# Annex: Chapter 2

## LEVERAGING/DELEVERAGING COMPONENTS – BALANCE SHEET AND INTERCONNECTEDNESS

The purpose of this annex is to provide a mathematical framework developed by Hyun S. Shin (2009). In summary, this shows how the unwinding of systemic leverage (or deleveraging) can be separated into two components: balance-sheet shrinking (due to haircuts/ shedding of assets) and reduced interconnectedness within the financial system (due to shorter collateral chains).

xi = market value of bank i’s total liabilities

yi = market value of bank i’s assets that can be pledged as collateral

ei = market value of bank i’s equity

ai = market value of bank i’s assets

πji = proportion of j’s liabilities held by i, and j are nonbank j’s funding to bank i

${d}_{i}=1-\left(\frac{{e}_{i}}{{a}_{i}}\right)$ is the ratio of debt to total assets

Noting that the total assets of bank i are given by

and from a simple accounting identity, it follows that the total debt can be computed by multiplying the totals assets with the leverage ratio:

Let x = [x1xn], y = [y1yn], and ∆ = diag[d1dn] and rewriting the previous equation in vector form:

Solving for x and using Taylor series