<p> CHAPTER 6</p><p>THE STRUCTURE OF INTEREST RATES</p><p>CHAPTER OVERVIEW AND LEARNING OBJECTIVES</p><p> This chapter discusses the relationship between security-specific factors and the interest rates on their debt securities. It builds on the material of Chapter 4 which discusses how market-based factors such as real rates and the inflation rates affect the level of interest rates in the economy.  The chapter explains the relationship between term to maturity and interest rates and extends the discussion into the role of the yield curve in the business cycle. Three theories that seek to explain the shape of the yield cure are identified.</p><p> Additionally, other security-specific factors such as 1) default risk, 2) tax treatment, 3) marketability, and 4) options on debt securities – call, put, or conversion – also affect the yield of a given security. The impact of these factors is analyzed here.</p><p> While there are many other variables, these factors affect the yield of a security more significantly than others. It is important to not only know what these factors are but also how changes in these factors influence the yield of a security.</p><p>CAREER PLANNING NOTE: KEEP UP WITH CURRENT EVENTS</p><p>The last three chapters and the many to come are discussing and describing what happens each working day in financial markets and institutions. Your text and this Study Guide give you a good understanding of what goes on in the financial markets. Many finance and economics graduates take the first step of their working career in a financial institution. All business firms need to understand what the economy is like and how their firm may be affected by changes in interest rates.</p><p>Finance is alive, dynamic, and happening each moment of the day and night! Fortunes are made and lost; and the employment opportunities in the field are unlimited. How can you keep in touch? Pick up and read the daily Wall Street Journal (WSJ) or business periodicals like Business Week, Forbes, Fortune, or Smart Money. You are also not restricted these days to printed material. Get on the web. Visit finance related sites like CNNMoney, Yahoo Finance or Bloomberg. As many of you spend time on the computer, a few minutes of scanning these websites for articles and news of interest will soon prepare you for your classes and the future as well.</p><p>If you are interested in a particular area of finance, such as banking, or any other industry, such as housing or chemicals, be aware of the several monthly or daily periodicals that you should be reading. Anticipate the economic conditions of "your" industry in the next two or three years, when you will be looking for a job and beginning your career. If you think you are studying now, wait until then! Meanwhile, get ready! Read!</p><p>FINDING IT IN THE WALL STREET JOURNAL 56 There are many "look it up" assignments in this Study Guide. Your time on the assignment should be spent on the assigned material and not thumbing page by page through the paper. The WSJ has three tables of contents to assist you. The first is found at the bottom of the front page. The left-hand section on the front page titled “What’s News” makes references to major feature articles appearing that day in all sections of the paper. The second table of contents, titled "Index To Businesses," is located on the first inside page of Section B. All firms with significant references in that day's WSJ are listed, along with the page of the article. If you are following a firm for an investment project or employment opportunities, a brief review of this listing can save considerable time and keep you "tuned in" at the same time. The third table of contents is located at the bottom-left of front page of Section C, titled "Money & Investing Index." All the regular and frequently appearing columns and tables are listed. Use these tables of contents for more effective use of your precious time. If you are subscribing to Journal, the Interactive Edition is also available to you. The Interactive Edition includes an opportunity to create a "Personal Journal" set up so that specific industry and company information will be compiled in your Personal Journal for your review.</p><p>One table of information in the third section that is of relevance to this chapter is the table titled “Bonds”. This table contains the daily prices and yields of U.S. and foreign government bonds as well as U.S. corporate bonds.You can use this information to plot the yield curve on any day. </p><p>TOPIC OUTLINE AND KEY TERMS</p><p>I. Interest rate changes and differences between interest rates can be explained by several variables.</p><p>A. Term to Maturity B. Default Risk C. Tax Treatment D. Marketability E. Callability</p><p>II. Term Structure of Interest Rates </p><p>A. Relationship between interest rates and term-to-maturity. Term structure may be studied visually by plotting a yield curve at a point in time.</p><p>1. A graphical plot of yield vs. maturity for securities that are similar in all other aspects is called the yield curve. 