Have you ever found a U.S. dime minted before 1965 in your change?

We ask, because 1964 was the last year in which U.S. coins with a face value of 10 cents or more were produced by the U.S. Mint were mostly made out of silver [1]. During the next 10 years, the price of silver rose, then skyrocketed in the mid-to-late 1970s, giving Americans a strong incentive to pluck any these silver coins out of circulation [2] because they became worth more than their face values.

By the 1980s, it was extraordinarily rare to find any pre-1965 silver coin in your change. Today, it has become exceedingly rare.

But next week, it may become a just a bit more commonplace.

As part of National Coin Week, which runs from 21 April 2019 through 28 April 2019, coin dealers across the U.S. plan to return up to one million vintage or collectible coins back into circulation, where they may very well turn up in your change. What they hope will be The Great American Coin Hunt will be on!

Not all the coins they put back into circulation in this Willy Wonka-fashion will be silver. The dealers will release a variety of coins dating back as far as the 1800s into the wild, so to speak, where anyone paying cash and getting change back may be in for a neat surprise.

If that's you, happy hunting!

Notes

[1] Before 1965, U.S. dimes, quarters, and half-dollars were minted from an alloy known as "junk silver", which is 90% silver and 10% copper.

[2] For an anecdotal history of the disappearance of silver-based coinage in the U.S., check out this discussion thread at the Metal Detecting Forum.

From time to time, we test out hypotheses that make for interesting discussion. For example, let's consider the impact of two recent 737 crashes on Boeing's stock price (NYSE: BA).

Here, we had the idea that if we compared Boeing's stock price with that of another company whose fortunes are closely tied to Boeing, we could perhaps get an indication of how investors are pricing the company's potential legal liability for the Lion Air Flight JT610 crash on 29 October 2018 in Indonesia, which killed 189 people, and the Ethiopian Airlines Flight ET302 crash on 10 March 2019, which killed 157 passengers and crew, by comparing Boeing's stock price with that of Spirit Aerosystems (NYSE: SPR). The following chart shows the trajectory of both companies' stock prices from 15 October 2018 through 12 April 2019, where we've also indicated the timing of the two crashes.

This comparison is particularly relevant because Spirit Aerosystems manufactures the fuselage of Boeing's 737 MAX aircraft. Our hypothesis was that because the cause of the two recent Boeing 737 MAX crashes have been attributed to faulty sensors and software malfunctions, Boeing's stock would show a relatively greater decline than Spirit Aerosystems, where because the fuselage is a mechanical structure, we reasoned that Spirit would only be exposed to reductions in Boeing's production of the 737 MAX, while Boeing would bear the additional burden of legal liability related to the two crashes linked to the sensor and software failures.

It sounds plausible, and in the initial reaction of both companies' stock prices to each crash, it also looks plausible, but there is some strange synchronization going on between the stock prices of the two companies, which you can see in the next chart as we've tracked each company's stock price as a percentage of the peak market closing values both companies simultaneously reached on 1 March 2019.

It's no surprise to find that the two companies' stock prices are closely synchronized to each other, with their changes over time typically falling within a few percentage points of each other. It's also not strange to see that Boeing's share price did indeed bear a heavier burden than Spirit Aerosystems' stock price did in the immediate aftermath of both 737 MAX crashes.

What is a bit strange is how they've gone about recoupling to each to other's performance after each crash event. Following the Lion Air crash, we find that Boeing's stock price surged after 27 November 2018 to catch up to the level of Spirit Aerosystems' stock within the matter of a week. Which kind of makes sense because at that time, it seemed that Boeing would avoid having exposure to any additional legal liability from that crash.

But that pattern was reversed after the Ethiopian Air crash, where Spirit Aerosystems' stock price was the one that moved to recouple with Boeing's relative stock price performance after 2 April 2019, which it did by dropping down to Boeing's level. We suspect that may have been speculation on the part of investors that production of the 737 MAX aircraft would be reduced as news that the grounding of the troubled aircraft would extend longer than expected emerged. Although Boeing had resisted it, the situation with so many aircraft on the ground and order cancellations forced its hand as the production cut announcement for the 737 MAX finally came on 8 April 2019.

Boeing should still has greater exposure to legal liability from the crashes, so it will be interesting to see how its relative stock price performance evolves over time as those costs become better quantified. As long as it does, we should see it as a persistent relative underperformance when compared with Spirit Aerosystems' stock price.

At least until the legal issues are finally in Boeing's rear view mirror. We'll see how that hypothesis works in practice.

If you're a member of Starbucks (Nasdaq: SBUX) customer loyalty rewards program, the sad news today is that company's new rewards redemption scheme has gone into effect.

