (γP − γQ) = θ (ω1)−1 S(s) − ω2 ΣJ j=1 θj ξj + ω3 ΣJ j=1 θj γj (σj(hj) σj(hj)′)−1 μj(hj).  (Eq. (15))
  - (b − ι bτ) (AP − AQ) = ω3 ΣJ j=1 θj γj (σj(hj) σj(hj)′)−1.  (Eq. (16))
- Interpretation of terms in Eq. (15):
  - Positive relation to quantity of Treasury securities available, weighted by total absolute risk aversion.
  - Negative relation to aggregate leverage ratio (ΣJ j=1 θj ξj is the mean leverage constraint for the investor distribution).
  - Last term captures expected lifetime utility across investor buckets, adjusted by covariance with interest-rate risk factors (ω3).
- Slope of the price of risk (Eq. (16)) captures how changes in interest-rate risk factors impact expected excess returns via marginal investor variance-covariance of expected utility; amplified by ω3 and investors’ absolute risk aversion.
- Hedging demand impact on equilibrium excess return (as stated):
  - Hedging = ω3 ΣJ j=1 γj θj (σj(hj) σj(hj)′)−1 (X − μj(hj)).  (Eq. (17))
  - Thus, expected excess return is negatively related to (σj(hj) σj(hj)′)−1 X relative to conditional mean (σj(hj) σj(hj)′)−1 μj(hj) desired by investors in bucket j.

### Equilibrium implications, wealth effects, and risk aversion dynamics
- The portfolio position depends on constant relative risk aversion γj, but the price of risk also depends on wealth Wj and the distribution of absolute risk aversion γj / Wj.
- Holding relative risk aversion fixed, less wealthy investors hold portfolios with fewer longer-term Treasuries to reduce risk.
- Equilibrium return can be found as fixed point of Eqs. (15) and (16) with two investor types (γ = 10 and γ = 5).
  - In equilibrium with zero net supply among investors, the distribution must imply one investor is shorter and the other longer on long-term Treasuries; equilibrium cannot be left of point A or right of point B (from Fig. 6).
  - If S(s) sufficiently positive, both investors can be long long-term Treasuries, with more risk averse investor even longer.
- Wealth changes affect absolute risk aversion:
  - Wγ / W ∂γ W / ∂W = −1, so a percentage change in wealth leads to the same percentage decrease in the level of absolute risk aversion.
  - Example dynamic: an unanticipated increase in estimated level of yield curve in a low-rate environment causes larger percentage capital losses for more risk averse investors (γ = 10) than for less risk averse investors (γ = 5) when Wγ=5 > Wγ=10, widening the spread in absolute risk aversion and decreasing hedging behavior by the more risk averse investors relative to the less risk averse investors.

### Key conclusions and market-level findings
- Non-linearities in the stochastic discount factor (SDF) are critical for transmission of interest-rate risk across investors’ portfolio positions: movements away from the conditional mean of the expected SDF lead to declines in expected future cashflows.
- Empirical observation: during the November 2016 U.S. presidential elections, cashflow valuations increased as factors reverted back toward their mean.
- Portfolio implications from the no-arbitrage model:
  - When interest-rate risk factors are above long-term mean, investors go long long-term bonds anticipating future falls in factors and capital gains.
  - When factors are below their means, positions reverse as investors anticipate increases in factors.
- Equilibrium characterization:
  - The estimated no-arbitrage model of the term structure is an equilibrium where investors follow optimal portfolio rules with different degrees of absolute risk aversion.
  - More risk averse investors hold longer positions in long-term bonds relative to less risk averse investors and thus suffer larger capital losses from average interest-rate increases, amplifying absolute risk aversion.
- Twofold market impact of interest-rate risk exposure:
  - Changes in the entire yield curve influence market prices of risk non-linearly depending on current value of interest-rate risk factors relative to historical averages.
  - Changes in the yield curve influence portfolio positions by changing the distribution of investors’ absolute risk aversion in equilibrium, amplifying risk exposure of risk-averse or less wealthy investors.

*Source: wpiea2020053-print-pdf (selected excerpts).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020053-print-pdf.pdf_