# Time Value of Money

Some details on the numbers behind the NBN post:

To arrive at this \$10 per person per month figure I first divided \$46bn by the population of Australia which I have conveniently, for the maths, set at 23 million. That works out at \$2000 per head or, on average, \$5000 per household. Many commentators have asked, would you pay \$2000 upfront for super fast broadband? Probably not, but this isn’t really comparable with the usual monthly billing arrangement. So to convert the \$2000 to a monthly amount, I multiply by a discount factor of, in this case, the long-term GBR (Government Bond Rate) of 6%. This works out to be \$120/year or \$10/month.

Why multiply by the GBR? Well this is an example of reverse engineering the argument of the time value of money. How much is a perpetual stream of \$120/yr worth to someone whose costs of funds is the GBR (which what we, the people of Australia, collectively face…or at least the Government we have chosen to represent us does)? The answer is to discount each cashflow by the appropriate discount factor, and sum them to get the Present Value (PV). The details can be found in this Wikipedia article. For a perpetual constant cashflow stream the formula is particularly simple. The PV is just the cashflow divided by the discount rate, i.e. \$2000 = \$120/6%. Working this in reverse tells us that \$2,000 up-front is equivalent to \$120 monthly.

In the case of the NBN where we know the upfront cost (same as PV) is \$2000 we can work out the equivalent perpetual cashflow by multiplying the \$2000 by the discount rate of 6% which gives us \$120 per annum or \$10 per month. What this means is that if the Government wanted to raise the \$46bn to build the NBN upfront then the interest we would each pay is \$10/month (forever, unless we paid it back early).