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MODEL RISKS

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Model risks are present in both appraised price estimates and the assessment of realization risks:
Second order price risks. Model assumptions that prices change in a linear way with an underlying factor can lead to material differences when curvature is present. This occurs with bonds and can be taken account of by using the measure convexity. Callable bonds with embedded options exhibit negative convexity as prices approach the call price. Tranches of mortgage backed securities provide particular challenges depending on pre-payment rates.
These risks are also present with options where this is referred to as gamma risk where the rate of change does not simply vary with the underlying factor but may even reverse sign. Price distribution. Most methods used to assess downside risks from particular trading positions are dependent on price changes being essentially random and having a “normal” distribution. This is not always the case. Some distributions are skewed, with a “long tail” in one direction, while others have a distribution close to that of the normal distribution but have “fat tails”. Fat tails occur when the incidence of occurrences at the extremes is greater than would be expected from a normal distribution.
A further implicit assumption is that price changes are not serially correlated, that is that price changes in one period are independent of price changes in prior periods. This assumption does not always hold.

Reversion to mean. A further assumption often made is that of reversion to mean. This means that past relationships between factors will continue to apply in the future. Examples of these include the following:

Price volatility. Methods to determine the likelihood of losses in a given period exceeding a particular level depends on the volatility of prices of the instruments concerned. A common assumption is that future volatility levels will revert back to historic levels or if  they do change over time they change slowly.
Correlations. The level of potential losses for a portfolio of financial assets depends not only on price changes of individual instruments but the way in which such changes are related to one another. The prices of two instruments may tend to move together, or if the price of one instrument rises the price of the other tends to fall. These relationships are core to overall portfolio risks. These relationships are captured in a quantitative way through the use of correlation coefficients calculated on the basis of historic prices.

These correlation coefficients are assumed to remain stable over time. Risk premiums. The same considerations apply to the market equity risk premium and stock specific betas. Trading volumes. In estimating the time necessary to unwind a position the historic average daily trading volume is usually assumed.

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Risk-Averse Investors Have Demanded Higher Expected Rates

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We have assumed that investors are risk-neutral—indifferent between two loans that have the same expected rate of return. As we have already mentioned, in the real world, risk-averse investors would demand and expect to receive a little bit more for the risky loan. Would you rather invest into a bond that is known to pay off 5% (for example, a U.S. government bond), or would you rather invest in a bond that is “merely” expected to pay off 5% (such as my 6.63% bond)? Like most lenders, you are likely to be better off if you know exactly how much you will receive, rather than live with the uncertainty of my situation. Thus, as a risk-averse investor, you would probably ask me not only for the higher promised interest rate of 6.63%, which only gets you to an expected interest rate of 5%, but an even higher promise in order to get you more than 6.63%. For example, you might demand 6.75%, in which case you would expect to earn not just 5%, but a little more. The extra 12 basis points is called a risk premium, and it is an interest component required above and beyond the time premium (i.e., what the U.S. Treasury Department pays for use of money over time) and above and beyond the default premium (i.e., what the promised interest has to be for you to just expect to receive the same rate of return as what the government offers). 

Recapping, we know that 5% is the time-value of money that you can earn in interest from the Treasury. You also know that 1.63% is the extra default premium that I must promise you, a risk-neutral lender, to allow you to expect to earn 5%, given that repayment is not guaranteed. Finally, if you are not risk-neutral but risk-averse, I may have to pay even more than 6.63%, although we do not know exactly how much. 

If you want, you could think of further interest decompositions. It could even be that the time-premium is itself determined by other factors (such as your preference between consuming today and consuming next year, the inflation rate, taxes, or other issues, that we are brushing over). Then there would be a liquidity premium, an extra interest rate that a lender would demand if the bond could not easily be sold—resale is much easier with Treasury bonds.The risk premium itself depends on such strange concepts as the correlation of loan default with the general economy. We can preview the relative importance of these components for you in the context of corporate bonds. The highest-quality bonds are called investment-grade. A typical such bond may promise about 6% per annum, 150 to 200 basis points above the equivalent Treasury. The probability of default would be small—less than 3% in total over a ten-year horizon (0.3% per annum). When an investment-grade bond does default, it still returns about 75% of what it promised. For such bonds, the risk premium would be small—a reasonable estimate would be that only about 10 to 20 basis points of the 200 basis point spread is the risk premium. The quoted interest rate of 6% per annum therefore would reflect first the time premium, then the default premium, and only then a small risk premium. (In fact, the liquidity premium would probably be more important than the risk premium.) For low-quality corporate bonds, however, the risk premium can be important. Ed Altman has been collecting corporate bond statistics since the 1970s. In an average year, about 3.5% to 5.5% of low-grade corporate bonds defaulted. But in recessions, the default rate shot up to 10% per year, and in booms it dropped to 1.5% per year. The average value of a bond after default was only about 40 cents on the dollar, though it was as low 25 cents in recessions and as high as 50 cents in booms. Altman then computes that the most risky corporate bonds promised a spread of about 5%/year above the 10-Year Treasury bond, but ultimately delivered a spread of only about 2.2%/year. 280 points are therefore the default premium. The remaining 220 basis points contain both the liquidity premium and the risk premium—perhaps in roughly equal parts.

