# Investments

Introduction
Investments
An investment is a transaction today
Creation of value for ourselves
The modelling of time
Right and wrong reasoning
When the future sum X is sure
When future the sum X is risky
Summary
One of the first modern problems of probabilities
Why investors are risk averse

Introduction

Our review of accounting, general accounting and cost accounting, in order to prepare our understanding of finance, is now complete. We are ready to study investments, the main economic concept Finance is concerned with.

In order to study investments, today we introduce one of the fundamental mathematical tools of Finance: simple probability theory and random variables.

Modelling of risk
Probability theory is the mathematical tool used to model  risk, as everyone knows an important feature of investments. The lesson will introduce several concepts and ideas, one after the other. If you follow carefully, and miss no step in the presentation of the ideas, it should be reasonably easy. But if you miss one single step in the unfolding of ideas, things can become extremely confusing. So - as always - listen carefully and take good notes. The present ones are here to help you.

An endless source of confusion
Modern money is an endless source of confusion: people not versed in accounting and finance (and even some who are) tend to view any value as tangible value, any security as equivalent to "real value". Real value is something we can consume. The test is "would we take it to a desert island where we intend to live alone for a few months?". In accounting and in finance, we shall deal with lots of real values (goods, machines, services, land, buildings, etc.), but also with even more values that are only promises made by other people to us: when someone signs and hands us a piece of paper on which he wrote "I will pay you 1000 euros in one year from today", we then have "value" in our pocket. But this value is only the promise made by somebody else. We can do all sorts of things with this piece of paper, in particular we might pay somebody else with it (this is sometimes called "endorsing"). But it remains only a promise, resting in the end on the first issuer's creditworthiness. It so happens that modern money (banknotes and deposit accounts in credit) is a type of promissory value too, it is not "tangible value" (gold is considered by some as "real value" and by others as "signs" of value like paper, but more reliable for all sorts of - bad - reasons). Therefore we cannot exactly hoard modern money like we can hoard rice or houses. We must always be concerned with the question "what will be the value, in the future, of the promise, signed by so and so, I hold today in my pocket?" If this question seems abstract to you, ask your grand-parents who financed the transsiberian railway and were given Russian bonds ("emprunts russes") to explain it to you.

The effortless creation of promissory value, and the problems it causes

First of all, let's start with a description of what are investments, made with real value or with money.

Investments

The purpose of finance is to study investments: how to select good investments, and how to get the money to pay for them. The second part, "how to get the money to pay for them" (that is "how to finance investments"), we shall leave for later lectures in this course. During the first few lectures, we shall focus our study on which investments are good ones.

What is an investment?
At the most fundamental level (whether it is with money or without is irrelevant), an investment is the setting aside of some value today in order to produce and reap value in the future.

In other words, it can be viewed as the opposite to consumption. If, during a period [t1, t2], I consume all the value I produce (from my work or otherwise) like the cicada from the fable of La Fontaine "The Cicada and the Ant", I save nothing, I invest nothing. If, on the contrary, I produce Q, but consume only C, and C is less than Q (with some appropriate unit of  measurement), then S = Q - C is what I save, or equivalently what I "invest". I intend to consume S, or, better, to consume whatever S will be able to produce, in the future.

The latter case is the most customary definition of an investment: we set aside some value, and we make this value work, we don't just wait to consume it later.

Monetary investments
Since we live in societies where just about everything is measured in monetary terms, so it is with investments. Then, an investment is an expenditure of money today, in order to reap money in the future.

We hope to reap more than we invested.
This statement is more subtle than it looks. What we mean is that we hope that the value today of what we expect to reap, is higher than the amount of money we invest. Therefore we will need to figure out what is and how to compute the value today of a promise to receive something in the future.

The common concept of investment encompasses several concepts:

• the concept of money (one special form of value)
• the concept of time: in the words "in order to reap, in the future,..."
• the concept of risk: in the word "we hope, hopefully..."

It is because of this last concept of risk that we shall use elementary probability theory to model and study investments.

Two types of investments: physical, and financial
There are two ways to spend money into investments:

1. physical investments into projects: we buy land, construct a building, buy machinery, etc. run the project, and produce cash flows...

2. financial investments: we buy pieces of ownership of somebody else's project, or lend money to somebody else...

