A general equilibrium analysis of low nominal interest rates under banks balance sheet constraint

Transcription

1 A general equilibriu analysis of low noinal interest rates under banks balance sheet constraint Septeber 7, 08 Abstract This paper provides a fraework to exaine general-equilibriu behavior of extreely low or negative noinal interest rates in a noinal dynaic in particular, continuous-tie capital asset pricing odel under banks balance sheet constraint. It finds that a safe asset/wealth ratio is one of key factors to deterine equilibriu asset prices. The dynaics of extreely low or negative equilibriu noinal interest rates are subject to the banks balance sheet constraint. Specifically, during the continuation no-intervention phase, a higher level of the safe asset/wealth ratio tends to result in a lower possibly, negative noinal interest rate. When the ratio hits a ceiling caused by the banks balance sheet constraint, interventions such as bail-ins/outs and asset purchase progras lower the ratio, and in addition, underpin the noinal interest rate and the banks net worth. The ceiling produces an effective lower bound of noinal interest rates, which can be either negative or positive. Keywords: low noinal interest rates, lower bound of noinal interest rates, banks balance sheet constraint, safe assets. JEL Classification Codes: E43, E44, G, G.

2 Introduction Noinal interest rates have been low across any advanced econoies for a long tie since the 990s. After the recent global financial crisis, short-ter policy rates approached zero. Financial conditions were further loosened through forward guidance and asset purchase progras. Furtherore, negative interest rate policies were launched in several European countries and Japan. However, the April 08 Global Financial Stability Report IMF, 08 points out that periods of low interest rates and easy financial conditions ay lead to a decline in lending standards and increased risk taking. Brunnereier and Koby 08 also argue the existence of the reversal interest rate defined as the rate at which accoodative onetary policy reverses its intended effect and becoes contractionary for lending. In fact, ost recently, after the prolonged period of loose financial conditions, the Federal Reserve Board, the European Central Bank and the Bank of England are unwinding their accoodative onetary policies, so as to reduce the effect of the reversal. The Bank of Japan widened the range that it would allow yields on the 0-year Japanese governent bond to ove around zero in fro 0. percentage points to 0. percentage points on July 3, 08. The long-ter interest rates were then raised slightly in Japan. These observations otivate a question: how are extreely low or negative noinal interest rates generated and underpinned structurally in general equilibriu under banks capital constraint? The objective of this paper is to answer the question by providing a fraework to exaine general-equilibriu behavior of extreely low or negative noinal interest rates under banks balance sheet constraint in a noinal dynaic in particular, continuous-tie capital asset pricing odel. A representative investor lives on [0, T ], ranking his consuption based on stochastic differential utility. There is a representative bank, which serves as a vehicle that provides the investor with the opportunity to invest in a noinal safe asset and real assets and anages the assets on behalf of the investor so that the bank s action reflects the preference of the investor. The safe asset is noinal inside oney provided by the bank i.e., deposits, cash, reserves, etc., while the real assets are loans, physical assets, financial assets, derivatives, etc..

