T HE R AMSEY RULE AND YIELD CURVE MODELING : ECONOMIC AND FINANCIAL - - PowerPoint PPT Presentation

The economic framework The financial framework and progressive utilities Yield curve dynamics

THE RAMSEY RULE AND YIELD CURVE MODELING:

ECONOMIC AND FINANCIAL VIEWPOINTS. Caroline HILLAIRET, Ensae, CREST

Joint work with Nicole El Karoui and Mohamed Mrad (Sorbonne Université, Université Paris XIII) With the support of "Chaire Risques Financiers" and ANR "Lolita"

OICA Conference April 28th, 2020

1/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics

MOTIVATION

Modeling accurately long term interest rates is a crucial challenge ◮ Embedded long term interest rate risk in longevity-linked securities (maturity up to 30 − 50 years.) ◮ financing of ecological projects ◮ valuation of any other investment with long term impact. ◮ Because of the lack of liquidity for long horizon, the standard financial point of view cannot be easily extended. Economic point of view ◮ Extensive literature on the economic aspects of long-term policy-making (Ekeland, Gollier, Weitzman...), ◮ Often motivated by ecological issues (Gollier, Hourcade & Lecocq) ◮ How to discount the far-distant future? (Gollier) ◮ Based on the equilibrium theory

2/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics

THE UNDERLYING OPTIMIZATION PROBLEMS

The following utility optimization program has to be solved in both financial and economic frames; usually formulated on a given horizon TH. U(t, x) = ess sup

(π,c)∈A

E

• u(TH, X π,c

TH (t, x)) +

TH

t

v(s, cs)ds|Xt = x

• , a.s.

(1) ◮ Link between optimal wealth and consumption process (X ∗

t , c∗ t ) and

• ptimal discounted pricing kernel (Y ∗

t ) of the dual problem, given by the

first order relation Ux(t, X ∗

t ) = Y ∗ t = vc(t, c∗ t ),

− ˜ Uy(t, Y ∗

t ) = X ∗ t , −˜

vy(t, Y ∗

t ) = c∗ t .

◮ Dual optimization program

• U(t, y) = ess inf

Y

E

• ˜

u(TH, YTH ) + TH

t

˜ v(s, Ys)ds|Yt = y

• , a.s.

(2) ◮ dual conjugate utility ˜ u(y) = supx>0

• u(x) − yx)

3/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics

THE UNDERLYING OPTIMIZATION PROBLEMS

Financial and economic frameworks both rely on the same optimization problem, that determines the optimal discounted pricing kernel used to evaluate claims (and that is the cornerstone of the Ramsey rule) ◮ The financial point of view, based on No-Arbitrage assumption

assets, bank account, time-horizon, and utility functions are given exogenously, the problem is to characterize the optimal (self-financing) wealth-consumption plan.

◮ At the economic equilibrium

the optimal investment strategy πe is given (market clearing condition) the problem is to find (if they exist), two utility functions (U, v) and a consumption rate ce such that the pair (πe, ce) is optimal In general, the dynamics of the technology/risky asset as well as the short rate are endogenously determined

4/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The economic equilibrium The Ramsey rule

OUTLINE

1 THE ECONOMIC FRAMEWORK

The economic equilibrium The Ramsey rule

2 THE FINANCIAL FRAMEWORK AND PROGRESSIVE UTILITIES

The financial market Pathwise Ramsey rule and financial interpretation

3 YIELD CURVE DYNAMICS

Yield curve dynamics and their volatilities Infinite maturity yield curve

5/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The economic equilibrium The Ramsey rule

THE EQUILIBRIUM SPOT RATE

◮ Usual setting of equilibrium

Power utility functions Geometric Brownian motion for the risky security/technology dSt = St

• µtdt + σtdWt
• with deterministic coefficient µ and σ

◮ optimal investment π∗

t (x) = − (µt −rt (x))Ux (t,x) σ2

t xUxx (t,x)

together with the supply-equals-demand condition for risky security π∗ = 1 determines endogenously the risk free rate r ∗

t .

r ∗

t (x) = r(t, X ∗ t (x))

with r(t, x) = µt + σ2

t x Uxx(t, x)

Ux(t, x) . (3) ◮ link between the risk premium and relative risk aversion of the utility process U η(t, X ∗

t ) = −σt X ∗ t Uxx(t, X ∗ t )

Ux(t, X ∗

t )

= σtRr

A(U)(t, X ∗ t ).