2. A yield curve is a smooth line which shows the relationship between maturity and a security's yield at a point in time. 3. The yield curve may be ascending, flat, descending, or twisted. 4. Several theories explain the shape of the yield curve. The three main theories that are discussed in this chapter are the expectations theory, the liquidity premium theory, and the market segmentation theory.</p><p>57 B. The Expectations Theory</p><p>1. The expectations theory argues that the shape of the yield curve is determined solely by expectations of future interest rate movements and changes in these expectations lead to changes in the shape of the yield curve. </p><p>2. Investors are assumed to trade in a very efficient market with excellent information and minimal trading costs. Other theories discussed later presume less efficient markets.</p><p>3. The slope of the yield curve reflects investors' expectations about future interest rates. a. An ascending yield curve is formed when interest rates increase with maturity. Such yield curves are also called Normal Yield Curves. According to the expectations theory, an ascending yield curve reflects expectations of increasing interest rates in the future. b. Descending yield curves imply that short-term rates are higher than long-term rates. These curves are inverted yield curves and reflect expectations of lower interest rates in the future. c. A flat yield curve implies that interest rates are expected to be stable in the near future.</p><p>4. Long-term interest rates represent the geometric average of current and expected future (implied, forward) interest rates. Forward rates (which are expected future rates) may be calculated by observing two spot interest rates of differing maturity.</p><p>C. Liquidity Premium Theory </p><p>1. Long-term securities have (a) greater price variability and (b) less marketability than short-term securities. Consequently, investors require a premium for investing in less liquid securities.</p><p>2. Liquidity may be a more critical factor to investors at one point in time relative to another. As a result, liquidity premiums change over time!</p><p>4. Since the liquidity premium increases with maturity, it causes the observed market yield curve to be more upward sloping than that predicted by the expectations theory.</p><p>5. The liquidity premium suggests a more upward sloping yield curve than that predicted by the expectations theory. It means that the liquidity premium theory can explain why the yield curve slopes upward most of the time.</p><p>D. Market Segmentation and Preferred Habitat Theories of the Term Structure </p><p>Maturity preferences may affect security prices, explaining variations in yields by time.</p><p>1. Investors (lenders) and borrowers (issuers) choose securities with maturities that satisfy their forecasted cash needs. As a result, the yield curve is determined by the supply of 58 and the demand for securities at or near a particular maturity.</p><p>2. The choice of short-term vs. long-term maturities is pre-determined according to need rather than expectations of future interest rates. a Pension funds and life insurance companies generally prefer long-term investments that match their long-term liabilities. Commercial banks may prefer short-term investments to coincide with their short- term liabilities. b Borrowers will also issue securities with maturities that match their needs.</p><p>3. Under this theory, investors will not shift out of their preferred maturities even if offered higher yields.</p><p>4. A variation of the market segmentation theory is the preferred habitat theory. Under this theory, investors may be willing to move out of their preferred maturity in return for a premium if expectations of yields dictate they do so. </p><p>5. The segmentation theory can explain twists, spikes, and discontinuities in the yield curve, while the preferred habitat theory explains why the yield curve is usually a smooth line without discontinuities.</p><p>E. Which theory is right?</p><p>1. The expectations and liquidity premium theories are important long-term influences on term structure and find support among economists. 2. Short-term analysis of yield curves supports the preferred habitat theory and is favored by market participants.</p><p>F. Yield Curves and the Business Cycle</p><p>1. Interest rates are directly related to the level of economic activity. a. An ascending yield curve notes the market expectations of economic expansion and/or inflation. b. A descending yield curve forecasts lower rates possibly related to slower economic growth or lower inflation rates. 2. Security markets respond to updated new information and expectations and reflect their reactions in security prices and yields.</p><p>G. Yield Curves and Financial Intermediaries (FI)</p><p>1. The current and expected slope of the yield curve is important to FIs that bear maturity intermediation risk. 2. Depository institutions traditionally did well in periods with positively sloped yield curves - borrow short (deposits) and lend long (mortgages) at a higher rate.