economics_of_money_banking__financial_markets
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11 Appendix 1 to Chapter 5
To see that securities should be priced so that their expected return-beta combination should lie on the security market line, consider a security like S in Figure 2, which is below the security market line. If an investor makes an investment in which half is put into the market portfolio and half into a risk-free loan, then the beta of this investment will be 0.5, the same as security S. However, this investment will have an expected return on the security market line, which is greater than that for security S. Hence investors will not want to hold security S and its current price will fall, thus raising its expected return until it equals the amount indicated on the security market line. On the other hand, suppose there is a security like T which has a beta of 0.5 but whose expected return is above the security market line. By including this security in a well-diversified portfolio with other assets with a beta of 0.5, none of which can have an expected return less than that indicated by the security line (as we have shown), investors can obtain a portfolio with a higher expected return than that obtained by putting half into a risk-free loan and half into the market portfolio. This would mean that all investors would want to hold more of security T, and so its price would rise, thus lowering its expected return until it equaled the amount indicated on the security market line.
The capital asset pricing model formalizes the following important idea: An asset should be priced so that is has a higher expected return not when it has a greater risk in isolation, but rather when its systematic risk is greater.
Arbitrage Pricing Theory
Although the capital asset pricing model has proved to be very useful in practice, deriving it does require the adoption of some unrealistic assumptions; for example, the assumption that investors can borrow and lend freely at the risk-free rate, or the assumption that all investors have the same assessment of expected returns and standard deviations of returns for all assets. An important alternative to the capital asset pricing model is the arbitrage pricing theory (APT) developed by Stephen Ross of M.I.T.
In contrast to CAPM, which has only one source of systematic risk, the market return, APT takes the view that there can be several sources of systematic risk in the economy that cannot be eliminated through diversification. These sources of risk can be thought of as factors that may be related to such items as inflation, aggregate output, default risk premiums, and/or the term structure of interest rates. The return on an asset i can thus be written as being made up of components that move with these factors and a random component that is unique to the asset ( i):
Ri 1i (factor 1) 2i (factor 2) … ki (factor k) i |
(10) |
Since there are k factors, this model is called a k-factor model. The 1i ,…, ki describe the sensitivity of the asset iÕs return to each of these factors.
Just as in the capital asset pricing model, these systematic sources of risk should
be priced. The market price for each factor j can be thought of as E(Rfactor j ) Rf, and hence the expected return on a security can be written as:
E(Ri) Rf 1i [E(Rfactor 1) Rf] … ki [E(Rfactor k) Rf] |
(11) |
Models of Asset Pricing 12
This asset pricing equation indicates that all the securities should have the same market price for the risk contributed by each factor. If the expected return for a security were above the amount indicated by the APT pricing equation, then it would provide a higher expected return than a portfolio of other securities with the same average sensitivity to each factor. Hence investors would want to hold more of this security and its price would rise until the expected return fell to the value indicated by the APT pricing equation. On the other hand, if the securityÕs expected return were less than the amount indicated by the APT pricing equation, then no one would want to hold this security, because a higher expected return could be obtained with a portfolio of securities with the same average sensitivity to each factor. As a result, the price of the security would fall until its expected return rose to the value indicated by the APT equation.
As this brief outline of arbitrage pricing theory indicates, the theory supports a basic conclusion from the capital asset pricing model: An asset should be priced so that it has a higher expected return not when it has a greater risk in isolation, but rather when its systematic risk is greater. There is still substantial controversy about whether a variant of the capital asset pricing model or the arbitrage pricing theory is a better description of reality. At the present time, both frameworks are considered valuable tools for understanding how risk affects the prices of assets.
appendix 2
to chapter
5
Applying the Asset Market
Approach to a Commodity
Market: The Case of Gold
Both models of interest-rate determination in Chapter 4 make use of an asset market approach in which supply and demand are always considered in terms of stocks of assets (amounts at a given point in time). The asset market approach is useful in understanding not only why interest rates fluctuate but also how any assetÕs price is determined.