6/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The economic equilibrium The Ramsey rule

THE EQUILIBRIUM : CONS

Traditional approaches, based on the theory of general equilibrium, are not always flexible enough to apprehend the long-term and to adapt to the uncertain evolution of the economic or financial environment. ◮ Cons of the usual setting of equilibrium (power utilities, geometric Brownian motion for optimal discounted pricing kernel)

the way that preferences of multiple agents aggregate is a difficult task, and it is unlikely that the aggregate utility could be modeled by a simple function : the heterogeneity of economic actors is often downplayed in concrete applications, that use a simplified version of the theory not flexible framework

• ptimal processes/choices are linear w.r.t. their initial conditions : can

induce an underestimation of extreme risks equilibrium rate r does not depend on the wealth of the economy

7/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The economic equilibrium The Ramsey rule

THE EQUILIBRIUM : PROS

◮ Pros : of the usual setting of economic equilibrium

For a complete Markovian market, it is the only frame compatible with the existence of an equilibrium (see recent work of El Karoui and Mrad):

the pricing kernel is a geometric Brownian motion the utility process is a mixture of dynamic power utilities. the market risk premium is deterministic the equilibrium portfolio is a mixture of geometric Brownian motion

the dependency of the rate on the time-horizon TH of the optimization problem is in fact artificial, since the utility is part of the processes that should be determined at equilibrium. the expression for the interest rate (3), together with the dynamics of the wealth process dX ∗

t = (µtX ∗ t − c∗ t )dt + X ∗ t σtdWt

→ the equilibrium poses the problem in a natural forward formulation

◮ Our approach: adopt a forward formulation with stochastic utility:

can incorporate the possibility of changes in agent preferences over time or the uncertain evolution of the economic or financial environment. generate more complex pricing kernel capture more phenomena, particularly with regard to aggregation of heterogeneous agents and extreme risks.

8/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The economic equilibrium The Ramsey rule

RAMSEY RULE IN ECONOMICS

A link between consumption rate and discount rate at the economic equilibrium ◮ compute today of a long term discount rate R0(T). ◮ Based on "representative agent" (Ramsey, Gollier) ◮ Small perturbation around the economic equilibrium Data and parameters ◮ u utility function (concave, increasing) typically u(t, c) = e−βtc1−θ/(1 − θ) ◮ θ= risk aversion coefficient ◮ β pure time preference parameter ◮ c aggregate consumption rate, typically geometric Brownian motion cT = c0 exp((g − 1

2σ2)T + σWT) with g consumption growth rate.

Ramsey rule : R0(T) = − 1

T ln E

u′(T,cT )

u′(0,c0)

• . (u′= marginal utility)

9/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The economic equilibrium The Ramsey rule

◮ "Historical" Ramsey rule (Ramsey, 1928) : cT = c0 exp(gT) R0(T) = β + θg, (4)

β pure time preference parameter, θ risk aversion, g growth rate.

◮ No consensus among economists about the parameter values that should be considered

Example : Stern review on climate change (2006), with θ = 1, g = 1.3%, β = 0.1% → R0(T) = 1.4%. Or θ = 1.2, g = 2%, β = 0.1% → R0(T) = 2.5%

◮ Economic rates are very sensitive to the rate of preference for the present β, which can be viewed as the intensity of an independent exponential random horizon ◮ If cT = c0 exp((g − 1

2σ2)T + σWT) the Ramsey rule still induces a flat

curve R0(T) = β + θg − 1 2θ(θ + 1)σ2. (5)

10/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

OUTLINE

1 THE ECONOMIC FRAMEWORK

The economic equilibrium The Ramsey rule

2 THE FINANCIAL FRAMEWORK AND PROGRESSIVE UTILITIES

The financial market Pathwise Ramsey rule and financial interpretation

3 YIELD CURVE DYNAMICS

Yield curve dynamics and their volatilities Infinite maturity yield curve

11/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

THE FINANCIAL MARKET

Financial framework: No arbitrage approach with exogenously given spot rate. ◮ Filtered probability space (Ω, F = (Ft)t≥0, P). ◮ N-dimensional Brownian motion Market Parameters : Incomplete market ◮ M risky assets, M ≤ N. ◮ (rt)t≥0, (ηt)t≥0, (σt)t≥0 adapted processes. ◮ rt ≥ 0 spot rate. ◮ ηt N-dimensional risk premium vector. ◮ σt volatility process M × N. σtσtr

t invertible.

