Maths prize for 29 September 2015: profit and loss


This was the first-ever maths prize to have no entries, other than 23 June 2015 when students were worn out after exams. Some fortnights previously, no-one got a prize because no-one wrote down their solution, or half-solution, but this is the first time no student has had any solution at all.

The lesson, evidently, is we should stick with the simply-stated problems requiring mathematical ingenuity, and not bring in the more real-life-type problems which involve wading through masses of information to extract the key figures needed for a good calculation.

To tackle this problem, you needed no maths more advanced than compound-interest calculation. It was a “Fermi problem” in the sense that it’s about getting a good guess for an answer from patchy information.

Here’s the problem, and, below, my answer.

Mr Thomas writes: Over the summer I was in Brisbane, Australia. While I was there, dock workers (“wharfies”) at the container terminal operated in the Port of Brisbane by Hutchison, the world’s biggest container-terminal firm, were sacked: 41 of them, including all the union delegates and health and safety reps, and all the more experienced workers, of 84 operations (70) and maintenance (14) workers. The total workforce in the terminal, including managers, IT staff, and security, was 100.

Hutchison simultaneously contracted-out all its customers to other container terminals, and said:

1. It was making losses.

2. It made more losses the more containers it moved.

But it published no comprehensive figures. The wharfies are far from dunces (one has an engineering degree, many have studied large parts of the content of A level maths and further maths for trade qualifications), but they asked me to check out the financial figures.

Neither Hutchison nor the Port of Brisbane will disclose how much rent the port charges Hutchison. Hutchison will not disclose how much it charges shipping lines to move containers. What we have to go on is:

1. Newspaper reports that Hutchison’s total investment in the terminal, which opened in 2013, has been A$250 million.

2. Estimates from the wharfies that Hutchison has been moving about 95,000 teu (twenty-foot equivalent units) of containers per year, and that the wharfies’ average income is about A$90,000 a year.

3. Estimates from the wharfies that container terminals usually charge about A$250 per teu to shipping lines, but Hutchison have been quoting $140 per teu to some lines.

4. Published information from the Port of Brisbane that its total income from wharfage fees is about A$50 million a year, that its total container throughput is about 1 million teu per year, and that it levied about A$30 million in per-teu charges on shipping lines; word-of-mouth information from someone in the port office that rent on quay space charged to terminal operators accounts for the bulk of the rest of the A$50 million; knowledge that there are 12 berths in the port and Hutchison rents two of them.

5. A briefing by the Siemens corporation on crane life cycle costs: click here. The briefing tells us that terminal operators calculate the fixed costs of their equipment as if they had borrowed money to buy the equipment on a 20-year loan at interest.

6. A report from the Australian Competition and Consumer Commission on the container terminal business in Australia: click here.

Your task: to use this information to make a well-explained assessment of Hutchison’s claims that they were making losses; and that they were making more losses the more containers they moved.

This is the answer I gave to the wharfies:

Hutchison have said they spent $250 million on setting up the Brisbane terminal. According to a study in Port Technology magazine (, cranes are usually depreciated over 20 years.

The study’s illustrative figures assume a 6% interest rate. That makes $20 million a year for depreciation on $250 million (You can work that out by doing a compound-interest calculation, or by scaling up the Port Technology magazine’s figures for a single crane).
The Port of Brisbane’s accounts report $50 million a year in wharfage charges. About $30 million of that is per-teu charges. So, about $20 million is wharf leasing charges, or maybe $3 million for Hutchison’s two berths.

Add on $1 million a year for management and security, minimal maintenance, and other charges which Hutchison have to pay even if they move not a single container. Just having the terminal brings Hutchison fixed costs of around $24 million a year before they pay a cent in wages for operations.

Divide $24 million by 95,000 teu moved (per year), and you get an average cost to Hutchison per teu of about $250.

The ACCC report (October 2014) found that “in real terms, per unit labour costs have fallen by 39.7 per cent, from $94.73 in 1998–99 to $57.08 in 2013–14″.

So the extra cost to Hutchison of moving containers, or top of their fixed costs which they have to pay before moving a single container, would be something like $57 per teu.

70 operations workers at $90,000 a year divided by 95,000 teu makes an average labour cost of about $66 per teu. That’s a bit more than the $57 figure given by the ACCC (but, so the wharfies explained to me, you expect a higher figure, because a wharf with relatively low traffic has bigger gaps between ships, and smaller ships; also, the layout of the Hutchison terminal is poor).

Even charging $140 per teu, which is a very low rate, Hutchison are making a profit of over $80 when they move a teu container. The more containers they move, the more profit.

The problem is that those profits are more than wiped out by their fixed costs, so Hutchison are probably telling the truth when they say they are making losses overall.