In the former case we are the main entrepreneur. In the latter case, we only buy stocks, or bonds, but the entrepreneur is usually somebody else; we only have some financial contract with him.

The case where we buy apartments to rent them is closer to a financial investment than to a physical one. Indeed, there is usually no interference between our proprietary know-how and the running of a flat-rental activity. One must also note that often, investing into flats for rent is primarily a speculative investment expecting (from urbanistic considerations) capital gains or at least value protection.

Anyway, for what we want to study today, the two types of investments, physical and financial, make no difference.

Standard graphical representation
For an investment the proceeds of which we shall reap in one year, the standard representation is this:

One year

If the investment will produce cash flows over several years, the representation is this:

Several years

An investment is a transaction today

It is important to pay attention to this point: an investment is a transaction today
That is, today we give out a sum of money P, and, in a financial investment, we receive, today, a piece of paper, a title of ownership, a sign, an electronic recording or whatever, entitling us to some future cash flows. What we buy, in our transaction, is the piece of paper, the sign. We buy it, and receive it, today. Often, this piece of paper is called S (for "security"). Do not mix it up with the notation for a sum of money.

As you see, we shall mostly use the implicit description of financial investments, but all the considerations of today's lecture apply undistinctively to physical and financial investments.

A contract
The "piece of paper" is actually a financial contract: it entitles us to some future cash flows. The simplest such financial contract is the one year bond: if we become the bond holder, we give some money to somebody else today, and we receive today the bond. The bond (signed by the other party borrowing the money) says: in one year the other party will refund us and on top of this will pay us an interest. The other party is called the bond issuer, or simply the borrower; and we become the lender or the bond holder. Yet we say  that we buy the bond. And there is a very big bond market worldwide, with plenty of submarkets by type of bonds, type of issuers, etc. Secondly, this bond market has a fundamental conceptual division in two parts: the primary market (where we can buy new bonds), and the secondary market (where we can trade bonds issued in some past and already held by some bond holders).

A link between the present and the future
It can be said that an investment, and more specifically the contract representing the investment, is a link between the present and the future. This link is at the heart of finance.

More complicated contracts
There exist plenty of more complicated bonds, or more complicated financial contracts. Many of them pertain to the field of derivatives.

A security. Its value today
The piece of paper S, which we buy today, is called a security. It has a value for us today. This value is not the simple sum of the future cash flows. For instance, in the second example above, the value today of the security S, which we buy for a price P = 100, is not 50 + 80 + 40 = 170.

The case of a physical investment
In the case of a physical investment, the value, to us today, of the series of future cash flows, is usually not the quantity of money P we invest, either. More on this fundamental point later.

Personal values [this item can be skipped]
Obviously, S should have a value to us, today, greater than the price P (= 100) which we pay. Otherwise we would not buy S! This is one of the important aspects of a transaction: the personal values to each parties are different:

• If you buy, for \$670, one Google share, it is because you prefer one Google share to \$670 in cash in your pocket. But one Google share is no longer money. You cannot pay at the supermarket with it; you need first to sell it back, with a profit or with a loss.

• When a firm sells for 100 € a product, recorded in its stocks at 70 €, it makes a profit because it receives something (100 € in money, if it is a cash sale) worth more to the firm than what it gives out (a product worth, to the firm, 70 €)

The question of "what is the value of S, today, for us?" is a central question of finance. Yet, in this course, we shall not dig into the subtle subject of personal value, and we shall stick to the standard approach. For us the personal value of something is the amount of money we are willing to pay for it. In other words, we shall equate personal value for an agent and the money an agent pays for the concerned object.

The above considerations glide over the various deep questions raised by the standard theory of money: for instance in a financial contract, the value of what we buy is the value of the money we give. But money looses value (and this has nothing to do with inflation!) in the sense that it doesn't produce value, whereas bonds do - but only if we trust that the sovereign authorities will not change the interest rates... And bonds are not money, although it is likely that in the future they will be more and more money-like. The theory of money is in need of an overhaul as sweeping as the theory of relativity.

What makes finance fun: immediate creation of value
This is what makes finance fun: we invest money today, and we will get some future cash flows in the years to come; are they worth more (for us), today, than what we invest? We shall see the various techniques finance has developed to try and answer this question. Some of the techniques are highly mathematical (stochastic calculus) and quite nice from a mathematical point of view. They belong to "Modern Financial Theory" (MFT).