12/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

THE REPRESENTATIVE AGENT

◮ Representative agent, strategy (π, c).

c(.) : consumption rate, ct = ρtXt π(.) : fractions of the wealth invested in the risky asset. We set κt := σtr

t πt

◮ Constraints on the portfolio ⇒ Incompleteness of the market. κt ∈ Rt where Rt adapted subvector spaces in RN. ◮ Self financing dynamics of wealth process with risky portfolio κ and consumption rate c is given by dX κ,ρ

t

= X κ,ρ

t

[(rt − ρt)dt + κt(dWt + ηtdt)], κt ∈ Rt. (6) Remark : κt.ηt = κt.ηR

t

where ηR is the “minimal” risk premium.

13/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

DISCOUNTED PRICING KERNEL AND DUALITY

◮ An Itô semimartingale Y ν is called a discounted pricing kernel if for any admissible X κ,ρ, Y ν

. X κ,ρ .

+ .

0 Y ν s ρsX κ,ρ s

ds is a local martingale. ◮ ⇒ there exists ν ∈ R⊥ such that dY ν

t = Y ν t [−rtdt + (νt − ηR t ).dWt],

νt ∈ R⊥

t ,

Y ν

0 = y

(7) ◮ Y ν is the product of the “minimal” discounted pricing kernel Y 0 (ν = 0) by the orthogonal density martingale Lν

t = L⊥ t = exp

t

0 νs.dWs − 1 2

t

0 ||νs||2ds

• .

The discounted pricing kernels are the cornerstone of the Ramsey rule ◮ In the economic framework: r and η are determined endogenously at equilibrium ◮ In the financial framework r and η are exogenous, ν is determined at the optimum.

14/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

PRIMAL AND DUAL PROGRESSIVE UTILITY

◮ Necessity to adjust the optimisation criterion ◮ The decision has to be sequential based on learning from the past, and from the environment evolution to revise the optimal policy over the time (Lecocq and Hourcade) ◮ Progressive utilities first introduced by Musiela & Zariphopoulou, existence and characterization studied by El Karoui & Mrad ◮ (U, V) progressive utility system satisfying a dynamic programming principle called market-consistency.

for any admissible wealth process X κ,ρ, with consumption rate c = ρX κ,ρ, U(t, X κ,ρ

t

) + t

0 V(s, cs)ds is a positive supermartingale.

there exists an optimal strategy for which it is a martingale.

◮ Market-consistency for the dual progressive utility system ( U, V)

for any admissible Y ν (with ν ∈ R⊥) U(t, Y ν

t ) +

t V(s, Y ν

s )ds is a

submartingale there exists an optimal process Y ∗ for which it is a martingale.

15/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

PATHWISE RAMSEY RULE

◮ Pathwise relation between optimal consumption rate c∗ and the optimal discounted pricing kernel Y ∗ c∗

t

= −˜ Vy(t, Y ∗

t )

i.e. Vc(t, c∗

t ) = Y ∗ t , 0 ≤ t ≤ T

c = −˜ Vy(0, y) i.e. Vc(0, c) = y Vc(t, c∗

t )

Vc(0, c) = Y ∗

t (y)

y , 0 ≤ t ≤ T with Vc(0, c) = y. (8) ◮ The Ramsey rule leads to a description of the equilibrium interest rate as a function of the optimal discounted pricing kernel Y ∗, ∀t < T Re

t (T−t)(y) := −

1 T − t ln E Vc(T, c∗

T(c))

Vc(t, c∗

t (c))

• Ft
• = −

1 T − t ln E Y ∗

T (y)

Y ∗

t (y)|Ft

• 16/ 33

Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

UTILITY INDIFFERENCE PRICING

Valuation of a non-replicable claim ζT ◮ Utility indifference price of a positive claim ζT delivered in T: it is the cash amount for which the investor is indifferent from investing or not in the claim

without the claim UT (x) := sup

(κ,c)∈Ac E[U(T, X κ,ρ T

) + T V(s, cs)ds]. (9) with the claim Uζ,T (x, q) := sup

(κ,c)∈Ac E[U(T, X κ,ρ T

− q ζT ) + T V(s, cs)ds]. (10)

◮ The utility indifference price is the cash amount pq(x, ζT, q) determined by the relationship Uζ,T(x + pq(x, ζT, q), q) = UT(x). (11)

17/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

MARGINAL UTILITY PRICES (DAVIS PRICE)

When the agents are aware of their sensitivity to the unhedgeable risk, they can try to transact for only a little amount in the risky contract ◮ Davis price or marginal utility price, ◮ corresponds to the zero marginal rate of substitution pu

t (x) := lim q→0

∂ˆ pq

t

∂q (x, ζT, q). Marginal utility price at time t of the claim ζT: pu

t (x) = E[ζT Y ∗ T (y)

Y ∗

t (y)|Ft],

y = Ux(t, x).