One asset that has fascinated people for thousands of years is gold. It has been a driving force in history: The conquest of the Americas by Europeans was to a great extent the result of the quest for gold, to cite just one example. The fascination with gold continues to the present day, and developments in the gold market are followed closely by financial analysts and the media. This appendix shows how the asset market approach can be applied to understanding the behavior of commodity markets, in particular the gold market. (The analysis in this appendix can also be used to understand behavior in many other asset markets.)
Supply and Demand in the Gold Market
Demand Curve
The analysis of a commodity market, such as the gold market, proceeds in a similar fashion to the analysis of the bond market by examining the supply of and demand for the commodity. We again use our analysis of the determinants of asset demand to obtain a demand curve for gold, which shows the relationship between the quantity of gold demanded and the price when all other economic variables are held constant.
To derive the relationship between the quantity of gold demanded and its price, we again recognize that an important determinant of the quantity demanded is its expected return:
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where |
Re expected return |
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price of gold today |
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expected price of gold next year |
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expected capital gain |
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In deriving the demand curve, we hold all other variables constant, particularly the expected price of gold next year Pet 1. With a given value of the expected price of gold next year Pet 1, a lower price of gold today Pt means that there will be a greater
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Supply Curve
Market
Equilibrium
Applying the Asset Market Approach to a Commodity Market: The Case of Gold 2
appreciation in the price of gold over the coming year. The result is that a lower price of gold today implies a higher expected capital gain over the coming year and hence a higher expected return: Re (Pet 1 Pt)/Pt. Thus because the price of gold today (which for simplicity we will denote as P) is lower, the expected return on gold is higher, and the quantity demanded is higher. Consequently, the demand curve Gd1 slopes downward in Figure 1.
To derive the supply curve, expressing the relationship between the quantity supplied and the price, we again assume that all other economic variables are held constant. A higher price of gold will induce producers to mine for extra gold and also possibly induce governments to sell some of their gold stocks to the public, thus increasing the quantity supplied. Hence the supply curve Gs1 in Figure 1 slopes upward. Notice that the supply curve in the figure is drawn to be very steep. The reason for this is that the actual amount of gold produced in any year is only a tiny fraction of the outstanding stock of gold that has been accumulated over hundreds of years. Thus the increase in the quantity of the gold supplied in response to a higher price is only a small fraction of the stock of gold, resulting in a very steep supply curve.
Market equilibrium in the gold market occurs when the quantity of gold demanded equals the quantity of gold supplied:
Gd Gs
With the initial demand and supply curves of G d1 and G s1, equilibrium occurs at point 1, where these curves intersect at a gold price of P1. At a price above this
F I G U R E 1 A Change in the
Equilibrium Price of Gold
When the demand curve shifts rightward from G1d to G d2—say, because expected inflation rises—equilibrium moves from point 1 to point 2, and the equilibrium price of gold rises from P1 to P2.
Price of Gold P |
Gs |
1 |
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P2 |
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P1 |
Gd |
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Gd |
1 |
Quantity of Gold G
3 Appendix 2 to Chapter 5
equilibrium, the amount of gold supplied exceeds the amount demanded, and this condition of excess supply leads to a decline in the gold price until it reaches P1, the equilibrium price. Similarly, if the price is below P1, there is excess demand for gold, which drives the price upward until it settles at the equilibrium price P1.
Changes in the Equilibrium Price of Gold
Shift in the Demand Curve for Gold
Changes in the equilibrium price of gold occur when there is a shift in either the supply curve or the demand curve; that is, when the quantity demanded or supplied changes at each given price of gold in response to a change in some factor other than todayÕs gold price.