In our view, Modern Financial Theory has failed
In our view, MFT has failed. The story of LTCM, advised by two Nobel prizes in Economics (Merton and Scholes), which eventually lost billions of dollars for its subscribers and the banks that were asked to wipe out LTCM losses, is an illustration of this failure. Other techniques are more "pragmatic" and intuitive, although they require a lot of in depth analyses: they are those practiced by Warren Buffett. His success has proved that he holds some truth.

Yet we need to understand simple probabilistic techniques
In this course, we shall learn simple traditional mathematical techniques, like the "Discounted Cash Flow" (DCF) analysis, which was developed in the 1930's; we shall get a glimpse of a part of Modern Financial Theory, called Capital Asset Pricing Model (CAPM). And we shall learn in later lectures how to find the money to finance investments.

Creation of value for ourselves:

Comparison between the money we pay or spend today, and the value for us today of the stream of future cash flows we are buying

• In a physical investment, the value today of a stream of future cash flows, usually is not the initial payment we make.

• Whereas in a financial investment usually the price we pay today is the value of the future cash flows.

This is the reason why, perhaps surprisingly, it is only in physical investments that we can create value, more or less for sure (technically, with a positive expectation), while in financial investments the theory, as it goes (that is the Modern Financial Theory, or MFT), says that what we buy we pay exactly its value for, therefore we don't create any value for ourselves. We may only make money from fluke.

Modern Financial Theory
The MFT, which is taught in business schools all over the world, like here, was developed mostly in the 50's and 60's of last century. There are early beginnings (in the 30's, Williams, Keynes,..); then seminal papers by Markowitz and by Roy in 1952; then further improvements by Tobin (58), Sharpe (65); and finally in the 70's by Black and Scholes.

This theory is powerful, but also has limitations. It doesn't explain well phenomena like bubbles, day-trading, the success of superinvestors like Buffett and others from the "Graham school"; etc.

The modelling of time

In daily life, as well as in Physics, time is represented as a continuous parameter, although in advanced Physics, it is a debatable question. After all energy was represented by Planck on a discrete scale. At first, that was just in his equations, because he needed that to solve the ultraviolet catastrophe riddle! Yet, it became reality as far as we can talk meaningfully of it. But for the sake of simplicity, in Finance, we shall first of all represent it as discrete. The calculations and reasoning, at first, will be simpler.

Therefore, there will be the dates: today, in one year, in two years, etc.

Also, as a consequence, the shortest time duration will be "one year".

Right and wrong reasoning in finance:

Do not compare values of things at different times
When starting in Finance, we have a tendency to want to compare values today to values in the future. For instance the sentence "a dollar today is worth more than a dollar tomorrow" suggests this type of reasoning. But it is a mistake.

In Finance we always compare two values at the same time, usually today.

Vague sentences
What the sentence "a dollar today is worth more than a dollar in one year" means is this: we actually compare a dollar today to the promise, held today, to receive a dollar in one year. And indeed, the former is worth more than the latter.

Similarly, in a paragraph below, we shall compare a value of 1000 euros today, to 1200 euros in one year. What is really meant by this is: how does 1000 euros today compare to the promise, today, to receive 1200 euros in one year. If we can exchange one for the other, is it a good deal? We shall see that it depends...

Deep misunderstandings
Careless formulation can lead to deep misunderstanding. The extreme example is the sentence: "Finance looks at the time value of money." Nobody with a properly cabled mind can readily understand this sentence, because, aside from sounding very intelligent, it is essentially meaningless garble.

Finance, like accounting, is full of misunderstandings. A beautiful one, seen everyday on TV, is when anchormen or anchorwomen say sententiously "the bank so and so is strong because it has a high level of capital (in French "fonds propres")". Most TV watchers, as well as the speakers, picture the "fonds propres" on the asset side of the balance sheet of banks, and think "well, if they have a lot of "fonds propres", they will be able to withstand a run, so there is no risk for my deposits". Such a view of banks, which is very common, and stems from the age-old perception of money as precious metal or tangible value, is profoundly mistaken.

