A Production-Based Model for the Term Structure U Wharton School of the University of Pennsylvania U Term Structure Wharton School of the University 1 / 19
Production-based asset pricing in the literature General equilibrium with endogenous capital U Term Structure Wharton School of the University 2 / 19
Production-based asset pricing in the literature General equilibrium with endogenous capital "Pure" production-based: Term Structure Wharton School of the University 2 / 19
Production-based asset pricing in the literature General equilibrium with endogenous capital "Pure" production-based: Firm s return function of investment, productivity... Cochrane 1991) Term Structure Wharton School of the University 2 / 19
Production-based asset pricing in the literature General equilibrium with endogenous capital "Pure" production-based: Firm s return function of investment, productivity... Cochrane 1991) "Complete" production-based pricing Cochrane 1988, 1993, Belo 2010, Jermann 2010) Term Structure Wharton School of the University 2 / 19
What is done Present a production-based model for pricing nominal bonds U Term Structure Wharton School of the University 3 / 19
What is done Present a production-based model for pricing nominal bonds Examine implied term structure quantitatively and analytically U Term Structure Wharton School of the University 3 / 19
Findings Match average and standard deviation of longer term yields Term Structure Wharton School of the University 4 / 19
Findings Match average and standard deviation of longer term yields Time-varying premiums, partially match Fama-Bliss Term Structure Wharton School of the University 4 / 19
Findings Match average and standard deviation of longer term yields Time-varying premiums, partially match Fama-Bliss Depreciation rates are important for term premium Term Structure Wharton School of the University 4 / 19
Real Model, 1 Uncertainty: s s 1, s 2 ), current realization s t, history s t Term Structure Wharton School of the University 5 / 19
Real Model, 1 Uncertainty: s s 1, s 2 ), current realization s t, history s t Firms solve max {I,K } t=0 {Kj P s t) F s t 1 )} j 1,2), st) s t 2 j=1 H j Kj s t 1 ), I j s t ) ) s.t. K j s t ) = K j s t 1 ) 1 δ j ) + I j s t ), s t, j, Term Structure Wharton School of the University 5 / 19
Real Model, 1 Uncertainty: s s 1, s 2 ), current realization s t, history s t Firms solve max {I,K } t=0 {Kj P s t) F s t 1 )} j 1,2), st) s t 2 j=1 H j Kj s t 1 ), I j s t ) ) s.t. K j s t ) = K j s t 1 ) 1 δ j ) + I j s t ), s t, j, F...) = 2 j=1 A j s t ) K j s t 1 ) Term Structure Wharton School of the University 5 / 19
Real Model, 1 Uncertainty: s s 1, s 2 ), current realization s t, history s t Firms solve max {I,K } t=0 {Kj P s t) F s t 1 )} j 1,2), st) s t 2 j=1 H j Kj s t 1 ), I j s t ) ) s.t. K j s t ) = K j s t 1 ) 1 δ j ) + I j s t ), s t, j, F...) = 2 j=1 A j s t ) K j s t 1 ) { bj H j...) = ν Ij j s t ) /K j s t 1 )) } ν j + cj K j s t 1 ) Term Structure Wharton School of the University 5 / 19
Real Model, 2 First-order conditions 1 = s t+1 P s t+1 s t) R I j s t, s t+1 ) for j = 1, 2 with and R I j s t, s t+1 ) FKj s t,s t+1 ) H j,1 s t,s t+1 )+1 δ j)q j s t,s t+1 ) q j s t ) q j s t ) Ij s t ) ) νj 1 = H j,2...) = b j K j s t 1 ) ) U Term Structure Wharton School of the University 6 / 19
Real Model, 3 Recovering state prices [ R I 1 s t, s 1 ) R1 I st, s 2 ) R2 I st, s 1 ) R2 I st, s 2 ) ] [ P s1 s t ) P s 2 s t ) ] = 1 Term Structure Wharton School of the University 7 / 19
