Will the worm reach the end of the rubber band?

A worm is at the beginning of an infinitely flexible rubber band which has a length of 1 kilometre.

During a one second interval, the worm advances one centimetre. However, the rubber band's length increases to 2 kilometres. The worm is slightly pulled forward as a consequence of this.

In the following second, the worm advances another centimetre. The rubber band's length increases to 3 kilometres. And so on.

It seems pretty obvious that the worm never reaches the end of the rubber band. However, it does! Why?

In the 1st second the worm covers 1/100,000 of the rubber band's distance. This is 1cm divided by 100,000 cm (1 km).

In the 2nd second it covers 1/200,000, in the 3rd second it covers 1/300,000 , etc.

The sum of the proportions covered is 1/100,000 (1 + 1/2 + 1/3 + 1/4 + 1/5 + ...); will this sum ever reach 1 (100% of the distance). It surprisingly does.

The so-called harmonic series (1 + 1/2 + 1/3 + 1/4 + 1/5 + ...) is fascinating. It seems that if one keeps summing smaller and smaller numbers, the sum always needs to converge to some finite number. If one sums the first 65,536 terms of the series using Excel, the sum appears to converge to around 11.67. But if we keep adding numbers, by the time we are adding 1/1,000,000 the sum has already increased to 14.39. By the time 1/2,000,000 is added, the sum has kept increasing to 15.09. And it can get as large as we want. The proof is here.

The worm will take more than the estimated age of the Universe to reach the end of the rubber band (which by that stage will be larger than the diameter of the known Universe), but it will eventually get there.

How to create value 101

EnhanceValue plc has a market capitalisation of £40m. With 1m outstanding shares, this equates to a share price of £40. The company has annual earnings of £2m, leading to Earnings Per Share (EPS) of £2 (which is earnings of £2m divided by 1m shares) and a (slightly demanding) P/E ratio of 20 (which is the £40 share price divided by EPS of £2).

Target plc has the same earnings and number of shares, but a lower P/E ratio of 12, so it trades at £24 per share (£2 x 12) and therefore has a market cap of only £24m (£24 x 1m shares).

C. Real Acquirer, the CEO of EnhanceValue plc, is planning a value-enhancing takeover of Target plc. The plan is to pay for Target using EnhanceValue's stock, a move widely welcomed by analysts as 'immediately accretive'. In fact, EnhanceValue will need to issue only an additional 600,000 shares (£24m/£40). After the acquisition's completion, the combined entity will have 1.6m shares and earnings of £4m (£2m + £2m), or £2.50 per share. This is a significant improvement of £0.50 per share. The market is expected to value the 'improved' EnhancedValue plc at £50 per share (EPS of £2.50 x P/E ratio of 20), leading to a market cap of £80m (£50 x 1.6m). The total shareholder value created is expected to be £16m (£80m-£40m-£24m), to be added to C. Real Acquirer's already outstanding track record.

However, a group of activist shareholders, led by hedge fund manager E.P.S. Thiruvananthapuram, is opposing this move. He claims the the shareholders of EnhanceValue plc would be much better off with a share buyback programme and that Mr. Acquirer's acquisition frenzy has failed to create any value. "It should be obvious for everybody that while the acquisition of Target plc does increase Earnings Per Share to £2.50, this only happens because Target trades on a lower P/E multiple. It is a mathematical trick. The market will understand that there are no synergies between the two entities and therefore the price of EnhanceValue will be unchanged at £40 and its value will be £64m - possibly less after paying all those fat investment banking fees-, while the implied P/E ratio will now be 16 instead of 20".

Instead, Mr. Thiruvananthapuram thinks that EnhanceValue should undertake an aggressive share buyback programme. He reckons that Mr. Acquirer should return £20m to shareholders, by buying back 500,000 shares (or 50% of all outstanding shares) at £40 per share. To pay for this, EnhanceValue would raise £20m from the debt markets at 5% interest rate. This would lead to an annual after-tax interest cost of £700,000 (assuming a marginal tax rate of 30%). Earnings will therefore be reduced from £2m to £1.3m but now there are only 500,000 shares outstanding, so Earnings Per Share are expected to skyrocket from £2 to £2.60 (which is £1.3m divided by 500,000).