When the future sum X is sure

(Let's go back to investments with a payoff in one year.)

Usually - we all know that - the future is not sure. If we invest a sum P today into buying a security S, or - equivalently - into setting up a project, the cash flow X that S will produce in one year is not a sure figure.

There is one special case where X is a sure figure (or a figure as sure as possible in the world we live in): it is when we lend money, for a short period of time (a few days up to one year), to the government of a large economically well developed nation. The typical example is the United States. Over the last twenty years, at least, the US government has run budget deficits, and therefore has become a large borrower.

It borrows by selling short term Treasury Bonds in dollars. And, when they mature (when the time comes to pay them back, after one year), it borrows new bonds to pay for the old ones, and their interest, plus the new deficit.... The current US public debt is of the order of \$8000 billion. This debt is made of many kinds of bonds, a large part of them are short term bonds. A part of this debt is held by foreigners (between 1/4 and 1/3, figure to be checked). Here is a list of foreign governments holding US Treasury Bonds: US Bond Holders. See also article on China exports.

The interest rates on the bond markets, for the various types of bonds, result from supply and demand of money, between lenders and borrowers, and from risk appreciation. But for very short term (a few hours) bonds, the rates are fixed by the central banks managing each currency. In Europe, the central bank managing the euro is the European Central Bank (ECB), located in Frankfurt. In the US, the central bank, created in 1913, is the Federal Reserve System (the Fed). It manages the dollar; for instance, it adjusts the quantity of dollars in the US economy according to some economic objectives; and it decides the very short term rate. To simplify the exposition of the concepts, let's consider the very short rate corresponds to the one year rate. It is denoted r0.

At the moment (late October 2007), the short term rate in the US is r0 = 4,75%. That is, you can lend \$1000 dollars to the US government; you receive today a TB (Treasury Bond); and, in one year, the US government will take back (or just cancel) your TB and give you \$1047,5. It is as simple as that. And it is "sure" - at least as sure as what sure means.

Four years ago, the short term rate in the US was 1% (see article). But, since the US gov borrows a lot, it must keep its TB attractive, so it jacked up the rate. The Fed also does this to keep inflation at bay in the US. These are vast questions that we shall study step by step.

US risk free rate

In the euro zone, lending and borrowing take place (mostly) in euros. The short term rate r0 of the ECB (European Central Bank), in late October 2007, is 4%. If you lend to one of the governments of the euro zone 1000 € for one year, in one year you will receive 1040 €.

Euro risk free rate

For further discussions, see article.

From now on, we shall become a more theoretical, and consider an economic space with one currency, and no longer think of the dollar zone, or the euro zone, or the yen zone etc. The rate r0 is called the risk-free rate (it is different in different currency zones).

If we lend money to a risk free borrower, we get the risk free rate. Therefore, lending today a sum P, we will get, in one year, a sum

X = P x (1 + r0)

When the future sum X is risky

When the future sum X is not sure, we say it is risky. That is, the actual sum we shall get in one year will be one value out of a collection of possible values. For instance, if you buy today one Google stock, pay a price P = \$670 (it was \$312 two years ago) , and sell your stock in one year; in one year, you will get a value that may be \$700, or \$800, but also may be \$600, or \$500, etc. It is not sure. The same is true with a physical investment: you invest today a sum P into a physical investment, expecting a cash flow in one year (or, more realistically, a stream of cash flows over the next few years); you don't know what exactly these cash flows will be. Physical investments are never risk-free.

Let's look at a simple investment of 1000. Suppose, to start with, that the payoff in one year is 1200, for sure:

Great ! This a sure investment with a profitability of 20%. It's only a classroom example, unfortunately. At present the risk-free rates, in all the strong currencies of the world, are less than 5%.

Suppose, now, instead of receiving for sure 1200 in one year, we will receive a sum X that may be 1200, but may be more, and may also be less. 1200 is only a "middle value". In a short while, we shall put a more precise concept on this idea of middle value. We could represent the investment as this:

To represent risk in finance, we use simple probability theory and in particular the concept of random variable (abbreviation: RV). We say that the payoff X which we shall receive in one year is a random variable which will take an actual value (it is called "an outcome") out of a possible range of values.