3 Because of the safe asset, the dynaics of the econoy are subject to two constraints. First, the bank is subject to its balance sheet constraint, in that the bank s net worth should be positive. Note that there occurs no inforation-based depletion of a scarce reserve fro the banking syste, in contrast to the odel of Diaond and Dybvig 983. Second, there exists a cash-in-advance constraint of the inside oney, in that the investor can withdraw his funds fro the safe asset, not directly fro the real assets, for the purpose of consuption. The ain findings of this paper are as follows. A safe asset/wealth ratio is one of key factors to deterine equilibriu asset prices. The dynaics of extreely low or negative equilibriu noinal interest rates are subject to the bank s balance sheet constraint. Specifically, during the continuation no-intervention phase, a higher level of the safe asset/wealth ratio tends to result in a lower possibly, negative noinal interest rate. When the safe asset/wealth ratio hits a ceiling caused by the bank s balance sheet constraint, interventions such as bail-ins/outs and asset purchase progras lower the ratio, and in addition, underpin the noinal interest rate and the bank s net worth. The ceiling produces an effective lower bound of noinal interest rates. Three strands in previous literatures are related to this paper. First, the financial engineering literature studies shadow rate odels to exaine low noinal interest rates Black, 995; Gorovoi and Linetsky, 004. The shadow interest rate can be negative, and a noinal interest rate is a positive part of the shadow rate owing to the option to convert to currency at zero policy rates. In addition, the lower bound of noinal interest rates can be pushed down below zero Goodfriend, 000; Buiter and Panigirtzoglou, 00; Agarwal and Kiball, 05. These odels are applied to studies of fixed incoe and onetary policy, such as ter structure of interest rates Ki and Singleton, 0; Bauer and Rudebusch, 03; Christensen and Rudebusch, 05; Wu and Xia, 05 and stance of onetary policy Bullard, 0; Krippner, 03. However, the literature has not studied general-equilibriu behavior of noinal interest rates. The constraint could be set to be either tighter or looser, depending on actual data. Skeie 008 exaines the case that withdrawals of a noinal safe asset take the for of electronic payents through the banking account, as in a odern banking syste. 3

4 The second strand is the acroeconoic literature on low noinal interest rates. Zero lower bound has been assued in ost of the literature. There are a recent few exceptions. Bech and Malkhozov 06 detect no change in the transission to arket rates in a siple data analysis, but argue great uncertainty in a prolonged tie of negative rates. Brunnereier and Koby 08 study an effective lower bound in the for of reduced banks profitability and capital constraint. Specifically, they ebed their banking odel in a New Keynesian odel and show that, with sticky prices, an interest cut triggers a deand boost, which raises banks interest incoe. Wu and Zhang 07 connect a shadow rate odel to a New Keynesian odel by replacing the policy rate entering the IS curve with the shadow rate. Still, those odels do not construct a capital asset pricing odel in general equilibriu. Third, in the literature on asset pricing with financial interediaries, the odel of He and Krishnaurthy 03 is related to this odel, in the sense of showing that the capital of financial interediaries plays an iportant role in deterining asset arket equilibriu. In their odel, there are two groups of agents in the econoy: households and specialists. The households face an optial portfolio allocation proble between purchasing equity in the interediaries and the riskless bond. The specialists axiize their own utilities by anaging the interediaries in which the households invest and investing in the risky assets on behalf of the households. On the other hand, this odel invokes the assuption that the bank s action reflects the preference of the investor. Still, the bank is not a veil here, because the bank s capital structure changes the balance between noinal and real shocks and influences equilibriu asset prices. In addition, their odel does not focus on the behavior of shadow rates in an extreely low interest rate environent. This paper bridges a gap aong the three strands by providing a capital asset pricing odel of shadow rates in general equilibriu under banks balance sheet constraint. The safe asset as the inside oney influences equilibriu asset prices. The shadow rates are endogenously driven by the safe asset/wealth ratio in equilibriu, and the bank s balance sheet constraint produces a lower bound of noinal interest rates. The noinal interest rate and the arket price of risk are coputable in general equilibriu. 4

5 Model Tie is t [0, T ] a given terinal tie T > 0. The filtered probability space is Ω, F, F = {F t } 0 t T, P, where F is generated by three-diensional standard Brownian otion B = B B B 3 defined on the probability space. B represents a real GDP shock; B and B 3 represent noinal shocks. B, B and B 3 are independent each other. A representative investor lives on [0, T ] and consues a single consuption good. There is a representative bank financial expert, which serves as a vehicle that provides the investor with the opportunity to invest in a noinal safe short-ter risk-free asset and real assets and anages the assets on behalf of the investor so that the bank s action reflects the preference of the investor in an environent of full inforation. Let the noinal safe asset and the real assets be denoted by D and W, respectively. The investor invests his whole funds X in the bank, with the initial funds X0 = x 0 > 0 given. X := W D call X wealth. For notational convenience, define the real asset/wealth ratio as ρ := W. Correspondingly, let X := ρ denote the safe asset/wealth ratio. The transaction between the safe asset D and the real assets W is carried out at singular, proportional costs λ, λ. Specifically, letting ξ t and ξ t denote cuulative purchase and sale, respectively, of the real assets W fro/to the safe asset D up to t, the purchase of the real assets dξ leads to a decrease λ dξ in the safe asset; the sale of the real assets dξ leads to an increase λ dξ in the safe asset. The real assets W are loans, physical assets, financial assets, derivatives, etc.. In particular, in this odel, they consist of two types of assets: one real risk-free asset, the price of which is denoted by P 0, and two real risky assets, the excess returns of which are denoted by R. The return of the real risk-free asset is dp 0 P 0 = r dt. For siplicity, assue that r is a constant. This assuption iplies that the real yield curve 5