The reason why that's sad news is because the program's rewards have become much less generous. Previously, a rewards member could redeem 125 points (or stars) to get any item off Starbucks' menu for "free", with the best bang for the buck coming from redeeming points for either upgraded or high-priced menu items, such as handcrafted beverages, sandwiches and salads.

That changes today because the company is seeking to reduce its costs from financially literate rewards program members, who took advantage of the program's previous redemption terms to receive the maximum value they could from the points they accumulated through regular purchases and periodic promotional deals. Starbucks' new rewards program scheme makes redeeming points for higher-priced menu items more costly:

Starbucks is overhauling its rewards program so that customers will start earning rewards faster. Quicker access to freebies means the coffee chain is also essentially eliminating its Gold Level status by default.

Previously, you had to reach Gold Level — 300 stars — to be able to redeem a free drink or food item for an additional 125 stars. Starting April 16, customers can cash in their Starbucks stars as soon as they get them. Like before, 2 stars will be earned for every dollar spent....

For 25 stars you can ask for an extra shot of espresso, pump of flavor, or opt for any dairy substitute at no extra cost; 50 stars lets you walk out of your local Sbux with a brewed hot coffee, tea, or baked good, on the house; 150 means a free handcrafted drink, hot breakfast, or yogurt parfait; 200 stars, and lunch is on Starbucks. And if you ever make it to 400 stars without cashing in first, there's an option for "select merchandise" or packaged coffee.

But not all of Starbucks' Rewards stars are equal. In the following table, we did the math to identify where Starbucks' rewards members can get the biggest bang for their rewards points when redeemining them to buy the indicated items off of Starbucks' menu. If you're wondering about the particular items shown, most prices and items could be found for sale at Starbucks' first retail store, located across from Pike Place Market in Seattle, Washington, on 16 April 2019, while others may be found at other retail locations in the United States.

We find that the sweet spot for redeeming Starbucks rewards for loyalty program members seeking maximum value has moved from the high-end of the company's menu offerings to the company's bakery goods. Not uncoincidentally, Starbucks' bakery goods tend to carry the highest profit margins of all the items on its menu, so that's a change that may benefit the company's bottom line.

Meanwhile, the strategy of up-sizing drinks would appear to continue providing greater value in redeeming reward points than ordering beverages in smaller volume cups. The values in the table above apply for "grande"-sized beverages, which fall in the middle of the beverage options on Starbucks' menu - "short" or "tall"-sized beverages would deliver less value than what these figures indicate, while "venti" or "trente"-sized drinks would deliver greater value for their new reward redemption value of 150 points. It is unclear at this time if Starbucks Rewards members would be required to redeem an additional 25 points to upgrade a Café Latte to a Café Vanilla Latte on top of the 150 points it will now take to purchase the base beverage (of any size) in Starbucks' new rewards scheme.

The worst deal, by far, would be to redeem points to add flavored syrups to their regularly ordered beverage by itself. We think the only reason anyone would voluntarily choose to do this would be to use up small totals of accumulated points that are at direct risk of expiring before they might otherwise be used.

The rewards program change seems primarily aimed at boosting Starbucks' profit margins, which have been declining since early 2017. Starbucks is expected to close 150 of its U.S. stores in 2019 to help boost its profitability in the U.S. At the same time, Starbucks has focused considerable attention on growing overseas, particularly in China. Starbucks' stock price has generally risen since mid-2018, largely on the strength of its international expansion.

Image Credit: Kal Visuals

The S&P 500 (Index: SPX) continued its upward trajectory in the second week of April 2019, where it is now within 24 points, or 1%, of its previous peak of 2,930.75 set back on 20 September 2019.

The thing that boosted the market the most during the week was the news on Friday, 12 April 2019 that China's exports to global markets has rebounded to a five-month high, suggesting relative improvements in the economic health of importing countries.

The same news story revealed that China's own imports have continued to decline however, indicating that nation's economy is still slowing. For stock markets, that bad news was offset in part by the ongoing expansion of the Chinese government's efforts to stimulate its economy, where there are indications they are gaining some traction.

Our roundup of those headlines, and other market-moving headlines, is straight ahead....

Monday, 8 April 2019

• Oil gains up to 2% to five-month high as Libyan output threatened
• Bigger stimulus developing in China:
• China to relax residency curbs, boost infrastructure in new urbanization push
• China to step up bank reserve ratio cuts to help small firms
• S&P 500, Nasdaq edge higher as earnings season looms

Tuesday, 9 April 2019

• Oil dips on global growth worry, possible output rise
• Global economy cooling, coordinated stimulus may be needed: IMF
• Despite differences, U.S., German yield curves flash slowdown warnings
• U.S. job openings hit 11-month low; quits rate stagnates
• Fed's Clarida: U.S. jobless rate may have room to fall without inflation
• Wall Street drops on U.S. trade tensions with EU, IMF global outlook
• Trump slams EU in aircraft dispute, pushes tariffs on $11 billion of imports