Nothing else has changed in EnhanceValue and the company still has the same growth opportunities, so the same multiple (or P/E ratio) should apply. "This is unlike management's plan where clearly an inferior company is being acquired. Under our plan EnhanceValue is worth 20 times £2.60 per share, or £52, compared to the current price of £40. This is true value creation", concludes Mr. Thiruvananthapuram.

How many years does it take to double something?

If a variable is growing at r% per annum, the number of years it takes for it to double is the logarithm of 2 in the base 1+r... but in a surprising mental math helper, y=log(2,1+r) is very well approximated by y=70/r. This result is surprising stable for large ranges of r, say 1% to 200%.

So, if the population of a country grows at 1% p.a., it takes 70 years to double, if it grows at 5% p.a., it only takes 14 years!

The approximation error (actual minus estimate) is also relatively stable at 0.25 to 0.3 years (3 to 4 months) for r>5%. So, to look cool, mention that "if sales keep increasing at 23% per annum, it will take 3 years and 4 months for this company to double its size"!

McProblem

If Chicken McNuggets come in packs of 4, 6, 9, 10, 20, and 50, what is the highest number of McNuggets that cannot be bought using a combination of packs?

Calculating returns

The famous job interview question (which many people miss)...

If something goes up by 50% and then decreases by 50%, how much was the overall change? It's quite annoying to realise that it's not zero after the interview and then get dinged. It's also quite annoying to find out that after losing 75% in one day in, say, online gaming shares, they will have to increase fourfold or 300% before one breaks even. What about the investor that buys £1,000 and gets a £100 profit (or a 10% return), only to lose those £100 shortly afterwards due to an annoying negative 9% return? Quite confusing!

All these problems could be solved if continuous returns were used. Continuous compounding is like receiving interest in your bank account at every millisecond. A great proposition which unfortunately does not happen often in reality. Computationally, if one receives an annual return of "R" on an initial investment of "I" compounded over "M" periods, the final investment value is:

I * [1+(R/M)] ^M

It's not difficult to imagine M becoming larger and larger and the thing becoming I * e^R. That is actually the definition of the constant "e".

In the world of continuous compounding, if something goes up 50% and then down 50%, we are back to square zero. My online gaming shares would have gone down by 139% (which is ln(250)-ln(1,000)) and they need to go up 139% for me to break even. The £100 profit on the £1,000 investment would mean a return of 9.5% and subsequent loss would represent a return of minus 9.5%. Furthermore, multiple year returns become basic calculations. Five consecutive years of 10% returns are no longer equal to a confusing 61% return, but rather a straightforward 50%.

Of course, there are some annoying aspects as well. A doubling of one's initial investment is no longer a nice 100% round return, but a cryptic 69.3147% continuous return. And, if one loses all the initial investment, there is no other way to put it than an infinitely negative return, which, well, might be a more accurate representation of such tragedy.

Life expectancy

Life expectancy in country XYZ is 73 years for men and 77 for women. This means that:

a) A five-year old child with unknown gender can expect to live 70 years
b) A 68-year old female can expect to live for an additional 9 years
c) A 90-year male on average should have died 17 years ago
d) All of the above

Whatever your view

"Whatever your view" says an advertisement of a leading UK financial betting service.

In the middle age, humans would invest in an asset with a view that it would increase in value. Then, they came up with a way of investing in an asset with a view that it will decrease in value and called it 'shorting'. One just needs to borrow the asset from somebody that owns it, sell it, and then at some point buy it cheaper and return it to original owner.

Then, at some point, humans invented instruments whose value changes with an asset's expected volatility over a period and called them 'options'. Moreover, asset volatility is the only parameter that is both unknown and unhedgeable in an option. (Smart) humans could now trade volatility - invest in instruments that increase or decrease in value according to the market's expectation of volatility. Well, their value also fluctuates with time, but that's part of the game. By the way, using options they can also make bets on the volatility of the volatility ('vvol') of an asset. They just need to buy and sell different amounts of options maturing at different times.

More or less at this point in time, humans also came up with a way to have a view on correlation between assets. They created baskets of assets and options markets around them. (Even smarter) humans can trade correlation as the difference between the value of an option on a basket and the value of a basket of options. How elegant.

0.999999999999...=1?

More on the topic here.

