Hopefully, the reader will agree that a rather substantial amount was accomplished in prior sections of this work. Yet he or she may not have recognized a rather remarkable aspect of the analysis to this point: probability played no direct role whatever! Instead, most of the results followed from the law of one price.
The importance of this law should not be underestimated. Whenever it is violated, "the same thing" can be obtained at two different prices. Better yet, it can be purchased for one price and sold at a higher price. In some instances it may take a clever analyst to determine how to construct "the same thing" synthetically via combinations of marketable instruments. Nonetheless, the stakes are high enough to discover how to do so, for the result will be an arbitrage, with concomitant prospects of increased wealth.
Modern markets populated by investors and investment professionals driven by greed and cupidity are unlikely to the characterized by large and persistent violations of the law of one price. Thus results that rely on it are likely to be quite robust. In the real world, characterized as it is by transactions costs, differences in information, and so on, prices of things that are "almost the same" may not be equal, but they are nonetheless likely to be close to one another
Unhappily, there is no such thing as the "law of one probability". Thus Investor X may believe that there is a 40% probability that the weather will be good (and hence a 60% chance that it will be bad), while Investor Y believes that there is a 50% probability that it will be good (and hence a 50% probability that it will be bad). Their beliefs will undoubtedly influence their attitudes towards the associated time-state claims, and hence the equilibrium prices for such claims. But markets can clear without investors reaching any agreement concerning such probabilities.
Despite this, much of modern financial theory is built around the notion that there is a single set of probabilities for various outcomes. In some cases it is simply assumed that all investors agree concerning such probabilities. In others, the probabilities utilized for calculations are assumed to be those of a "consensus of well-informed investors". Often the latter characterization represents a rationale for a model built on the foundations of the former (more extreme) assumption.
To proceed, we join the theorists who make such heroic assumptions. In particular, we assume that all individuals in our simple economy agree on the probabilities associated with future states of the world. For our initial examples, we assume that they believe our two states are equally probable. Representing probabilities by the vector prob:
Bad Good
Weather Weather
0.50 0.50
Not surprisingly, as long as the states of the world that have been enumerated are mutually exclusive and exhaustive (i.e., one and only one will occur), the sum of such probabilities must equal 1.0. Other than this, little can be said ex cathedra, since the concept applied here is that of subjective probability (people's beliefs about the relative likelihoods of various events), not that of objective probability (with its accompanying notion that there is a "true" set of such likelihoods).
Whatever the source of such probabilities, we assume for now that they exist and that all market participants agree both about them and about the results of computations involving them -- to which we now turn.
The expected value of an uncertain variable is obtained by weighting every possible outcome by the associated probability. Equivalently, it is a probability-weighted average of the possibilities.
The one-period return from a security is the change in its value plus any distributions received at the end of the period, all divided by the initial value. Consider the Stock that we have been following that will increase in value by 26% if the weather is good but decrease by 4% if the weather is bad. Its return vector r is:
Good Weather 0.26 Bad Weather -0.04
The expected return on the Stock will thus equal prob*r, or 0.11 (11.0 percent).
We can, of course, compute expected returns for a number of securities in one stage. Assume that a Bond offers a guaranteed return of 5% and that the Market portfolio consists of 60% Stocks and 40% Bonds. Consider the matrix R of returns for the Stock, the Market and the Bond shown below:
Stock Market Bond
Bad Weather -0.04 -0.004 0.05
Good Weather 0.26 0.176 0.05
As is so often the case in life, as one goes across the columns, good news (a better return in bad times) accompanies bad news (a poorer return in good times).
The expected returns are e = prob*R:
Stock Market Bond
0.110 0.086 0.050
Not surprisingly, the expected return of the riskless bond is its certain return. For each of the other choices, the expected return is neither the highest possible value nor the lowest. Rather, it is a "middle value". Note that for the Market and the Stock, under no circumstances can the actual return equal the expected value. No matter what, the actual value will deviate from the expectation. Moreover, in this case the magnitude of the deviation from the expected return is larger for the Stock than for the Market. In this sense, the Stock is riskier than the Market, which is in turn riskier than the Bond.
The difference between the expected return on a security or portfolio and the "riskless rate of interest" (the certain return on a riskless security) is often termed its risk premium. Underlying the terminology is the notion that there should be a premium (higher expected return) for bearing risk. As we will see, however, there is no reason why such premia should be associated with all types of risk.
An equivalent definition of a risk premium is: the expected excess return on a security or portfolio, where excess return is the difference between an actual return and that of a riskless security.
