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Working paper / Digital edition
Asset Value and Securitization under Heterogeneous Beliefs
Abstract
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Example 1
One asset. Two tranches.
Three equilibrium price functions.
Keep dividends, beliefs, and the attachment point fixed. See when the junior tranche pays in each equilibrium.
Complete manuscript
The motivating example · Section 2
Two claims on the same resale value
The senior tranche receives up to the attachment point. The junior tranche receives what is left. Their payoffs add up to the asset’s next-period resale price.
01 · Divide the resale value
Senior first. Junior receives the remainder.
For an attachment point of 2.25 and a resale price of q, the senior payoff is min(q, 2.25). The junior payoff is max(q − 2.25, 0).
This slider illustrates the payoff rule. It does not solve for a new equilibrium.
Senior payoffJunior payoff
02 · Price the claims separately
Different buyers value different parts most.
In the limiting example, theory A assigns equal probabilities to all three next-period states. Theory B assigns probability one to staying in the current state. At current state m, B values the senior tranche more and A values the junior tranche more.
At the tranching equilibrium, the resale prices in states (l, m, h) are (1.05, 2.25, 3.00). The table shows each theory’s expected tranche payoff discounted by the gross return of 2.
Willingness to pay at state m
Each tranche is priced at its highest valuation. The row maxima can come from different theories.
03 · Feed the security prices back into the asset
The asset price includes the value of issuing the claims.
At state m, the discounted declared dividend is 2 / 2 = 1. The senior and junior prices are 1.125 and 0.125. Together they support an asset price of 1 + 1.125 + 0.125 = 2.25, compared with 2 without tranching.
The same equilibrium resale prices determine tomorrow’s tranche payoffs. An equilibrium must satisfy both sides of this feedback.
Read the calculation in Section 2.2 →Declared dividend / R1.000+ Senior price1.125+ Junior price0.125= Asset price at m2.250
These values describe the limit as ε tends to zero. The limiting transition matrix has zero entries. The multiplicity example below instead has full support. Explore Example 1 →
Example 1 · Full-support beliefs
Three prices. The same economy.
Dividends, beliefs, the interest rate, and the attachment point stay fixed. Select an equilibrium to see how the asset’s resale price changes the junior tranche’s payoff.
States l, m, hDividends (0, 3, 5) / 12Gross return 13 / 12Junior attachment 3
Asset resale price by next-period state
The dashed line is the junior attachment point of 3.
Check the equilibrium prices and the underlying beliefs
Theory A assigns probability 1/3 to each next-period state at every current state. Theory B uses the transition matrix below, which is also the objective process. All its entries are positive.
The three vectors are (848, 930, 978) / 327, (259, 285, 297) / 97, and (2201, 2425, 2523) / 803. They are known fixed points from Example 1, not the output of a procedure that finds every equilibrium.
At each current state, add the highest discounted valuation of each tranche to the discounted declared dividend. The result is the underlying asset price. Zero-payoff tranches can have tied valuations.
Read Example 1 and its exact prices →Without tranching
The same economy has a unique equilibrium, (2.593, 2.844, 2.991), equal to the lowest of the three vectors above. Here, adding a senior and a junior tranche creates multiplicity. This is an example, not a claim that every tranching creates multiple equilibria.
From Proposition 1 to the code
Finding the lowest and highest equilibria
Apply the price-forming operator repeatedly. The lower sequence starts at zero and rises. The upper sequence starts at the bound from Proposition 1 and falls. Their limits are the least and greatest equilibrium price functions.
Lower sequenceUpper sequence
Inputs, stopping rule, and numerical accuracy
The browser displays precomputed, validated results. It does not solve new parameter combinations. The algorithm computes only the two extreme equilibria. It does not find the middle fixed point displayed above or enumerate all equilibria.
Download the inputs and stored iterations (JSON) ↓ · Run the Python code →Open research
Read the argument.
Inspect the calculation.
The manuscript, the examples, and the computational method are available together. The web reader is generated from LaTeX. The explorer uses the existing Python companion.
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Source repository
The manuscript and the digital-edition source.
Open the notebook in Colab
Visible inputs and algorithm, with no local Python installation required.
Read the solver
compute_extreme_equilibria and the price-forming operator.
Build and verify this edition
Reproducible exports, source hashes, conversion checks, and scope.
What the solver does
lower = zero price in every state
upper = M_T in every state
repeat for each sequence until the stopping rule is met:
next = price_operator(economy, current)
return lower limit, upper limitThis is a guide to the algorithm, not a replacement for the executable implementation. The code accepts at most three states and three tranches. Numerical agreement of the extremes is not, by itself, a proof of uniqueness.
Aggregate realized excess payoffs in the paper are defined using unit tranche purchases and objective long-run frequencies. They are not a measure of total welfare or actual portfolio holdings.