Why Have Interest Rates Fallen Far Below the Return on Capital

Transcription

1 Why Have Interest Rates Fallen Far Below the Return on Capital Magali Marx Banque de France Benoît Mojon Banque de France François R. Velde Federal Reserve Bank of Chicago Macroeconomic and Financial Imbalances and Spillovers, June 9 Why Have Interest Rates Fallen Far Below the Return on Capital 1

2 The decrease of real interest rates Real rates (%) Hamilton et al 1yr MA, US King Low (indexed bonds) US 1yr ex-ante real rate Hamilton et al 1yr MA, France Laubach-Williams r * Why Have Interest Rates Fallen Far Below the Return on Capital

3 Which does not reflect the evolution of capital return Real return on capital (%) US EA large 4 countries EA Japan Why Have Interest Rates Fallen Far Below the Return on Capital 3

4 The usual suspects Low rates have been loosely tied to secular stagnation A number of potential explanations have been cited: productivity slowdown changing demographics (population slowdown, increased longevity) change in the price of investment goods tightening of borrowing constraint shortage of safe assets rising inequality Why Have Interest Rates Fallen Far Below the Return on Capital 4

5 Our goal quantitative assessment of the various factors cited embed them in a single, tractable model explain both the evolution of capital return and risk-free rate this means having risk, and attitudes toward risk, in the model Why Have Interest Rates Fallen Far Below the Return on Capital 5

6 Related literature low rates: King and Low (14); Hamilton et al. (16); Holston et al. (16) safe assets: Coeurdacier et al. (15); Caballero et al. (8); Caballero and Farhi (14) deleveraging: Eggertsson and Krugman (1); Korinek and Simsek (16); Farhi and Werning (13) secular stagnation: Bean et al. (15); Rachel and Smith (15) demographics: Carvalho et al. (16); Gagnon et al. (16) risk: Kozlowski et al. (15); Hall (16) return on capital: Caballero et al. (17) Why Have Interest Rates Fallen Far Below the Return on Capital 6

7 The Model add risk to Eggertsson and Mehrotra (14) and Coeurdacier et al. (15). time is discrete, infinite 3-period OLG structure (y, m, o) population N t, growth rate g L recursive preferences with Epstein-Zin-Weil utility function capital and labor (supplied inelastically), age-specific productivities (e y, 1, ) output Y = K α (AL) 1 α productivity A: trend growth g A + shock with variance σ (only source of risk) growth in price of investment g I Why Have Interest Rates Fallen Far Below the Return on Capital 7

8 Preferences Epstein and Zin (1989) Weil (199) recursive preferences: V t = U(c t, E tv t+1) = ( c 1 ρ t + β ( (E tv 1 γ t+1 ) 1 γ 1 ) 1 ρ) 1 ρ 1 Why Have Interest Rates Fallen Far Below the Return on Capital 8

9 Preferences Epstein and Zin (1989) Weil (199) recursive preferences: V t = U(c t, E tv t+1) = CES functional form applied to ( c 1 ρ t + β ( (E tv 1 γ t+1 ) 1 γ 1 ) 1 ρ) 1 ρ 1 time: ( c 1 ρ t + β ( t+1) 1 ρ) 1 ρ 1 ρ: inverse of intertemporal elasticity of substitution risk: (E tv 1 γ t+1 ) 1 γ 1 γ: risk aversion when ρ = γ standard time-additive preferences tension between high γ required to match asset pricing low ρ required to match consumption growth with interest rates Why Have Interest Rates Fallen Far Below the Return on Capital 8

10 Budget constraints young borrow from middle-aged up to a fraction θ of their t + 1 labor income we focus on equilibria where this binds no other frictions (e.g., price stickiness) middle-aged lend to young, buy capital from old, invest old collect returns, sell depreciated capital market-clearing: c y t = b y t+1 + wtey t b y t+1 θtet(wt+1/rt+1) c m t+1 b m t+ + p k t+1k m t+ = w t+1 R t+1b y t+1 c o t+ = (p k t+(1 δ) + r k t+)k m t+ R t+b m t+ g L,t b y t+1 + bm t+1 = g L,t+1 b y t+ + bm t+ = Why Have Interest Rates Fallen Far Below the Return on Capital 9