18/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

FINANCIAL INTERPRETATION OF THE EQUILIBRIUM YIELD CURVE

◮ Aim: Give a financial interpretation of E

• Y ∗

T (y)

Y ∗

t (y)|Ft

• (for t < T) in terms of

price of zero-coupon bonds ◮ Yield curve Rt(T − t) and zero-coupon bond price B(t, T) are linked by B(t, T) = exp(−(T − t)Rt(T − t)). ◮ Replicable Bond B(t, T) is computed by the minimal risk neutral pricing rule B(t, T) = E[

Y 0

T

Y 0

t |Ft] = EQ[e−

T

t

rsds|Ft] : for replicable bond,

equilibrium interest rate and market interest rate coincide. ◮ For non replicable Bond B∗(t, T) denotes the marginal indifference utility price (or Davis price) B∗(t, T)(y) = E[Y ∗

T (y)

Y ∗

t (y)|Ft] = E[Vc(T, c∗ T(c))

Vc(t, c∗

t (c)) |Ft].

Equilibrium interest rates and marginal utility interest rates are the same. Nevertheless, this last curve is valid only for small trades.

19/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

AGGREGATION AND WEALTH DEPENDENCY

Aggregation of power utilities = aggregation of discounted pricing kernels ◮ natural modeling in the context of heterogeneity of the investors ◮ compatible with the existence of an equilibrium (see El karoui & Mrad) ◮ more flexible model while being still tractable ◮ setting in which the yield curves may depend on the wealth of the economy Aggregation of N investors ◮ (constant) relative risk aversion parameters θ1 < · · · < θN. ◮ endowed at time 0 with a proportion αi of the initial global wealth x B∗(0, T)(y) = N

i=1 y θi (y)B∗,θi (0, T)

y = N

i=1(αix)−θi B∗,θi (0, T)

N

i=1(αix)−θi

with y θi (y) = uθi

x (αix) = (αix)−θi

◮ Asymptotic behavior for small and large wealth lim

y→0 B∗(0, T)(y) = Bθ1 0 (T) and

lim

y→+∞ B∗(0, T)(y) = BθN 0 (T).

20/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

AGGREGATION AND WEALTH DEPENDENCY

Aggregation of a continuum of heterogeneous investors indexed by a (constant) relative risk aversion θ, with different weights m(dθ) ◮ the marginal utility bond curve B∗(t, T)(y) is a normalized mixture of individual bond curves, based on the orthogonal martingales L⊥,∗,θ

t

, B∗(t, T)(y) =

• B∗,θ(t, T)(y θ)

L⊥,∗,θ

t

(y θ)

• L⊥,∗,θ

t

(y θ)m(dθ) m(dθ). ◮ The marginal utility spot forward rate f ∗(t, T)(y) = −∂T ln B∗(t, T)(y) is a normalized mixture of individual spot forward rates curve based on the martingales Y ∗,θ

t

(y θ)B∗,θ(t, T)(y θ) f ∗(t, T)(y) =

• f ∗,θ(t, T)(y θ)

Y ∗,θ

t

(y θ)B∗,θ(t, T)(y θ)

• Y ∗,θ

t

(y θ)B∗,θ(t, T)(y θ)m(dθ) m(dθ).

21/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

INDIVIDUAL YIELD CURVE FOR DIFFERENT VALUES OF THE RISK

AVERSION

FIGURE: Individual yield curve R∗,θ (δ) for different values of the risk aversion θ

22/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

INDIVIDUAL AND AGGREGATE YIELD CURVE SPREAD

Spreads between the different rate curves and the market yield curve R0

0(δ) :

spread = R∗,θ (δ) − R0

0(δ).