</p><p>III. Default risk: Differences in interest rates may be explained by relative levels of default risk.</p><p>59 A. Default risk is the probability of the borrower not honoring the security contract.</p><p>1. Losses may range from "interest a few days late" to a complete loss of principal.</p><p>2. Risk-averse investors want adequate compensation for expected default losses.</p><p>B. Investors require a default risk premium (above risk-free or less risky securities) for added risk assumed.</p><p>1. DRP = i - irf 2. The default risk premium (DRP) is the difference between the promised or nominal rate and the yield on a comparable (same term) risk-free security (Treasury security). 3. Investors are satisfied if the default risk premium is equal to the expected default loss.</p><p>C. Default risk premiums increase (widen) in periods of recession and decrease in economic expansion.</p><p>1. In good times, risky security prices are bid up; yields move nearer those of risk- free securities. 2. With increased economic pessimism, investors sell risky securities and buy "quality," widening the DRP.</p><p>D. Credit rating agencies measure and grade relative default risk among DSUs and their securities.</p><p>1. Cash flow, level of debt, profitability, and variability of earnings are indicators of default riskiness. 2. As conditions change, rating agencies alter ratings of businesses and governmental debtors. 3. Bonds in the top four rating categories – Aaa to Baa (Moody’s) or AAA to BBB (S&Ps) – are called investment grade bonds. Bonds rated below Baa or BBB are called speculative grade ( or junk) bonds.</p><p>IV. Tax Treatment: The taxation of security gains and income affects the yield differences among securities. Investors are interested in their after-tax return on investments.</p><p>A. The after-tax return, iat, is found by multiplying the pre-tax return by one minus the investor’s marginal tax rate.</p><p> iat = ibt (1-t)</p><p>B. Municipal bonds issued by state and local governments are usually exempt from federal income taxes.</p><p>C. Treasury securities are exempt from state income taxes.</p><p>60 V. Differences in marketability affect interest yields.</p><p>A. Marketability - The costs and speed with which investors can resell a security.</p><p>1. Cost of trade 2. Physical transfer cost 3. Search costs 4. Information costs</p><p>B. Securities with good marketability have higher prices (and demand) and lower yields. This is because higher marketability lowers the risk to investors.</p><p>VI. Varied option provisions may explain yield differences between securities.</p><p>A. An option is a contract provision which gives the holder the right, but not the obligation, to buy, sell, or convert an asset at some specified price within a defined future time period.</p><p>B. A call option permits the issuer (borrower) to call (redeem) the obligation before maturity.</p><p>1. Borrowers will "call" if interest rates decline. 2. Investors in callable securities bear the risk of losing their high-yielding security. 3. With increased call risk, investors demand a call interest premium (CIP).</p><p> a. CIP = ic - inc b. A callable bond, ic, will be priced to yield a higher return (by the CIP) than a similar non-callable, inc bond.</p><p>C. A put option permits the investor (lender) to sell the bond or terminate the contract at a designated price before maturity.</p><p>1. Investors are likely to "put" their security or loan back to the borrower during periods of increasing interest rates. The difference in interest rates between putable and non-putable contracts is called the put interest discount (PID).</p><p> a. PID = ip - inp b. The yield on a putable bond, ip, will be lower than the yield on a similar non-putable bond, inp, by the PIP.</p><p>D. A conversion option permits the investor to convert a security contract into another security, usually common stock.</p><p>1. Convertible bonds generally have lower yields, icon, than non-convertibles, incon. 2. The conversion yield discount (CYD) is the difference between the yields on convertibles relative to non convertibles.</p><p>3. CYD = icon - incon. Investors accept the lower yield on convertible bonds because they have an opportunity for increased rates of return through conversion.</p><p>61 VII. Co-movement of Interest Rates - Interest rates tend to move up and down together.</p><p>A. Securities of varying maturity, default risk, and contract provisions tend to have some substitutability among investors.</p><p>B. Yield differences will be generally maintained by investors.</p><p>C. Co-movement increases if investors are willing and able to invest and trade a variety of securities.</p><p>COMPLETION QUESTIONS</p><p>1. Expectations of future interest rates may explain why interest rates vary by ______.</p><p>2. An ascending yield curve reflects investors' expectations that future short-term interest rates will be ______.</p><p>3. Long-term rates represent a geometric average of current______-term interest rates and expected ______-term interest rates.