Why do Risk Premia exist?

When I entered this industry several years ago, I blew my first real interview because I didn’t understand the concept of a risk premium. Ironically, I later heard through the grapevine that I performed well during interview, but I was struggling with my unwillingness to invest as my analysis suggested that I should, and I later wrote an email explaining my discomfort involving stuff from my old probability studies in physics. It apparently was this follow-up letter that struck the interviewer as strange, and he decided to look elsewhere. In retrospect, what was missing from the analysis was a better understanding of risk premiums.

The interviewer asked me how much I should be willing to pay today for a contract where he would pay me $100 in six months if the temperature were above the historical median for that day. The expected value of that bet is fairly simple: $50, and I knew enough from economics to know that the $50 needed to be discounted to the present to come up with the contract’s present value. Assuming that treasuries were yielding 4%, the contract would seem to be worth $49 today.

Although I felt comfortable with the analysis, I struggled with the fact that, were I pressed in real life, I would not have been willing to pay $49 for that contract. I was unemployed at the time, and my savings were especially needed, and so I would not have been willing to put that money at risk. That instinct turned out to be the right one, but at the time I did not understand why.

To make the contract attractive, the price would need to be less than $49. If the contract had simply been to pay me $50 (and I was certain it would be honored, and I didn’t need the money before then, and there were no other attractive investment), then $49 would have been more or less the right price. But the interviewer wasn’t going to pay $50; he was going to pay $100, or possibly $0. In other words, there was risk. Risk made me uneasy, even more so, given my conditions at the time. That uneasiness deserves compensation, and it turns out to be appropriate to compensate.

Why is there a risk premium?

A substantial part of financial analysis is about estimating the right premium to compensate an investor for taking on investment risk. To figure a risk premium correctly, it helps to understand why there should be one in the first place.

It turns out that a risk premium is a natural outcome of the economic principle of diminishing utility. Without the jargon, risk premia come from the fact that, although winning $200 is twice as much as just winning $100, and definitely better, it is less than twice as satisfying.

A utility function tells us how “satisfying” it is to have a particular quantity of money or combination of goods. It converts an observable quantity like “dollars in the bank” to the “degree of satisfaction” those dollars confer. Utilities are notoriously difficult to work with in practice, because they cannot be quantified directly, and one person’s utility function, even if it could be observed, may not be comparable to another person’s function (this causes difficulties in welfare economics because it implies that there is no aggregable social welfare function for public policy makers to maximize).

There are two things we know do about utility functions: 1) having more wealth should be preferable to having less (at extreme wealth levels even this might not be true, but that is a subject for a different post), and 2) as one acquires more, additional wealth provides less additional satisfaction, an effect called “decreasing marginal utility.”

There is a rational reason to expect decreasing marginal utility: if I give you $10,000, you are likely to be happy and use it for the things that matter most on your list of wants and needs. If I hand you a second $10,000, you are likely to be even happier, but the things you use it for will be less essential to you than with the first $10,000 you gained, or else you would have attended to them with the first $10k, rather than wait for the second (which presumably you did not know was coming).

The mathematical implication of an increasing utility function with decreasing marginal utility is that a graph of utility vs. wealth should look roughly like figure 1, always growing with wealth, but flattening out and growing more slowly as wealth increases.

[ Click for Figure 1 ]

<img src=“http://www.bruce.chadwick.org/blogAssets/RiskPremiumP1/RiskPremiumFig1.jpg”>

The curvature of the utility curve is important, because it is ultimately the reason for the risk premium.

Let’s assume now that we are about to enter a risky investment. To keep it simple, we will have one of two outcomes, with equal likelihood. Either we will get a high payoff, labeled as H on Figure 2, or we will get a low payoff, labeled L on the same figure. Since both payoffs have an equal likelihood of occurring, the expected payoff, E(x), is simply the average of the two outcome, and sits exactly between the the positive and negative outcomes (shown as (x+∂) and (x-∂) on the figure).