6 is flat. The excess returns of the two risky assets are characterized by dr = µ R dt σ R dbt. The value of the first risky asset is the one of the real GDP, the excess return of which is characterized by dr = µ y dt σ y dbt where σ y = σ y 0 0 where σ y > 0, while the second risky asset is a contingent clai i.e., options, swaps, etc. that is of zero net supply. Let ψ denote the asset allocation ratio on the risky assets in the real assets W. The process of the real assets are characterized by dw t = W t r dt ψ µ R dt σ R dbt dξ dξ. On the other hand, the safe asset is noinal inside oney provided by the bank i.e., deposits, cash, reserves, etc.. This is the nueraire in the econoy. Assue that the investor can withdraw his funds fro the safe asset D, not directly fro the real assets W, for the purpose of consuption, denoted by Ct. Thus, there exists a cash-in-advance constraint of the inside oney. Let i denote the noinal rate of return on the safe asset. The price index process, denoted by P, is characterized by dp P = ε dt σp ρ dbt. With regard to the process of the expected inflation rate ε, defining ν := ρε, ν is ean-reverting, characterized by dν = κθ ν dt σ ν dbt where κ and θ are the speed of the ean-reversion and the long-run ean, respectively. The 6

7 process of the safe asset in real ters is characterized by ddt = Dt it εt dt Dtσ P dbt Ct dt λ dξ λ dξ. In addition, assue that the bank is subject to a balance sheet constraint, in that its net worth should be positive: X D. Note that the constraint could be either tighter or looser, depending on actual data. Assue also that the real su of the noinal safe asset and the real risk-free asset is of zero net supply in equilibriu. In this econoy, the safe asset serves not only as a unit of account and a ediu of exchange but also as a store of value especially in sluggish real arkets, as in the traditional onetary theory. The three functions coexist and influence one another in the econoy. As a result, the spread between the noinal interest rate i and the real interest rate r is endogenously deterined in equilibriu in this odel. For a stopping tie τ T, the investor ranks his consuption process {Ct; 0 t τ} based on the following stochastic differential utility process V : for a consuption process {Ct; 0 t τ}, dv t = fct, V t dt Σt dbt; γ ρ ϑ W τ V τ = γ where f is the Duffie-Epstein utility aggregator: fc, v = β γv log c γ log γv where β > 0 and γ > 0 represent the rate of tie preference and the coefficient of coparative risk aversion, respectively Duffie and Epstein, 99. The contract of the safe asset D is terinated at τ or T, and the investor then receives only the real assets at an additional proportional cost ρ ϑ where 0 < ϑ < and β < ϑ. The cost structure takes such 7

8 particular for as to obtain explicit solution here, but could be generalized. Note that the elasticity of interteporal substitution EIS is unity here. There exists a unique V t for each well-defined consuption process {Ct; 0 t τ}. In addition, V 0 is strictly increasing and concave in {Ct; 0 t τ} Skiadas, Optiization The investor s optiization proble is forulated as follows: V 0 = sup E { C,ψ, ξ, ξ, τ } 0 s.t. dw t = W t r dt ψ µ R dt σ R dbt ddt = Dt it εt dt σp ρ dbt τ T λ dξ λ dξ, νt = κ θ νt dt σ ν dbt. fcs, V s ds e βτ T dξ dξ, Ct dt ρ ϑ W τ T γ γ Recall that dξ and dξ are interventions the purchase and sale of the real assets W fro/to the safe asset D, respectively. Assue that the consuption and the asset allocation ratio {Ct, ψt; t [0, τ T ]} are Markovian controls. This proble is a singular control proble as to the details of its atheatical regularities, see e.g. Øksendal and Sule 007, Ch.5. First, we look at the phase of no intervention. Defining C = φx, dx X = ρdw W ρdd D = ρr ρψ µ R ρi ε φ dt ρψ σ R σ P dbt. 8