11 Production Y t = (N t k m t ) α [A t(e y t N t + N t 1)] 1 α N t k m t : capital (chosen by current old in the previous period) e y t N t + N t 1: labor (of young and middle-aged) Competitive factor markets: w t = (1 α)a 1 α t kt α rt k = αa 1 α t kt α 1 both written in terms of the capital/labor ratio k t defined as k t Nt km t kt m e y = t N t + N t 1 g L,t 1 (1 + e y t g L,t ). Why Have Interest Rates Fallen Far Below the Return on Capital 1

12 Solution strategy only the middle-aged have an intertemporal problem how much to save in what form: bonds or capital write the middle-aged s Euler equation and substitute equilibrium quantities quantity of bonds determined by young s constraint Euler equation also relates risk-free rate R and return to capital R k we derive a law of motion expressed in terms of R or equivalently k Why Have Interest Rates Fallen Far Below the Return on Capital 11

13 Solution strategy () Middle-aged FOCs: γ ρ ] (ct m ) ρ = β [E t(ct+1) o 1 γ] 1 γ E t [(c t+1) o γ Rt+1 k γ ρ (ct m ) ρ = β [E t(ct+1) o 1 γ] 1 γ [ E t (c o t+1 ) γ] R t+1. Define R m t+1 = α tr k t + (1 α t)r t+1 and express budget constraints as W t = Y t ct m ct+1 o = Rt+1W m t. Portfolio choice: set α t so that ( ) E t(rt+1 m γ )R t+1 = E t R t+1 γ m Rt+1 k Saving decision: Y t = ( ) 1 + (βφ tr 1 ρ 1 t+1 ) ρ W t Then use market clearing to express Y t, W t, R m t+1 in term of the aggregate capital stock Why Have Interest Rates Fallen Far Below the Return on Capital 1

14 Law of motion ( ) (βφ t) 1/ρ R 1 1/ρ t+1 (1 θt 1 )(1 α)( Rk t δ)k t ã t g I }{{}}{{} saving rate = g L,t α(1 + ey g L,t+1 ) }{{} capital ( 1 + income ) 1 δ (1 α)θ t(1 g I ) ξ t R t+1 }{{} bonds kt+1 } {{ } investments Why Have Interest Rates Fallen Far Below the Return on Capital 13

15 Law of motion ( ) (βφ t) 1/ρ R 1 1/ρ t+1 (1 θt 1 )(1 α)( Rk t δ)k t ã t g I }{{}}{{} saving rate = g L,t α(1 + ey g L,t+1 ) }{{} capital ( 1 + income ) 1 δ (1 α)θ t(1 g I ) ξ t R t+1 }{{} bonds kt+1 } {{ } investments overlapping generations saving only done out of labor income borrowing constraint disappears if θ =, e y = risk φ t: precautionary saving, acts like discount factor distortion ( 1) 1/ξ t: portfolio choice Why Have Interest Rates Fallen Far Below the Return on Capital 13

16 Risk terms The factors φ t and ξ t are with ξ t = Et(ut+1 γ ã t+1) E t(u γ t+1 ) [ ] (γ ρ)/(1 γ) 1 γ φ t = E tu t+1 Etu γ ρ t+1 v t u t+1 α(1 + e y g L,t+1 )ã t+1 + (1 α)θ t ã t+1 A 1 α t+1 E ta 1 α t+1 only functions of (moments of) the exogenous process A t+1 when δ 1, φ t involves R t+1 as well. Why Have Interest Rates Fallen Far Below the Return on Capital 14