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Risk Neutrality (and Risk Aversion Preview)

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Fortunately, the expected value is all that we need to learn about statistics until we will get to Part III (investments). This is because we are assuming—only for learning purposes—that everyone is risk-neutral. Essentially, this means that investors are willing to write or take any fair bet. For example, you would be indifferent between getting $1 for sure, and getting either $0 or $2 with 50% probability. And you would be indifferent between earning 10% from a risk-free bond, and earning either 0% or 20% from a risky bond. You have no preference between investments with equal expected values, no matter how safe or uncertain they may be. If, instead, you were risk-averse—which you probably are in the real world—you would not want to invest in the more risky alternative if both the risky and safe alternative offered the same expected rate of return. You would prefer the safe $1 to the unsafe $0 or $2 investment. You would prefer the 10% risk-free Treasury bond to the unsafe corporate bond that would pay either 0% or 20%. In this case, if I wanted to sell you a risky project or bond, I would have to offer you a higher rate of return as risk compensation. I might have to pay you, say, 5 cents to get you to be willing to accept the project that pays off $0 or $2 if you can instead earn $1. Or, I would have to lower the price of my corporate bond, so that it offers you a higher expected rate of return, say, 1% and 21% instead of 0% and 20%. Would you really worry about a bet for either +$1 or −$1? Probably not. For small bets, you

are probably close to risk-neutral—I may not have to offer even one cent to take this bet. But what about a bet for plus or minus $100? Or for plus and minus $10,000? My guess is that you would be fairly reluctant to accept the latter bet without getting extra compensation. For large bets, you are probably fairly risk-averse—I would have to offer you several hundred dollars to take this bet. However, your own personal risk aversion is not what matters in financial markets. Instead, it is an aggregate risk aversion. For example, if you could share the $10,000 bet with 100 other students in your class, the bet would be only $100 for you. And some of your colleagues may be willing to accept even more risk for less extra money—they may have healthier bank accounts or wealthier parents. If you could lay bets across many investors, the effective risk aversion would therefore be less. And this is exactly how financial markets work: the aggregate risk absorption capability is considerably higher than that of any individual. In effect, financial markets are less risk averse than individuals. 

We will study risk aversion in the investments part of the blog. There, we will also need to define good measures of risk, a subject we can avoid here. But, as always, all tools we learn under the simpler scenario (risk-neutrality) will remain applicable under the more complex scenario (risk-aversion).

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Why is the Yield Curve not Flat?

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There is no necessary reason why capital should be equally productive at all times. For example, in agrarian societies, capital could be very productive in summer (and earn a rate of return of 3%), but not in winter (and earn a rate of return of only 1%). This does not mean that investment in summer is a better deal or a worse deal than investment in winter, because cash in winter is not the same—not as valuable—as cash in summer, so the two interest rates are not comparable. You could not invest winter money at the 3% interest rate you will be able to invest it with 6 months later. 

But although seasonal effects do influence both prices and rates of return on agricultural com-modities, and although the season example makes it clear that capital can be differently produc- tive at different times, it is not likely that seasonality is the reason why 30-year Treasury bonds in May 2002 paid 5.6% per annum, and 6-month Treasury notes paid only 1.9% per annum. So why is it that the yield curve was so steep? There are essentially three explanations: 

1. The 30-year bond is a much better deal than the 1-year bond. This explanation is highly unlikely. The market for Treasury bond investments is close to perfect, in the sense that we have used the definition. It is very competitive and efficient. If there was a great deal to be had, thousands of traders would have already jumped on it. So, more likely, the interest rate differential does not overthrow the old tried-and-true axiom: you get what you pay for. It is just a fact of life that investments for which the interest payments are tied down for 30 years must offer higher interest rates now. 