9 Thus, dρ ρ = ρ r i ε ψ µ R φ dt ρψ σ R σ P ρψ σ R σ P ψ σ R ρψ σ R σ P dbt. Assue that there exists a C,,, [0, T ] R 3 function J satisfying V t = Jt, x, ρ, ν. The Hailtonian-Jacobi-Bellan HJB equation is as follow: 0 = fc, J J t J x x ρr ρψ µ R ρi ε φ ρ r i ε ψ µ R φ J ρ ρ ρψ σ R σ P ρψ σ R σ P ψ σ R J ν κθ ν φψ J xx x σ R σ P Jρρ ρ ρψ σ R σ P σ ν Jνν J xρ xρ ρψ σ R σ φψ P σ R σ P J xν x φψ σ R σ P σ ν J ρν ρ ρψ σ R σ P σ ν. We try log γj = t log ρ qtν nt γ log x. Fro the first-order conditions, φ = ψ = β γ γ, γρ ρ µ R qσ R σ ν σ R σ P γρ γ γρ ρ σ R σ R γρ ρ γ γρ ρ. In addition, we characterize a triplet r, i, η as follows. Since f c = β γ J C, df c f c = dv V d/c /C dv V d/c /C. 9

10 Since C = φx, dc C = β ρψ σ R σ P = γ dt dbt ρr ρψ µ R ρi ε β ρr ρψ µ R ρi ε φ dt dt ρψ σ R σ P dbt. Thus, d/c /C = dc C ρψ σ R σ P dt. Also, dv V = f v β dt ρψ σ R σ P qσ ν γ ρψ σ R σ P dbt. Owing to the optiality condition f v dt dfc f c = r dt η dbt see e.g. Skiadas 008, 0 = ρψ µ R ρr i ε ρψ σ R σ P η, η = ρ γρ ψ σ R γσ P qσ ν. Assue that a 3 3-atrix σ J := σr is invertible, although σ R is not. Note that, σ P since σ R is not invertible, the noinal safe asset is not a redundant asset. ψ and the triplet r, i, η are well-defined in the arkets. Next, we look at the phase of the interventions. With regard to the purchase of the real assets dξ, λ J D J W 0 0

11 and, with regard to the sale of the real assets dξ, λ J D J W 0. Define α := dξ x and α := dξ x. We can guess that there exist two thresholds ρ L, ρ H of ρ triggering the interventions as follows: for α, α,, q, n, J t, x, ρ, ν = J t, x λ α, ρ L α λ α, ν t exp log ρ qtν nt γ log x γ J t, x λ α, ρ H α λ α, ν 0 < ρ < ρ L ρ L ρ ρ H ρ H < ρ <. Correspondingly, define H := ρ L and L := ρ H. 4 Equilibriu 4. Characterization of equilibriu The econoy is said to be in equilibriu if the optiization holds and the arkets are cleared. The arket-clearing conditions hold: ψ = ρ are characterized as follows: recalling = ρ, 0. Thus, equilibriu asset prices µ y r i ε = σ y σ P η, η = γ σ y γσ P qσ ν. Taking all things together in equilibriu, β = 0, τ T = ϑ γ, q β κq = γ, qτ T = 0, n βn = γr Kt