Our analysis of the determinants of asset demand in the chapter provides the factors that shift the demand curve for gold: wealth, expected return on gold relative to alternative assets, riskiness of gold relative to alternative assets, and liquidity of gold relative to alternative assets. The analysis of how changes in each of these factors shift the demand curve for gold is the same as that found in the chapter.
When wealth rises, at a given price of gold, the quantity demanded increases, and the demand curve shifts to the right, as in Figure 1. When the expected return on gold relative to other assets risesÑeither because speculators think that the future price of gold will be higher or because the expected return on other assets declinesÑgold becomes more desirable; the quantity demanded therefore increases at any given price of gold, and the demand curve shifts to the right, as in Figure 1. When the relative riskiness of gold declines, either because gold prices become less volatile or because returns on other assets become more volatile, gold becomes more desirable, the quantity demanded at every given price rises, and the demand curve again shifts to the right. When the gold market becomes relatively more liquid and gold therefore becomes more desirable, the quantity demanded at any given price rises, and the demand curve also shifts to the right, as in Figure 1.
Shifts in the Supply Curve for Gold
Study Guide
The supply curve for gold shifts when there are changes in technology that make gold mining more efficient or when governments at any given price of gold decide to increase sales of their holdings of gold. In these cases, the quantity of gold supplied at any given price increases, and the supply curve shifts to the right.
To give yourself practice with supply and demand analysis in the gold market, see if you can analyze what happens to the price of gold for the following situations, remembering that all other things are held constant: 1) Interest rates rise, 2) the gold market becomes more liquid, 3) the volatility of gold prices increases, 4) the stock market is expected to turn bullish in the near future, 5) investors suddenly become fearful that there will be a collapse in real estate prices, and 6) Russia sells a lot of gold in the open market to raise hard currency to finance expenditures.
Applying the Asset Market Approach to a Commodity Market: The Case of Gold 4
Application
Changes in the Equilibrium Price of Gold Due to a Rise in
Expected Inflation
To illustrate how changes in the equilibrium price of gold occur when supply and demand curves shift, letÕs look at what happens when there is a change in expected inflation.
Suppose that expected inflation is 5% and the initial supply and demand curves are at G1s and G1d so that the equilibrium price of gold is at P1 in Figure 1. If expected inflation now rises to 10%, prices of goods and commodities next year will be expected to be higher than they otherwise would have been,
and the price of gold next year Pte 1 will also be expected to be higher than otherwise. Now at any given price of gold today, gold is expected to have a greater rate of appreciation over the coming year and hence a higher expected capital gain and return. The greater expected return means that the quantity of gold demanded increases at any given price, thus shifting the demand curve from Gd1 to Gd2. Equilibrium therefore moves from point 1 to point 2, and the price of gold rises from P1 to P2.
By using a supply and demand diagram like that in Figure 1, you should be able to see that if the expected rate of inflation falls, the price of gold today will also fall. We thus reach the following conclusion: The price of gold should be positively related to the expected inflation rate.
Because the gold market responds immediately to any changes in expected inflation, it is considered a good barometer of the trend of inflation in the future. Indeed, Alan Greenspan, the chairman of the Board of Governors of the Federal Reserve System, at one point advocated using the price of gold as an indicator of inflationary pressures in the economy. Not surprisingly, then, the gold market is followed closely by financial analysts and monetary policymakers.
C h a p t e r |
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6 |
The Risk and Term Structure |
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of Interest Rates |
PREVIEW
In our supply and demand analysis of interest-rate behavior in Chapter 5, we examined the determination of just one interest rate. Yet we saw earlier that there are enormous numbers of bonds on which the interest rates can and do differ. In this chapter, we complete the interest-rate picture by examining the relationship of the various interest rates to one another. Understanding why they differ from bond to bond can help businesses, banks, insurance companies, and private investors decide which bonds to purchase as investments and which ones to sell.
