Model

This is two-type agent, continuous time model of financial intermediaries. He and Krishnamurthy (2013)

Agents are either households or specialists.

Distinction will be made clear once we understand the market structure

Market structure

• One risk-free short run bond in zero net supply
  • All agents can freely buy this asset
  • Specialists can also short
• One risky asset whose dividends evolve as GBM
  • Only specialists can purchase this asset, however they can do so in behalf of households via an intermediary mechanism to be explained shortly
  • Net supply normalized to 1

Technology (how markets work)

Each period between t and t + dt is split into 5 mini-periods:

1. At t each specialist is randomly matched with a household to form and intermediary
2. Specialists allocate all w_t of their wealth to buy equity in intermediary. Households allocate a part, H_t, of their wealth to purchase equity
3. Specialists take w_t + H_t and allocates all of it between the risk free bond and risky. There are not restrictions in buying or shorting either asset
4. Returns are realized and distributed according to equity shares. Agents consume out of net wealth
5. At t + dt the match is broken and new one is formed

Note about choice of matching structure:

• Alternative would be standard Walrasian market.
  • Did this in 2012 paper.
  • Found with this market structure the specialists charge fees that rise in financial crisis – this is counterfactual.

Agents

• Households:
  • An overlapping generation of agents.
  • At each time t a unit mass of time t agents is born with wealth w_t^h (evenly distributed based on end of period wealth of previous generation – means they don’t need to track the wealth distribution)
  • They live between period t and t + dt, during which time they receive labor income that is constant fraction of risky asset dividends. (NOTE: without this it is possible to arrive in state where HH sector vanishes)
  • Authors assume a fraction λ can only invest in risk-free asset. This generates the + side of the zero net supply condition, making it possible for other agents to take a levered position in risky asset. HH cannot short the bond
  • Choose period t consumption and asset positions to maximize convex combination of log of consumption and expectation of log of continuation wealth (utility and bequest motive both log form)
• Unit mass of identical specialists. Each one:
  • Infinitely lived.
  • Operates a single intermediary (represent decision makers of bank, hedge fund, extc.)
  • Chooses sequence of consumption and portfolio shares in risky asset (acting as intermediary) to maximize the expected present discounted value of a CRRA utility function of consumption subject to…
  • Budget constraint: dw_t =  − c_tdt + w_tr_tdt + w_t(dR_t(α_t^I)−r_tdt), where α_t^I is intermediary share in risky asset and dR_t(α_t^I) is the associated return

Equilibrium outcomes

In this section we discuss the key outcomes in the paper.

Friction

Model contains one key financial friction that drives most results:

• Households are willing to invest no more than a constant fraction m of specialist wealth as equity in the intermediary: H_t ≤ mw_t for some constant m > 0
• Constrains intermediary’s ability to raise outside equity financing.
• Interpretation:
  • Managers usually have significant wealth tied in their own funds (aligns incentives)
• Equilibrium effect:
  • Effectively creates a boundary x^c such that if specialist wealth relative to value (price) of risky asset x = w/P falls below x_c, they are constrained in how much equity they can raise.
    • In constrained region we have H_t = mw_t
    • If unconstrained H_t < mx_t and value of m doesn’t impact decisions
  • Adds leverage effect such that when specialist is constrained he can’t raise enough capital in intermediary via equity, therefore must take levered position by shorting risk free asset and holding a very large position in risky asset (see figure 2)

The main asset pricing impact of this friction can be seen in how the risk premium (return on expected risky asset less risk free return) changes as a function of specialist wealth. Before getting there we need to understand three details

Equity premium

Risk aversion

First, note that specialist CRRA parameter γ is calibrated to be greater than 1 (i.e. specialists are more risk adverse than households). This causes all households who can to invest all their wealth in risky equity

Intuitive reasoning:

• Recall market clearing: zero net supply risk free asset and positive net supply risky asset ⇒ intermediary always holds more than 100% of wealth in risky asset
• Household is less risk adverse, so they would like to hold more risky asset than specialist.
• To hold more than specialist, who already holds more than 100%, household would have to be able to short bond, but they can’t. So they get as close as possible by spending all their wealth to purchase risky equity.

Risky asset position vs. specialist wealth

Second, the relationship between intermediary position in risky asset and specialist wealth is very asymmetric.

Let αⁱ = risky asset holdings/total assets. Then the relationship beween α^I and w_t can be decomposed in two parts:

• In unconstrained region (specialist wealth relatively high), α^I nearly independent of w_t
• In constrained region; strong, non-linear inverse relationship between α^I and w_t

This is driven by the leverage effect outlined above: a binding constraint forces specialists to short bond - ratcheting up αⁱ.

Risk premium vs. specialist wealth

Finally, the equilibrium risk premium is increasing in αⁱ.

Combining these points we see how the model delivers a state dependent risk premium. This result is different from similar results obtained in the literature on two fronts:

1. This model has “standard” CRRA utility. Others modify the utility function to introduce state dependence (Campbell & Cochrane (1999), Barberis, Huang, and Santos (2001))
2. The relationship between capital (wealth) and the risk premium is very asymmetric: might provide important window for studying crises

Dynamics

Wealth distribution is mean reverting. To see it notice what happens in the two tails:

• Specialist wealth very low -> risk premium high -> specialist wealth expected to increase -> wealth distribution mean reverts
• • Household wealth very low -> risk premium low -> aggregate household return similar to aggregate specialist return -> extra labor income for HH pushes their wealth up -> wealth distribution mean reverts
• That the equity premium is very high when intermediary leverage is high means that specialist wealth exhibits strong mean aversion.
• When household wealth is relatively low, so is the risk premium. Thus, as household’s save their labor income, their wealth is expected to grow, causing the wealth distribution to mean revert from the other side

References