12 γ σy σ y ρ µ y γ γ ρ ρ ρ γ σ y γσ P qσ ν ρ ρ γ ρ ρ σ y σ P, nτ T = 0. where Kt is a deterinistic function of t, defined as Kt := β γ logβ γ log γ γ qκθ γ γ γ σ P σ P. q σ ν σ ν q γ σ P σ ν To obtain approxiate solutions, we focus on the dynaics in the neighborhood of H = ρ L for the finite period [0, T ], by iposing ρ ρ equations ODEs. = H ρl in the above ordinary differential Fro the ODE of, γ and γ is increasing in tie. With regard to α and ρ L or, H, owing to the value-atching and the sooth-pasting conditions, α ρ L = α γ λ α, γλ γ ρ L. Suppose that α = Thus, α > γλ γλ γ γ ρ L, i.e., λ J ρ L. Then, when ρ L > 0, α D J ρ L > λ α γ. W < 0. Hence, ρ L = 0. Or, equivalently, H =. That is, the bank s balance sheet constraint binds when, and only when, the safe asset/wealth ratio hits the ceiling. Thus, α = and ρ H or, L, since ρ α λ α α = φ = γλ. On the other hand, with regard to α =< 0 for any ρ at the sale of the real assets, ρ H = φ. Thus, β γ γ. Since φ is increasing up to β ϑ <, ρ H is decreasing up to β ϑ. Therefore, τ > T, due to the dynaics of ρ L and ρ H. Owing to the boundary conditions, t = ϑ γ e βt t > 0 and qt = T e βκs t γ s ds. The dynaics t

13 of the optial controls are shown in Figure. Figure : Policy controls Safe short-ter asset D d H =: Bank s B/S constraint 45 degree line Asset purchases, bail-outs/ins Slope d:=-r d L =ft: Cash-in-advance line Total assets X = W D 4. Equilibriu asset prices With regard to the equilibriu asset prices, the noinal interest rate i and the arket price of risk η are: i = r ε µ y σ y σ P η = r ε µ y σ y σ P γ η = γ σ y γσ P qσ ν. σ y γσ P qσ ν, Obviously, the Fisher equation does not hold true: i r ε. The interest rate i can be interpreted as a shadow rate, which is explored first in the seinal paper of Black 985. It can take on negative values, and the bank s balance sheet constraint produces the lower 3

14 bound of the noinal interest rates. Looking at the stochastic differentials of i and η, di = dε µ y σ y σ P γ σ y γσ P qσ ν γ σ y µ y σ y σ P γσ P qσ ν { σy σ P β σ P σy = dε i r ε σy σ P σ y [ ] i r ε σy σ P σ y d { σy σ P β σ P σy d σy σ P σy = d σy β dη = σ P σy } γ σ ν dt d γ σ y σ P σ y [ ] σy σ P σ y d } σ ν dt, d σ ν dq γ σ ν dt d where, noting d = dρ, the process of is characterized by d [ ] d = dρ ρ ρ d[ρ] = = ρr i ε µ y σ y ρσp dt ρ ρ ρ φ σ y ρσp σ P dt σ y ρσp dbt, ρ ρ ρ and, noting ν = ρε and dν = κθ ν dt σ ν dbt, the process of ε is characterized by ν dε = d ρ = dν ν d dν d 4

15 = κθ ν dt σ ν dbt ν ρr i ε µ y ρ ρ φ σ y ρσp σ P dt σ y ρσp dbt ρ ρ ρ σ y ρσp σ ν dt ρ = κ ρr i ε µy θ ν ν σ y ρσp σ ν dt β σ y ρσp σ P ρ ρ ρ σ y ρσp ρ σν ρ ρ ϑ e βt t dbt. Thus, the dynaics of the noinal interest rate and the arket price of risk are driven by the expected inflation rate process ε and the safe asset/wealth ratio process. The ain findings of this paper are as follows. The safe asset/wealth ratio = ρ is one of key factors to deterine equilibriu asset prices in this odel. The dynaics of extreely low or negative equilibriu noinal interest rates are subject to the bank s balance sheet constraint. Specifically, noting that there is the ter ν d in dε, the arginal effect of i on the noinal interest rate i is r ε ν σ y σ P σ y. Assue that the covariance between inflation and real GDP is saller than the variance of real GDP: σ y σ y > σ P σ y, and that the expected inflation ν is zero or slightly positive. Let us look at the case of either i r ε or i r ε. Recalling that = ϑ γ e βt t > 0, i r ε ν σ y σ P σ y > 0. During the continuation no-intervention phase, a higher level of results in a lower noinal interest rate i. As the safe asset/wealth ratio hits the ceiling H = caused by the bank s balance sheet constraint, the intervention lowers the safe asset/wealth ratio, and in addition, underpins the noinal interest rate and the bank s net worth. The intervention can be interpreted as bail-ins/outs and asset purchase progras. The ceiling H = produces an effective lower bound of noinal interest rates, denoted by i L. The lower bound i L can be either positive or negative: i L = r ε µ y σ y σ P γσ y γσ P qσ ν. 5

