Ranjit Dighe
Lecture notes to accompany Cecchetti's Chapter 5 ("Understanding Risk")

Last revised (very slightly) on 3-April-2007.

In these notes:
I. Introducing risk
II. Measuring risk
III. Sources of risk: idiosyncratic and systematic
IV. Risk and diversification


Risk is defined as the uncertainty of an asset's return over a given period. We usually think of risk as involving the possibility of a loss or a major hardship, like losing money in the stock market or being stranded on a deserted stretch of highway with a flat tire. But more generally risk just means uncertainty, a range of possible outcomes, not all of which are equally good.

Almost all people are risk averse (they have risk aversion), in that they hate to lose more than they love to win. A risk-averse person dislikes being exposed to risk, and will always refuse an even-money bet(such as a coin toss with $20 at stake) to the point of paying to avoid such bets. Buying insurance is a sign of risk aversion. Someone who is not risk-averse is either risk-neutral (indifferent between taking or not taking even-money bets) or risk-loving (will always accept such bets -- e.g., habitual gamblers).
-- If returns on two assets are equal, risk-averse people will prefer the lower-risk asset
---- increase in an asset's risk (relative to other assets)
--> decreased demand for that asset,
      increased demand for all other assets


Conceptually, an asset's risk is the volatility of the asset's return. We can measure that volatility, or variability, of the asset's return as the
* variance (average squared deviation from mean return) of the asset's yearly returns,
or as the
* standard deviation (typical deviation from mean, or average, return; equal to the square root of the variance).
* An alternative that involves only simple arithmetic is the average (absolute-value) deviation from the mean return, which produces a result not far from the standard deviation. The standard deviation is the preferred measure, however (one reason why is that it gives more weight to outliers, or large deviations. If we're risk-averse, then it makes sense to give extra weight to outliers.)

Two of the most crucial pieces of information about any financial asset are its mean return (relates to expected return) and the standard deviation of its year-to-year returns.
-- If the expected return is just a simple average of past returns, then it's very easy to compute. Alternatively, one might put a lot more weight on recent returns, or incorporate other information into one's expectations. The expected value of an investment is defined as the probability-weighted sum of the possible values of an investment. The expected return on an investment is the expected value of future payouts, minus what you paid for the investment.  (To put it in percentage terms, we would then divide that amount by what you paid for the investment, then take it to the 1/n power (to get an annualized return), and multiply by 100%.)

Old bonus question, long since answered:
Q: Suppose that a slot machine costs $1 to play once, and that the player has a 1 in 5,000 chance of a $1,000 payout, a 1 in 500 chance of a $100 payout, a 1 in 50 chance of a $10 payout, and otherwise nothing.  What is the expected payout from playing that slot machine?  Show your work.
A: Expected payout, or Expected value = (Probability of $1000 payout)*($1,000 payout) + (Probability of $100 payout)*($100 payout) + (Probability of $10 payout)*($10 payout)
= (1/5000)*($1000) + (1/500)*($100) + (1/50)*($10)
= $0.20 + $0.20 + $0.20
= $0.60.
(That's the expected payout.  Since it costs $1 to play, your expected return from playing is minus forty cents.

Ex.: A company's stock has had the following yearly returns over the past five years: 5%, 15%, 10%, 2%, 8%

--> Mean return = simple average of those = (5+15+10+2+8)% / 5 = 40% / 5 = 8%

Deviations from mean = difference each year's return and the mean return (yearly return minus mean return)
= -3%, 7%, 2%, -6%, 0%
(calculated as 5% - 8%, 15% - 8%, 10% - 8%, 2% - 8%, 8% - 8%)

Absolute deviations from mean = absolute values of deviations
= 3%, 7%, 2%, 6%, 0%

Average absolute deviation = simple average of absolute deviations
= (3+7+2+6+0)% / 5
= 18% / 5
= 3.6%
The standard deviation will be pretty close to this number, and only a bit more complicated to compute. It's the square root of the variance, which is the average squared deviation from the mean. So let's first compute the (population) variance, by taking the squares of those deviations from the mean (which were 3%, 7%, 2%, -6%, 0%), add them up, and divide by 5 (the number of observations):

= (9% + 49% + 4% + 36% + 0%) / 5
= 98% / 5
= 19.6%.

Taking the square root of that gives us the standard deviation, which is

So the standard deviation is very much like the average absolute deviation, except it tends to be a bit bigger. That first step of squaring all those deviations means that large deviations become magnified, and the final step of taking the square root of the average does not entirely undo that magnification.
-- For example, consider two more stocks with an 8% mean return and yearly returns over a five-year period of (a) 4%, 12%, 4%, 12%, 8% and (b) 0%, 16%, 8%, 8%, 8%. The standard deviation is 3.6% for the first, 5.1% for the second.

{Nice to know: On an Excel spreadsheet, the command for the mean or average is =AVERAGE(range of cells or numbers), the command for standard deviation is =STDEV(range...), and the command for variance is =VAR(range...).}

Another useful measure of risk is value at risk, which is defined as the worst possible loss over a specific time horizon at a given probability. One can look at past returns and outcomes to form a prediction of how large a loss could occur and how likely it is to occur.
-- Ex.: If you have $10,000 and want to invest it in an index fund of the stock market over the next year, then you might want to know your odds of losing half of your investment. If a 50% loss in a one-year period has occurred in four of the past 100 years, then you could say you have a 4% probability of losing $5,000 (i.e., 50% of $10,000).
-- Value at risk is a helpful concept because it explains why risk-averse (or even risk-neutral) people would do things like go to casinos or play the lottery, where the expected return is negative and you're likely to lose money. The answer is that such people tend to budget just a small amount of money for casino gambling or lottery tickets, so they're not putting a lot at risk.