FIGURE: Individual and aggregate yield curve spread

23/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

AGGREGATE YIELD CURVE SPREAD DEPENDING ON THE WEALTH x

FIGURE: Aggregate yield curve spread depending on the wealth x

24/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics The financial market Pathwise Ramsey rule and financial interpretation

AGGREGATE YIELD CURVE SPREAD DEPENDING ON INITIAL PROPORTION

PARAMETERS α

FIGURE: Aggregate yield curve spread depending on the initial proportion parameters α

25/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics Yield curve dynamics and their volatilities Infinite maturity yield curve

OUTLINE

1 THE ECONOMIC FRAMEWORK

The economic equilibrium The Ramsey rule

2 THE FINANCIAL FRAMEWORK AND PROGRESSIVE UTILITIES

The financial market Pathwise Ramsey rule and financial interpretation

3 YIELD CURVE DYNAMICS

Yield curve dynamics and their volatilities Infinite maturity yield curve

26/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics Yield curve dynamics and their volatilities Infinite maturity yield curve

HEATH-JARROW-MORTON FRAMEWORK

We consider both the economic and the financial viewpoints We focus on the volatility family of the zero-coupon bonds Heath-Jarrow-Morton framework ◮ it characterizes the dynamics of the yield curve. ◮ this characteristic is determined directly by the martingale property of the process (Y ∗

t (y)B∗(t, T)(y))t∈[0,T]

◮ volatility of Y ∗

t (y) : S∗(y) := ν∗(y) − ηR(y) (resp. (−ηR(y)) for Y 0)

In the economic framework ηR(y) is endogenous, in the financial setting ηR is exogenous (and usually taken independent of y).

◮ volatility of B∗(t, T)(y) denoted by Γ∗(t, T)(y).

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The economic framework The financial framework and progressive utilities Yield curve dynamics Yield curve dynamics and their volatilities Infinite maturity yield curve

SPOT RATES AND WEALTH DEPENDENCY

◮ Taking the logarithm derivatives of (Y ∗

t (y)B∗(t, T)(y))t∈[0,T] w.r.t. T gives

the spot forward rates f ∗(t, T)(y) = −∂T ln B∗(t, T)(y) is f ∗(t, T)(y) = f ∗(0, T)(y)− t γ∗(s, T)(y).(dWs+(S∗

s(y)−Γ∗(s, T)(y))ds).

with γ∗(t, T)(y) := ∂TΓ∗(t, T)(y) assumed to be locally bounded ◮ The spot rate rt(y) = lim

T→t f ∗(t, T)(y) is given by

rt(y) = f ∗(0, t)(y) − t γ∗(s, t)(y).(dWs + (S∗

s(y) − Γ∗(s, t)(y))ds),

and its dynamics is given by drt(y) = ∂2f ∗(t, t)(y)dt − γ∗(t, t)(y).

• dWt + S∗

t (y)dt

• .

◮ This implies that for exogenous spot rate r that does not depend on y, γ∗(t, t) and ∂2f ∗(t, t) + γ∗(t, t).ν∗

t (y) should not depend on y.

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A GAUSSIAN FRAMEWORK

The Vasicek (Hull and White) model (in incomplete market) ◮ we assume deterministic volatilities S∗(y) and Γ(., T)(y), and

Γ∗(t, T)(y) = Γ∗(T−t)(y) = σ(y) (1 − e−a(y)(T−t)) a(y) , γ∗(t, T) = σ(y)e−a(y)(T−t)

σ(y) is the diffusion vector, a(y) is the mean reversion speed. ◮ It follows the dynamics for the spot rate r drt(y) = a(y)(bt(y) − rt(y))dt + σ(y).(dWt + S∗

t (y)dt)

with bt(y) = ∂2f ∗(0,t)(y)

a(y)

+ f ∗(0, t)(y) + ||σ(y)||2

2a(y)2 (1 − e−2a(y)t).

◮ The initial spot forward rates curve t → f ∗(0, t)(y) satisfies a Vasicek model (corresponding to b(y) constant in time) if ∂2f ∗(0, t)(y) + a(y)(f ∗(0, t)(y) − b(y)) + ||σ||2

2a(y) (1 − e−2a(y)t) = 0

◮ If the spot rate r does not depend on y, then σ is independent of y, and the drift parameter a(y)(bt(y) − rt) is equal to σ.S∗(y) plus a term that does not depend on y.

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INFINITE MATURITY YIELD CURVE, FORWARD CASE

B∗(t, T)(y) B∗(0, T)(y) = exp t rs(y)ds+Γ∗(s, T)(y).(dWs+S∗

s(y)ds)−1

2||Γ∗(s, T)(y)||2ds

• ◮ Infinite maturity yield curve lt(y) :=

lim

T→+∞ R∗ t (T, y).