We first look at why bonds with the same term to maturity have different interest rates. The relationship among these interest rates is called the risk structure of interest rates, although risk, liquidity, and income tax rules all play a role in determining the risk structure. A bondÕs term to maturity also affects its interest rate, and the relationship among interest rates on bonds with different terms to maturity is called the term structure of interest rates. In this chapter, we examine the sources and causes of fluctuations in interest rates relative to one another and look at a number of theories that explain these fluctuations.
Risk Structure of Interest Rates
Default Risk
Figure 1 shows the yields to maturity for several categories of long-term bonds from 1919 to 2002. It shows us two important features of interest-rate behavior for bonds of the same maturity: Interest rates on different categories of bonds differ from one another in any given year, and the spread (or difference) between the interest rates varies over time. The interest rates on municipal bonds, for example, are above those on U.S. government (Treasury) bonds in the late 1930s but lower thereafter. In addition, the spread between the interest rates on Baa corporate bonds (riskier than Aaa corporate bonds) and U.S. government bonds is very large during the Great Depression years 1930Ð1933, is smaller during the 1940sÐ1960s, and then widens again afterwards. What factors are responsible for these phenomena?
One attribute of a bond that influences its interest rate is its risk of default, which occurs when the issuer of the bond is unable or unwilling to make interest payments when promised or pay off the face value when the bond matures. A corporation suffering big losses, such as Chrysler Corporation did in the 1970s, might be more likely
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C H A P T E R 6 The Risk and Term Structure of Interest Rates |
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Annual Yield (%)
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F I G U R E 1 Long-Term Bond Yields, 1919–2002
Sources: Board of Governors of the Federal Reserve System, Banking and Monetary Statistics, 1941Ð1970; Federal Reserve: www.federalreserve.gov/releases/h15/data/.
www.federalreserve.gov /Releases/h15/update/
The Federal Reserve reports the returns on different quality bonds. Look at the bottom of the listing of interest rates for AAA and BBB rated bonds.
to suspend interest payments on its bonds.1 The default risk on its bonds would therefore be quite high. By contrast, U.S. Treasury bonds have usually been considered to have no default risk because the federal government can always increase taxes to pay off its obligations. Bonds like these with no default risk are called default-free bonds. (However, during the budget negotiations in Congress in 1995 and 1996, the Republicans threatened to let Treasury bonds default, and this had an impact on the bond market, as one application following this section indicates.) The spread between the interest rates on bonds with default risk and default-free bonds, called the risk premium, indicates how much additional interest people must earn in order to be willing to hold that risky bond. Our supply and demand analysis of the bond market in Chapter 5 can be used to explain why a bond with default risk always has a positive risk premium and why the higher the default risk is, the larger the risk premium will be.
To examine the effect of default risk on interest rates, let us look at the supply and demand diagrams for the default-free (U.S. Treasury) and corporate long-term bond markets in Figure 2. To make the diagrams somewhat easier to read, letÕs assume that initially corporate bonds have the same default risk as U.S. Treasury bonds. In this case, these two bonds have the same attributes (identical risk and maturity); their equilibrium prices and interest rates will initially be equal (P c1 P T1 and i c1 i T1 ), and the risk premium on corporate bonds (i c1 i T1 ) will be zero.
1Chrysler did not default on its loans in this period, but it would have were it not for a government bailout plan intended to preserve jobs, which in effect provided Chrysler with funds that were used to pay off creditors.
122 P A R T I I Financial Markets
Price of Bonds, P
(P increases ↑)
P c1
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Interest Rate, i |
Price of Bonds, P |
Interest Rate, i |
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F I G U R E 2 Response to an Increase in Default Risk on Corporate Bonds
An increase in default risk on corporate bonds shifts the demand curve from D c1 to D c2. Simultaneously, it shifts the demand curve for Treasury bonds from D T1 to D T2. The equilibrium price for corporate bonds (left axis) falls from P c1 to P c2, and the equilibrium interest rate on corporate bonds (right axis) rises from i c1 to i c2. In the Treasury market, the equilibrium bond price rises from P T1 to P T2, and the equilibrium interest rate falls from i T1 to i T2. The brace indicates the difference between i c2 and i T2, the risk premium on corporate bonds. (Note: P and i increase in opposite directions. P on the left vertical axis increases as we go up the axis, while i on the right vertical axis increases as we go down the axis.)
