TCS Daily

One Price for Risk

By Arnold Kling - August 3, 2005 12:00 AM

Editors note: this is the fourth in a series of essays on financial markets and the economy.

4. One Price for Risk


"The CAPM world that Fischer imagined back in 1969 is a world much simpler than the one we live in. It is a world of only two financial instruments, risky equity and riskless debt, in which households hold a diversified market portfolio of the equity issued by firms, and borrow and lend among themselves in order to achieve their desired risk exposure. It is a world of index mutual funds and uncontrolled banking. It is a world in which risk is the price of reward, and economic fluctuation the price of economic growth and expansion. It is a libertarian world..."
-- Perry Mehrling, Fischer Black and the Revolutionary Idea of Finance, p 93


In a previous essay, I set up a metaphor in which the economy is a Food Court and capital investment consists of funding research and testing of new recipes. In a subsequent essay, I described the Modigliani-Miller theorem which, in this context, says that the value of starting a risky project to develop a new recipe is not affected by whether the project is funded by issuing debt or by issuing stock. In this essay, I describe how the Capital Asset Pricing Model (CAPM) determines the risk premium for a recipe project. I then describe Fischer Black's "CAPM world," to which Perry Mehrling refers.


A risk premium is the additional expected return on an investment that is demanded by investors in order to fund the project. An investor has the option of investing in risk-free short-term debt and earning a low but certain interest rate. The investor can be induced to take a chance on funding a recipe idea that may flop, but only if on average the return from investing in that recipe will be higher than the risk-free interest rate. The amount of additional return needed to entice the investor to make the risky investment is called the risk premium.


Our uninformed intuition, what I call "folk finance," suggests that the risk premium on a stock, or on a recipe project, should depend primarily on the variance of that firm's or recipe's prospects. A project with extremely uncertain returns, such as an attempt to grow meat in a lab, should have a higher risk premium than a project with relatively certain returns, such as an attempt to develop a recipe for sesame noodles without using peanuts. This intuition is wrong.


"Folk finance" also suggests that people with different risk preferences will choose different stock portfolios. A conservative investor will choose the sesame noodles project, and an aggressive investor will choose the cultured meat project. This intuition is wrong.


Finally, "folk finance" says that while conservative investors will choose diversified stock portfolios in order to reduce risk, aggressive investors will choose portfolios consisting of only a few stocks, in order to earn higher expected returns. This intuition is wrong. In fact, one story that circulated at MIT when I was there was that when a student in one of Fischer Black's classes tried to argue that you could earn higher returns by holding fewer stocks, Black said nothing, but simply turned around and wrote on the board "NO!" in huge letters.


Explaining CAPM


The Capital Asset Pricing Model rejects folk finance by showing how investors can use two tools to manage risk: variation in personal leverage; and diversification.


We saw the effect of variation in personal leverage when we looked at Modigliani-Miller. We pointed out that if a firm tries to use debt finance to create a more aggressive risk profile, then the firm's shareholders can "undo" the firm's actions by increasing the share of interest-bearing assets in their personal portfolios. From the individual's point of view, a portfolio consisting of a large share of interest-bearing securities and a small investment in a high-risk stock can be just as conservative as a portfolio that consists of a small share of interest-bearing assets and a large investment in a low-risk stock.


Diversification is another way for an individual investor to manage risk. Suppose that you have some of your portfolio in securities that earn the short-term risk-free interest rate, and the rest of your portfolio invested in the stock of a single firm, which has one recipe-development project. You choose your portfolio to balance risk and return. If you wanted more return and were willing to take more risk, you would shift more of your investment to the recipe-development project, and conversely. The rate at which you can trade off risk for return by varying the shares in your portfolio is called the price of risk.


Now, suppose another recipe-development project opportunity comes along. By shifting a portion of your portfolio into this new recipe-development project, you can gain the benefits of diversification. If diversification reduces your expected return, you can offset that by cutting back on your investment in risk-free securities and putting more into the "market portfolio" that combines the two recipe projects. Adding diversification to your portfolio lowers the price of risk, allowing you to have both higher returns and lower risk.


If the covariance between the two recipe projects is low, meaning that they are unlikely to both fail at once, then the diversification benefits are high, and you can shift a lot of funds out of the risk-free asset and into the market portfolio. If the covariance is high, then the diversification benefits are low, and the second recipe does not cause such a large portfolio shift.

The risk that remains in the portfolio with two assets is called market risk. There is always some market risk, because covariance is not zero, and diversification is imperfect. At this point, we can treat the combination of two assets as if it were one risky asset -- the market portfolio. The price of risk is measured by the trade-off between risk and return as one increases or decreases the proportion of funds invested in the market portfolio vs. the risk-free asset.


Next, suppose that a third recipe project comes along. Once again, relative to the market portfolio, the third recipe project offers diversification benefits. So we add it to the market portfolio, and we reduce our holdings of risk-free securities. The diversified portfolio becomes the new market portfolio, and the price of risk falls once again. This process goes on and on, until we have added every economically viable recipe project to the market portfolio.


No matter how many risky recipe projects there are in the market portfolio, there is still just one price of risk. That price is the amount of expected return that we give up by shifting a small portion of our portfolio from the market portfolio to the risk-free asset.


