Math Is Fun Forum

Hi boys and girls

I recently encounter an interesting task on studying fund rating methodologies. And I fund Lipper's Preservation Measure is

Sum(Min(0,ri))/T or Sum(Min(0,ri))/N*(T/N)

What it actually does is turn all the positive return r's to 0 and compute the average.

If we assume r normally distributed as N(u,s²)

The negative expectation can be modeled as  E- =∫r*pdf dr   over (-∞,0)

I came up with the answer

E-= u*N(-u/s)-(s/√2π)*exp(-u²/2s²)

But Michael Stutzer in his paper Mutual Fund Ratings: What is the Risk in Risk-Adjusted Fund Returns? derived an approximation as

Could you check this out and tell me why the difference? Thanks!

Click Here and Try for yourself!

Many individuals, looking at a body of water, may be struck by its beauty or wonder about its swimmability. When MIT’s Daniel Nocera looks, he sees, among other things…fuel.

And not just any fuel, but potentially green and basically limitless fuel.

Nocera’s perspective reflects his long-time goal of achieving a variation on what plants do: aided by sunlight, split water into oxygen and hydrogen. Then, he wants the gases to be stored for use as electricity is needed.

Very little water would be required. “The average household uses about 30 kilowatt hours of electricity a day,” notes the MIT chemistry professor. “To obtain the fuel to generate that amount would take roughly five liters of water.” And because the oxygen and hydrogen recombine, the system could be closed: you could truck one to your desert hideaway along with a water supply and be set for years.

That’s the vision. And if realized, not only would we have a green technology whose “feedstocks” — water and sunlight — are everywhere, we’d be on our way to overcoming a tough obstacle to a big ramp-up in solar power use, which is storing the sun’s energy for when it’s dark or cloudy.

Of course, realizing the dream will itself be very tough. But Nocera and his main co-worker on the project, postdoctoral associate Matthew Kanan, recently overcame one of the biggest hurdles: separating out the oxygen in water in much the way plants do.

The basics of the proposed system are straightforward. Using solar panels, you generate electricity. That power can directly meet your home’s electrical needs some of the time. Or, it can run a device called an electrolyzer — the guts of which are the pair’s main contribution to the system — which breaks down H2O molecules, two at a time, into a molecule of oxygen and two molecules of hydrogen. You store the gases in tanks. They can then be used to run a fuel cell, an electricity producer that’s like a battery in generating current electrochemically but that requires a fuel supply. You use the electricity to power — well, whatever you want.

To make the system work financially and otherwise, though, Nocera needed low-cost watersplitting catalysts that work in environmentally benign conditions. It’s a goal the faculty member has been pursuing since he began his career more than 20 years ago.

Nocera started with foundational research — for example, studying plant processes to understand more about them. More recently, he’d made progress on specific scientific phenomena undergirding the reaction chemistry needed to split water. “But the problem was,” says Nocera, “that we hadn’t done anything on the oxygen side. I knew we had to get O2 for my system to work as a closed one.”

Last winter, Nocera and Kanan obtained the result they wanted. Among the keys to success were two chemicals: phosphate, widely used in fertilizers among other products; and the element cobalt, whose features include the fact that positively charged particles of the metal readily dissolve in water.

Using an apparatus that resembles a high school chemistry lab set up for splitting water, the researchers dissolved the cobalt ions and phosphate. They then sent current through the electrodes. The cobalt built up on the positively charged electrode — and off those deposits bubbled molecules of oxygen.

It was a chemist’s version of a Eureka moment, though Nocera says the pair didn’t start popping champagne corks. “We immediately started thinking about the months of experiments we would have to do to verify the results,” he says.

Those experiments worked, but a vast amount is still to be done. One goal is to solve the chemical structure of the catalyst that allows the oxygen to emerge — a quest that seems in reach thanks to the researchers’ access to a high-energy instrument for probing matter.

Nocera and Kanan have other goals, too. One example: find alternatives to the pricey platinum that’s the catalyst on the negative electrode.

Then there’s the overall cost issue. In its current form, his system would go for at least $20,000, and maybe a lot more. That’s too much to make it competitive, especially since Nocera sees emerging countries as a key market. “These systems have got to be cheap, cheap, cheap,” he says.

Nocera thinks that’s doable. A startup is forming to further the system’s technological development. The MIT group, he notes, continues to focus on the science.

Meanwhile, the work has already had one positive impact. His system has become a conversation-starter, with the subject being what he calls personalized energy. “Whether people are for it or against it,” says Nocera, “they’re talking about it. And that’s the first step toward making it happen.”