</p><p>4. A ______premium may be added by investors in order for them to purchase long-term bonds.</p><p>5. When securities of varied terms are not acceptable substitutes for investors and institutions in the short run, the ______theory may explain the shape of the yield curve.</p><p>6. The default risk premium on a corporate bond may be computed by subtracting the yield on a ______bond from that of the ______bond.</p><p>7. An increase in the required default risk premium of a security will lead to a(n) ______in the price of the security.</p><p>8. Default risk premiums narrow/widen in recession periods and narrow/widen in periods of economic prosperity.</p><p>9. The relevant interest rate to investors is the ______-tax return.</p><p>10. A option is an option of the lender to sell the security back to the issuer; a _ option is an option of the borrower to pay off the debt.</p><p>62 TRUE-FALSE QUESTIONS</p><p>T F 1. A yield curve plots coupon yields by maturity.</p><p>T F 2. Treasury and corporate security yields may be plotted together when generating a yield curve.</p><p>T F 3. One area of expectations affecting yields by maturities is anticipated inflation.</p><p>T F 4. The expectations theory states that long-term rates represent the market estimate of the average of current and future short-term rates.</p><p>T F 5. If interest rates are expected to increase in the future, one would expect to see an upward sloping yield curve </p><p>T F 6. According to the preferred habitat theory, investors may move out of their preferred maturities in response to expected yield premiums.</p><p>T F 7. The default risk premium compensates the holder of the risky security for the risk assumed.</p><p>T F 8. Liquidity premiums cause an observed yield curve to be less upward sloping than that predicted by the expectations theory</p><p>T F 9. The higher the marginal tax bracket of the investor, the less the attraction of municipal bonds.</p><p>T F 10. Bonds with call options are likely to have lower yields than non- callable bonds.</p><p>MULTIPLE-CHOICE QUESTIONS</p><p>1. The term structure of interest rates a. describes the relationship between maturity and yield for similar securities. b. ranks security yield according to the default risk structure. c. describes how interest rates vary over time. d. describes the pattern of interest rates over the business cycle.</p><p>2. The yield curve is a plot of a. maturity changes as risk changes. b. yields by varied risk-taking of varied bond issuers. c. yields by maturity of securities with similar default risk. d. interest rates over time past.</p><p>63 3. Applying the expectations theory, a bank depositor has the option of purchasing a one-year CD at 7 percent and a 9 percent two-year CD. If indifferent between the two, the depositor must expect one-year CDs one year from now to have a rate of a. 8 percent. b. 11 percent. c. slightly over 11 percent. d. 12 percent.</p><p>4. The yield differentials between a AAA corporate bond and a BAA corporate bond of the same maturity may be explained by a. marketability. b. tax treatment. c. default risk. d. term to maturity.</p><p>5. A call option on a bond a. increases the price an investor may pay. b. decreases the price an investor may pay. c. decreases the yield required by investors. d. has no effect on price or yield.</p><p>Use the data below to answer questions 6 through 9. Yield Treasury Bill (6 month) 6.41% Treasury Bill (1 year) 6.57% Treasury Bill (2 year) 7.02% Treasury Note (10 year) 7.58% Treasury Bond (30 year) 7.43% Corporate Bond (10 year AA) 9.25% Municipal Bond (10 year AA) 7.12% Expected Annual Inflation Rate 3.50%</p><p>6. The slope of the yield curve for U.S. Treasury securities indicates a. declining interest rates in the future. b. increasing interest rates in the future. c. increasing prices on U.S. Treasuries in the future. d. constant interest rates in the future.</p><p>7. The default risk premium on the ten-year corporate bond is a. 1.67% b. 2.13% c. 2.84% d. 1.82%</p><p>64 8. The expected real return on a one-year Treasury bill is a. 3.50% b. 10.07% c. 3.76% d. 3.07%</p><p>9. The yield difference between the corporate and municipal bond may be best explained by the fact that a. the muni has lower default risk. b. the capital gain income on a municipal bond is tax-free. c. the interest income on each is federal tax exempt. d. the interest income on the municipal bond is federal tax exempt.</p><p>10. Which of the following bonds probably has the highest call premium included in its yield? a. a low coupon, short-term corporate note in an increasing rate market b. a high coupon rate bond in a falling interest rate market c. a high coupon rate bond in a rising interest rate market d. a low coupon rate bond in an increasing interest rate market.</p><p>11. A downward sloping yield curve indicates that future short-term rates are expected to ______and outstanding security prices will ______. a. fall; rise. b. fall; fall. c. rise; rise. d. rise; fall.