17 Risky steady state to account for risk in a tractable way, we appeal to the concept of risky steady state : exogenous trends as in the data productivity shock is assumed i.i.d. in the law of motion, ã t set at its mean, ã t+1 is stochastic agents take into account the uncertainty Why Have Interest Rates Fallen Far Below the Return on Capital 15

18 Risk and borrowing constraint When δ = 1, ρ < 1, θ =, e y = (no young): φ t γ(1 ρ)σ 1 ξ t 1 Why Have Interest Rates Fallen Far Below the Return on Capital 16

19 Risk and borrowing constraint When δ = 1, ρ < 1: φ t γ(1 ρ) α (1 + e y g L,t+1 ) (α(1 + e y g L,t+1 ) + (1 α)θ t) σ 1 ξ t α(1 + e y g L,t+1 ) 1+γ σ α(1 + e y g L,t+1 ) + (1 α)θ t Why Have Interest Rates Fallen Far Below the Return on Capital 16

20 Risk and borrowing constraint When δ = 1, ρ < 1: law of motion: φ t γ(1 ρ) α (1 + e y g L,t+1 ) (α(1 + e y g L,t+1 ) + (1 α)θ t) σ 1 ξ t α(1 + e y g L,t+1 ) 1+γ σ α(1 + e y g L,t+1 ) + (1 α)θ t 1 + (β φ t ) 1/ρ R 1 1/ρ θ,σ + t+1 1 (1 θt 1 )(1 α) Rk t+1 ã t ( ) 1 = g L,t α(1 + ey g L,t+1 ) + (1 α)θ t ξ t θ,σ + g I k t kt+1 Why Have Interest Rates Fallen Far Below the Return on Capital 16

21 Long run determinants of the bond interest rate r and the return on capital r K δ = 1, ρ = 1: Observable factors productivity growth g A evolution of working age population g L trend in investment price g I Unobservable factors borrowing constraint θ variance of the shock on the trend of productivity σ. Why Have Interest Rates Fallen Far Below the Return on Capital 17

22 Long run determinants of the bond interest rate r and the return on capital r K δ = 1, ρ = 1: Observable factors productivity growth g A evolution of working age population g L trend in investment price g I Unobservable factors borrowing constraint θ variance of the shock on the trend of productivity σ. r = r + (g L 1) + (g A 1) 1 (g I 1) + cθ + γu( θ σ, σ ) + r K = r + γv( θ σ, σ ) + The wedge between r and r K is only affected by θ and σ Why Have Interest Rates Fallen Far Below the Return on Capital 17

23 Empirical strategy our targets are the risk-free rate and the return on capital we segregate the usual suspects into the observables: productivity, demographics, price of investment the less observables : borrowing constraint, productivity risk Three steps: 1 input the observables, set θ and σ constant to match the levels of the targets input the observables, compute θ to match the risk-free rate, keep σ constant 3 input the observables, compute σ to match the risk-free rate, keep θ constant 4 input the observables, compute θ and σ to match both targets repeat for US and Euro area (and the world) then stare at the pictures... caveats we interpret the generations loosely (1-year averages) risk-free rates before the 198s are less meaningful (financial repression etc), so we focus on 199s to present Why Have Interest Rates Fallen Far Below the Return on Capital 18

24 Model calibration and data sources Parameters T length of period (years) 1 β discount factor.98 T α capital share.8 γ risk aversion 1 ρ inverse of IES.8 δ capital depreciation rate.1 T e y relative productivity of young.3 Factors g L,t growth rate of population -64 US, EA (France), China, Japan: OECD g I,t investment price growth DiCecio (9) g A,t productivity growth US: Fernald (1), Euro: NAWM model R t real interest rate US: Hamilton et al. (16), France Rt k return on capital US, EA: our calculations à la Gomme et al. (15) ã t productivity shock ln(ã) is a i.i.d. N( σ /, σ ) Free parameters θ borrowing constraint on young σ variance of ã t Why Have Interest Rates Fallen Far Below the Return on Capital 19