Two types of risk:

(1) idiosyncratic (firm- or industry-specific, nonsystematic) -- unique to the individual firm or industry; can be diversified away
-- Ex.: Nike stock, Philip Morris (tobacco) stock, New York municipal bonds
-- Studies have shown that to eliminate nearly all of the company-specific, or non-systematic, risk in a stock portfolio, you need own maybe 30-40 stocks. The average mutual fund holds 130.

(2) systematic -- cannot be diversified away
-- Ex.: stocks have systematic risk, because you can never be certain what will happen to those companies, and fluctuations in the current interest rate will raise or lower their resale prices (PDVs). Long-term bonds have systematic interest-rate risk as well.
-- Risk that is unique to a particular class of asset, as opposed to a particular firm or industry, classifies as systematic risk. You can diversify by holding assets of many different types (e.g., stocks, bonds, real estate, cash), but you won't be able to eliminate the risk from your portfolio entirely.

Asset risk = idiosyncratic risk + systematic risk

In a well-diversified portfolio, there is no idiosyncratic (company- or industry-specific) risk. All of the risk is systematic, arising from the inherent riskiness of the individual components (stocks, bonds, etc.) of that portfolio.

The standard measure of systematic risk: beta: measures the sensitivity of an asset's return to changes in the average return on the entire market. Beta is a numerical measure (a regression coefficient, to be precise) of the mutual relationship between market return and asset's return. If beta is 1, then an increase in the market return of, say, 10% means that the asset will typically gain 10% as well (a market index fund has a beta of 1). If beta is 2, then a 10% increase in the market return typically means a 20% increase in the asset's return. In an "up" market, a high beta means a high, above-average return; but in a "down" market, high betas mean bigger-than-average losses. Some assets have negativebetas, meaning that they do poorly when the market does well and vice versa. Some examples:
-- A stock-market index fund will have a beta of 1, because it holds the exact same stocks that go into the market average.
-- Technology stocks and other new-industry stocks tend to have betas > 1.
-- Utility stocks and those of long-established companies tend to have betas < 1.
-- Long-term bonds seem to have a beta close to 0. (Stocks and bonds are substitutes, but both are affected about the same by things like changes in market interest rates.)
-- Precious metals have a beta of less than 0. Their prices tend to go up when stock prices are down.

The greater an asset's risk, the greater the return it must offer to induce people to hold it and hence the greater its risk premium (the extra return on a risky asset, relative to the return on a risk-free asset like a Treasury bill). This concept will be explored more fully when we cover chapter 7 ("The Risk and Term Structure of Interest Rates").
-- Ex.: Rock star David Bowie issued $55 million worth of bonds in early 1997. His capacity to repay was pretty good, because he had a large and steady stream of income from royalties, album sales, etc. It's unclear whether those royalties will rise or fall in the future, but it's a fairly safe bet that they'll be enough for him to make his bond payments. Still, it's not a completely safe bet, so compared with Treasury bonds of the same maturity length (10 years), the Bowie bonds should pay a higher interest rate. As indeed they did: the Bowie bonds paid 6.9% interest, and Treasury bonds at the time paid 6.4% interest. The difference, 0.5% (i.e., 50 basis points), was the risk premium on the Bowie bonds.
---- Side note: Why did Bowie issue those bonds in the first place? Don't know. He may have wanted the money to finance some big new investment project, or, as the book suggests, he might just have wanted the sure thing of having $55 million right now instead of waiting for the money to trickle in the form of royalties and other income. So the book is suggesting that Bowie himself is risk-averse, because he gets $55 million now and will pay his bondholders out of that large-but-uncertain stream of future royalty income. If his royalties are less than expected and he can't make the bond payments, that's a bigger problem for the bondholders than for him.)


How can we avoid risk and still earn a decent rate of return? One popular solution is diversification, ordiversifying your portfolio. Diversification, in keeping with the old cliche "Don't put all of your eggs in one basket," is the art of putting your eggs in several different baskets.

DIVERSIFICATION: holding a variety of (risky) assets
--> reduces the overall riskiness of your portfolio
. When one asset tanks, others might not.
-- Ex.: Stocks may have the highest rate of return, but, to avoid risk, you should
(1) hold a variety of stocks (e.g., through a mutual fund);
(2) hold several other assets, too (bonds, money-market accounts, ...).

Key point: Diversification reduces the riskiness of your portfolio even if all of your assets carry equal risk.
-- Ex.:
* 2 beer companies: Samuel Adams (high-end), Beast (low-end). Both issue stock.
* 2 states of the economy: good, bad. Each occurs with 50% probability (or half the time).
* You have $1000 to invest

Samuel Adams does well when the economy is good ("procyclical"), Beast does well when the economy is bad ("countercyclical"). In the different states of the economy, the returns on the two stocks are:

State of the economy Expected Return

Good Bad
Samuel Adams 20% -10% .5 * 20% 

+ .5 * (-10%) = 5%

Beast -10% 20% .5 * (-10%)

+ .5 * (20%) = 5%

The two have the same expected return, but both are risky: With either one, there's a 50% chance that you could lose 10% of your investment.
--> Solution: Invest $500 in each one.
* Expected return is still 5%, but there is no risk!
The return is 5% in both states of the economy.  We can add another line to the table:

State of the economy Expected Return

Good Bad
½ Samuel Adams,

½ Beast

5%* 5%** 5%

* .5 * 20% + .5 * (-10%) = 5%

** .5 * (-10%) + .5 * (20%) = 5%

-- In this example, the returns on Samuel Adams and Beast are perfectly negatively correlated with each other (when one goes up 20%, the other one always goes down 10%). In the real world, there is less-than-perfect, but still some negative correlation among assets (e.g., gold and real estate perform well when stocks perform badly), so there is ample scope for risk reduction by diversification.