In continuous time, a pricing kernel w.r.t. probability space (Ω, F , P) is defined as an adapted process Λt, such that ΛtPt is a martingale for any asset with price Pt, where we
do not consider dividends2. Then we have
Et[d(ΛtPt)] = 0
In practice, it is easier to work with growth rate, so we can express the above into Et[ dΛt Λt +dPt Pt +dΛt Λt · dPt Pt ] = 0 For risk free rate asset Pt = P0ert, we get
rt= −Et[
dΛt
Λt
]/dt For risky assets with return
dPt Pt ∆ = dRt= µtdt + σtdBt We get µt− rt= −Et[ dΛt Λt · dPt Pt ]/dt = −Covt( dΛt Λt ,dPt Pt )/dt where the second equality comes from the fact that the term
Et[ dΛt Λt ] · Et[ dPt Pt ] = o(t)
Thus if an asset has higher return correlation with the SDF, the required expected return should be lower. This is intuitive because higher SDF value means it is more valuable to have a good return at that time. Any asset that caters to a higher SDF value should demand lower expected return since it is a good hedge.
To connect the above to consumption, we can work with CRRA utility u(ct) =
c1−γt 1 − γ
with discount rate ρ, which implies
Λt = e−ρtc −γ t
is a pricing kernel. Thus the risk free rate is r = ρ + γEt[ dct ct ]/dt − 1 2γ(γ + 1)( dct ct )2/dt
Thus with higher consumption growth rate, the interest rate will be higher, and but the interest rate is lower with more consumption growth volatility. With log utility, we can use the above formula with γ = 1, which yields
r = ρ + Et[ dct ct ]/dt − (dct ct )2/dt
To get a term structure, we need to price longer term asset. For example, denote rt,t+∆
as the interest rate of a bond that matures at t + ∆, starting from t. Then we should have
Et[e∆rt,t+∆e−ρ∆(
ct+∆
ct
)−γ] = 1 which should imply the long-term interest rate.
8
Continuous Dynamic General Equilibrium
Typically, this type of model involves the following steps:
• Solve the individual optimization problem, and get optimal consumption and port- folio choices.
• Use equilibrium condition to get differential equations.
8.1
Solution to
Brunnermeier and Sannikov
(2014)
In the paper, both households and bankers are risk neutral. Bankers have better investment technology, higher discounting (thus less patient), and nonnegative consump- tion, while households can have negative consumption to guarantee the risk-free rate is just the discounting rate, which greatly simplifies the model. Basic notations: (1) Price of capital q(η). (2) Value to wealth ratio θ(η). (3) Fraction of risky assets held by bankers ψ(η).
• Households.
– If households ever hold a positive fraction of the risky asset, the expected return of the risky asset held by households and the risk free asset should be the same, i.e.
Et[drkt] = rdt
Otherwise, households hold zero fraction of the risky asset, and ψt= 1 (bankers
hold all the risky asset). Household consumption in this model is flexible to pin down risk free rate rt= ρh, and clears both the consumption and risk free
asset markets, because households can transform between consumption and risk free assets.
• Bankers
– If bankers ever hold a positive fraction of the risky asset, then the expected return of the risky asset should satisfy the risk premium equation
Et[drkt] dt − r = −σ θ t(σ + σ q t) where σθ
t ≤ 0. The risk premium comes from precautionary motive because
bankers suffer losses exactly when the investment opportunities are good, i.e. Zt ↑, ηt ↓, and θt ↑ (marginal value of wealth increases). Thus either ψt = 0,
or the above equation holds. When either ψt = 0 or ψt = 1, we know the
portfolio choice for both households and bankers. When ψt ∈ (0, 1), both
return equations should hold, and thus we can put them together and get a − a
q(η) + δ − δ + (σ + σ
q
t)σtθ = 0
which can be used to solve ψ. After solving ψ, we can get q00 and θ00, and thus a system of ordinary differential equations.
• Boundary conditions.
– First, when η = 0, the economy will stay at η = 0 forever, and thus only households price the assets. This will result in an asset price q(0) = q.
– Second, when η = η∗, by definition the household will consume, and the Bellman equation implies θ(η∗) = 1.
– Third, we note that the optimal instantaneous consumption at η∗ is to push ηt back to η∗, because otherwise the slope of value function is greater than 1
and the additional consumption makes bankers lose value. This means that the point η∗is a reflection boundary of the system. By the standard arguments for reflection boundary, we should have θ0(η∗) = q0(η∗) = 0.3 The intuition is as follows. If ηt= η∗, then ηt+ε can be approximated by η∗− Aσηt
√
dt, where A = p2/π, because the change is the absolute value of a normal distribution with mean 0 and variance (σtη)2dt. Then starting from ηt = η∗, we have q(ηt+dt) ∼
q(η∗) + q0(η∗)Aσtη√dt. Thus the loss pert unit of time dt is q0(η∗)Aσtη√dt, which means the average loss rate goes to infinity if q0(η∗) 6= 0. Similarly, the drift of θ(ηt) will be infinite at η∗ if θ0(η∗), violating the finite drift solution in
the paper.
– Finally, when η → 0, the asset price is absorbed at q. Thus bankers can take infinite leverage with small wealth and generate an infinite rate of return, leading to θ(0) = ∞.
– The above five boundary conditions are used to solve two second order ODEs, with one endogenous boundary.
Explanation on “hedging demand”: In the future, there are states where marginal utility of wealth is high or low. Return on capital is correlated with the marginal utility
of wealth. If return on capital is high when marginal utility on wealth is low, then inter- mediaries are demanding higher risk premium. This hedging demand will endogenously restrict the leverage of the intermediary.
Why household discount is the same as the interest rate? Technically, the HJB equa- tion results in the marginal utility of wealth to households always equal to 1, which implies the interest rate equals the discount rate. Intuitively, households can freely transform from consumption to wealth, thus making sure the discount rate is the same as the inter- est rate. It is thus very important to have intermediaries with nonnegative consumption, which prevents the “first best” solution, where intermediaries consume minus infinity only at the beginning by borrowing from households, while households consume continuously into the future.
As shown in the paper, with log utility, the solution is much simpler. The main reason is that portfolio choice and consumption decisions are no longer dependent on individual indirect utility functions, which greatly simplifies discussions. Without log utility, even under risk neutrality, we need to describe the marginal utility process for wealth, which requires solving differential equations.
In my paper, if I allow bankers to be risk neutral, then the model has bankruptcy, which could cause riskiness of bank debt, and closer to reality. However, the cost is to introduce another function θ(w, λ) to be solved.