Assumptions and constraints

Nature of the randomness

For each individual asset and liability class on the reported balance sheet, and the annual revenue and expenditure flows, the continuously compounded single-period returns are modeled as independent and identically distributed (IID) variables.

Thus, the model assumes that the variance of returns is constant over time with no serial correlation (ie, future returns are not correlated with past returns). The model assumes different means and variances of returns for different asset and liability classes and allows for cross-sectional correlation in the distribution of returns. We assume that the correlation coefficients remain constant over time.

The continuously compounded returns are assumed to be drawn from the normal distribution. This means that single-period gross simple returns are distributed as IID lognormal (Campbell, Lo, and MacKinlay, 1997). Given that the actual distributions may have different properties to the normal distribution (eg, fatter tails), it may be worthwhile to consider modeling with alternative probability distributions in future work.

Expectations of future government tax and spending

We define the tax asset (spending liability) as the discounted sum of expected revenue (expenditure) cash flows. We compute this by first computing annual government revenue and expenditure using the stochastic process described above. Then expected values for future revenue and expenditure are derived by assuming revenue and expenditure will grow at rates determined by separate long-term fiscal projections, assuming constant terminal growth rates. That is, once we have computed single-period revenue or expenditure (ie, or ) we can compute the expected values in periods as:

Capital budget identity

There is an accounting identity whereby the sum of net capital injections to each asset and liability cannot exceed the government's operating balance in any one time-period. That is:

The Crown may borrow to increase the capital of one of its asset holdings or reduce a liability, but this is equivalent to a negative net capital injection into debt, so the identity holds. The identity can be rewritten:

In words, this says ‘debt repayment = operating balance - other net capital injections'.

Debt is the residual

For asset and liability classes on the reported balance sheet, excluding debt, we model net capital injections to be a constant proportion of their total values. We set a parameter which defines a constant proportional capital injection where or a constant proportional capital withdrawal (ie, dividend to the Crown) where . The net capital injection into debt is the residual to maintain the capital budget identity.