16 When i is sufficiently far downwards fro r ε that a negative spread i r ε < 0 ultiplied by overwhels the positive ters ν i is reversed, so that i is reverting to r ε. σ y σ P σ y, the effect of on 5 Conclusion This paper exained general-equilibriu behavior of extreely low or negative noinal interest rates in a noinal dynaic in particular, continuous-tie capital asset pricing odel under the bank s balance sheet constraint. We have founded that the safe asset/wealth ratio is one of key factors to deterine equilibriu asset prices. The dynaics of extreely low or negative equilibriu noinal interest rates are subject to the bank s balance sheet constraint. Specifically, during the continuation no-intervention phase, a higher level of the safe asset/wealth ratio tends to result in a lower possibly, negative noinal interest rate. When the ratio hits a ceiling caused by the constraint, interventions such as bail-ins/outs and asset purchase progras lower the ratio, and in addition, underpin the noinal interest rate and the bank s net worth. The ceiling produces an effective lower bound of noinal interest rates. Several points reain unsolved in this paper. EIS was assued to be unity in this odel. However, it is well known that the assuption is irrelevant in practice. The odel with nonunity EIS would require further nuerical approxiations to obtain explicit solutions. In addition, the fraework of this paper should be applied to elaborate nuerical studies of effective lower bounds, ter structures of interest rates, liquidity shortage, and ultiple currencies. They are future work in this line of research. References [] Agarwal, R. and Kiball, M. 05 Breaking through the zero lower bound, IMF Working Paper, No.5/4. 6

17 [] Bauer, M.D., and Rudebusch, G.D. 06 Monetary policy expectations at the zero lower bound, Journal of Money, Credit and Banking, 48: [3] Bech, M. and Malkhozov, A. 06 How have central banks ipleented negative policy rates? BIS Quarterly Review, [4] Black, F. 995 Interest rates as options, Journal of Finance, 50: [5] Brunnereier, M., and Koby, Y. 08 The reversal interest rate, Working paper. [6] Buiter, W. and Panigirtzoglou, N. 00 Liquidity traps: How to avoid the and how to escape the, in W. Vanthoor and J. Mooij Eds., Reflections on Econoics and Econoetrics, Essays in Honour of Martin Fase, De Nederlandsche Bank, [7] Bullard, J. 0 Shadow interest rates and the stance of U.S. onetary policy, Speech at Center for Finance and Accounting Research. Annual Corporate Finance Conference. Olin Business School, Washington University in St. Louis. [8] Christensen, J.H.E., and Rudebusch, G.D. 05 Estiating shadow-rate ter structure odels with near-zero yields, Journal of Financial Econoetrics, 05, 3: [9] Diaond, D.W., and Dybvig, P.H. 983 Bank Runs, Deposit Insurance, and Liquidity, Journal of Political Econoy, 9: [0] Duffie, D., and Epstein, L. 99 Stochastic Differential Utility, Econoetrica, 60: [] Goodfriend M. 000 Overcoing the zero bound on interest rate policy, Journal of Money, Credit and Banking, 3: [] Gorovoi, V., and Linetsky V. 004 Black s odel of interest rates as options, eigenfunction expansions and Japanese interest rates, Matheatical Finance, 4:

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