Real Model, 3 Recovering state prices [ R I 1 s t, s 1 ) R1 I st, s 2 ) R2 I st, s 1 ) R2 I st, s 2 ) ] [ P s1 s t ) P s 2 s t ) ] = 1 so that state prices depend on I1 s t ) K 1 s t 1 ), I 2 s t ) K 2 s t 1 ), λi 1 s t+1 ), λ I 2 s t+1 ), A j s t+1 )) Term Structure Wharton School of the University 7 / 19
Nominal bonds Assume λ P z t ), with z t z 1, z 2 ) Term Structure Wharton School of the University 8 / 19
Nominal bonds Assume λ P z t ), with z t z 1, z 2 ) Assume investment and technology not contingent on inflation. For instance, P s 1 s t) = P s 1 s t, z t ) = P s1, z 1 s t, z t ) + P s1, z 2 s t, z t ) Term Structure Wharton School of the University 8 / 19
Nominal bonds Assume λ P z t ), with z t z 1, z 2 ) Assume investment and technology not contingent on inflation. For instance, P s 1 s t) = P s 1 s t, z t ) = P s1, z 1 s t, z t ) + P s1, z 2 s t, z t ) Inflation not directly priced. For instance, P s 1, z 1 s t Prs, z t ) = 1,z 1 s t,z t ) P s 1, z 2 s t, z t ) = 1 Prs 1,z 1 s t,z t )+Prs 1,z 2 s t,z t ) Prs 1,z 1 s t,z t ) Prs 1,z 1 s t,z t )+Prs 1,z 2 s t,z t ) ) P s 1 s t ), and ) P s 1 s t ) Term Structure Wharton School of the University 8 / 19
Nominal bonds Assume λ P z t ), with z t z 1, z 2 ) Assume investment and technology not contingent on inflation. For instance, P s 1 s t) = P s 1 s t, z t ) = P s1, z 1 s t, z t ) + P s1, z 2 s t, z t ) Inflation not directly priced. For instance, P s 1, z 1 s t Prs, z t ) = 1,z 1 s t,z t ) P s 1, z 2 s t, z t ) = 1 Prs 1,z 1 s t,z t )+Prs 1,z 2 s t,z t ) Prs 1,z 1 s t,z t ) Prs 1,z 1 s t,z t )+Prs 1,z 2 s t,z t ) If inflation and investment independent V $1) t ) P s 1 s t ), and ) P s 1 s t ) s t ) {, z t = P s 1 s t) + P s 2 s t)} ) 1 E λ P st, z t Term Structure Wharton School of the University 8 / 19
Table 1: Parameter values Parameter Symbol Value Investment rates λ I s 1 ), λ I s 2 ) 0.9497, 1.1109 Serial correlation 0.2 Relative freq. of low 0.8 Inflation rates λ P z 1 ), λ P z 2 ) 1.0169, 1.0763 Serial correlation 0.8 Relative freq. of low 1.9 Depreciation rates δ E, δ S 0.112, 0.031 Relative value of cap. K E /K S 0.6 Adjustment cost par. b E, b S, c E, c S so that q 1.5 Adjustment cost curv. ν E, ν S 2.2385, 4.1080 Marginal prod. of cap. A E, A S so that R E, R S 1.04515, 1.05773 U Term Structure Wharton School of the University 9 / 19
Table 2: Equity returns and short term yields Model Data E r M y 1)) % 4.64 4.64 σ r M,r ) % 17.13 17.13 E y 1)) % 5.29 5.29 σ y 1)) % 2.98 2.98 Yields, y, are from Fama and Bliss, defined as ln price)/maturity, stock returns are the logs of value-weighted returns from CRSP, r M,r is the stock return deflated by the CPI-U. All data is 1952-2010. U Term Structure Wharton School of the10 University / 19
Table 3: Term structure Maturity years) 1 2 3 4 5 Nominal yields Mean - Model % 5.29 5.44 5.58 5.72 5.86 Mean - Data % 5.29 5.49 5.67 5.81 5.90 Std - Model % 2.98 2.73 2.51 2.33 2.17 Std - Data % 2.98 2.93 2.85 2.80 2.75 Real yields Mean - Model % 1.68 1.84 2.00 2.15 2.31 Std - Model % 2.06 1.92 1.81 1.71 1.62 U Term Structure Wharton School of the11 University / 19
Table 4: Fama-Bliss excess ) return regressions rx n) t+1 = α + β f n) t y 1) t + ε n) t+1 Maturity years) 2 3 4 5 Model - β.3050.3906.5144.6135 Data - β.7606 1.0007 1.2723.9952 Yields are from Fama and Bliss 1952-2010, rx n) t+1 of a n-period discount bond, f n) t is the excess return p n) t ), is the 1 period yield. is the forward rate, p n 1) t p n) t the log of the price discount bond, and y 1) t U Term Structure Wharton School of the12 University / 19