</p><p>12. Suppose we consider a yield curve that has taken into consideration both the expectations theory and the liquidity premium theory. Assume the yield curve is initially downward sloping. If liquidity premium theory is no longer important, the yield curve you would expect to see would be: a. more steeply downward sloping b. more upward sloping c. less steeply downward sloping d. none of the above.</p><p>13. According to expectations theory, an investor who believes that interest rates are likely to decrease in the near future would a. would invest in short-term securities immediately. b. would invest in long-term securities immediately. c. would sell long-term securities from her portfolio. d. would sell short-term securities from her portfolio.</p><p>14. According to expectations theory, if the market believes that interest rates are expected to increase in the near future, a. borrowers would immediately increase their supply of short-term securities. b. investors would immediately increase their demand for long-term securities. c. borrowers would immediately increase their supply of long-term securities. d. neither borrowers nor investors would do anything until the interest rates actually increased. 65 15. According to expectations theory, if the market believes that interest rates are likely to decrease in the near future, it would lead to:</p><p> a. An increase in the demand for short-term securities. b. An increase in the demand for long-term securities. c. A decrease in the supply of short-term securities. d. An increase in the supply of long-term securities.</p><p>SUPPLEMENTARY MATERIALS</p><p>A) Plotting a Yield Curve</p><p>Open the web page http://finance.yahoo.com/bonds. You will see a table that contains the yields of U. S. government securities. a) Plot the yield to maturity (Y-axis) by maturity (X-axis) and fit a yield curve to the plot of points. (Hint: Plot a scatter diagram and fit a line to the points.) b) Interpret the yield curve using the expectations theory. What is the market's forecast of future short-term interest rates?</p><p>B) Computing Forward Rates</p><p>You will be asked "What's going to happen to interest rates?" many times in the future. If any one of us could predict future interest rates consistently over time, we would easily become rich and famous! Few persons have been able to forecast future interest rates consistently even though many business economists are paid well to do so. </p><p>If interest rate expectations are a significant factor in the trading of short- and long-term treasuries, then we can calculate expected future interest rates as implied by two actual market interest rates. The text has given us an excellent background, but below is a formula for calculating forward (expected) interest rates, f, given any two observed (actual) interest rates, R, of different maturities.</p><p> n 1/ k  1t rn   tnk f k   nk  1 1t rnk   where t+n-kfk = the k-year forward rate beginning (n - k ) years from today.</p><p> k = time period between two actual observed rates and also the maturity of the forward security in years.</p><p> tRn = actual observed interest rate at time t for an n period security (years to maturity)</p><p>66 Example: An investor is interested in purchasing one of the following government bonds, each of which yields (annual compounding):</p><p>Term Market Yield 2 year 6.50% 5 year 7.25%</p><p>Should the investor buy the two- or five-year bond? Of course it depends on what interest rates will be two years from now. A $100 investment in the five-year bond will yield (1.0725)5 = $141.90 in five years. Purchase of the two-year bond would give the investor (1.065)2 = $113.42 at the end of the two years and the need to reinvest the proceeds for three more years. At what rate does he need to reinvest to accumulate a sum of $141.90 by the end of the fifth year? We need to solve for the three-year forward rate, two years from now, that allows a $113.42 investment to grow to $141.90. Or, what are expected three-year rates two years from now as revealed by the current yield structure of interest rates?</p><p>Using our formula above where k = 3,</p><p>1/3 1/3 1/ 3 1 r 5  1.07255  1.4190 f  t 5 1  1  1  7.75% t2 3  2   2    1t r2    1.065  1.1342</p><p>The expected one-year interest rate two years from now is 7.75 percent. Investing $113.42 at 7.75 percent for three years equals $141.90 at the end of five years. This is the same as the total return from the five-year bond (1.0725)5 = $141.90. The three-year forward rate of 7.75 percent is the forecasted or expected three-year rate two years from now as indicated by today's actual market rates.</p><p>Assignment:</p><p>1) In July 2002 the yield curve for U.S. Treasury securities was the following:</p><p>Treasury Yield 3 month 1.71% 6 month 1.73% 9 month 1.73% 1 year 1.83% 2 year 2.60% 3 year 3.18% 4 year 3.59% 5 year 3.90% 6 year 4.09% 7 year 4.36%</p><p>Calculate the implied one-year forward (expected) rates over the next six years. What are the expectations of future one-year rates one, two, three, four, five, and six years from July, 2002? How accurate was the market in predicting interest rates for the next few years? </p><p>67 2) Calculate the expected 90-day (3-month) rates 3 months, 6 months, and 9 months from now using the rates above. (Hint: Use the same formula, but with fractions of the year: 1/4 for 3 months, 2/4 for 6 months, etc.; then annualize when finished.) Prove your answers.