Appendix 1: Structure of the model
Model variables
The balance sheet is decomposed into asset and liability classes (denoted by subscript i hereafter). The classes are based on the different entities that manage assets or liabilities on behalf of the Crown.
In addition, we model the primary balance with a notional asset class for the present value of primary revenue and a notional liability class for the present value of future primary spending.
The value of each asset or liability class i at time t is denoted as 
, withassets taking positive values and liabilities taking negative ones. Net worth at time t is denoted 
. The particular measure of net worth (eg, GAAP or comprehensive) depends on the choice of asset and liability classes.
| Symbol | Definition |
|---|---|
 |
Value of asset or liability class i at time t. |
 |
Crown net worth at time t. Note we can choose which measure of net worth (eg, GAAP or comprehensive) by the choice of asset and liability classes. |
 |
Crown primary revenue at time t. |
 |
Crown primary expenditure at time t. |
 |
Net capital injection into asset or liability class i at time t. A negative value signifies a withdrawal of capital (ie, dividend for an asset class and borrowing for a liability class). |
 |
Net capital injection ratio. It is assumed that the net capital injection is a constant proportion of the value of an asset or liability class. |
| δ | Discount rate used to compute present value of primary revenue and expenditure. |
 |
Mean return (compounded continuously) for asset or liability class i on the reported balance sheet. |
 |
Expected growth in  at time t. |
 |
Expected growth in  at time t. |
 |
Standard deviation of return to asset or liability class. |
 |
Correlation coefficient of asset or liability classes i and j. |
 |
Random number drawn from standard normal distribution with correlation structure . |
 |
Expected value operator. |
Modeling the dynamics
For the reported balance sheet, we assume that the value of each asset and liability class will evolve over time with a stochastic rate of return and net of any capital injections or dividends. That is: ‘Change in value of asset or liability class = average return (drift) + stochastic element (volatility) + net capital injection'.
For primary revenue and spending, we also assume that growth occurs in a stochastic fashion: ‘Change in revenue or expenses = expected growth rate + stochastic element (volatility)'.
We can then compute the value of the primary revenue asset or spending liability by computing the present value of expected future cash flows.
We can represent these dynamics as follows in the notation of stochastic differential equations, where W denotes a Wiener process (ie, Brownian motion):