In our example, the expected return on the Market portfolio is 8.60%, and the market risk premium is 3.60% (8.60% - 5.00%). The associated risk can be measured by the likely divergence between the actual return and the expected return. If the weather is Good, the difference will be 9.00% (17.6% - 8.6%). If the weather is bad, the difference will be -9.00% (-0.4% - 8.6%). Since the two outcomes are equally likely, the absolute value of the divergence will be 9.00%.
The market portfolio provides an excess return of 12.60% (3.60 + 9.00) if the weather is good and an excess return of -5.40% (3.60 - 9.00) if the weather is bad. The potential gain over the riskless rate (12.60%) is thus 2.33 times as large as the potential loss relative to the riskless rate (5.40%). This may seem particularly generous, given the assumption that the two situations are equally likely. However, such a relationship is not uncommon in actual markets.
Since Stocks and Bonds are issued by firms, the Market Portfolio represents a package of claims on all the productive assets of such firms. Equivalently, it is the set of claims that would be held if only equity financing had been utilized by firms.
Given the use of Bonds and Stocks as financing vehicles by a firm, an investor can create an "all-equity" version of the firm synthetically, by holding a portfolio with its bonds and stocks in market value proportions. Conversely, if a firm is financed solely by equity, an investor who would have preferred the payment pattern associated with a "levered" stock could create the latter synthetically by borrowing money, then using both the borrowed money and his or her own funds to buy stock in the unlevered firm. These relationships form the basis for the original Modigliani-Miller theorem: in a world of the sort we are analyzing, "home-made leverage" (borrowing) can serve as a substitute for "firm-made leverage" (borrowing); hence, corporate financing decisions do not matter.
In our example, one can, of course, create pure securities synthetically. The security price vector ps and payoff matrix Q can be written as:
ps:
Bond Stock
1.00 1.00
Q:
Bond Stock
Good 1.05 1.26
Bad 1.05 0.96
The state prices, given by p = ps*inv(Q) are:
Good Bad
0.2857 0.6667
giving value-relatives vr = 1 ./ p:
Good Bad
3.50 1.50
and returns (vr-1):
Good Bad
2.50 0.50
A security that promises to pay off only if the weather is good will provide a return equal to 250% of the amount invested if the weather is good, and -100% if it is not. A security that promises to pay off only if the weather is bad will return 50% if the weather is bad and -100% if it is not. The return matrix Q is thus:
Good Bad
Security Security
Good 2.50 -1.00
Bad -1.00 0.50
and the expected return vector e = prob*Q is:
Good Bad
Security Security
0.75 -0.25
The "Good Security" has an expected return of 75%, while the "Bad Security" has an expected return of minus 25%.
Clearly, the Good Security has a very high risk premium: 75% - 5%, or 70%. This might not seem too surprising, given the extreme risk involved. However, the Bad Security actually has a negative risk premium (i.e. a "risk discount"): -25% - 5% = -30%. Yet it too is very risky.
Obviously, this world is not one in which just any kind of risk is rewarded with a risk premium.
What is going on here?
Part of the answer to the question can be found by comparing the forward prices of pure securities with the probabilities that the associated states will occur. Recall that the forward price for a pure security is the amount that one must agree today to pay at a future date. As we argued earlier, arbitrage will ensure that this is simply the current price for the contingent claim plus interest for the period in question. But this implies that the ratio of a current atomic price to the sum of such prices will be the same as the ratio of the corresponding forward price to the sum of such prices.
As always, the sum of the forward prices is 1.0. As indicated earlier, this must be the case, since by purchasing one unit of every time-state claim, one is guaranteed a payment of $1 at the future date. Clearly the cost of obtaining this, if paid at the future date, must be $1. Thus forward prices, like probabilities, sum to 1.0.
Since the sum of a set of forward prices for a given date must equal 1.0, it follows that the atomic forward price vector f will equal p./sum(p). In this case:
Good Bad
0.30 0.70
Now, compare the forward prices for the states with their probabilities:
Good Bad
Forward Price 0.30 0.70
Probability 0.50 0.50
The forward price for a state need not equal its probability. In our example, one such price (Good) is lower, and the other (Bad) is higher than the probability that the state will actually occur. Note, however, that since both sums must equal 1.0, if any price is below its associated probability, at least one other must be above its associated probability.
