# Explaining Risk Premium

*The purpose of this article is to simply explain to readers what a risk premium is and how it effects asset prices. I’m posting this because it is a very important concept for understanding the economy that I will need to reference in future posts when commenting on current events. The video below does a good job introducing the concept, but you may skip it if you’d like as a more precise explanation will follow.
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Risk premium is an incredibly important concept in conceptualizing asset prices and understanding finance. In essence the risk premium is the additional payment an individual requires to compensate them for holding* (or purchasing)* an asset* (or good)* whose future value is uncertain.

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If it helps we can think of the difference between a bond and a stock. If an investor buys $100 of a bond yielding 5% annually in one year they are guaranteed to have $105 . If an investor buys $100 worth of stock the future price *(the value to the investor)* of the stock is uncertain. To compensate for this uncertainty the investor will demand a higher return *(or compensation)* for holding the stock.

Now of course the investor doesn’t know the true future value of the stock, but what they do have are expectations about what possibilities that price *(value)* might be. If the investor expects that the stock value will never be above $105, then they should always purchase a bond because they know that this guarantees them $105 . The concept of risk premium denotes the price (which will always be above $105) that an investor will need to expect in order to take the gamble and hold a stock instead of a bond.

We can express it mathematically as follows:

With asset 1 being the bond and asset 2 is the stock. R1 and R2 are the returns for each. T=x expresses that this is the value at some future time “x”, and the E expressing the fact that the value of the stock at T=x is only an expectation (*its true value is uncertain*). What this equation tells us is that if we assume the asset values are equal to the investment ($100), then the compensation for the uncertain payout of the stock will have to be greater than the return for the bond by a certain amount:

The actual expected value that the stock must be expected to attain in the future, in order to compensate for the risk, will vary based on the individual preference for risk. But it will always have to be greater than the sure thing.

The importance of this concept is paramount when understanding trade, finance, and the price of any asset. This is because prices will change constantly in the market place, and so the price one initially pays always has the possibility to vary in the future. This aspect of pricing inevitably leads to asset price bubbles. In paradoxical manner: as the future is always uncertain, and as prices increase so does uncertainty about true and expected value (in the same moment just as uncertainty increases so too do prices). If you can wrap your head around all these concepts and the previous paradox, you will have understand quite a bit about how finance works, bubbles; and how people make money with information, risk, and rising and falling prices.

*Note: Models of risk and asset pricing can get very complicated but understanding this basic concept probably gives you 90% of the information needed to understand the bulk of whats actually important in the literature and reality.*

The risk premium analysis can illustrate the risk when it has something to compare against a “risk-free” asset. In most cases, we assume it is the treasuries. Given the debt that we are piling on (in fact most developed countries have more debt than emerging economies) wouldn’t we need to start looking for a more risk-free asset to compare?

I guess what I am asking is in the real world if people cannot use treasuries as a baseline for risk analysis, what do we compare against?

Thanks

Great point.

I know treasuries are not “risk free” but I wanted concepts most people would understand.

If you look at the rate of interest on long term Treasury-bills (government bonds), it is actually quite low (especially if you believe the hub-bub about high-inflation). This is an indicator that most investors feel these treasuries are risk-free (the U.S. won’t default on the loans). Furthermore, it also indicates most investors aren’t too concerned about long-term inflation… but that is another point in and of itself.

Traditionally U.S. treasuries have been considered by the financial world to be risk-free investments, although you are correct in pointing out that in reality there are no risk-free investments (and I think this is a really interesting point when it comes to asset prices and feeds into the problematization of price-bubbles and uncertainty I mention at the end).

Typically cash is also viewed as a risk-free asset, but of course as its value is relative to the price of other goods inflation is a risk to holding cash. All this being said there is a bright side to things which makes this model useful.

You do not need assets to be risk free to calculate a risk premium. The risk premium expresses the difference in percieved risk between two assets based on their value and rate of return. You can think of it as the premium (price demanded) for holding one risky asset over another risky asset. If you look at the second equation you will see what I mean. If you assume both assets are risky, you can compare the rate of returns between them. If one asset has a rate of return of 7% while the other has a rate of return of 9% the market risk premium of the second asset would be 2%. This is what the market has decided is a rate of risk compensation sufficient to make an investor willing to hold the second asset given the expectation of risk. Of course between individuals risk premiums and information vary. This is where a lot of interesting things happen in the financial world.

I hope this explanation is clear. Let me know if you have any other questions.

Note: We are assuming the risks for each of the assets are independent (no shared risk). Typically for financial assets this does not hold (a downturn effects multiple assets/asset classes), and there are many interesting economic models that do interesting things with risk covariance. However, the initial concept of risk premium still gives you a good deal information and is conceptually necessary for interpreting the more developed models.