the ramsey rule and yield curve modeling: economic and ... · 9/ 33 caroline hillairet, ensae,...

The economic frameworkThe 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 ConferenceApril 28th, 2020

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



MOTIVATION

Modeling accurately long term interest rates is a crucial challengeI Embedded long term interest rate risk in longevity-linked securities

(maturity up to 30− 50 years.)I financing of ecological projectsI valuation of any other investment with long term impact.I Because of the lack of liquidity for long horizon, the standard financial

point of view cannot be easily extended.

Economic point of viewI Extensive literature on the economic aspects of long-term policy-making

(Ekeland, Gollier, Weitzman...),I Often motivated by ecological issues (Gollier, Hourcade & Lecocq)I How to discount the far-distant future? (Gollier)I Based on the equilibrium theory




THE UNDERLYING OPTIMIZATION PROBLEMS

The following utility optimization program has to be solved in both financialand economic frames; usually formulated on a given horizon TH .

U(t , x) = ess sup(π,c)∈A

E(

u(TH ,Xπ,cTH

(t , x)) +

∫ TH

tv(s, cs)ds|Xt = x

), a.s. (1)

I Link between optimal wealth and consumption process (X∗t , c∗t ) and

optimal discounted pricing kernel (Y ∗t ) of the dual problem, given by thefirst order relation

Ux (t ,X∗t ) = Y ∗t = vc(t , c∗t ), −Uy (t ,Y ∗t ) = X∗t ,−vy (t ,Y ∗t ) = c∗t .

I Dual optimization program

U(t , y) = ess infY

E(

u(TH ,YTH ) +

∫ TH

tv(s,Ys)ds|Yt = y

), a.s. (2)

I dual conjugate utility u(y) = supx>0

(u(x)− yx)




THE UNDERLYING OPTIMIZATION PROBLEMS

Financial and economic frameworks both rely on the same optimizationproblem, that determines the optimal discounted pricing kernel used toevaluate claims (and that is the cornerstone of the Ramsey rule)

I The financial point of view, based on No-Arbitrage assumptionassets, bank account, time-horizon, and utility functions are givenexogenously,the problem is to characterize the optimal (self-financing)wealth-consumption plan.

I At the economic equilibriumthe optimal investment strategy πe is given (market clearing condition)the problem is to find (if they exist), two utility functions (U , v) and aconsumption rate ce such that the pair (πe, ce) is optimalIn general, the dynamics of the technology/risky asset as well as the shortrate are endogenously determined




The economic equilibriumThe Ramsey rule

OUTLINE

1 THE ECONOMIC FRAMEWORKThe economic equilibriumThe Ramsey rule

2 THE FINANCIAL FRAMEWORK AND PROGRESSIVE UTILITIESThe financial marketPathwise Ramsey rule and financial interpretation

3 YIELD CURVE DYNAMICSYield curve dynamics and their volatilitiesInfinite maturity yield curve





THE EQUILIBRIUM SPOT RATE

I Usual setting of equilibriumPower utility functionsGeometric Brownian motion for the risky security/technology

dSt = St(µt dt + σt dWt

)with deterministic coefficient µ and σ

I optimal investment π∗t (x) = − (µt−rt (x))Ux (t,x)

σ2t xUxx (t,x)

together with the

supply-equals-demand condition for risky security π∗ = 1 determinesendogenously the risk free rate r∗t .

r∗t (x) = r(t ,X∗t (x)) with r(t , x) = µt + σ2t xUxx (t , x)

Ux (t , x). (3)

I link between the risk premium and relative risk aversion of the utilityprocess U

η(t ,X∗t ) = −σtX∗t Uxx (t ,X∗t )

Ux (t ,X∗t )= σt Rr

A(U)(t ,X∗t ).





THE EQUILIBRIUM : CONS

Traditional approaches, based on the theory of general equilibrium, are notalways flexible enough to apprehend the long-term and to adapt to theuncertain evolution of the economic or financial environment.