In this example, equilibrium has been achieved when prices are such that a payment of $1 if the weather is good is "cheap" -- it only costs $0.30 (forward) to obtain a payment with an expected value of $0.50 (0.50*$1). On the other hand, a payment of $1 if the weather is bad is "expensive" -- it costs $0.70 (forward) to obtain a payment with an expected value of $0.50 (0.50*$1).
Why are people willing to pay these prices? The answer is not too surprising. Other things equal, one would prefer to have goods and services when there are not as many available. Thus payments under bad conditions are more highly prized than those under good conditions. In this case, the society produces less when the weather is Bad than when it is Good. All contingent claims are not equal. The fewer there are, the more valuable another one will be.
This aspect of our example captures an important feature of most economies. To see this, consider an alternative scenario in which each of the forward prices of the securities is $0.50, and thus equal to the probability of the associated state.
As before, the riskless rate of interest would be 5.0%. What would be the expected return on the Market Portfolio? Assume that it still offers payments of $1.26 if the weather is Good and $0.96 if the weather is bad. The state prices would each equal 0.50/1.05, or 0.4762. We thus have:
p:
Good Bad
0.4762 0.4762
q:
Good 1.26
Bad 0.96
price = p*q:
1.0571
prob:
Good Bad
0.50 0.50
expected value = prob*q:
1.11
expected return = (expected future value/price) - 1:
0.05
Thus the expected return on the market would equal 5% -- the riskless rate of interest. There would be no risk premium at all!
In fact, in a world of this sort, there would be no risk premium on any security -- every single one would have an expected return equal to the riskless rate, as we will show.
Consider the vector f of forward prices. Given a cash flow vector c, we may calculate an associated forward value fv:
fv = f*c
The forward value of a set of (contingent) cash flows is an amount agreed upon in the present that must be paid for the set at a specified date in the future. In our case, there is only one future period, so the payment date coincides with the date at which cash flows (if any) will be received.
The expected value of c is calculated by multiplying each possibility by its probability, then summing. As before, assume that the probabilities of the states are included in vector prob. Then the expected value ev will be given by:
ev = prob*c
We know that arbitrage will insure that the present value pv of a set of claims will equal its forward value discounted at the riskless rate of interest. Thus:
pv = fv/(1+i)
The expected value relative evr for an investment is its expected value divided by its present value (price):
evr = ev/pv
But this can be stated in terms of the forward value, using the arbitrage relationship between present and forward values:
evr = (ev/fv)*(1+i)
The expected value relative for a riskless security is, of course, 1+i. Thus the expected value relative for an investment will be greater than that for a riskless security if ev/fv is greater than one. In such cases the investment will provide a risk premium. If ev/fv is less than one, the expected value relative for the investment will be less than that for a riskless security, and the investment will "provide" a risk discount. An investment for which the expected value is equal to the forward value will offer an expected value relative equal to that of a riskless security and will provide neither a risk premium nor a risk discount.
This shows why the risk premium will be zero for every security or portfolio if the forward price for each state equals the associated probability (i.e. f equals prob). In such a world, the expected value (ev) will equal the forward value (fv) in every case, and every expected value relative will equal 1+i. Hence the expected return on every security will equal the riskless rate of interest and there will be no risk premia.
For there to be a market risk premium, some atomic forward prices must differ from the probabilities of the associated states.
The expected return on an investment is:
(ev/pv) -1
while its risk premium is:
((ev/pv) - 1) -i
or
(ev/pv) - (1+i)
where i is the riskless rate of interest.
The arbitrage relationship between a present value and an associated forward value insures that the risk premium for an investment will also equal:
(ev/fv)*(1+i) - (1+i)
or
((ev/fv) - 1) * (1+i)
As shown earlier, an investment will have a risk premium if ev/fv is greater than 1.0, a risk discount if ev/fv is less than 1.0, and neither if ev/fv equals 1.0. Clearly, it is crucial to understand the determinants of differences in ev/fv across atomic securities in order to understand the nature of risk premia.
For an atomic security, the expected value will equal the probability of the associated state. Thus the vector of probabilities, prob is the vector of expected values. Moreover, the vector of atomic forward prices f is the vector of forward values. Thus the vector of ev/fv values can be computed directly via the formula:
prob ./ f
Note that since both vectors must sum to 1.0, either all the ratios will equal 1.0, or some will be below 1.0 and others above it.
