Assets and Liability Impact on ROE
Overview
One of the ways to change the return on equity (ROE) is to change the amount or quality of assets on the
books.
Altering Total Assets
Given the fundamental accounting equation
{% Assets = Liabilities + Equities %}
a company can only change total assets by changing either its liabilities (debt) or its equity.
Rate of Change of ROE
Taking the
derivative
of the
roe breakdown
with respect to assets, gives the rate of
change of ROE to changes in assets.
{% \Delta Return \, on \, Equity / \Delta Assets = %}
{% [ \Delta(Average \, Asset \, Income \times Total \, Assets)/\Delta Assets %}
{% - \Delta(Average \, Interest \, Expense * Total \, Debt)/\Delta Assets %}
{% - \Delta(Average \, Operating \, Expense \times Total \, Assets)/\Delta Assets ] %}
{% \times (1 - Tax \, Rate)/ Equity %}
First Term of ROE Rate
The first term is
{% \frac{\Delta Average Asset Income}{\Delta Assets} \times Assets + Average Asset Income %}
The change in ROE comes from two effects. The first effect is that as assets goes up, it may change the average asset
income. This can be both because new assets may have a different average income that currently help assets, but also,
over time the demand for assets will drive down the rate of return on those assets, so this effect is dependent on
the elasticity of the asset demand.
The second effect is just the effect of adding more assets and receiving the additional income. If the bank
can be assumed to be small enough such that its asset purch
Second Term of ROE Rate
The second term above is the effects of debt. If the assets are purchased entirely with equity, then
there is no effect and this term is zero. On the other hand, if the assets are purchased entirely with debt,
then we have.
{% \Delta Assets = \Delta Debt %}
In which case, the second term becomes.
{% - \frac{\Delta Average Interest Expese}{\Delta Debt} \times Debt - Average Interest Expense %}
Similar to the two terms given above for the asset effect.
Third Term of ROE Rate
{% - \frac{\Delta Average Operating Expese}{\Delta Assets} \times Assets - Average Operating Expense %}
Common Assumptions
A common assumption is that the elasticities above are all zero, that is the average income, average interest expense
and average operating expense does not change with changing asset and debt sizes (or the changes are two small
to be material).
When the assets are fully funded by debt, we get the intuitive expression that each extra dollar in assets yields the following
change in ROE
{% (Average Asset Income - Average Interest Expense - Average Operating Expense) %}
{% \times (1 - Tax \, Rate)/ Equity %}