I Cons of the usual setting of equilibrium (power utilities, geometricBrownian motion for optimal discounted pricing kernel)

the way that preferences of multiple agents aggregate is a difficult task, andit is unlikely that the aggregate utility could be modeled by a simple function :the heterogeneity of economic actors is often downplayed in concreteapplications, that use a simplified version of the theorynot flexible frameworkoptimal processes/choices are linear w.r.t. their initial conditions : caninduce an underestimation of extreme risksequilibrium rate r does not depend on the wealth of the economy





THE EQUILIBRIUM : PROS

I Pros : of the usual setting of economic equilibriumFor a complete Markovian market, it is the only frame compatible with theexistence of an equilibrium (see recent work of El Karoui and Mrad):

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

the dependency of the rate on the time-horizon TH of the optimizationproblem is in fact artificial, since the utility is part of the processes thatshould be determined at equilibrium.the expression for the interest rate (3), together with the dynamics of thewealth process dX∗t = (µt X∗t − c∗t )dt + X∗t σt dWt→ the equilibrium poses the problem in a natural forward formulation

I Our approach: adopt a forward formulation with stochastic utility:can incorporate the possibility of changes in agent preferences over time orthe uncertain evolution of the economic or financial environment.generate more complex pricing kernelcapture more phenomena, particularly with regard to aggregation ofheterogeneous agents and extreme risks.





RAMSEY RULE IN ECONOMICS

A link between consumption rate and discount rate at the economicequilibriumI compute today of a long term discount rate R0(T ).I Based on "representative agent" (Ramsey, Gollier)I Small perturbation around the economic equilibrium

Data and parametersI u utility function (concave, increasing) typically

u(t , c) = e−βtc1−θ/(1− θ)

I θ= risk aversion coefficientI β pure time preference parameterI c aggregate consumption rate, typically geometric Brownian motion

cT = c0 exp((g − 12σ

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

Ramsey rule : R0(T ) = − 1T lnE

[ u′(T ,cT )u′(0,c0)

]. (u′= marginal utility)





I "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.I No consensus among economists about the parameter values that

should be consideredExample : 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%

I Economic rates are very sensitive to the rate of preference for thepresent β, which can be viewed as the intensity of an independentexponential random horizon

I If cT = c0 exp((g − 12σ

2)T + σWT ) the Ramsey rule still induces a flatcurve

R0(T ) = β + θg − 12θ(θ + 1)σ2. (5)




The financial marketPathwise Ramsey rule and financial interpretation

OUTLINE








THE FINANCIAL MARKET

Financial framework: No arbitrage approach with exogenously given spotrate.I Filtered probability space (Ω,F = (Ft )t≥0,P).I N-dimensional Brownian motion

Market Parameters : Incomplete marketI M risky assets, M ≤ N.I (rt )t≥0, (ηt )t≥0, (σt )t≥0 adapted processes.I rt ≥ 0 spot rate.I ηt N-dimensional risk premium vector.I σt volatility process M × N. σtσ

trt invertible.





THE REPRESENTATIVE AGENT

I Representative agent, strategy (π, c).c(.) : consumption rate, ct = ρt Xtπ(.) : fractions of the wealth invested in the risky asset. We set

κt := σtrt πt

I Constraints on the portfolio⇒ Incompleteness of the market.κt ∈ Rt where Rt adapted subvector spaces in RN .

I Self financing dynamics of wealth process with risky portfolio κ andconsumption rate c is given by

dXκ,ρt = Xκ,ρ

t [(rt − ρt )dt + κt (dWt + ηtdt)], κt ∈ Rt . (6)

Remark : κt .ηt = κt .ηRt where ηR is the “minimal” risk premium.





DISCOUNTED PRICING KERNEL AND DUALITY

I An Itô semimartingale Y ν is called a discounted pricing kernel if for anyadmissible Xκ,ρ, Y ν

. Xκ,ρ. +

∫ .0 Y ν

s ρsXκ,ρs ds is a local martingale.