Why should one atomic risk premium differ from another? And if there are differences, what might explain them? As in other realms of economics, we would expect market prices to adjust until there is good news to go with every piece of bad news. This suggests that higher risk premia (good news) should be associated with states in which additional goods and services are of less value (bad news). But when are additional amounts of consumption of less value? When the amount available for consumption is large. Hence, states of plenty should have high risk premia (ev/fv, or prob/f values) while states of scarcity have low risk premia (ev/fv or prob/f values).
In a one-good economy, the notion of aggregate output is unambiguous. In such a case, if aggregate output in state s1 exceeds that in s2, then the ratio of probability to forward price will be higher for state s1 than for state s2. If the aggregate output in two states is the same, then the ratio of probability to forward price should be the same for the two states. We deal later with cases involving multiple goods.
Consider the following example:
c prob' f' f'-prob'
40 0.25 0.35 0.10
50 0.25 0.30 0.05
60 0.25 0.20 -0.05
70 0.25 0.15 -0.10
----- ---- ----
1.00 1.00 0.00
Note that the transposes of the last three vectors have been shown for convenience.
The expected value for the market portfolio (c) is given by prob*c; it is 55.0. The forward value for the market portfolio is given by f*c; it is 51.5. The market portfolio thus has an expected value relative equal to (55.0/51.5)*(1+i). Assume that the interest rate is .06 (6%). Then the market portfolio has an expected value relative of 1.132 and hence a risk premium of 0.132-0.06, or 0.072 (7.2%).
In this case, prob*c is 55.0 and f*c is 51.5. Thus (f-prob)*c must be -3.5. Moving from vector prob to vector f as a multiplier of c lowered the value of the product, and hence implied the presence of a risk premium.
Consider the results of multiplying a vector x times c, where x can equal vector prob, vector f, or something in between. We wish to move from x=prob to x=f by a series of small steps. To see how this might be done, consider the final vector in the table: f'-prob', each entry of which equals the sum of all the steps to be taken. Since the entries in this vector must sum to zero, some will be positive and others negative. Moreover, the sum of the positive numbers must equal the sum of the negative numbers. Finally, given our assumptions about atomic risk premia and the ordering of states by increasing aggregate consumption, the positive numbers will all precede the negative numbers.
Given these relationships, we can move from x=prob to x=f by a series of steps, each of which involves adding a fixed amount (e.g. 0.05) to an entry in x for one state of the world and subtracting an equal amount from one or more entries for states of the world in which aggregate output is larger. But each such step must lower the value of x*c, since the amount chosen is first multiplied by one level of output, and then by one or more larger levels of output. with the first amount added to the total product and the second (larger) amount subtracted from it.
Since each such step will lower the product of the two vectors, the total effect of all such steps must lower the product. This gives the key result that:
If atomic risk premia increase with aggregate output,
(1) the expected value for the market portfolio will exceed its
forward price, and
(2) the expected return on the market portfolio will exceed the
riskless rate of interest.
In simpler terms:
If atomic risk premia increase with aggregate output, there will be a market risk premium.
While there may be other explanations for such a risk premium, the diminishing value of added consumption as more consumption becomes available appears to be by far the most plausible cause. Indeed, it is tempting to conclude that:
If there is a market risk premium, atomic risk premia increase with aggregate output.
We hereby yield to that temptation.
Over the long run, portfolios comprising large numbers of risky securities tend to provide higher returns than do short-term riskless deposits. This is consistent with the existence of a market risk premium in the sense that we have used the term. We consider this strong presumptive evidence that on average, people consider additional consumption more valuable in states of scarcity than in states of plenty.
Not all atomic securities offer risk premia (ev/fv>1). Some offer risk discounts (ev/fv<1). But all are risky. Thus there is not a simple relationship between risk and expected return. This is true not only for atomic securities, but also for more traditional ones.
The key issue in determining the presence and magnitude of a risk premium is the distribution of the value of a security's cash flows across various states. If more of its value comes from states with high ev/fv (probability / forward price) ratios than from those with low ev/fv (probability / forward price) ratios, the security will generally provide a risk premium. But since states with high ev/fv ratios are generally associated with times of abundance, this is equivalent to saying that an investment which pays more in good times and less in bad times will generally offer a risk premium. Indeed, the greater the extent to which an investment is a "fair weather friend" (bad news), the greater will be its expected return (good news).
This is an important corollary of the general theorem that in a competitive economic market, bad news is likely to accompany good news. Investors demand higher expected returns from securities that are likely to fail them when they most need help.
To summarize, some risky securities will provide a risk premium. However, the premium will be associated with the risk of doing badly when times are bad. There is no reason to expected higher returns to be associated with any type of risk -- just "bad times risk".