Interest-rate pegs and central bank asset purchases: Perfect foresight and the reversal puzzle

Interest-rate pegs and central bank asset purchases: Perfect foresight and the reversal puzzle Rafael Gerke Sebastian Giesen Daniel Kienzler Jörn Tenhofen Deutsche Bundesbank Swiss National Bank The views expressed in these slides are those of the authors and not necessarily those of the Deutsche Bundesbank or the Swiss National Bank. Geneva, August 216

Motivation and contribution After effective lower bound on policy rate has been reached: unconventional monetary policy like QE or forward guidance Policy consensus: QE and forward guidance are expansionary The problem with QE is it works in practice, but it doesn t work in theory. (Ben Bernanke) Fixing nominal interest rate leads to peculiar outcomes in standard New Keynesian models under perfect foresight (Cochrane, 215, García-Schmidt and Woodford, 215 & Carlstrom, Fuerst, and Paustian, 215) Focus here: Reversal puzzle

Outline Introduction A model with financial intermediaries QE and interest-rate peg Interest-rate peg: perfect foresight Interest-rate peg: extended path Conclusion

Key features of the model Non-linear version of (non-standard) New Keynesian model with financial intermediaries: Carlstrom, Fuerst, and Paustian (214) Households face loan-in-advance constraint: investment must be financed by corresponding bonds more Time-varying distortion in capital accumulation decision Financial intermediaries (FIs): Use accumulated net worth and (short-term) deposits to finance purchases of long-term (investment and government) bonds. Hold-up problem limits ability to attract deposits: need net worth, but face net-worth adjustment cost. Market segmentation: short- and long-term bonds more Typical New Keynesian features: Habit formation in consumption Price and wage stickiness (à la Calvo) with indexation Investment adjustment cost

Modeling QE and interest-rate peg QE program is implemented via an AR(2) process for the market value of long-term bonds available to FIs, B t : B t = B (1 ρ 1+ ρ 2 ) ) ρ1 ) ρ2 ss ( Bt 1 ( Bt 2 ε t. (1) Can trigger QE by a single (negative) shock and the rest is perfectly anticipated by rational agents. Resulting (inverse) hump shape well suited to represent plausible QE program (bonds held by FIs decline and return back only gradually). Interest-rate peg is implemented via sequence of binary dummy variables, D TR t : ( R t = Dt TR R ss + 1 D TR t ) [ ( ) τy ] 1 ρ (R t 1 ) ρ R ssπ τ Π Yt t (2) Y t 1 Analyze effects of QE under different assumptions about path of the policy rate: Standard Taylor rule Interest-rate peg for P periods, subsequently revert to Taylor rule and under different assumptions regarding information structure of the agents: Perfect foresight about interest-rate peg Each period, agents surprised by interest-rate peg ( extended path )

QE and interest-rate peg (perfect foresight): -4 periods of pegged rates.1 Aggregate Output in % Dev from ss Inflation in %P Dev from ss (annualized).1.5.5 5 1 15 2 25 5 1 15 2 25 Policy Rate in %P Dev from ss (annualized).6.4.2 2 4 6 x Long Rate in %P Dev from ss (annualized) 1 3 5 1 15 2 25 8 5 1 15 2 25

QE and interest-rate peg (perfect foresight): 5-1 periods of pegged rates Aggregate Output in % Dev from ss Inflation in %P Dev from ss (annualized).2.4.2.4.6.6 5 1 15 2 25.8 5 1 15 2 25 5 1 15 2 Policy x 1 3 Rate in %P Dev from ss (annualized) 5 1 15 2 25 2 1 Long x 1 3 Rate in %P Dev from ss (annualized) 5 1 15 2 25

QE and interest-rate peg (perfect foresight): 19-25 periods of pegged rates Aggregate Output in % Dev from ss Inflation in %P Dev from ss (annualized).2.15.2.1.1.5 5 1 15 2 25 5 1 15 2 25 2 x Policy Rate in %P Dev from ss (annualized) 1 3.5 Long Rate in %P Dev from ss (annualized) 4.1.15 6 5 1 15 2 25 5 1 15 2 25

QE and interest-rate peg (perfect foresight): source of reversal Obviously, the reversal is related to the peg. Note: Peg does not give rise to indeterminacy, since eventually the Taylor rule kicks-in and stabilizes the economy after end of peg (locally) determinate equilibrium after peg. Thus, determinate equilibrium after the peg imposes terminal conditions for the period of the peg initial levels of variables & the path of variables during the peg depend on terminal conditions. This is true in a world without endogenous state variables (i.e. lagged variables). (However, a priori, the path of variables during the peg is not necessarily determinate! See for instance, Cochrane, 215, The New Keynesian liquidity trap: Here, we focus on the (non-explosive) equilibrium)