I ⇒ there exists ν ∈ R⊥ such that

dY νt = Y ν

t [−rtdt + (νt − ηRt ).dWt ], νt ∈ R⊥t , Y ν0 = y (7)

I Y ν is the product of the “minimal” discounted pricing kernel Y 0 (ν = 0)by the orthogonal density martingaleLνt = L⊥t = exp

( ∫ t0 νs.dWs − 1

2

∫ t0 ||νs||2ds

).

The discounted pricing kernels are the cornerstone of the Ramsey ruleI In the economic framework: r and η are determined endogenously at

equilibriumI In the financial framework r and η are exogenous, ν is determined at

the optimum.





PRIMAL AND DUAL PROGRESSIVE UTILITY

I Necessity to adjust the optimisation criterionI 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)

I Progressive utilities first introduced by Musiela & Zariphopoulou,existence and characterization studied by El Karoui & Mrad

I (U,V) progressive utility system satisfying a dynamic programmingprinciple called market-consistency.

for any admissible wealth process Xκ,ρ, with consumption rate c = ρXκ,ρ,U(t ,Xκ,ρt ) +

∫ t0 V (s, cs)ds is a positive supermartingale.

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

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

for any admissible Yν (with ν ∈ R⊥) U(t ,Yνt ) +∫ t

0 V (s,Yνs )ds is asubmartingalethere exists an optimal process Y∗ for which it is a martingale.





PATHWISE RAMSEY RULE

I Pathwise relation between optimal consumption rate c∗ and the optimaldiscounted pricing kernel Y ∗

c∗t = −Vy (t ,Y ∗t ) i.e. Vc(t , c∗t ) = Y ∗t , 0 ≤ t ≤ Tc = −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)

I The Ramsey rule leads to a description of the equilibrium interest rate asa function of the optimal discounted pricing kernel Y ∗, ∀t < T

Ret (T−t)(y) := − 1

T − tlnE

[Vc(T , c∗T (c))

Vc(t , c∗t (c))

∣∣Ft

]= − 1

T − tlnE

[Y ∗T (y)

Y ∗t (y)|Ft

]





UTILITY INDIFFERENCE PRICING

Valuation of a non-replicable claim ζT

I Utility indifference price of a positive claim ζT delivered in T : it is thecash amount for which the investor is indifferent from investing or not inthe claim

without the claim

UT (x) := sup(κ,c)∈Ac

E[U(T ,Xκ,ρT ) +

∫ T

0V (s, cs)ds]. (9)

with the claim

Uζ,T (x , q) := sup(κ,c)∈Ac

E[U(T ,Xκ,ρT − q ζT ) +

∫ T

0V (s, cs)ds]. (10)

I The utility indifference price is the cash amount pq(x , ζT , q) determinedby the relationship

Uζ,T (x + pq(x , ζT , q), q) = UT (x). (11)





MARGINAL UTILITY PRICES (DAVIS PRICE)

When the agents are aware of their sensitivity to the unhedgeable risk, theycan try to transact for only a little amount in the risky contractI Davis price or marginal utility price,I corresponds to the zero marginal rate of substitution

put (x) := lim

q→0

∂pqt

∂q(x , ζT , q).

Marginal utility price at time t of the claim ζT :

put (x) = E[ζT

Y ∗T (y)

Y ∗t (y)|Ft ], y = Ux (t , x).





FINANCIAL INTERPRETATION OF THE EQUILIBRIUM YIELD CURVE

I Aim: Give a financial interpretation of E[

Y∗T (y)

Y∗t (y)|Ft

](for t < T ) in terms of

price of zero-coupon bondsI 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)).

I Replicable Bond B(t ,T ) is computed by the minimal risk neutral pricing

rule B(t ,T ) = E[Y 0

TY 0

t|Ft ] = EQ[e−

∫ Tt rsds|Ft ] : for replicable bond,

equilibrium interest rate and market interest rate coincide.I 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.