Wednesday, 10 April 2019

• Oil rallies as U.S. gasoline inventory draw offsets crude build
• Mortgage rates driving mortgage market: U.S. mortgage applications retreat from 2-1/2 year peak
• Citing global slowdown, Fed sees no changes to rates in 2019
• Bigger trouble developing in China:
• March mobile phone shipments to China fall 6 percent as economy slows
• Exports are about strength of other nation's economies, imports tell story about China's domestic economy: China's March exports seen rebounding, imports falling again: Reuters poll
• Wall Street ekes out gains as investors shrug off Fed minutes

Thursday, 11 April 2019

• Oil prices slip as OPEC mulls output hike, U.S. stocks build
• U.S., China agree to establish trade deal enforcement offices: Mnuchin
• China stimulus gaining traction? China producer inflation picks up for first time in nine months, eases deflation worries
• Fed must 'tread carefully' given market signals on outlook
• Bullard: Would need downside 'deviation' to inflation, growth for rate cut
• Fed's Clarida: U.S. slowing, but expansion will 'almost certainly' break record
• U.S. economy 'healthy' from a monetary policy perspective: Fed's Williams
• Fed's Kashkari: Slight rise in U.S. inflation would not be alarming
• The S&P 500 ends flat as investors await bank earnings

Friday, 12 April 2019

• Oil rises 1 percent on tightening crude supply, upbeat economic data
• Signs of improving world economy: China March exports rebound to five-month high but imports fall more than expected
• Bigger stimulus developing in China: To spur rural development, China to send millions of students on 'volunteering' trips
• China data, JPMorgan results lift stocks, riskier assets
• S&P 500 closes near record high as earnings season begins in earnest

How many positives and negatives for the U.S.' markets and economy did Barry Ritholtz find during the second week of April 2019? Click through to find out!...

We have recently breaking news from the world of maths, where David Harvey and Joris van der Hoeven have posted a new paper describing a computational method they developed that may have reached the theoretical speed limit for performing the multiplication of very large numbers.

In the following video, Harvey describes what that speed limit is, or rather, the minimum number of calculations required to reach the correct product for two large numbers that are being multiplied:

If the method is proven, it represents a large step forward in speeding the solution of large number multiplication by reducing the number of required calculations. We built the following tool to get a sense of how much smaller the theoretical limit of N log (N) [N multiplied by the natural logarithm of N] is when compared to the N² operations required by the traditional method of multiplying two numbers with the same number of digits. If you're reading this article on a site that republishes our RSS news feed, click here to access a working version of this tool!

For the default example, where we're multiplying two numbers with 10,000 digits each, you can see why this development is exciting because it would be possible to reduce the number of individual operations needed to arrive at the product from 100,000,000 to 92,104, a computational efficiency gain of nearly 99.91%.

Which is to say that you can not only get to the answer much more quickly, you would also greatly reduce the amount of energy that computers consume in reaching the product of the two very large numbers being multiplied.

Writing at The Conversation, Harvey describes what he and van der Hoeven have achieved, where the secret to reaching the theoretical peak efficiency for multiplying very large integers is to use multidimensional Fast Fourier Transforms (FFTs):

A few weeks ago, Joris van der Hoeven and I posted a research paper describing a new multiplication algorithm that finally reaches the N log (N) holy grail, thus settling the “easy” part of the Schönhage–Strassen conjecture.

The paper has not yet been peer-reviewed, so some caution is warranted. It is standard practice in mathematics to disseminate research results before they have undergone peer review.

Instead of using one-dimensional FFTs — the staple of all work on this problem since 1971 — our algorithm relies on multidimensional FFTs. These gadgets are nothing new: the widely-used JPEG image format depends on 2-dimensional FFTs, and 3-dimensional FFTs have many applications in physics and engineering.

In our paper, we use FFTs with 1,729 dimensions. This is tricky to visualise, but mathematically no more troublesome than the 2-dimensional case.

If you want to find out more about FFTs, Better Explained has one of the gentler introductions to the math of Fast Fourier Transforms.

Meanwhile, Harvey recognizes the current generation of van der Hoeven's and his algorithm is limited in how it can be used effectively:

The new algorithm is not really practical in its current form, because the proof given in our paper only works for ludicrously large numbers. Even if each digit was written on a hydrogen atom, there would not be nearly enough room available in the observable universe to write them down.

On the other hand, we are hopeful that with further refinements, the algorithm might become practical for numbers with merely billions or trillions of digits. If so, it may well become an indispensable tool in the computational mathematician’s arsenal.

So it's not quite yet the perfect way to multiply, but it initially appears to be much closer than what anyone else has achieved, and with a little more work, could very well become the "perfect way to multiply" very big numbers!