The distribution of the first digit

Consider the powers of 2 (1, 2, 4, 8, 16, 32, 64, ...), or more specifically, the first digit of these powers. Can we expect all digits (1-9) to appear more or less with the same frequency (11.1%)?

It doesn't happen. From the first 1,000 powers, 30% start with the digit 1 (this is easily verifiable, for example with a simple Excel spreadsheet), 18% start with 2, 12% with 3, 10% with 4, 8% with 5, 7% with 6, 6% with 7, and 5% with the digits 8 and 9. Strange!

Another experiment: pick a newspaper and look at all numbers that appear on it. Record the first digit of every single number. Repeat the experience with stock prices, national statistical publications and company annual reports. A very similar pattern exists. A lot of numbers start with the digit 1. And the higher the digit, the lower the frequency.

Are these two experiments related? More on the topic here.

The old convertible bond fallacy...

FT: "Issuers like them because the interest payments are lower than they are on conventional bonds, since they are subsidised by the value of the equity option. They can also be more advantageous than issuing straight equity, because if the company’s share price does not rise, then the bondholders are unlikely to convert the bonds into shares and the company will have, in effect, raised finance at a subsidised rate and avoided dilution to existing shareholders"

Better than debt and better than equity. Are there any free lunches?

It is an old fallacy. Convertibles are better than debt when things go badly and bonds are not converted (a company got debt with a cheap coupon) and better than equity when things go well and bonds get converted (the company ended up issuing equity at much lower cost and avoiding the usual negative reaction from the market).

It is the same as insuring half of a house with the argument that it is better than no insurance if it catches fire, and better than full insurance if nothing happens.

Clearly, if shareholders know that things will go well, they'd rather issue debt instead of convertibles and keep the upside for themselves. If they know that things will go badly, they'd rather issue equity instead of convertibles and share the pain.

Private equity - the new kings of capitalism?

"Private equity is a broad term that refers to any type of equity investment in an asset in which the equity is not freely tradable on a public stock market. Categories of private equity investment include leveraged buyouts, venture capital, growth capital, angel investing, mezzanine capital and others"
Investordictionary.com

FT:

"The data is murky, but LBOs and venture capital may account for a fifth of equity capital employed in Europe and the US. JP Morgan thinks it accounts for a similar chunk of current merger and acquisition volume. This activity is now focused on buying quoted companies. (...)This shift from public to private ownership will accelerate. The industry has raised $250bn this year, according to Private Equity Intelligence. Assume this is leveraged up by four times, and it implies almost enough fire-power to take the entire French market private"

"In a perfectly efficient economy, private equity would have only one, modest advantage: legitimate access to privileged information. Since they buy entire companies, private equity buyers can get access to the books. In the real, inefficient world another advantage exists: management. Private equity, it is argued, bypasses the agency problem inherent in public markets. Its managers are incentivised, put under pressure by high gearing, and free from red tape and the quarterly earnings shackle"

"Only some businesses can outperform or be relatively efficient. Talented managers are by definition scarce. (...) That leaves the remaining, claimed, advantage that private equity understands leverage better than public equity. Nobel prize winners Modigliani and Miller showed that, in a zero tax world, a company's enterprise value is not altered by its level of debt. In the real world, debt creates a tax shield, but the value of this is often grossly exaggerated. Moving a typical company's debt from 25 per cent to 75 per cent of EV creates a tax shield worth about only 10 per cent of EV. Similarly the argument that private equity has just been quick to adapt to low real interest rates is, again, overstated. The roughly 200 basis point fall in real sovereign interest rates in the last decade, assuming constant interest cover, justifies a moderate increase in gearing. Furthermore, lower risk-free rates also reduce the cost of equity too"

"Finally, often ignored, are private equity's disadvantages. Illiquidity is one. This is hard to quantify but might add 1 per cent to the cost of capital. Most glaring, however, is the fact that private equity has to pay more for assets than public equity investors: to wrestle a business from the market, it has to pay a premium for control, despite having no industrial synergies in most cases. As an asset class, this is a pretty major drawback"

"So, private equity has soft advantages and some hard disadvantages. Leverage allows it to take more risk, not to generate better returns. It should probably not outperform public markets. Does it?"

