Variations on coin flipping

In this section, we consider the mother of all random experiments: coin flipping. This is a good way to get acquainted with probabilities, which are not necessarily relative frequencies, as well as with investments under uncertainty. We illustrate plain coin flipping first, and then some more interesting variations on the theme.

Plain coin flipping Assuming that we flip a fair coin, what is the probability of its landing head? Probably, you will not flip the coin one million times to conclude that this probability is 0.5. The reasoning you are following is, in fact, based on some form of symmetry that is exploited whenever you think about gambling by simple mechanisms such as throwing dice, picking cards at random, or spinning a roulette. Hopefully, you start to see that there may be different concepts of probability, a topic that we will discuss a bit in Section 5.1. The very nature of probabilities has been the subject of heated debate, but let us leave such philosophical issues aside and assume that you play a simple lottery. A fair coin is flipped: You win $10 when it lands head, and you lose $5 when it lands tail. If the game is repeated a large number of times, how much money should you win with each flip, on the average? Looking back at how we computed average demand in Section 1.2.1, you might suggest the idea of multiplying each possible payoff outcome by its probability:

We are using a new notation here, as E[·] denotes the expected value of a random variable. Indeed, we are not taking averages based on observed data, looking backward; rather, we are stating something about the future, looking forward.

It would be nice to play that game a large number of times without having to pay for the privilege of flipping the coin. Still, a basic law of economics says that there is no free lunch; hence, someone will ask you for some money to play the game. How much money would you be willing to pay exactly? Suppose that someone says that, in order to play a fair game, the price is $2.5; note that, if you pay that amount, the expected overall profit is

which does sound fair. Would you accept? Maybe yes, as the worst that can happen is losing $7.5, which will not change your life. But let’s scale the game up a million times: You may win $10,000,000, you may lose $5,000,000. Would you still be willing to pay $2,500,000 for playing the game? I guess that the answer is no; probably, you would not play the game even for free.

What you have seen is an example of how risk aversion affects our behavior. This is a fundamental ingredient of any decision under uncertainty, including investments in financial assets or new products. Decision making under uncertainty and risk aversion are the subject. For now, let us consider a simple investment decision that looks quite similar to coin flipping.

Fancy coin flipping Suppose that you are the lucky manager of a movie company, which has just signed a contract with a new and promising director.¹² The contract provides that you may produce zero, one, or two movies with this director, during the next 2 years. To clarify, we could produce a movie now, investing some money immediately and hopefully collecting some payoff in, say, one year. At the end of the first year, we could produce a new movie,¹³ collecting revenues at the end of the second year. One decision we have to make is whether we should produce those movies or not. In pondering the decision, note that any advance money you have paid to the good director is gone; in other words, it is a sunk cost. If, on second thought, you feel that producing a movie with him will turn out a disaster, you should not suffer from the sunk cost syndrome, i.e., the tendency to go on at any cost since you have already spent money on your endeavor.¹⁴ On the contrary, you should only consider the money you are about to spend now to produce the movie, and the prospects for future revenue.

To spice things up, let us say that there is another dimension, as we may select one out of two production and marketing plans, a conservative and an aggressive one. We may invest more money in well-known actors and/or special effects, as well as in marketing the movie around the world.

1. Production/marketing plan A costs 2500 whatever monetary unit you like. This is the money you have to spend for the privilege of flipping the coin. We have some uncertainty about the success of our pet group. If the movie is successful, revenue after one year will be 4400 the same whatever; if it’s a flop, you make nothing. To consider a case as close as possible to coin flipping, and to simplify calculations, let us assume that the probability of a success is 50%. Of course, coin flipping is a rather crude model of uncertainty for such a situation, and you should really ponder how to come up with a set of sensible scenarios along with their probabilities. By the way, the astute reader will suspect that here we should work with yet another, more subjective concept of probability. The symmetry of coin flipping or dice throwing does not apply and if this is an unknown director, looking at the past relative frequencies case might just make no sense. This happens whenever you deal with a brand new product.¹⁵
2. Production/marketing plan B is more aggressive and expensive, as it costs 4000, but it increases revenue by 50% in the case of a hit, without changing probabilities (hence, we have two possible outcomes: 6600 or 0, with 50% probability). Arguably, a different marketing plan could also change success/flop probabilities, but for the sake of simplicity we will neglect this.

