December 16, 2003

Fun with stats (and the M6 Toll)

Patrick Crozier | Road Pricing

Now that the M6 Toll road has opened (don’t say you hadn’t heard) it gives us the opportunity to answer a few questions. Like, for instance, will it make money? Just for fun I did some calculations based on the numbers where available and my own guesses where not.

Costs

Building the road: £900m
Assumed interest rate: 5%
Length of concession: 53 years – call that 50 years
Assume interest payment constant (yes, I know)
Assume costs of maintenance, collecting tolls negligible

Annual expenditure = Interest payment + capital payment.

Capital payment = £900m/50 = £18m/year

Interest payment = £900m × 5% = £45m/year

So, Annual expenditure = £63m ≈ £60m

Income

Assume only cars use the road (probably not so far of the mark as it happens)
Assume 300 days in the year (weekends count as one day because fewer people moving about)
Toll = £2

Number of cars per day to break even = Annual expenditure/Number of days x Toll

= 60m/300 × 2

= 100,000 a day

But, 180,000 vehicles use the M6 every day. M6 was designed to take no more than 72,000 per day. So, it seems reasonable to assume much the same is true of the M6 toll and, of course, if it exceeds this figure it is in danger of wrecking its unique selling point (convenience). In other words, on my calculations MEL is going to go bust.

Except it probably isn’t. They reckon it is a licence to print money and let’s face it they’ve got lots of experience in this field. I’d love to know where I’ve gone wrong though.

Trackbacks

Relevant?
Whether this transaction --- Camino Colombia auctioned --- says anything about the viability of "private roads" I have no idea. The Camino Colombia Toll Road, the only private toll road in the state of Texas, was sold, auction-style, Tuesday morning
City Comforts Blog on January 8, 2004

Comments

You assumed that capital and interest payments remain constant over the term of the loan. This isn't correct. In the early years of a loan, the repayments are mostly interest; in the late years, they are mostly capital.

The correct formula for calculating repayments, assuming constant interest and continuous compounding, is:

R = r X / (1 - exp(-r T))

where R is the repayment rate, r is the continuously compounded interest rate, X is the initial size of the loan and T is the term of the loan.

Plugging in your figures gives a figure of about £47.5 millions, or 79,000 cars per day, in a 300 day year.

I think it's more usual to calculate the present value of the future income stream and compare that to the capital costs. The formula for this is

V = I (1 - (1+r)^-(T+1))/(1 - (1+r)^(-1))

where V is the present value, I is the annual income, r is the AER (rather than the continuous rate), T is the term of the loan. This formula assumes annual, rather than continuous compounding, so will give a slightly different result.

Plugging in your figures gives about 19.5 I. To cover the capital payments, I would have to be at least £46.2m/year or 77,000 cars per day, in a 300 day year.

If you assumed a 365 day year, those numbers would come down to 65,000 and 63,000 cars/day respectively. So I reckon the road will pay its way after all.

PS. I nearly made a fool of myself by pointing out that 60m/300 * 2 is 400,000 not 100,000. You should have put brackets in your formula, ie it should have been 60m/(300 * 2). A typical working day involves me drowning in dozens of sheets of paper covered in algebra and, trust me, it's really, really important. Notational mistakes like that have the potential to wreck entire PhD theses. Or, in your case, the potential to wreck important road building projects.

Posted by Andy Wood on December 16, 2003

Andy beat me by a few minutes! I used a spreadsheet and got this:

Assuming there is a capital repayment each year the interest charge would decline over the 50-year period. That would mean that the average annual repayment would be in the region of £49.3 million giving a daily vehicle target of around 82,000. The fare goes up to £3 after the first 10 million cars (in about 4 months at 82k per day) and at that point the daily breakeven would be about 55,000 cars.

Posted by David Farrer on December 16, 2003

I thought I posted a comment here yesterday. (Perhaps I pressed "Preview" and not "Post", saying similar things to Andy). Note though that this is equity financed. The required rate of return on an equity investment is larger than on a debt investment (for both reasons of risk and taxation reasons) which would actually mean that the number of cars would have to be greater for investors to want to continue to hold the stock - which is sort of the equity investor's equivalent of "break even". I just asked a friend of mine (an Australian fund manager who holds a lot of MIG stock) what number he is using, and the answer he gave me was just under 9%. He considers the stock to be a bargain at the present share price. How present market capitalisation compares with the actual cost of building the road is another question to which I do not know the answer, but it is easily looked up. And there is the complication about the fact that the listed entity owns a number of other (mostly Australian) toll roads. However, my understanding is that the M6 toll is responsible for something like two thirds of the company's market value.

I am not answering the question here, as that would require me to do actual work, but just raising a couple of issues.

Posted by Michael Jennings on December 16, 2003

This would be a trip down memory lane if the memory lane in question hadn't been rendered a muddy wilderness by 20 years of neglect.

exp(x) means e to the power x, yes?

It's a fun exercise but I suppose we are very much frustrated by the unknowns: what will be the mix of the traffic? What effect will that have on capacity? What will happen at weekends? How significant will maintenance be?

If I've got my sums correct (based on Andy's first formula) a 9% return would demand 137,000 cars a day at £2 or 91,000 a day at a £3 toll.

Posted by Patrick Crozier on December 16, 2003

Unfortunately, you're all wrong. The interest part is corrected well but the assumptions weren't only unrealistic on the income side but also on the expense side.

How to calculate the likely cost of resurfacing 25 or 30 years in is beyond me though.

I suspect that the most likely outcome is that after some time, the tolls will increase if income turns out to be insufficient.

Posted by TM Lutas on December 18, 2003

After a little googling, I think I can provide an estimate of maintenance costs. On 4th July 2002, in a written answer to Don Foster MP, the transport minister David Jamieson stated:

...in the financial year 2002-03 the Agency plans to spend some £740 million on maintaining the motorway and trunk road network. This compares to £692 million last year, which included the resurfacing of 1793 lane kilometres of carriageway and remedial works on 487 bridges.

The total road network maintained by the Highways Agency is about 9,400 km long, but I don't know what the average number of lanes is. This accounts for single carriageways and two, three and four lane dual carriageways and motorways. Three lanes seems to me to be a reasonable guess, so that makes about 28,000 lane km in total.

Assuming the costs were entirely for resurfacing, this amounts to about £26,000 per lane km year (using 2002-03 figures). The M6 Toll is 6 lanes wide (ie 3 lanes in either direction) and 30 miles or 48 km long, giving an annual resurfacing cost of about £7.6 millions. Applying Friedman's First Law (which is really a rule of thumb) that it costs the government about twice as much to do anything as it costs anyone else, this gives a final figure of about £3.8m/year.

With a 5% interest rate, that would imply that the total maintenance costs over the fifty years would be about 8% of the initial costs, or, in the worst case, 16%, if, for some reason, the government really is as efficient as the private sector in maintaining roads. If the average width of a road is four lanes rather than three, maintenance could cost about 6% of the initial building costs.

Our estimates of the traffic flow required to break even varied by about 6%, depending on the assumptions about interest calculations, and assuming a 300 day year as opposed to a 365 day gives an even larger variation.

So, for the purposes of his post, Patrick's assumption that maintenance costs are negligible seems to me to be a perfectly reasonable one.

Posted by Andy Wood on December 18, 2003

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