25 The inputs 4 Productivity growth (g A 1) 3 Working age pop. growth (g L 1) Risk free rate (r) Capital return (r K ) US EA World Why Have Interest Rates Fallen Far Below the Return on Capital

26 Impact of observable factors, in the US Observable factors explain about 1.4% from 199 to 14 Observed and simulated rates Observed and simulated risk premium Borrowing ratio θ. Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 1

27 Impact of observable factors, in the EA Observable factors explain about 1.8% from 199 to 14 Observed and simulated rates Observed and simulated risk premium Borrowing ratio θ. Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital

28 Impact of the borrowing constraint, in the US. A tighter constraint can account for the fall in the risk-free rate and.8% increase of the risk premium Observed and simulated rates Observed and simulated risk premium Borrowing ratio θ. Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 3

29 Impact of the borrowing constraint, in the EA. A tighter constraint can account for the fall in the risk-free rate and.7% increase of the risk premium Observed and simulated rates Observed and simulated risk premium Borrowing ratio θ. Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 4

30 Impact of risk, in the US. A higher risk perception can account for the fall in the risk-free rate and the increase in the risk premium Observed and simulated rates Observed and simulated risk premium Borrowing ratio θ. Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 5

31 Impact of risk, in the EA. A higher risk perception can account for the fall in the risk-free rate and the increase in the risk premium Observed and simulated rates Observed and simulated risk premium Borrowing ratio θ. Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 6

32 Impact of risk and the borrowing constraint, in the US. With higher risk perception data are consistent with non decreasing debts Observed rates -5 Observed risk premium Borrowing ratio θ Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 7

33 Impact of risk and the borrowing constraint, in the EA. With higher risk perception data are consistent with non decreasing debts 1 5 Observed rates -5 Observed risk premium Borrowing ratio θ Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 8

34 Borrowing constraint and risk, in the US. 1 5 Borrowing ratio θ..1 Observed and simulated rates Debt/income.3 Observed and simulated risk premium 1 6 Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 9

35 Borrowing constraint and risk, in the EA. 1 5 Borrowing ratio θ..1 Observed and simulated rates Debt/income.4 Observed and simulated risk premium 1 6 Productivity Risk (std) σ..1. Why Have Interest Rates Fallen Far Below the Return on Capital 3

36 Global perspective Impact of observable factors Observed and simulated rates Observed and simulated risk premium Borrowing ratio θ. Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 31

37 Global perspective Impact of the borrowing constraint Observed and simulated rates Observed and simulated risk premium Borrowing ratio θ. Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 3

38 Global perspective Impact of risk Observed and simulated rates Observed and simulated risk premium Borrowing ratio θ. Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 33

39 Global perspective Impact of risk and the borrowing constraint Observed rates -5 Observed risk premium Borrowing ratio θ Productivity Risk (std) σ Why Have Interest Rates Fallen Far Below the Return on Capital 34

40 Conclusion usual suspects aren t enough deleveraging story increased (perception of) risk can account for the patterns but it s a residual more work to be done on longevity inequality (through a bequest motive) exogenous supply of safe assets Why Have Interest Rates Fallen Far Below the Return on Capital 35

41 Additional slides Why Have Interest Rates Fallen Far Below the Return on Capital 36

42 Sensitivity to γ (US) Observed and simulated rates Observed and simulated risk premium Borrowing ratio. Productivity Risk (std) γ = 1 γ = 5 Why Have Interest Rates Fallen Far Below the Return on Capital 37

43 Sensitivity to γ (EA) Observed and simulated rates 1 Observed and simulated risk premium Borrowing ratio. Productivity Risk (std) γ = 1 γ = Why Have Interest Rates Fallen Far Below the Return on Capital 38

44 The inputs for the US 3 1 Productivity growth Working Age Population 1 Relative investment price Debt/GDP (%) safe assets private total Why Have Interest Rates Fallen Far Below the Return on Capital 39