◮ when T is large, R∗

t (T)(y) behaves as

R∗

0 (T)(y)−

t Γ∗(s, T)(y) T .dWs+ t ||Γ∗(s, T)(y)||2 2T ds+ t Γ∗(s, T)(y) T .S∗

s (y)ds

◮ thus we have to study together the behavior of 1

T

t

0 Γ∗(s, T)(y).dWs, 1 T

t

0 ||Γ∗(s, T)(y)||2ds and 1 T

t

0 Γ∗(s, T)(y).S∗ s(y)ds for a fixed t.

◮ In the forward case, the infinite maturity yield curve lt(y) is

infinite if lim

T→+∞ Γ∗(t,T)(y) T

exists and is not equal to zero dt ⊗ dP a.s. Otherwise, lt(y) = l0(y) + t

g2

s (y)

2

ds with g2

t (y) :=

lim

T→+∞ ||Γ∗(t,T)(y)||2 T

. So, the long run interest rate lt is still a nondecreasing process in time starting from l0, constant if gt ≡ 0 for all t.

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The economic framework The financial framework and progressive utilities Yield curve dynamics Yield curve dynamics and their volatilities Infinite maturity yield curve

INFINITE MATURITY YIELD CURVE, BACKWARD CASE

In the backward case, ν∗,H

s

may also depends on TH, thus the infinite maturity yield curve lt(y) is ◮ the same as in the forward case if lim

TH→+∞ ||Γ∗,⊥(s,TH)|| ||ν∗,H

s

||

= ∞ ◮ if lim

TH→+∞ ||Γ∗,⊥(s,TH)|| ||ν∗,H

s

||

= 0, lt(y) = l0(y) + t

0 hs(y)ds with

hs(y) := lim

TH→+∞ ||Γ∗,R(s,TH)(y)||2 2TH

+ Γ∗,⊥(s,TH).ν∗,H

s

TH

◮ otherwise lt(y) = l0(y) + t

0 hs(y)ds with

hs(y) := lim

TH→+∞ ||Γ∗(s,TH)(y)||2 2TH

+ Γ∗,⊥(s,TH).ν∗,H

s

TH

. Example: for power utility and Gaussian market ν∗,H

t

= −(1 − θ)Γ∗,⊥(t, TH), θκ∗,H

t

+ (1 − θ)Γ∗,R(t, TH) = ηR

t .

31/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics Yield curve dynamics and their volatilities Infinite maturity yield curve

CONCLUSION

◮ Compare financial and economic points of view for long term yield curve modeling ◮ Common threads: Ramsey rule and discounted pricing kernel ◮ Dynamic utility framework allows capture more phenomena, particularly with regard to aggregation of heterogeneous agents and extreme risks ◮ Discussion on backward/forward approach ◮ Focus on the dependency on the wealth of the economy. Paper available on Hal https://hal.archives-ouvertes.fr/hal-00974815

32/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints

The economic framework The financial framework and progressive utilities Yield curve dynamics Yield curve dynamics and their volatilities Infinite maturity yield curve

BIBLIOGRAPHY

[1] I. Ekeland. Discounting the future: the case of climate change. Lecture Notes 2009. [2] C. Gollier. The consumption-based determinants of the term structure of discount rates. Mathematics and Financial Economics, 1(2):81-101, July 2007. [4] E. Jouini, J.M. Marin, and C. Napp. Discounting and divergence of opinion. Journal of Economic Theory, 145:830-859, 2010. [5] F. Lecocq and J.C. Hourcade. Le taux d’actualisation contre le principe de précaution? Leçons à partir du cas des politiques climatiques. [6] N. El Karoui, M. Mrad. An Exact Connection between two Solvable SDEs and a Non Linear Utility Stochastic PDEs, SIAM Journal on Financial Mathematics (2013). [7] N. El Karoui, C. Hillairet, M. Mrad. Construction of an aggregate consistent utility, without Pareto optimality. Application to Long-Term yield curve Modeling, International Symposium on BSDEs, Springer (2018) [8] M. Musiela and T. Zariphopoulou. Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model. Advances in mathematical finance, (2010).

33/ 33 Caroline HILLAIRET, Ensae, CREST Ramsey rule: economic and financial viewpoints