[ Click for Figure 2 ]

<img src=“http://www.bruce.chadwick.org/blogAssets/RiskPremiumP1/RiskPremiumFig2.jpg”>

Now look at the utilities for this scenario. Just as we can compute expected wealth as the average of H and L’s wealth, we can also compute expected utility by averaging the utility expected at H and L. This utility is also exactly halfway between H’s and L’s utility, and is shown by point A on Figure 2. But point A (the expected utility with risk) delivers less utility (i.e. satisfaction) than the one would get if we were just guaranteed E(x). The fact that we are either going to have (x+∂) or (x-∂) at the end of the investment means that our investment’s expected utility E(U) is less than the utility of the expected outcome U( E[x] ), simply because there is risk involved.

In other words, the expected utility of a risky bet is less than the utility of the bet’s expected value. Mathematically,

        E(U) < U( E(x) )

[ Click for Figure 3 ]

<img src=“http://www.bruce.chadwick.org/blogAssets/RiskPremiumP1/RiskPremiumFig3.jpg”>

Figure 3 goes a step further, and shows that this risky bet has the same expected utility as another wealth outcome, xc, sometimes called the “certainty equivalent” wealth. The certainty equivalent wealth (point C on Figure 3) represents the level of wealth that - if guaranteed - provides the same degree as satisfaction as the expected utility of the risky option. Importantly, because of the way the utility function curves, the certainty equivalent wealth, xc, is always less than the expected wealth, E(x).

Since an investor is really trying to maximize expected utility, not expected wealth, the value of the risky bet to that investor is actually xc, the certainty equivalent, and not E(x), the expected value, because both the certainty equivalent and the risky investment produce the same expected utility.

As a result, it is xc, the certainty equivalent, not E(x), that should be discounted to the present to account for the time value of money, because PV(xc) is the value that, if invested today without any risk, will produce the same expected satisfaction to the investor.

And how do we get from E(x), which we can hope to compute from expected outcomes, to xc, the non-risky equivalent? We discount E(x) by a risk premium to estimate xc.

How much discounting is necessary? If one could observe the true utility curve, and therefore calculate xc, the risk premium would be:

        Risk Premium = [ E(x) / xc ] - 1

Mathematically, xc is is the inverse utility function, applied to the investment’s expected utility xc = U-1( E(U) ). On first blush, it would look like the inverse utility function simply “undoes” the utility calculation based on the investment’s performance, but if we remember that E(U) < U( E(x) ), we can see again that xc < E(x).

Getting at the risk premium

Obviously, as nice as it would be to use mathematics to calculate the true risk premium, the problem is that we can’t really observe utility functions in practice, and therefore there is no formula that can truly tell us what risk premium we should really use for our own investments. As investment analysts, however, we might be able to sense our own utility functions, even if it is difficult to map the functions of others.

For example, we can ask ourselves how much we would feel comfortable paying for an investment if we were to receive the investment result by day’s end, rather than at the end of the investment horizon. Doing this focuses our analysis solely on the risk aspect of the investment, rather than confounding the risk with the time aspect. Although an answer by this method is admittedly determined subjectively, it does give us a sense of the appropriate discount to use for the risk premium. We should not be surprised if our “comfortable price” turns out to be less than the investment’s expected value; indeed, we should expect it. How much less gives us the risk premium.

As mentioned, the advantage of this method is that it separates the time component of an investment’s discount factor from its risk component. Normally, when risk factors are estimated, they are “added on top of” the discount factor for time value (a.k.a the risk-free rate). This encourages investment analysts to project the risks further into the future, making risks seem more abstract than they really are. By thinking in terms of an investment result that, while unknown, will be known later today, we can focus our minds specifically on the risk, rather than conflating it with the discount for time. The suggested mechanism is:

1) Evaluate risks as quantitatively as possible, and determine expected value E(x).

2) Determine a risk discount, based on the price, P, you would be comfortable paying for the investment, assuming that the result becomes known and delivered later that same day. Then the risk premium, RP = E(x) / P.

3) Discount P to the present, using the risk-free rate (RFR) as the discount factor for the time value of money.

Working backwards, we see that the discount rate is then constructed as: Discount Rate = RFR + RP .

Future posts on risk premia

This analysis of the origin of risk premia has some additional implications that I want to cover in a future post. Most importantly, it suggests that risk premia may change not simply a result of asset volatility (though it responds to that too), but also can be the result of changing utility curves. The implications of this for investment strategy are many, and I will return to them in a later post.