QE and interest-rate peg (perfect foresight): source of reversal However, in the present model there are several endogenous state variables with lagged variables there is nothing to anchor the terminal conditions. Initial level of variables depends on terminal level of variables but terminal level of variables depends on initial level of variables. This interaction is likely to be the source of the reversal. Thus, in contrast to, say, the standard New Keynesian model the reversal cannot be easily overcome by (just) shutting down inflation and/or wage indexation (as in CFP, 215). more

QE and interest-rate peg (perfect foresight) Length of pegs for which inflation reversals occur for different parameter values Length of interest rate pegs 3 28 26 24 22 2 18 16 14 12 1 8 6 4 2.1.2.3.4.5.6.7.8.9 1 Values of ι p

QE and interest-rate peg (perfect foresight) Length of pegs for which inflation reversals occur for different parameter values more Length of interest rate pegs 3 28 26 24 22 2 18 16 14 12 1 8 6 4 2.2.4.6.8 1. 1.2 1.4 1.6 1.8 2 2.2 2.4 Values of ψ N

QE and interest-rate peg (perfect foresight): analytical explanation Consider linearized version of the model: Γ Y t = Γ 1 Y t 1 + Φε t + Ψη t (3) Employ QZ decomposition, define Z Y t w t, and partition into explosive and non-explosive part: [ ] [ ] [ ] [ ] [ ] Λ 11 Λ 12 w 1,t Ω = 11 Ω 12 w 1,t 1 Q + 1 (Φε t + Ψη t ) (4) Λ 22 w 2,t Ω 22 w 2,t 1 Q 2 Manipulate unstable part further, iterating forward, and taking expectations leads to: { } w 2,t = E t J n 1 Ω 1 22 Q 2Φε t+n (5) n=1 where α 11 δ 11 J Ω 1 22 Λ 22 = α 22 δ 22... α jj δ jj (6)

QE and interest-rate peg (perfect foresight): analytical explanation For the duration of the peg, P: E t (ε t+n ) P also influences J n 1 Some of the diagonal elements of J could be complex, define α jj δ jj z jj : z jj = a + bi = r (cos φ + i sin φ) (7) Consider powers of J and polar form: z k jj = r k (cos kφ + i sin kφ), for k =,..., P 1 (8) Solution to the system may involve trigonometric functions which depend on length of interest-rate peg: the longer the duration of the peg, the farther we move along those functions complex eigenvalues may induce cyclical effects Shutting down inflation indexation might be enough to eliminate reversals in canonical New Keynesian model, but probably not sufficient in more elaborate versions of such a model: additional backward-looking elements

QE and interest-rate peg (extended path) Modify relative importance of forward- and backward-looking elements in the model to get rid of complex eigenvalues does not work necessarily. For example: sticky information may not be sufficient Alternative here: reduce forward-lookingness of agents by employing extended path (only J, P irrelevant) Fully captures non-linearities and solves model in every period for all remaining periods. Basically, the idea is to use same strategy as for simulating the perfect foresight model with a sequence of unexpected shocks (Adjemian, 216). EP creates a stochastic simulation as if only shocks of the current period were random. Agents expect future shocks to be zero (and are certain of that): constantly surprised here by the continuing peg. Does not entirely skip expectation operator but imposes certainty equivalence (only approximation): E [f (z)] = f [E (z)] (9) Does not suffer from curse of dimensionality and can handle non-differentiability.

QE and interest-rate peg (extended path): with inflation indexation (-1 periods of pegged rates).3 Aggregate Output in % Dev from ss Inflation in %P Dev from ss (annualized).2.2.1.1 5 1 15 2 25 5 1 15 2 25.2.15 Policy Rate in %P Dev from ss (annualized) 1 x Long Rate in %P Dev from ss (annualized) 1 3.1 2.5 5 1 15 2 25 3 5 1 15 2 25

Discussion and concluding remarks Standard (purely forward-looking) New Keynesian model: Adding inflation indexation may introduce feedback-loop/complex-valued explosive eigenvalues: reversals. One potential way out: modify model (e.g. sticky information). More elaborate versions of New Keynesian model: There is no obvious modification of the model that prevents a reversal; e.g. here reversals may occur even without indexation in goods and labor market. There are elements in the model, which we cannot easily substitute without completely changing the model. Alternative strategy to eliminate feedback-loop & reversals: reduce forward-lookingness by using extended path. Caveats: Perfect foresight and extended path are polar cases. Might be preferable/more realistic to use less extreme information structure (e.g. stochastic extended path, Adjemian and Juillard, 213). But currently very high computational hurdles...