"Any private equity firm worth its salt can produce impressive internal rate of return (IRR) statistics. However these lack rigour. The valuations are produced internally. Failing funds may choose not to report. And IRR calculations are flawed. They assume dividends paid out to investors can be reinvested at a fund's overall return rate (rarely the case). This also means they are flattered by paying dividends early"

"Various academics have tried to overcome this. The most famous study by Steve Kaplan and Antoinette Schoar looked at 746 mature or liquidated funds between 1981 and 2001. It concluded that, after fees, some funds do well consistently, but overall returns were in line with the S&P 500. Most other research backs this up. Furthermore, these returns are not risk adjusted. Private equity volatility is not disclosed meaningfully. But with gearing 3-4 times public market levels it would be astonishing if it were not relatively high"

Dear Economist - www.timharford.com

Dear Economist,
How many gifts should I register on my wedding list to optimise my total utility?
Claire Song, via e-mail
--------------------------------------------------------------------------------------------
Dear Claire,
The wedding list reflects a rare piece of honesty in our social dealings: the admission that you do not expect your guests to choose particularly apt gifts. If only we could adopt the same candour when it comes to Christmas and birthdays the world would surely be a better place.
Nevertheless, the wedding list remains fraught with potential inefficiencies and you have evidently been thinking about that. If the list is too expansive you risk guests choosing the least preferred options: you will get the frilly lavatory roll holders while the quality saucepan set will go unpurchased. (I was married not so very long ago - I feel your pain.)
On the other hand, if the wedding list is too small you may find that the gifts run out and the guests decide to pick something a bit more “original” - obviously a disaster. Equally bad, you may find that willing guests don’t buy a gift at all.
The solution is a little labour-intensive but probably worth the effort. You need to release your wedding list in several tranches. Start with a selection of high-priority stuff and keep an eye on progress. When the choice is starting to wear a little thin, add the B-list gifts. If they, too, start to be snapped up, then unveil the C-list. Modern technology makes this fairly easy to do.
Of course, this is still a hassle. For my own wedding I planned to dispense with the gift list and instead charge for admission. That seemed much simpler all round, but my fiancee vetoed the idea.
I am not sure why.

The random hats

A group of 100 men, all wearing hats, throw them to the air at the same time, letting them fall, and then pick one random hat from the floor.

Some of them will pick their own hat, and some will pick the wrong hat. But what is the EXPECTED number of men that pick the RIGHT hat (their own)?

Obviously, 50%? (50 men on average end up with the right hat, 50 with the wrong one)

Let's try with a smaller group - 4 men. Will 2 men on average get back their own hat?

Listing all the 24 (4!) possible, equally likely, permutations of the hats, it is not difficult to find that:

• Only in 1 case (4%), they all get the right hat back
• In 6 cases (25%), 2 of them get the right hat back
• In 8 cases (33%), only 1 of them gets the right hat back
• In 9 cases (38%), NONE of them gets the right hat back

On average, only 1 gets the right hat! What's going on here?

A drunkard's walk

A drunk man stands half a step from the edge of the cliff, and drunkenly 2/3 of the time takes a step away from the edge, and 1/3 of the time towards. What is the probability that he falls?

Typos

Two colleagues correct the same document for typos independently. The first one finds a total of X typos and the second finds a total of Y typos. In Z cases, they find the same typo. How many typos can we expect to remain uncorrected?

At first inspection this problem looks impossible to solve. It is understandable that the two colleagues were not able to spot all the typos, but how can we assess how many, if they were not identified?? However, the question has more information than it initially seems.

Let's call A the total number of typos (which is unknown), p the probability that the 1st person finds a typo and q the probability that the 2nd person finds a typo. Through simple statistical logic it is not difficult to see that there is a relationship between all these quantities:

X=pA ; Y=qA ; Z=pqA

It then becomes obvious that XY=ZA, or that A=XY/Z

The number of expected uncorrected typos is then E=A-(X+Y-Z) (Z needs to be deducted from the sum of X and Y to calculate the number of unique typos identified by both persons)

Replacing A by XY/Z, E can be expressed as XY/Z - (X+Y-Z), or in a more simplified format,

E=(X-Z)(Y-Z)/Z

This means that the expected number of uncorrected typos is simply:

(Number of unique typos identified by 1st person) x (Number of unique typos identified by 2nd person) / (Number of typos identified by both persons)

The future(s) price of oil

Two colleagues discuss a business plan.