Both plans look like a coin flipping experiment, but in this case we should consider time or, to be more precise, the time value of money. Let us digress a bit in order to clarify this issue.

A little digression: time value of money Would you prefer to receive $100 right now or $100 in one year? Well, easy answer, and you surely concur that $100 in one year is not the same as $100 today. But what if you have to choose between $100 right now and $106 in one year? In order to make the two amounts comparable, the standard approach is to take money in the future and discount it back to now. To do so, we apply a discount rate, which is related to an interest rate.

Say that you may invest money at an annual risk-free interest rate of 5%; if you invest $100 now at that rate, in one year you will own

The general rule is that to evaluate money in one year, you multiply money now by (1 + r), where r is the prevailing interest rate. If you leave that money invested for 2 years, and interest on interest is earned, the money in 2 years will be

Note the effect of compounding; without that, in 2 years you would get only $110, i.e., the initial capital plus twice the interest payment. Now, if you see things the other way around, how much are $106 in one year worth now? A little thought should suggest the following rule: Discount money in the future using the interest rate, and compare it with money now:

which suggests the opportunity of waiting one year to grab almost one extra dollar.

This is what cash flow discounting is all about. We will see a bit more about financial mathematics later.¹⁶ In practice, the real issue is estimating the uncertain cash flows and choosing a suitable discount rate. This may be difficult, as the discount rate should take investment risk into account. For now, let us assume that we discount cash flows with an annual discount rate of 10%.

Fig. 1.7 Graphical representation of alternative plans for producing and marketing a movie.

Back to fancy coin flipping One sensible starting point in deciding whether we should trust the good director with our money is to find out how much we expect to make using each alternative marketing plan to produce one movie. Let us consider plan A. We pay 2500 for sure now, and there is a 50–50 chance of making 4400 or 0 in one year. Taking discounting into account, the money we expect to make is

This −500 is the (expected) net present value (NPV) of our investment. A basic rule of investment analysis says that we should invest in projects with positive NPV. Unfortunately, the idea of adopting marketing plan A to produce a movie now does not seem quite promising. What about using the bolder plan B? Well, repeating the analysis yields the following NPV:

an even bleaker perspective. A useful way of visualizing lotteries is illustrated in Fig. 1.7, which depicts a scenario tree. The leftmost node corresponds to “now,” i.e., the current state of the world, where we have to decide whether we should spend 2500 or 4400 to start one of the marketing plans. The two nodes on the right correspond to possible states of the world in the future. In our case, we have just two possible scenarios, success or flop, which are associated to given payoffs and probabilities. Clearly, this a two-state scenario tree is a very crude representation of uncertainty; in practice, scenario trees may be definitely richer.

It seems there is no hope, but wait: We just considered one movie! What if we produce two over the next 2 years? If one game of coin flipping is a bad lottery, can we make it any better by just repeating it? To illustrate the point with plain coin flipping, what is the probability of getting two heads in a row, or any other possible outcome? The possible head–tail sequences when you flip a coin twice are

Fig. 1.8 Repeating the fancy coin flipping experiment twice.

The usual symmetry argument suggests that none of these four outcomes is more likely than the others; hence, the probability of each sequence should just be  (note once again that probabilities add up to 1). A deeper reasoning should reveal that, unless the coin has some memory, the result of the second flip has nothing to do with the result of the first one; the two events are independent We will learn that the joint probability of two independent events is just the product of their probabilities. For instance, we obtain the following equation, which confirms the previous result:

Now, what should we conclude if we consider the movie production problem as no more than a fancy variation on coin flipping? The situation is depicted in Fig. 1.8, where we produce two movies using plan A (incidentally, we ignore inflation and assume that future production and marketing costs will be the same as today). We have four equally likely scenarios: success–success, success–flop, flop–success, flop–flop. Each scenario has probability 0.25. We should discount cash flows occurring in 1 and 2 years; we should also consider the net cash flow in one year, resulting from collecting revenues (if any) and investing in the new movie. For the success–success scenario, the NPV is as follows:

Please carry out the calculation for the other three scenarios, and verify that the expected NPV is

The result is negative again, but this is hardly a surprise; if flipping a coin once has a negative expected profit, repeating that gamble twice will not produce a positive result, because we are replicating the same experiment. In fact, a much smarter way to obtain the result is by noting that we are just carrying out the experiment with plan A twice in a row, and we may use the result of Eq. (1.4). We have just to discount the expected NPV for the second movie back from the end of year 1 to now:

By the same token, if we try mixing plans A and B in some sequence, we will hardly get any better, if our problem is just a fancy variation of coin flipping.