Appendix: Financial intermediaries hold-up problem and sticky net-worth FIs use accumulated net worth and (short-term) deposits to finance purchases of long-term (investment and government) bonds: B t + F t = D t + N t = L t N t (1) Hold-up problem limits ability to attract deposits: need net worth FI can choose to default on repayment to depositors Depositors can only seize the share (1 µ t ) of FI s assets Binding time-t incentive compatibility constraint is given by: { [( ) ]} { } P t R L E t Λ t+1 Rt+1 L P t t+1 P t+1 Rt d 1 L t + 1 = Φ t L t E t Λ t+1 Rt d, (11) P t+1 where Λ t is marginal utility of consumption, Rt L is the gross yield on long-term bonds, Rt d denotes the rate on deposits, L t is FI s leverage, and Φ t is a stochastic process exogenously influencing the financial friction (µ t ). FIs choose dividends, div t, and net worth, N t, to maximize value function subject to hold-up problem and budget constraint: div t + N t [1 + f (N t )] P [( ) ] t 1 Rt L Rt 1 d L t 1 + Rt 1 d N t 1, (12) P t ( ) where f (N t ) ψn Nt N ss denotes net-worth adjustment cost. back 2 N ss

Appendix: Households loan-in-advance constraint Households maximize standard utility subject to budget constraint, law of motion for physical capital and so-called loan-in-advance constraint: P k t I t Q t (F t κf t 1 ) P t = Q t B I t P t, (13) where Pt k is the real price of capital, I t denotes investment in physical capital, P t is the aggregate price level, F t 1 denotes household s nominal liabilities at time t, Bt I = (F t κf t 1 ) is the time-t issuance of new investment bonds, and Q t denotes the time-t price of newly issued bonds. Investment must be financed by corresponding bonds, which are Woodford (21)-type perpetuities (coupons decay exponentially: s + 1 periods later pay κ s ). Time-varying distortion in capital accumulation decision back

Appendix: Government fiscal and monetary policy Fiscal policy is passive: no government expenditure and lump-sum taxes move to support interest payments on government debt Monetary policy follows standard Taylor rule: R t = (R t 1 ) ρ [R ssπ τ Π t ( Yt Y t 1 ) τy ] 1 ρ (14) back

Total value of government bonds held by FIs Market Value for Long Term Bonds in % Dev from ss.1.2.3.4.5.6.7.8 5 1 15 2 25

QE and standard Taylor rule D TR t =, t. No difference between perfect foresight and extended path. Aggregate Output in % Dev from ss.1.5 5 1 15 2 25 Policy Rate in %P Dev from ss (annualized).1.1.5.2.1.1 5 5 1 15 2 25 Investment in % Dev from ss 5 1 15 2 25 Ex ante Real Rate in % Dev from ss 5 1 15 2 25 Net Worth in % Dev from ss x 1 3 5 1 15 2 25 5.5 1 1.5 Inflation x 1 3 in %P Dev from ss (annualized) 5 1 15 2 25 Long Rate in %P Dev from ss (annualized) x 1 3 5 1 15 2 25 Inv. Bonds in % Dev from ss x 1 3Real 15 1 5.2.4.5.1.15 5 1 15 2 25 Fin. Distortion in % Dev from ss 5 1 15 2 25 Leverage in % Dev from ss 5 1 15 2 25

Appendix: QE and interest-rate peg (perfect foresight): what about inflation indexation? (-1 periods of pegged rates) Aggregate Output in % Dev from ss Inflation in %P Dev from ss (annualized).5.1.5.1 5 1 15 2 25.1 5 1 15 2 25 Policy Rate in %P Dev from ss (annualized).4.2.2 5 1 15 2 25 5 5 x Long Rate in %P Dev from ss (annualized) 1 3 5 1 15 2 25 back

Appendix: QE and interest-rate peg (perfect foresight) Length of pegs for which inflation reversals occur for different parameter values back Length of interest rate pegs 3 28 26 24 22 2 18 16 14 12 1 8 6 4 2.2.4.6.8 1. 1.2 1.4 1.6 1.8 2 2.2 2.4 Values of ψ I

Appendix: QE and interest-rate peg (perfect foresight) Length of pegs for which inflation reversals occur for different parameter values back Length of interest rate pegs 3 28 26 24 22 2 18 16 14 12 1 8 6 4 2 3 4 5 6 7 8 9 1 11 12 13 14 15 Duration (in years) implied by κ