"- What assumptions should I make about the price of oil in 3 and 6 months? It's pretty high right now... I am sure it can only go down..."
"- Come on, we have to be more data-driven... you need to find some hard facts to base your assumptions on... there is an easy solution though! Why don't you open the Financial Times and look at the FUTURE PRICE OF OIL?"
"- That sounds a good idea... let me see... here it is! They predict the FUTURE PRICE OF OIL to go further up! How is this possible?... Well, they are the experts... I will change the business plan."

The truth is more complicated than this.

There is no way we can know the future price of anything. Otherwise, everybody would try to take advantage of this information and the current price would immediately drop or increase to this known "future price". What we know is the FUTURES price of several assets, i.e., the price for delivery in the future, contracted today. This has somehow to be related to the current spot price. For example, if the 6-month futures price is too high, somebody will buy the asset today, immediately contract its sale in 6 months at this too "expensive" price, carry the asset for 6 months (at a certain cost) and make a riskless profit. To avoid these and other similar arbitrage opportunities, the futures price of an asset has to be equal to the spot price plus the cost of carry.

The cost of carry of an asset is normally positive. For instance, for a stock index it is the risk-free rate minus the dividend yield (cost of financing the asset purchase minus income received). The S&P500 futures contract is always above the current level of the S&P500 (unless it's October 19th, 1987 and everybody is panicking) because of this, not because somebody is seeing in a crystal ball that markets will go up.

But the whole truth is again more complicated than this.

In most commodity markets, most of the time the futures price is lower than the spot price (the market is in backwardation), which doesn't seem to make any sense, since the cost of carry of a commodity (financing plus storage) is always positive.

Keynes has argued that backwardation is the normal state of a commodity market because typically sellers of commodity futures are producers who want to hedge their revenue, and buyers are speculators willing to take price risk. These speculators require a risk premium, which means that the expected future (not futures) price of the asset they will own must be higher than the futures price they will pay - i.e., the market must be in backwardation. Modern Portfolio Theory has refined this view by assuming that commodities with positive beta must have a positive risk premium (and therefore be in backwardation), whilst commodities with negative beta should have futures prices higher than spot prices (i.e., they must be in contango). For example, this would explain why gold, which has negative beta, is normally in contango.

Contango or backwardation also depend on demand and supply dynamics. As TheStreet.com reports, "Crude oil, like all commodities whose deliveries are constrained logistically - think of the capacities of field production, tankers, pipelines, loading jetties, etc. - should see its front-month price rise more in a bull market. When you need it, you need it now, not six months from now [i.e., oil has a positive "convenience yield"]. And producers looking to lock in the current high prices cannot sell futures for immediate delivery in excess of their production capacity, so they have to sell the back months. The combination of buyers buying now and sellers selling later creates backwardation."

However, oil has recently moved into contango, affecting some trading strategies and maybe a few business plans as well.

Finance for good

FT, 08/05/2006:

"Polite conversation for some, the weather is a matter of survival for others. Ethiopia suffered droughts in 1965, 1984 and 2002. At worst, such events kill millions. At best, emergency aid prevents deaths but does not stop desperate farmers selling productive assets such as tools or livestock. The United Nations World Food Programme has turned to derivatives to address this problem (...) Under the WFP's pilot scheme it will pay Axa a $930,000 annual fee in return for a maximum payout of $7.1m. Unlike insurance, the payout is not related to loss. Axa will stump up if Ethiopian rainfall is below a defined level. Derivatives redistribute risk, for a price. Has the WFP got value for money? Based on 30 years of data it estimates a 10 per cent chance of a payout in any year, compared to a fee equivalent to 13 per cent of the payout value. Superficially that does not look attractive."

How is it possible that the WFP is paying an annual premium of 13% for this 3-year option on Ethiopian rainfall when the probability of a payout each year is only 10%?

And, for a socially oriented conclusion, the FT adds:

"The WFP's pilot is logical. It creates an economic synergy in the form of a risk uncorrelated to insurance companies' existing exposures. The alternative is disastrously slow. The Axa contract should allow the WFP to start disbursements to farmers within six weeks of it identifying that a harvest has failed. Conventional aid might take three-five months to arrive, by when the damage to farmers' livelihoods has occurred."

e^(pi * i) = -1

e is approximately equal to 2.718 and is defined as the limit of (1+(1/n))^n, when n goes to infinity. That is why, for example, the exponential function is used to calculate continuously compounded interest.

pi is approximately equal to 3.14 and is defined as the ratio between the perimeter of a circumference and its diameter.

i is a complex number defined as the square root of -1.