But is the movie production case really like coin flipping? In Section 1.2.2 we learned about conditional probabilities: If the test is positive, this does change your probability of being ill. If the first movie is a success (or a flop), can we say that producing the next one will just be another coin flipping experiment? Probably not, as the result of the first trial tells us something about market reaction to our product. In fact, we need an assessment of conditional probabilities, i.e., the probability that the second movie is a success (or a flop), conditional on the fact that the first one has been a success (or a flop). Reasonably, if the first movie is a success, the probability that the second one is a success as well is greater than 0.5. By the same token, if we have a flop in the first trial, we raise our expectation for another flop with the second movie. Coins are supposed to have no memory, but the tendency to stack movie sequels one after another suggests that moviegoers are not memory less coins. Estimating those conditional probabilities may be quite hard, but let us consider a very extreme and overly simplified case, just to illustrate the point. Assume that if the first attempt is a success, the second one will certainly be as well (i.e., the conditional probability of a second hit after the first one is 1); on the other hand, assume that after a flop, we will have another flop for sure. In such a case, it is easy to see that if the first movie is a success, we should certainly produce the second one; on the other hand, if the first movie is a flop, we will just cut our losses and forget about it. This is what managerial flexibility is all about; we do not plan everything in advance, but we adapt our decisions when we gather additional information and revise our beliefs.

One sensible idea is to be conservative at the first trial and using plan A for the first movie; if we are lucky, and discover that the director can deliver a successful movie, we will be more aggressive and take plan B next year. This strategy is depicted in Fig. 1.9. In the figure, we do not clearly distinguish nodes at which we make a decision and nodes at which we observe results, but we will see a clearer representation of a dynamic decision-making process under uncertainty, where we deal with decision trees. If we calculate the expected NPV, we obtain

Fig. 1.9 Fancy coin flipping with memory.

Please beware of a common mistake. The last cash flow in the expression, 6600, occurs with probability 0.5, not 0.25. In order to evaluate the probability of each node in the scenario tree we must take the product of the conditional probabilities along the path leading from the root of the scenario tree (the “now” node) to that node. In our case, 0.5 × 1 = 0.5. This positive result shows that maybe our investment in the director was not so bad. Clearly, we should assess how much this conclusion depends on the probabilities that we have assumed. This process of checking the influence of uncertain data on the decision suggested by a model is called sensitivity analysis.

If you want to generalize a bit, a situation like this is common in risky research and development (R&D) projects. You could try a risky venture that, if successful, will pave the way to huge revenues by expanding the business. When analyzing investments, one should account for managerial flexibility and the possibility of learning over time by gathering new information. This approach leads to so-called real options;¹⁷ in the real options literature, our example is known as a growth option.

Coin flipping in the short or long run Before leaving coin flipping aside, it may be instructive to consider in more detail the idea of repeating bets. Suppose that you can play the plain coin-flipping game, where you may either win 10 million or lose 5 million of whatever monetary unit you like. The coin is fair and memoryless, and you can play the game for free, so that the expected payoff is 2.5 million.

1. Would you play the game once?
2. Would you play the same game 1000 times in a row?

The answer to the first question might be no, as the chance of being broke is far from negligible, even though the expected payoff is rich. But if you may repeat the game several times, you have the possibility of recovering your losses. In the long run, the bet is quite profitable indeed, but this is no solace if you can get out of the business after a losing streak of bad outcomes, without the time to recover.¹⁸ Again, risk aversion does play an important role and considering long-run, or expected, profits may not be always appropriate. In other words, the plain expected value of the payoff fails to take into account risk in the short term. In the movie production case, we have taken for granted that a proper discount rate can take risk aversion into account, but the issue is far from trivial and is the subject of quite some controversy in corporate finance. We investigate the role of risk in decision making under uncertainty.