AGGREGATION AND WEALTH DEPENDENCY

Aggregation of power utilities = aggregation of discounted pricing kernelsI natural modeling in the context of heterogeneity of the investorsI compatible with the existence of an equilibrium (see El karoui &

Mrad)I more flexible model while being still tractableI setting in which the yield curves may depend on the wealth of the

economyAggregation of N investorsI (constant) relative risk aversion parameters θ1 < · · · < θN .I endowed at time 0 with a proportion αi of the initial global wealth x

B∗(0,T )(y) =

∑Ni=1 yθi (y)B∗,θi (0,T )

y=

∑Ni=1(αix)−θi B∗,θi (0,T )∑N

i=1(αix)−θi

with yθi (y) = uθix (αix) = (αix)−θi

I Asymptotic behavior for small and large wealth

limy→0

B∗(0,T )(y) = Bθ10 (T ) and lim

y→+∞B∗(0,T )(y) = BθN

0 (T ).





AGGREGATION AND WEALTH DEPENDENCY

Aggregation of a continuum of heterogeneous investors indexed by a(constant) relative risk aversion θ, with different weights m(dθ)

I the marginal utility bond curve B∗(t ,T )(y) is a normalized mixture ofindividual bond curves, based on the orthogonal martingales L⊥,∗,θt ,

B∗(t ,T )(y) =

∫B∗,θ(t ,T )(yθ)

L⊥,∗,θt (yθ)∫L⊥,∗,θt (yθ)m(dθ)

m(dθ).

I 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 onthe martingales Y ∗,θt (yθ)B∗,θ(t ,T )(yθ)

f ∗(t ,T )(y) =

∫f ∗,θ(t ,T )(yθ)

Y ∗,θt (yθ)B∗,θ(t ,T )(yθ)∫Y ∗,θt (yθ)B∗,θ(t ,T )(yθ)m(dθ)

m(dθ).





INDIVIDUAL YIELD CURVE FOR DIFFERENT VALUES OF THE RISKAVERSION

FIGURE: Individual yield curve R∗,θ0 (δ) for different values of the risk aversion θ





INDIVIDUAL AND AGGREGATE YIELD CURVE SPREAD

Spreads between the different rate curves and the market yield curve R00(δ) :

spread = R∗,θ0 (δ)− R00(δ).

FIGURE: Individual and aggregate yield curve spread23/ 33

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




AGGREGATE YIELD CURVE SPREAD DEPENDING ON THE WEALTH x

FIGURE: Aggregate yield curve spread depending on the wealth x





AGGREGATE YIELD CURVE SPREAD DEPENDING ON INITIAL PROPORTIONPARAMETERS α

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




Yield curve dynamics and their volatilitiesInfinite maturity yield curve

OUTLINE








HEATH-JARROW-MORTON FRAMEWORK

We consider both the economic and the financial viewpointsWe focus on the volatility family of the zero-coupon bonds

Heath-Jarrow-Morton frameworkI it characterizes the dynamics of the yield curve.I this characteristic is determined directly by the martingale property of

the process (Y ∗t (y)B∗(t ,T )(y))t∈[0,T ]

I 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 ofy ).

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





SPOT RATES AND WEALTH DEPENDENCY

I Taking the logarithm derivatives of (Y ∗t (y)B∗(t ,T )(y))t∈[0,T ] w.r.t. T givesthe spot forward rates f ∗(t ,T )(y) = −∂T ln B∗(t ,T )(y) is

f ∗(t ,T )(y) = f ∗(0,T )(y)−∫ t

0γ∗(s,T )(y).(dWs+(S∗s (y)−Γ∗(s,T )(y))ds).

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

T→tf ∗(t ,T )(y) is given by

rt (y) = f ∗(0, t)(y)−∫ t

0γ∗(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

).

I 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 .