Table 5: Fama-Bliss excess return regressions No inflation risk ) rx n) t+1 = α + β f n) t y 1) t + ε n) t+1 Maturity years) 2 3 4 5 Model - β no inflation risk.4656.6101.7881.9465 Model - β real forward premium.4667.6039.7866.9473 Model - β benchmark.3050.3906.5144.6135 Data - β.7606 1.0007 1.2723.9952 U Term Structure Wharton School of the13 University / 19
Continuous-time Assume univariate dz with discount factor process dλ = r.) dt σ.) dz Λ with given returns for the two types of capital dr j R j = µ j.) dt + σ j.) dz, for j = 1, 2 Term Structure Wharton School of the14 University / 19
Continuous-time Assume univariate dz with discount factor process dλ = r.) dt σ.) dz Λ with given returns for the two types of capital dr j R j = µ j.) dt + σ j.) dz, for j = 1, 2 The absence of arbitrage implies that so that 0 = r + µ j σ j σ, for j = 1, 2 r = σ 2 σ 2 σ 1 µ 1 σ 1 σ 2 σ 1 µ 2 σ = µ 2 µ 1 σ 2 σ 1 Term Structure Wharton School of the14 University / 19
Capital return The return to a given capital stock equals ) A j c j ) νj 1 Ij,t ν j 1) 1 1 I νj j,t /K j,t δ j b j K j,t [ ) ] dt + ν j 1) λ I,j 1 + δ j + 1 2 ν j 2) σ 2 I,j }{{} µ j.) + ν j 1) σ I,j dz }{{} σ j.) U Term Structure Wharton School of the15 University / 19
Sharpe ratio At steady state, I /K = λ I 1 + δ, and with σ I,j = σ I, the Sharpe ratio is given by σ ss = µ j r = µ 2 µ 1 = R 2 R 1 + ν 1 + ν 2 3 σ j σ 2 σ 1 ν 2 ν 1 ) σ I 2 σ I with R = A c ) ν 1 + b λ I 1 δ) 1 1 ) λ I + 1 1 δ) ν ν U Term Structure Wharton School of the16 University / 19
ER) R 2 R 1 R f σr)
Dynamics of the short rate The short rate equals r = σ 2 σ 2 σ 1 µ 1 σ 1 σ 2 σ 1 µ 2 Term Structure Wharton School of the17 University / 19
Dynamics of the short rate The short rate equals Specializing to the case σ Ij = σ I r = σ 2 σ 2 σ 1 µ 1 σ 1 σ 2 σ 1 µ 2 r = ν 2 1 ν 2 ν 1 µ 1 ν 1 1 ν 2 ν 1 µ 2 Term Structure Wharton School of the17 University / 19
Dynamics of the short rate The short rate equals Specializing to the case σ Ij = σ I r = σ 2 σ 2 σ 1 µ 1 σ 1 σ 2 σ 1 µ 2 r = ν 2 1 ν 2 ν 1 µ 1 ν 1 1 ν 2 ν 1 µ 2 dr = µ r.) dt + σ r.) dz : at steady state, for σ I,j = σ I, and λ I,j and σ I constant, σ r ss = ν 2 1) ν 1 1) ν 2 ν 1 [ R 2 R 1 + δ 2 δ 1 ] σ I Term Structure Wharton School of the17 University / 19
Table 6: Term premium: continuous-time versus discrete-time model Cont.-time Discrete-time ) σ r σ E t r 2) t+1 y 1) t Benchmark.0024.0022 δ 1 = δ 2, R 1 = R 2, 0 0.00001 δ 1 = δ 2 0.00044.00036 R 1 = R 2, δ 1 =.112 > δ 2 =.0313.0017.0015 R 1 = R 2, δ 1 =.0313 < δ 2 =.112.0017.0018 U Term Structure Wharton School of the18 University / 19
Conclusion Two-sector q-theoretical model can do a good job replicating averages and volatilities of longer term US yields Term Structure Wharton School of the19 University / 19
Conclusion Two-sector q-theoretical model can do a good job replicating averages and volatilities of longer term US yields Time-varying term premiums are evidenced through Fama-Bliss regressions Term Structure Wharton School of the19 University / 19
Conclusion Two-sector q-theoretical model can do a good job replicating averages and volatilities of longer term US yields Time-varying term premiums are evidenced through Fama-Bliss regressions Even with homoscedastic investment and inflation, the market price of risk and the volatility of the short rate are naturally time-varying, driven by time-varying investment to capital ratios Term Structure Wharton School of the19 University / 19