These three constants are apparently completely unrelated and were developed at different stages of mathematics... but in reality, they are linked through a beautiful equation that seems to come directly from God.

e^(pi * i) = -1. More details here.

On gambling

Many people who do not invest in stock markets (or other financial markets) compare this activity to gambling.

Finance academics defend that investing in the stock markets carries a positive risk premium, i.e., that is has a positive expected value, unlike bets, which normally have a negative or zero expected value. Playing in the lottery is the typical example. The price that one pays for a lottery ticket is higher than the expected gain, as part of the proceeds are not converted to prizes, being rather utilised for different causes. So, investing in the stock market is speculation, not gambling. Speculation is defined as "the assumption of considerable risk in obtaining commensurate gain", whilst gambling is generally regarded as "betting on an uncertain outcome".

There are two issues with this.

The first is that perhaps gambling opportunities that carry a positive risk premium can be found. Suppose that for instance one finds a bet which has an extremely likely positive outcome, but with a very small return (for instance, betting on several strong football nations to win the World Cup). If the upside is +10% and the unlikely downside is -100%, this bet might not be much different from investing in a junk bond. The distribution of returns would have the same structural characteristics: positive risk premium, negative skewness (high likelihood of small gains, low likelihood of large losses), positive kurtosis (likelihood of extreme events higher than normal). Can these opportunities be found in potentially inefficient betting markets?

The second issue regards the utility function of rational investors. We know that risk-averse investors like mean, skewness and all other odd moments of the probability distribution of returns. Moreover, they dislike variance, kurtosis and all other even moments of the return distribution. Could the preference for positive skewness be so strong to rationally offset a small negative expected value? One can see casual empirical evidence of enthusiasm for outcomes where gains are big but unlikely and losses are small but likely - large lotteries and insurance are two examples. So maybe playing the lottery is a justifiable investment after all.

Deduction - Induction - Abduction

Deduction

1- Rule: All the beans from this bag are white.
2- Case: These beans are from this bag.
3- Result: These beans are white.

Induction

1- Case: These beans are from this bag.
2- Result: These beans are white.
3- Rule: All the beans from this bag are white.

Abduction

1- Rule: All the beans from this bag are white.
2- Result: These beans are white.
3- Case: These beans are from this bag.

What about this bet?

If in one year the temperature of London is equal or higher than today's, one gets paid £100, otherwise.

Suppose that there are no transaction costs, time value of money is 0%, participants are risk-neutral (they decide purely based on expected value) and temperature is a random walk process totally governed by constant annual volatility of 10%. This process implies that the temperature can go up and down in the future but in exactly one year's time it will most likely be where it is today.

How much should one pay today for this bet?

The logic of Mergers & Acquisitions

Joe C. Raider is attempting to acquire 100% of Shareholder Value, Inc. The value of Shareholder Value under current management can be anywhere between £0 and £100 per share, depending on the outcome of key R&D projects. All share values between £0 and £100 are equally likely.

Whatever the ultimate value under the current management, the company will be worth 50% more under Joe C. Raider’s management. For example, if Shareholder Value is worth £100/share under its current management, it would be worth £150/share to Joe C. Raider.

His offer must be made now, before the outcome of the projects are known. From all indications, Shareholder Value would be happy to be acquired, provided it is at a profitable price. They will decide on his bid after the results of the projects. Thus, Joe C. Raider will not know the value of Shareholder Value when submitting his offer, but they will know its value under current management when deciding whether or not to accept.

What price offer per share would Joe C. Raider tender for Shareholder Value’s stock?

Russian roulette

Two individuals play Russian roulette. The revolver, containing six chambers, is loaded with a single bullet, and the two duelists alternately spin the chamber and fire at themselves until one is killed.

Is this a fair game?

Hypergame

A game is called normal if it must terminate in a finite number of moves (for example, tournament chess or tic-tac-toe). The first move of hypergame is to state which normal game is to be played and the remaining moves consist of the moves of the normal game that has been named.

Is hypergame itself a normal game?