CAPM vs. Folk Finance


The first idea of folk finance is that high-variance projects necessarily require a higher rate of return than low-variance projects. Instead, what CAPM says is that the risk premium on a project depends on its diversification characteristics. If a high-variance project provides a lot of diversification benefit in the market portfolio, its risk premium actually will be quite low. For example, if the main risk in the economy is a rise in the cost of energy, then the risk premium on projects to develop alternative energy will be low, because such projects will provide diversification benefits. Investors will be willing to fund alternative fuels projects with relatively low expected returns, because of these diversification benefits. This could be true even though the alternative fuels projects themselves have highly uncertain outcomes.


The second idea of folk finance is that investors with different risk preferences will choose different recipe projects (different stocks in their portfolios). CAPM instead shows that there is always just one price of risk. Moreover, there is only one optimal way to choose more return and with higher risk: to increase your holding of the market portfolio and to reduce your holding of the risk-free security. Another way of putting this is that the shares of different stocks in the market portfolio is the same for everyone, because the weighting of stocks in the optimum portfolio is determined by their diversification benefits, which depend on covariance and not on any investor's risk preferences.


The third idea of folk finance is that you can earn higher returns from a concentrated portfolio of just a few stocks. CAPM shows that for any given desired level of expected return, a portfolio that combines the market portfolio and the risk-free asset will provide that expected return at lower risk than a concentrated portfolio.


This last point can be stated as follows: an investor is not rewarded for taking unnecessary risk. The risk of a concentrated portfolio is not necessary. You could get more reward at less risk by instead increasing your holdings of the market portfolio and reducing your holdings of risk-free securities. By concentrating your portfolio, you actually punish yourself by earning lower rewards relative to the risk you are taking.


In financial markets with CAPM, there is only one price of risk. In addition, there is an argument that there is only one interest rate. The differences that we observe between short-term rates and long-term rates reflect expectations about the future short term rate, as well as something that Fischer Black called "drift." While I understand the argument for one price of risk, and I know enough math to follow the formal proof, I cannot say the same about the argument that there is only one interest rate. I accept the proposition without fully grasping it.


The CAPM Economy


Fischer Black thought of CAPM not just as a model of the stock market but as a model of the entire economy. To see his point of view, consider that every source of value in the economy can be treated as a capital asset.


When you buy a home, you are purchasing a capital asset. This is also true for a car, or a dishwasher. Even a box of cereal may be thought of as an asset -- it just depreciates quickly as you consume the cereal. Your ability as a worker -- human capital -- is probably your most important asset. Most people earn more income from this asset than from their financial assets.


Next, imagine that all assets can be traded with ease. This may seem implausible, because there is no liquid market in which I can sell shares of my future labor income. On the other hand, when I run up a bill on my credit card, I am in some sense borrowing against my human capital. The credit card companies help turn human capital into a liquid asset.


Finally, imagine that everyone follows a CAPM strategy of holding a diversified portfolio of risks, rather than a concentrated portfolio. I find this a difficult story to tell, because I tend to think that we hold onto a lot of our idiosyncratic human capital risk. That is, if you obtain a degree in computer science, you retain most of the risk that computer science degrees will rise or fall in value. But pretend otherwise for now.


In the CAPM economy, no one is rewarded for taking unnecessary risk. The implications of this can be quite striking. In the 1970's, for example, nearly everyone believed that the Savings and Loan industry earned its profits by "riding the yield curve." The S&L's took in short-term deposits from consumers and made long-term mortgage loans. They thought that the spread between the long-term mortgage rate and the short-term deposit rate represented "profit."


In 1975, in a paper called "Bank Funds Management in an Efficient Market," Fischer Black attacked the idea that there was a profit from riding the yield curve. Instead, he insisted that riding the yield curve was taking unnecessary risk. It turned out, in fact, that short-term rates would soon rise quickly, and within ten years most of the S&L's became worthless, requiring a massive bailout and cleanup by the Federal government.


Stuff Happens


More controversially, Fischer Black suggested that in a CAPM economy, business fluctuations are the result of market risk, and there is nothing that the government can do about them. I call this the "stuff happens" model of the business cycle.


In a CAPM economy, the market portfolio is as diversified as possible, but some risk remains. When events turn out well, income and wealth are high, and everyone is happy. People with a high tolerance for risk, who have big exposures to the market portfolio and only small amounts invested in risk-free assets, are happiest of all. We call that a boom.


On the other hand, sometimes events do not turn out well. Stuff happens. In terms of our Food Court economy, the coffee craze unexpectedly evaporates, as everyone becomes enamored of specialty beers. As a result, huge losses are realized on all of the projects that were underway to come up with new exotic lattes. Then, everyone's wealth and incomes go down, and those with a high tolerance for risk lose the most. We call that a recession.


When stuff happens, there is nothing that the government can do about it. There is no way to recover sunk costs.


Most people believe that the government has some responsibility for managing the economy. From Fischer Black's perspective, the belief that government policies determine economic performance is what I call an attribution error.


Most people believe that the Federal Reserve is a powerful force in financial markets. Fischer Black did not see how this could be the case in a CAPM world. His logic influenced my essay, Can Greenspan Steer?. It was written nearly three years ago, but today the tendency for the bond market to move independently of Fed policy is, if anything, even more pronounced.


On the whole, however, I do not believe that we live in a CAPM economy. There are too many important differences between a world where all assets are liquid and tradable and the world as it exists. However, one way to look at financial innovation is that it is a process that works to close the gap between today's reality and the CAPM ideal. The next essay in this series will elaborate on that notion.


Arnold Kling is the author of Learning Economics


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