45 The inputs for the EA 3 1 Productivity growth Working Age Population Relative investment price Debt/GDP (%) safe assets private total Why Have Interest Rates Fallen Far Below the Return on Capital 4

46 Measures of risk Why Have Interest Rates Fallen Far Below the Return on Capital 41

47 References I Bean, S. C., C. Broda, T. Ito, and R. Kroszner (15): Low for Long? Causes and Consequences of Persistently Low Interest Rates, Geneva Reports on the World Economy 17, International Center for Monetary and Banking Studies. Caballero, R. J. and E. Farhi (14): The Safety Trap, Working Paper 1997, NBER. Caballero, R. J., E. Farhi, and P.-O. Gourinchas (8): An Equilibrium Model of Global Imbalances and Low Interest Rates, American Economic Review, 98, (17): Rents, Technical Change, and Risk Premia: Accounting for Secular Trends in Interest Rates, Returns on Capital, Earning Yields, and Factor Shares, NBER Working Papers 317, National Bureau of Economic Research, Inc. Carvalho, C., A. Ferrero, and F. Nechio (16): Demographics and Real Interest Rates: Inspecting the Mechanism, European Economic Review, 88, 8 6, si: The Post-Crisis Slump. Coeurdacier, N., S. Guibaud, and K. Jin (15): Credit Constraints and Growth in a Global Economy, American Economic Review, 15, DiCecio, R. (9): Sticky wages and sectoral labor comovement, Journal of Economic Dynamics and Control, 33, Eggertsson, G. B. and P. Krugman (1): Debt, Deleveraging, and the Liquidity Trap: A Fisher-Minsky-Koo Approach, The Quarterly Journal of Economics, 17, Why Have Interest Rates Fallen Far Below the Return on Capital 4

48 References II Eggertsson, G. B. and N. R. Mehrotra (14): A Model of Secular Stagnation, Working Paper 574, NBER. Epstein, L. G. and S. E. Zin (1989): Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework, Econometrica, 57, Farhi, E. and I. Werning (13): A Theory of Macroprudential Policies in the Presence of Nominal Rigidities, Tech. rep., MIT. Fernald, J. G. (1): A quarterly, utilization-adjusted series on total factor productivity, Working Paper Series 1-19, Federal Reserve Bank of San Francisco. Gagnon, E., B. K. Johannsen, and D. López-Salido (16): Understanding the New Normal: The Role of Demographics, Finance and Economics Discussion Series 16-8, Board of Governors of the Federal Reserve System. Gomme, P., B. Ravikumar, and P. Rupert (15): Secular Stagnation and Returns on Capital, Federal Reserve Bank of St Louis Economic Synopsis, 19. Hall, R. E. (16): Understanding the Decline in the Safe Real Interest Rate, Working paper 196, NBER. Hamilton, J. D., E. S. Harris, J. Hatzius, and K. D. West (16): The Equilibrium Real Funds Rate: Past, Present and Future, IMF Economic Review, 64, Why Have Interest Rates Fallen Far Below the Return on Capital 43

49 References III Holston, K., T. Laubach, and J. C. Williams (16): Measuring the Natural Rate of Interest: International Trends and Determinants, Working paper 16-11, Federal Reserve Bank of San Francisco. King, M. and D. Low (14): Measuring the World Real Interest Rate, working paper 19887, NBER. Korinek, A. and A. Simsek (16): Liquidity Trap and Excessive Leverage, American Economic Review, 16, Kozlowski, J., L. Veldkamp, and V. Venkateswaran (15): The Tail that Wags the Economy: Belief-Driven Business Cycles and Persistent Stagnation, Working Paper 1719, NBER. Rachel, L. and T. D. Smith (15): Secular Drivers of the Global Real Interest Rate, Staff Working Paper 571, Bank of England. Weil, P. (199): Nonexpected Utility in Macroeconomics, The Quarterly Journal of Economics, 15, 9 4. Why Have Interest Rates Fallen Far Below the Return on Capital 44