</p><p>3) Bond Rating Changes. From a recent issue of Standard and Poor's Credit Week or Moody's The Bond Record, what corporate security ratings have been changed recently and why? These two sources are excellent sources for keeping in touch with changes in default risk as assessed by two important rating services. </p><p>68 SOLUTIONS TO COMPLETION QUESTIONS</p><p>1. term or maturity</p><p>2. higher</p><p>3. short; short</p><p>4. liquidity</p><p>5. market segmentation</p><p>6. Treasury; corporate</p><p>7. decline</p><p>8. widen; narrow</p><p>9. after</p><p>10. put; call</p><p>SOLUTIONS TO TRUE/FALSE QUESTIONS</p><p>1. F A yield curve plots market yields by maturity.</p><p>2. F Default risk is held constant to study yields by time.</p><p>3. T Inflation expectations affect both the level and slope of the yield curve.</p><p>4. T Current long-term rates reflect expectations of future short-term rates.</p><p>5. T Investors expecting rates to go up will invest in short-term securities and shun long-term securities. This drives up the price of short-term securities and its yield down. At the same time long-term securities’ prices decrease driving up its yield.</p><p>6. F If yields increase sharply, security prices must be falling, indicating a movement away from a maturity range - market segmentation.</p><p>7. T Default risk premiums reflect the market's expected default losses from a large portfolio of securities of similar risk.</p><p>8. F Liquidity premiums actually cause the slope of the yield curve to be more upward sloping because liquidity premiums increase with maturity.</p><p>9. F The higher the marginal tax bracket, the greater the after-tax return compared to other taxable bonds. 10. F Investors demand a yield premium if the borrower has an option to terminate 69 their investment, usually when rates are low.</p><p>SOLUTIONS TO MULTIPLE-CHOICE QUESTIONS</p><p>1. a. Term structure of interest rates describes the relationship between maturity and yield for similar securities.</p><p>2. c. A graphical plot of yield vs. maturity for securities that are similar in all other aspects is called the yield curve.</p><p>3. c The two-year CD available at 9 percent represents the geometric average of current one-year CDs at 7 percent and an estimated one-year rate one year from now of slightly more than 11 percent.</p><p>Llet X equal the t+1f1, or one-year forward rate one year from today. </p><p>1.09 = (1.07) (X)</p><p>X = 11.04%</p><p>The geometric mean is a more accurate average with compound interest rates or growth rates than an arithmetic mean.</p><p>4. c The ratings on the bond refer to relative default risk between the securities.</p><p>5. b Investors will pay less, requiring a higher yield on a bond with a call option. </p><p>6. b The upward sloping yield curve reflects the market expectation of higher short- term rates in the future.</p><p>7. a The difference between a risk-free ten-year (holding maturity constant) and a risky corporate bond.</p><p>8. d A yield of 6.57 percent on the one-year bill less the expected loss in purchasing power of 3.5 percent.</p><p>9. d The market requires a lower yield on tax-free municipals.</p><p>10. b The high coupon bond in a falling rate market has the best chance of being called, thus investors want added returns for added call risk.</p><p>11. a. A descending yield curve forecasts lower rates possibly related to slower economic growth or lower inflation rates. Investors demand for short-term securities will decline driving down their prices.</p><p>12. a. The removal of liquidity premiums will decrease the yield at every maturity making the </p><p>70 yield curve to be more steeply downward sloping</p><p>13. b. Investors wanting to lock in on the current higher rates would invest in long-term securities immediately.</p><p>14. c. To avoid higher rates, borrowers would immediately increase their supply of long-term securities</p><p>15. b. Investors wanting to lock in on the current higher rates would invest in long-term securities immediately leading to an increase in the demand for long-term securities.</p><p>71 SELECTED SOLUTIONS TO ASSIGNMENTS</p><p>1. The expected one-year rate three years from July, 2002, using the forward rate calculator:</p><p>(1.0359 )4 t3 f1 = - 1 = 1.0483 - 1 = 0.0483 = 4.83% (1.0318 )3</p><p>Proof : (1.0359 )4 = (1.0318 )3 (1.0483 )1</p><p>2. The expected 90-day (three months) rate three months (one quarter) from today is</p><p>(1.0173 )0.5 f = - 1 t0.25 0.25 (1.0171 )0.25</p><p>1.00861 = = (1.00435 )4 - 1 1.00425</p><p>= 1.75%( annualized)</p><p>Proof: (1.0171).25 (1.0175).25 = (1.0173).5. Investing for six months (0.5 exponent) at 1.73% annualized is equivalent to investing in a three-month or one-quarter (0.25 exponent) rate of 1.71% followed by another quarter at 1.75%. </p><p>72</p>