When a security S promises a future payment X which is risky, with a "middle value" of, say, 1200, if we are offered to buy S today for a price P, we will require that the profitability of our purchase be more than r0, the profitability of risk-free securities. The price P, which we will accept to pay, is such that

or, equivalently,

Suppose, we are in an economic area where r0 = 5%. Then 1000 invested today in a risk free security would yield, in one year, 1050. Conversely, a risk free security, paying 1200 in one year, would cost today P = 1200 / (1 + 5%) = 1142,86

Check: P = 1142,86, and X in one year = 1200; the profitability is (1200 - 1142,86) / 1142,86 = 5%

But the point is that if X is no longer sure, but risky, then we won't accept to pay P = 1142,86 for the security S, we will pay less. This is a consequence of the fact that normal finance people are risk-averse. What we will pay depends upon the riskiness of X. The more risky, the more "average profitability" we want, and therefore the lower the price today. For instance, if the riskiness of X leads us to require a profitability of 30% (computed with the "middle value" of X), then we will only pay

P = 1200 / (1 + 30%) = 923

Who are risky borrowers?

• no risk: governments of large economically developed countries

• almost no risk: large corporations of the same countries

• average risk: small corporations, governments of stable developing countries

• big risk: junk bonds, governments of unstable developing countries (with monetary, social or political problems; to lend money to Ivory Coast, today, is highly risky)

Most countries, in their history, defaulted on their debt at one point or another. In modern history (after 1453), the first country to have defaulted is Philippe II of Spain in the late XVIth century. Various states of the US defaulted in the XIXth century. Argentina defaulted numerous times. Mexico defaulted in 1982, and again in 1994, etc.

And, as we saw, physical projects are always risky, some of them moderately, some of them very much. For a large corporation, well established in a market which is growing, to make an investment to increase capacity to adapt to the growing market is moderately risky. On the other hand, an investment to set up a new firm which will use a new technological process, or serve a new market, or produce a radically new type of product, is usually a highly risky project.

How do we measure risk? Answer: the risk of a security will be measured by the spread of the variability of the profitability of its payoff compared to its "middle value".

To formalize all these ideas more precisely, we need probability theory (for instance, what we introduced informally as "the middle value" of a risky sum will become the expectation of a random variable). So now we turn to the study of simple probability theory. Randomness is something we are all very familiar with, but it took about three centuries, from the first correct descriptions by Pierre de Fermat and Blaise Pascal in the mid XVIIth century, to put it on really clear grounds. It was done only around 1930 by the great Russian mathematician Andrei Kolmogorov (1903-1987).

The value today of a "security" S promising a sum of money X at a date t in the future (today being 0), depends upon three parameters:

1) obviously, the value X (in certain circumstances it is called the face value),

2) the time duration t

3) and the risk (or creditworthiness) of the signatory

We shall use the theory of probabilities to model risk.

One of the first modern problems of probabilities was posed, but not solved, by Luca Pacioli, in his book Summa written at the end of the XVth century. It referred to a game called balla between two players, which was essentially equivalent to tossing a coin, whose winner would receive the two equal stakes. The problem goes like this:

A and B are playing a fair game of balla. They agree to continue until one has won six rounds. The game, for some reason, has to be stopped when A has won five and B three. How should the stakes be divided? (source: Peter L. Bernstein, Against the Gods, p 43)

Another famous person who spurred the development of modern probabilistic ideas by Pascal and Fermat was Chevalier de Méré, in the XVIIth century: he had notice from a long experience of playing dice that in series of twenty four throws of a pair of dice, the double-six would come up (in at least one of the twenty four throws) about half of the times. He wanted to know why and  how to explain it.

Here is an illustration with a small example: suppose we have the choice between two offers

• offer 1: we receive 200 euros for sure,

• offer 2: we receive either 190 euros or 210 euros, each possibility with probability 50%.

Daniel Bernoulli in the early XVIIth century introduced the notion of utility. He explained that the extra 10 euros from 200 to 210 have a smaller utility to most people than the 10 euros from 190 to 200. (In other words, the curve of utility as a function of money is increasing but concave, like for instance the function x -> square root of x.)

In that case it can be shown that the utility of offer 1 is higher than the utility of offer 2. (It uses the notion of expectation, introduced in the second part of the lecture.)

So investors prefer offer 1 which has less uncertainty.

Break time