A GAUSSIAN FRAMEWORK

The Vasicek (Hull and White) model (in incomplete market)I 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.I 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 ).

I The initial spot forward rates curve t → f ∗(0, t)(y) satisfies a Vasicekmodel (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

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





INFINITE MATURITY YIELD CURVE, FORWARD CASE

B∗(t ,T )(y)

B∗(0,T )(y)= exp

( ∫ t

0rs(y)ds+Γ∗(s,T )(y).(dWs+S∗s (y)ds)−1

2||Γ∗(s,T )(y)||2ds

)I Infinite maturity yield curve lt (y) := lim

T→+∞R∗t (T , y).

I when T is large, R∗t (T )(y) behaves as

R∗0 (T )(y)−∫ t

0

Γ∗(s,T )(y)

T.dWs+

∫ t

0

||Γ∗(s,T )(y)||2

2Tds+

∫ t

0

Γ∗(s,T )(y)

T.S∗s (y)ds

I thus we have to study together the behavior of 1T

∫ t0 Γ∗(s,T )(y).dWs,

1T

∫ t0 ||Γ

∗(s,T )(y)||2ds and 1T

∫ t0 Γ∗(s,T )(y).S∗s (y)ds for a fixed t .

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

infinite if limT→+∞

Γ∗(t,T )(y)T exists and is not equal to zero dt ⊗ dP a.s.

Otherwise, lt (y) = l0(y) +∫ t

0g2

s (y)

2 ds with g2t (y) := lim

T→+∞||Γ∗(t,T )(y)||2

T .

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





INFINITE MATURITY YIELD CURVE, BACKWARD CASE

In the backward case, ν∗,Hs may also depends on TH ,thus the infinite maturity yield curve lt (y) is

I the same as in the forward case if limTH→+∞

||Γ∗,⊥(s,TH )||||ν∗,Hs ||

=∞

I if limTH→+∞

||Γ∗,⊥(s,TH )||||ν∗,Hs ||

= 0, lt (y) = l0(y) +∫ t

0 hs(y)ds with

hs(y) := limTH→+∞

||Γ∗,R(s,TH )(y)||22TH

+Γ∗,⊥(s,TH ).ν

∗,Hs

TH

I otherwise lt (y) = l0(y) +∫ t

0 hs(y)ds with

hs(y) := limTH→+∞

||Γ∗(s,TH )(y)||22TH

+Γ∗,⊥(s,TH ).ν

∗,Hs

TH.

Example: for power utility and Gaussian market

ν∗,Ht = −(1− θ)Γ∗,⊥(t ,TH), θκ∗,Ht + (1− θ)Γ∗,R(t ,TH) = ηRt .





CONCLUSION

I Compare financial and economic points of view for long term yield curvemodeling

I Common threads: Ramsey rule and discounted pricing kernelI Dynamic utility framework allows capture more phenomena, particularly

with regard to aggregation of heterogeneous agents and extreme risksI Discussion on backward/forward approachI Focus on the dependency on the wealth of the economy.

Paper available on Halhttps://hal.archives-ouvertes.fr/hal-00974815


https://hal.archives-ouvertes.fr/hal-00974815




BIBLIOGRAPHY

[1] I. Ekeland. Discounting the future: the case of climate change. Lecture Notes2009.

[2] C. Gollier. The consumption-based determinants of the term structure ofdiscount 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 deprécaution? Leçons à partir du cas des politiques climatiques.

[6] N. El Karoui, M. Mrad. An Exact Connection between two Solvable SDEs anda Non Linear Utility Stochastic PDEs, SIAM Journal on Financial Mathematics(2013).

[7] N. El Karoui, C. Hillairet, M. Mrad. Construction of an aggregate consistentutility, 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 backwardand forward dynamic exponential utilities in a stochastic factor model. Advancesin mathematical finance, (2010).


the ramsey rule and yield curve modeling: economic and ... · 9/ 33 caroline hillairet, ensae,...

Documents