Study Guide |
Two exercises will help you gain a better understanding of the risk structure: |
1.Put yourself in the shoes of an investorÑsee how your purchase decision would be affected by changes in risk and liquidity.
2.Practice drawing the appropriate shifts in the supply and demand curves when risk and liquidity change. For example, see if you can draw the appropriate shifts in the supply and demand curves when, in contrast to the examples in the text, a corporate bond has a decline in default risk or an improvement in its liquidity.
If the possibility of a default increases because a corporation begins to suffer large losses, the default risk on corporate bonds will increase, and the expected return on these bonds will decrease. In addition, the corporate bondÕs return will be more uncertain as well. The theory of asset demand predicts that because the expected return on the corporate bond falls relative to the expected return on the default-free Treasury bond while its relative riskiness rises, the corporate bond is less desirable (holding everything else equal), and demand for it will fall. The demand curve for corporate bonds in panel (a) of Figure 2 then shifts to the left, from D c1 to D c2.
At the same time, the expected return on default-free Treasury bonds increases relative to the expected return on corporate bonds, while their relative riskiness
C H A P T E R 6 The Risk and Term Structure of Interest Rates |
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declines. The Treasury bonds thus become more desirable, and demand rises, as shown in panel (b) by the rightward shift in the demand curve for these bonds from
D T1 to D T2.
As we can see in Figure 2, the equilibrium price for corporate bonds (left axis) falls from P c1 to P c2, and since the bond price is negatively related to the interest rate, the equilibrium interest rate on corporate bonds (right axis) rises from i c1 to i c2. At the same time, however, the equilibrium price for the Treasury bonds rises from P T1 to P T2, and the equilibrium interest rate falls from i T1 to i T2. The spread between the interest rates on corporate and default-free bondsÑthat is, the risk premium on corporate bondsÑhas risen from zero to i c2 i T2. We can now conclude that a bond with default risk will always have a positive risk premium, and an increase in its default risk will raise the risk premium.
Because default risk is so important to the size of the risk premium, purchasers of bonds need to know whether a corporation is likely to default on its bonds. Two major investment advisory firms, MoodyÕs Investors Service and Standard and PoorÕs Corporation, provide default risk information by rating the quality of corporate and municipal bonds in terms of the probability of default. The ratings and their description are contained in Table 1. Bonds with relatively low risk of default are called investment-grade securities and have a rating of Baa (or BBB) and above. Bonds with
Table 1 Bond Ratings by Moody’s and Standard and Poor’s
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Examples of Corporations with |
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Descriptions |
Bonds Outstanding in 2003 |
Aaa |
AAA |
Highest quality |
General Electric, Pfizer Inc., |
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(lowest default risk) |
North Carolina State, |
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Mobil Oil |
Aa |
AA |
High quality |
Wal-Mart, McDonaldÕs, |
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Credit Suisse First Boston |
A |
A |
Upper medium grade |
Hewlett-Packard, |
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Anheuser-Busch, |
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Ford, Household Finance |
Baa |
BBB |
Medium grade |
Motorola, AlbertsonÕs, Pennzoil, |
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Weyerhaeuser Co., |
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Tommy Hilfiger |
Ba |
BB |
Lower medium grade |
Royal Caribbean, Levi Strauss |
B |
B |
Speculative |
Rite Aid, Northwest Airlines Inc., |
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Six Flags |
Caa |
CCC, CC |
Poor (high default risk) |
Revlon, United Airlines |
Ca |
C |
Highly speculative |
US Airways, Polaroid |
C |
D |
Lowest grade |
Enron, Oakwood Homes |
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