# Does “risk premium” imply a difference in expected value?

I'm trying to understand the term "risk premium". I keep seeing statements like this (from Investopedia) "A risk premium is the investment return an asset is expected to yield in excess of the risk-free rate of return." The explanation then goes on to say "A risky investment must provide the potential for larger returns to compensate an investor for the risk of losing some or all of their capital."

It seems to me that these two sentences are not capturing the same concept. What I am hung up on is the use of the word "expected". Does "risk premium" refer to a difference in the expected value, as in the mean of the outcome distribution, for risk-free and risky investments? As in, as the variance of outcomes grows, the expected value of an investment should also grow (and this difference is the risk premimum?). If so, why should the expected value necessarily be higher for a risky investment? I see it said that a premium is required to get a generic "investor" to buy riskier assets, but this isn't necessarily true. Some people bet $50 bucks on a horse race and others bet their life savings. Or does "risk premium" merely refer to the fact that the potential upside must balance the potential downside-- which implies no change in the expected value? In which case, the above sentence would more accurately be "A risk premium is the investment return an asset is expected to yield (given the "success" of the investment) in excess of the risk-free rate of return." ## 2 Answers A more risky investment doesn't necessarily need to have a downside as in loss, just the possibility of earning less than a less risky asset. Let's walk through a simplified example - please tell me which parts are not clear. Suppose we have a choice of two investments. The "riskless" investment costs$1 now, and is certain to be worth \$2.50 in 1 year. The "risky" investment is known to have a 1/3 probability of being worth \$1.50 in 1 year, and 2/3 probability of being worth \$3. in 1 year. (Of course in real life you have a choice of many investments more or less risky; some can actually lose money; but you don't really know the probabiliy of their having any particular return. The possible returns may or may not even be bounded above or below.) Since we're given these made-up probabilities, the well-known formula for the "expected value" of this risky investment worth in 1 year is 1/3 * \$1.50 + 2/3 * \$3. = \$2.50 - same as the riskless asset (I made up the numbers to make them match).

So, what should the risky investment be worth today?

This is not a math or finance question. Rather it's a human psychology question. A pretty common job interview question is some version of "Suppose we throw a 6-faced die. If 1 comes up, you pay me \$2. Else (if 2..6 comes up), I pay you \$4. How much would you pay to play this game?" You can see that the expected value of the payoff is $$-\frac{1}{6}\times 2+\frac{5}{6}\times 4=1$$. But how much someone would be willing pay depends on their risk aversion.

Depending on the circumstances, including the size of the investment, the risky asset may be trading at lower or higher price than the riskless investment with same expected value. For example, in case of lottery tickets, lots of people pay a dollar or two for a lottery ticket whose expected value (the probability of winning $$\times$$ the jackpot) is less than the cost of the ticket (except for rare large jackpots). Likewise, many people have good time gambling at casinos, knowing that their winnings are likely not offset their losses. Under these circumstances, the investors are willing to pay more for a small chance of winning more money than keeping a riskless dollar in their pocket would.

"Negative" risk premium means that the risky investment trades at a higher price than a riskless investment with the same expected value in the future. Or equivalently if the risky and the riskless investment trade at the same price, then the expected value of the (smaller) risky investment in the future is less than that of the riskless investment.

Conversely, "positive" risk premium means that the risky investment trades at a lower price than a riskless investment with the same expected value in the future.

Lottery tickets are usually sold with negative risk premium, except for rare periods when no one wins for a while, and the jackpot becomes huge, and the risk premium becomes positive, and people who don't usually play lottery begin to buy tickets - but the price of the ticket does not change. Another example, if you drive through Connecticut, you may notice billboards advertising casinos on reservations featuring "loosest slots around" with 98% payout. Not a winning investment, even if you're comped all you can eat buffet and hotel room in consolation for losing money. What is the dollar value of a watery "free" drink from a cute underdressed cocktail waitress at a casino? I don't know, but some people seem to overprice it a lot.

But in other circumstances, the investors prefer certainty, and are averse to the possibility of earning less than the riskless investment, so the riskless investment is trading at a lower price than the risky one. This is an empirical observation. When people actually "invest", rather than "gamble" for entertainment purposes, they prefer positive risk premium. The larger the amount, the more risk averse they get. But concepts like "large" and "my utility of knowing that I won't earn less than the riskless investment" and "my utility of a chance to earn more than the riskless investment" vary for different market participants.

For example, insurance companies' business model is to find mispriced risks and to take the view that they are mispriced. E.g. most homeowners prefer buying fire insurance to self-insurance. They're willing to pay more for fire insurance to an insurance company than the expected value of self-insurance. A single fire event is traumatic and catastrophic for an individual home owner, but business as usual (meh) for a large insurance company.

For example, put options on stock indices usually trade at negative risk premium because investors buy them as hedge/insurance for tail risk, not as investment.

• Thank you for this explanation, which is perfectly clear and which I agree with-- but I don't think it answers my question, which is more or less "what is the meaning of risk premium?" I agree with your description of the context-dependence of human risk tolerance, but if "risk premium" means "increase in expected value for risk", then your lottery example would have a negative risk premium right? Given this reality of human psychology, why does, say, the Capital Asset Pricing Model include a "market risk premium"? investopedia.com/terms/c/capm.asp – Sean Mackesey Feb 3 at 6:07
• OK, I see. If you put the answer at the top of your response I can accept it (something like, "Risk premium refers to a difference in expected value, not merely balancing tails of a distribution. While it is possible for a risk premium to be negative, in practice it is usually positive.") – Sean Mackesey Feb 3 at 16:29

@DimitriVulis explained it well. The dynamic that is the source of your uncertainty here is that "risk premium" can mean slightly different things, when seen from slightly different perspectives. Much of the confusion around this subject stems from inconsistency in the use of these close and related but distinct concepts.

Start with the basic premise. If I can get say 1% from a 5 year government bond, then I would want more than this to investing in stocks. But how much more?

You might be talking about the historic excess return that stockholders earned over bondholders. [In the jargon, the "ex-post realised risk premium"] This can be measured, but is all in the past, so its relevance to the future is uncertain.

Or you might talking about the future excess return that investors should expect to receive in stocks over bonds, to compensate them from the risk. [In the jargon, the "ex-ante required risk premium"]. Which is of course different from the risk premium that investors might think they are likely to receive, given the relative prospects and valuations of both assets ("the ex-ante expected risk premium"). Which in turn might be different today to levels of excess return in the past that would have caused investors to change allocations. IE the expected (expected future) risk premium might have X% in the past; but y% today.

Confusing, innit ;-) The point being that the term is frequently used for any or all of the concepts above, without discrimination. So the related concepts can, and often do, appear inconsistent.

The consistent theme in all of this is that if riskier assets require higher returns than risk-free, then the expected future value of the riskier asset will be higher than for risk-free. And the expected present value of future cashflows will be lower for risky than for riskless. This difference in expected rates of returns means that the concept of "expected value" is essentially meaningless. You have to specify "expected value when". 105 of cash today might be worth an expected 110 in the future; while 100 of stock today might be worth an expected 120 in the future. Which has the higher value? ;-)

It's a good question, because the underlying concept it's not that complicated, really. But it is very easy to generate confusion and inconsistency, unless one is quite is quite precise what one means when one uses a concept, that can be used to address many different, related problems.