It is important that you recognize that your cash itself is not tied down if you invest in a 30-year bond, because you can of course sell your 30-year bond tomorrow to another investor if you so desire. 

2. Investors expect to be able to earn much higher interest rates in the future. For example, if the interest rate r0,1 is 2% and the interest rate r1,2 is 10%, then r0,2 = (1 + 2%) · (1 + 10%) ≈ 1 + 12%, or r2 = 5.9%. If you graph rT against T , you will find a steep yield curve, just as you observed. So, higher future interest rates can cause much steeper yield curves. 

However, I am cheating. This explanation is really no different from my “seasons” explanation, because I have given you no good explanation why investment opportunities were expected to be much better in May 2032 than they were in May 2002. I would need to give you an underlying reason. One particular such reason may be that investors believe that money will be worth progressively less. That is, even though they can earn higher interest rates over the long run, they also believe that the price inflation rate will increase. Inflation erodes the value of higher interest rates, so interest rates may have to be higher in the future merely to compensate investors for the lesser value of their money in the future.

However, the empirical evidence suggests that the yield curve is not a good predictor of future interest rates, except on the very shortest horizons (a month or less). So, the expectation of higher interest rates is not the most likely cause for the usually upward sloping curve in the real world. 

3. Long-term bonds might somehow be riskier than short-term bonds, so investors only want to buy them if they get an extra rate of return. Although we have yet to cover uncertainty more systematically, you can gain some intuition by considering the effects of changes in economy-wide interest rates on short-term bonds vs. long-term bonds. 

The empirical evidence indeed suggests that it is primarily compensation for taking more risk with long-term bonds than short-term bonds that explains why long-term bonds have higher yields than short-term bonds. That is, investors seem to earn higher expected rates of return on average in long-term bonds, because these bonds are riskier (at least in the interim).

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Investments, Projects, and Firms

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As far as finance is concerned, every project is a set of flows of money (cash flows). Most projects require an upfront cash outflow (an investment or expense or cost) and are followed by a series of later cash inflows (payoffs or revenues or returns). It does not matter whether the cash flows come from garbage hauling or diamond sales. Cash is cash. However, it is important that all costs and benefits are included as cash values. If you would have to spend more time to haul trash, or merely find it more distasteful than other projects, then you would have to translate these project features into equivalent cash negatives. Similarly, if you want to do a project “for the fun of it,” you must translate your “fun” into a cash positive. The discipline of finance takes over after all positives and negatives (inflows and outflows) from the project “black box” have been translated into their appropriate monetary cash values. 

This does not mean that the operations of the firm are unimportant—things like revenues, operations, inventory, marketing, payables, working capital, competition, etc. These business factors are all of the utmost importance in making the cash flows happen, and a good (financial) manager must understand these. After all, even if all you care about is cash flows, it is impossible to understand them well if you have no idea where they come from and how they can change in the future. 

Projects need not be physical. For example, a company may have a project called “customer relations,” with real cash outflows today and uncertain future inflows. You (a student) are a project: you pay for education and will earn a salary in the future. In addition, some of the payoffs from education are metaphysical rather than physical. If knowledge provides you with pleasure, either today or in the future, education yields a value that should be regarded as a positive cash flow. Of course, for some students, the distaste of learning should be factored in as a cost (equivalent cash outflow)—but I trust that you are not one of them. All such non-financial flows must be appropriately translated into cash equivalents if you want to arrive at a good project valuation! 

A firm can be viewed as just a collection of projects. Similarly, so can a family. Your family  may own a house, a car, tuition payments, education investments, etc.,—a collection of projects. This blog assumes that the value of a firm is the value of all its projects’ net cash flows, and nothing else. It is now your goal to learn how to determine these projects’ values, given cash flows. 

There are two important specific kinds of projects that you may consider investing in—bonds and stocks, also called debt and equity. As you will learn later, in a sense, the stock is the equivalent of investing to become an owner who is exposed to a lot of risk, while the bond is the equivalent of a lending money, an investment which is usually less risky. Together, if you own all outstanding bonds (and loans) and stock in a company, you own the firm: 

Entire Firm = All Outstanding Stocks + All Outstanding Bonds and Loans . 

This sum is sometimes called the enterprise value. Our blog will spend a lot of time discussing these two forms of financing—but for now, you can consider both of them just investment projects: you put money in, and they pay money out. For many stock and bond investments that you can buy and sell in the financial markets, we believe that most investors enjoy very few, if any, non-cash based benefits.

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